O X
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Ti Kik
Physics
f o r
T h e
I B
d I p l o m a
2014 edition
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Introduction and acknowledgements
Many people seem to think that you have to be really
closely to the recently revised International Baccalaureate
clever to understand Physics and this puts some people
syllabus. It aims to provide an explanation (albeit very
o studying it in the rst place. So do you really need a
brief) of all of the core ideas that are needed throughout
brain the size of a planet in order to cope with IB Higher
the whole IB Physics course. To this end each of the
Level Physics? The answer, you will be pleased to hear, is
sections is clearly marked as either being appropriate
‘No’. In fact, it is one of the world’s best kept secrets that
for everybody or only being needed by those studying at
Physics is easy! There is very little to learn by hear t and
Higher level. The same is true of the questions that can be
even ideas that seem really dicult when you rst meet
found at the end of the chapters.
them can end up being obvious by the end of a course of
I would like to take the oppor tunity to thank the many
study. But if this is the case why do so many people seem
people that have helped and encouraged me during the
to think that Physics is really hard?
writing of this book. In par ticular I need to mention David
I think the main reason is that there are no ‘safety nets’
Jones and Paul Ruth who provided many useful and
or ‘shor t cuts’ to understanding Physics principles. You
detailed suggestions for improvement – unfor tunately
won’t get far if you just learn laws by memorising them
there was not enough space to include everything. The
and try to plug numbers into equations in the hope of
biggest thanks, however, need to go to Betsan for her
getting the right answer. To really make progress you need
suppor t, patience and encouragement throughout the
to be familiar with a concept and be completely happy
whole project.
that you understand it. This will mean that you are able
to apply your understanding in unfamiliar situations. The
hardest thing, however, is often not the learning or the
Tim Kirk
October 2002
understanding of new ideas but the getting rid of wrong
and confused ‘every day explanations’.
This book should prove useful to anyone following a pre-
university Physics course but its structure sticks very
Third edition
Since the IB Study Guide's rst publication in 2002, there
have been two signicant IB Diploma syllabus changes.
The aim, to try and explain all the core ideas essential for
the IB Physics course in as concise a way as possible,
has remained the same.
I continue to be grateful to all the
teachers and students who have taken time to comment
and I would welcome fur ther feedback.
In addition to
the team at OUP
, I would par ticularly like to thank my
exceptional colleagues and all the outstanding students
at my current school, St. Dunstan's College, London.
It
goes without saying that this third edition could not have
been achieved without Betsan's continued suppor t and
encouragement.
This book is dedicated to the memory of my father,
Francis Kirk.
Tim Kirk
August 2014
I n t r o d u c t I o n
a n d
a c k n o w l e d g e m e n t s
iii
Contents
(Italics denote topics which are exclusively Higher Level)
Intensity
39
Superposition
40
1 measurement and uncertaIntIes
The realm of physics – range of magnitudes of
quantities in our universe
1
Polarization
41
The SI system of fundamental and derived units
2
Uses of polarization
42
Estimation
3
Wave behaviour – Reection
43
Uncer tainties and error in experimental measurement
4
Snell’s law and refractive index
44
Uncer tainties in calculated results
5
Refraction and critical angle
45
Uncer tainties in graphs
6
Diraction
46
Vectors and scalars
7
Two-source interference of waves
47
IB Questions – measurement and uncer tainties
8
Nature and production of standing (stationary)
waves
48
Boundary conditions
49
IB
50
2 mechanIcs
Motion
9
Graphical representation of motion
10
Uniformly accelerated motion
11
Projectile motion
12
Fluid resistance and free-fall
13
Forces and free-body diagrams
14
Newton’s rst law
15
Equilibrium
16
Newton’s second law
1
7
Newton’s third law
18
Mass and weight
19
Solid friction
20
Questions – waves
5 electrIcItY and magnetIsm
Electric charge and Coulomb's law
51
Electric elds
52
Electric potential energy and electric potential
Work
dierence
53
Electric current
54
Electric circuits
55
Resistors in series and parallel
56
Potential divider circuits and sensors
57
Resistivity
58
Example of use of Kircho's laws
59
Internal resistance and cells
60
2
1
Energy and power
22
Momentum and impulse
23
IB Questions – mechanics
24
Magnetic force and elds
6
1
Magnetic forces
62
Examples of the magnetic eld due to currents
63
IB Questions – electricity and magnetism
64
3 thermal PhYsIcs
Thermal concepts
25
Heat and internal energy
26
Specic heat capacity
27
Phases (states) of matter and latent heat
28
The gas laws 1
29
The gas laws 2
30
6 cIrcul ar motIon and graVItatIon
Uniform circular motion
65
Angular velocity and ver tical circular motion
66
Newton’s law of gravitation
67
IB Questions – circular motion and gravitation
68
Molecular model of an ideal gas
31
7 atomIc, nuclear and
IB Questions – thermal physics
32
PartIcle PhYsIcs
Emission and absorption spectra
69
Nuclear stability
70
4 waVes
Oscillations
33
Graphs of simple harmonic motion
34
Travelling waves
35
Wave characteristics
36
Electromagnetic spectrum
37
Investigating speed of sound experimentally
38
iv
c o n t e n t s
Fundamental forces
7
1
Radioactivity 1
72
Radioactivity 2
73
Half-life
74
Nuclear reactions
75
Fission and fusion
76
citr disrge
Structure of matter
77
citr rge
119
Description and classication of par ticles
78
IB Qestis – eetrgeti idti
120
Quarks
79
Feynman diagrams
80
118
12 QuanTum anD nuclEaR phYSIcS
pteetri eet
IB Questions – atomic, nuclear and par ticle physics
12
1
81
mtter wes
8 energY ProductIon
122
ati setr d ti eergy sttes
123
Energy and power generation – Sankey diagram
82
Br de  te t
124
Primary energy sources
83
Te Srödiger de  te t
125
Fossil fuel power production
84
Te heiseberg ertity riie d
Nuclear power – process
85
Nuclear power – safety and risks
86
Solar power and hydroelectric power
87
Wind power and other technologies
88
Thermal energy transfer
89
Radiation: Wien’s law and the Stefan–Boltzmann law
90
Solar power
91
The greenhouse eect
92
Global warming
93
IB Questions – energy production
94
te ss  deteriis
126
Teig, teti brrier d trs etig
teig rbbiity
127
Te es
128
ner eergy ees d rditie dey
129
IB Qestis – qt d er ysis
130
13 oPtIon a – rel atIVItY
Reference frames
131
Maxwell’s equations
132
Special relativity
133
Lorentz transformations
134
Velocity addition
135
Invariant quantities
136
Time dilation
137
9 WavE phEnomEna
Sie ri ti
95
Eergy ges drig sie ri ti
96
Dirti
97
Length contraction and evidence to suppor t
Tw-sre itereree  wes:
Yg’s dbe-sit exeriet
98
mtie-sit dirti
99
Ti re s
special relativity
138
Spacetime diagrams (Minkowski diagrams) 1
139
Spacetime diagrams 2
140
The twin paradox 1
140
100
Resti
101
Te Der eet
102
xes d itis  te Der eet
103
IB Qestis – we ee
104
Twin paradox 2
141
Spacetime diagrams 3
142
mss d eergy
143
Retiisti et d eergy
144
Retiisti eis exes
145
Geer retiity – te eqiee riie
146
Gritti red sit
147
Srtig eidee
148
crtre  setie
149
Bk es
150
10 fIElDS
pteti (gritti d eetri)
105
Eqitetis
106
Gritti teti eergy d teti
107
orbit ti
108
Eetri teti eergy d teti
109
Eetri d Gritti fieds red
110
IB Qestis – eds
111
IB Questions – option A – relativity
151
14 oPtIon B – engIneerIng PhYsIcs
11 ElEcTRomaG nETIc InDucTIon
Induced electromotive force (emf)
112
lez's w d frdy's w
113
atertig rret (1)
114
atertig rret (2)
115
Retiti d stig irits
116
cite
11
7
Translational and rotational motion
152
Translational and rotational relationships
153
Translational and rotational equilibrium
154
Equilibrium examples
155
Newton’s second law – moment of iner tia
156
Rotational dynamics
157
c o n t e n t s
v
Solving rotational problems
158
Thermodynamic systems and concepts
159
Work done by an ideal gas
160
The rst law of thermodynamics
16
1
Second law of thermodynamics and entropy
162
Heat engines and heat pumps
163
fids t rest
164
fids i ti – Beri eet
165
Beri – exes
166
vissity
167
fred sitis d rese (1)
168
Rese (2)
169
IB Questions – option B – engineering physics
1
70
15 oPtIon c – ImagIng
Image formation
1
7
1
Converging lenses
1
72
Image formation in convex lenses
1
73
Thin lens equation
1
74
Diverging lenses
1
75
Converging and diverging mirrors
1
76
The simple magnifying glass
1
77
Aberrations
1
78
The compound microscope and
astronomical telescope
Astronomical reecting telescopes
1
79
180
Radio telescopes
181
Fibre optics
182
Dispersion, attenuation and noise in optical bres
183
Channels of communication
184
X-rys
185
X-ry igig teiqes
186
utrsi igig
187
Igig tied
188
IB Questions – option C – imaging
189
16 oPtIon d – astroPhYsIcs
Objects in the universe (1)
190
Objects in the universe (2)
191
The nature of stars
192
Stellar parallax
193
Luminosity
194
Stellar spectra
195
Nucleosynthesis
196
The Her tzsprung–Russell diagram
197
Cepheid variables
198
Red giant stars
199
Stellar evolution
200
The Big Bang model
201
Galactic motion
202
Hubble’s law and cosmic scale factor
203
The accelerating universe
204
ner si – te Jes riteri
205
nesytesis  te i seqee
206
Tyes  sere
207
Te sgi riie d teti des
208
Rtti res d drk tter
209
Te istry  te uierse
2
10
Te tre  te uierse
2
11
Drk eergy
2
12
astrysis reser
2
13
IB Questions – astrophysics
2
14
17 aPPendIX
vi
c o n t e n t s
Graphs
2
15
Graphical analysis and determination of relationships
2
16
Gri ysis – griti tis
2
1
7
ANSWERS
2
18
ORIGIN OF INDIVIDUAL QUESTIONS
2
18
INDEX
2
19
1
M E A S U R E M E N T
A N D
U N C E R T A I N T I E S
Te ream o psics – rane o manitudes o
quantities in our uni erse
ORDERS Of MAgNITUDE –
RANgE Of MA SSES
INClUDINg ThEIR RATIOS
52
Physics
seeks
to
explain
RANgE Of lENgThS
Mass / kg
nothing
less
than
the
10
Size / m
radius of observable Universe
total mass of observable
26
10
Universe
itself.
In
attempting
to
do
this,
the
48
10
Universe
24
10
range
of
the
magnitudes
of
various
quantities
44
10
22
will
be
10
mass of local galaxy
huge.
radius of local galaxy (Mily ay
40
10
20
If
the
numbers
involved
are
going
to
10
(Milky Way)
mean
36
10
anything,
it
is
important
to
get
some
18
feel
10
disance o neares sar
32
for
their
relative
sizes.
To
avoid
‘getting
lost’
10
16
10
mass of Sun
among
the
numbers
it
is
helpful
to
state
them
28
10
14
10
to
the
nearest
order
of
magnitude
or
power
24
mass of Ear th
10
12
of
ten.
The
numbers
are
just
rounded
up
10
or
20
total mass of oceans
disance from ar  o Sun
10
down
as
10
appropriate.
10
total mass of atmosphere
16
disance from ar  o Moon
10
Comparisons
can
then
be
easily
made
8
because
10
12
working
out
the
ratio
between
two
powers
of
radius of e ar 
10
6
10
deees ar  of e
ten
is
just
a
matter
of
adding
or
subtracting
laden oil super tanker
8
10
4
10
whole
numbers.
The
diameter
of
an
elephant
4
10
m,
does
not
sound
that
much
ocean / iges mounain
atom,
alles building
2
10
10
10
larger
human
0
10
than
the
diameter
of
a
proton
in
its
0
nucleus,
10
mouse
15
5
10
m,
but
the
ratio
between
them
is
10
or
10
4
2
leng of ngernail
10
100,000
times
bigger.
This
is
the
same
ratio
as
10
8
grain of sand
icness of iece of aer
4
10
between
the
size
of
a
railway
station
blood corpuscle
(order
10
uman blood coruscle
12
6
2
of
magnitude
10
m)
and
the
diameter
of
10
the
bacterium
10
7
Earth
(order
of
magnitude
10
m).
16
aveleng of lig
8
10
10
20
10
electrons
10
haemoglobin molecule
10
diameer of ydrogen aom
24
12
10
proton
aveleng of gamma ray
10
28
14
10
electron
10
diameer of roon
32
16
10
protons
RANgE Of TIMES
RANgE Of ENERgIES
Time / s
20
10
Energy / J
Carbon atom
age of the Universe
44
18
railway
station
16
energy released in a supernova
10
10
age of the Ear th
34
10
10
14
10
age of species – Homo
30
12
Ear th
10
sapiens
10
energy radiated by Sun in 1 second
26
10
For
example,
you
would
probably
feel
10
10
very
typical human lifespan
pleased
with
yourself
if
you
designed
a
new,
8
10
22
10
1 year
environmentally
friendly
source
of
energy
energy released in an ear thuae
6
10
3
that
could
produce
2.03
×
10
J
from
0.72
18
1 day
kg
10
4
10
of
natural
produce.
But
the
meaning
of
energy released by annihilation of
these
2
14
10
10
numbers
is
not
clear
–
is
this
a
lot
or
is
it
0
little?
In
terms
of
orders
of
magnitudes,
1 g of atter
a
this
hear tbeat
10
10
energy in a lightning discharge
10
3
new
source
produces
10
joules
This
not
per
kilogram
2
10
period of high-frequency
of
produce.
does
compare
terribly
6
energy needed to charge a car
10
4
10
sound
battery
5
well
with
the
10
joules
provided
by
a
slice
of
6
2
10
10
8
bread
or
the
10
joules
released
per
kilogram
inetic energy of a tennis ball
8
10
of
passage of light across
2
petrol.
10
a room
during gae
10
10
You
do
NOT
need
to
memorize
all
of
energy in the beat of a y’s ing
the
6
10
12
values
shown
in
the
tables,
but
you
should
10
vibration of an ion in a solid
14
try
and
develop
a
familiarity
with
them.
10
10
10
period of visible light
16
10
14
10
18
10
20
passage of light across
10
18
10
energy needed to reove electron
an atom
fro the surface of a etal
22
10
22
10
24
passage of light across
10
a nucleus
26
10
M E A S U R E M E N T
A N D
U N C E R T A I N T I E S
1
Te SI sstem o undamenta and deri ed units
been
fUNDAMENTAl UNITS
Any
measurement
being
made
up
of
1.
the
number
2.
the
units.
Without
both
For
example
but
without
17
if
parts,
a
the
person’s
17
you
every
quantity
important
can
be
thought
of
as
(SI).
‘years’
measurement
age
might
the
months
saw
=
In
SI,
science
the
we
see
the
their
In
them,
nothing.
number
answers
There
for
are
or
be
17
years
a
not
quoted
situation
but
does
is
not
old?
statement
as
make
Mass
kilogram
kg
Length
metre
m
In
this
Are
of
to
the
Having
said
candidates
units
to
be
possible
this,
who
examination
many
xed
the
In
Time
second
s
case
you
ampere
A
mole
mol
kelvin
K
Electric
would
it
is
forget
really
to
surprising
SI
symbol
current
of
substance
include
the
to
units
in
(Luminous
intensity
understood,
systems
of
they
need
measurement
to
be
that
dened.
You
need
do
units
as
different
other
the
words,
all
all
other
combinations
the
fundamental
measurement
of
candela
cd)
order
other
list
of
units
units
are
speed.
The
know
use
the
them
precise
denitions
of
any
of
these
properly.
so
large
of
that
the
SI
magnitudes.
unit
In
(the
these
metre)
cases,
always
the
use
involves
of
a
large
different
measurements
of
does
to
to
have
are
units,
not
in
the
non
SI)
unit
derived
not
denition
of
is
very
common.
Astronomers
can
use
the
fundamental
units.
contain
a
speed
unit
(AU),
the
light-year
(ly)
or
the
parsec
(pc)
For
appropriate.
Whatever
the
unit,
the
conversion
to
SI
units
is
unit
simple
the
unit
like
as
for
of
follows
questions.
fundamental
expressed
example,
as
they
astronomical
units.
System
are
‘seventeen’
clear.
(but
be
units
SI
orders
can
International
base
sense.
DERIvED UNITS
Having
or
4.2
says
order
the
Quantity
Temperature
actually
use
fundamental
parts:
Amount
length
In
and
the
minutes,
know
and
two
developed.
units
can
arithmetic.
be
11
used
to
work
out
the
derived
unit.
distance
_
Since
speed
1
AU
1
ly
1
pc
=
1.5
×
10
m
15
=
=
9.5
×
10
m
time
16
=
3.1
×
10
m
units of distance
__
Units
of
speed
=
There
units
of
are
also
some
units
(for
example
the
hour)
which
are
so
time
common
that
they
are
often
used
even
though
they
do
not
form
metres
_
=
(pronounced
‘metres
per
second’)
part
of
into
equations
SI.
Once
again,
before
these
numbers
are
substituted
seconds
they
need
to
be
converted.
Some
common
unit
m
_
=
s
conversions
are
given
on
page
3
of
the
IB
data
booklet.
1
=
m
s
The
table
below
lists
the
SI
derived
units
that
you
will
meet.
1
Of
the
many
ways
of
writing
this
unit,
the
last
way
(m
s
)
is
the
SI
derived
unit
SI
base
unit
Alternative
SI
unit
best.
2
newton
Sometimes
particular
combinations
of
fundamental
(N)
kg
m
kg
m
s
units
1
pascal
are
so
common
that
they
are
given
a
new
derived
name.
(Pa)
2
2
s
N
m
N
m
For
1
example,
the
unit
of
force
is
a
derived
unit
–
it
turns
out
to
hertz
(Hz)
s
joule
(J)
kg
be
2
kg
m
s
2
.
This
unit
is
given
a
new
name
the
newton
(N)
so
that
m
2
s
2
1N
=
1
kg
m
s
.
2
watt
The
great
thing
about
SI
is
that,
so
long
as
the
numbers
that
(W)
into
an
equation
are
in
SI
units,
then
the
m
(C)
A
come
out
in
SI
units.
You
can
always
‘play
safe’
(V)
kg
m
3
s
ohm
all
the
numbers
into
proper
SI
units.
(Ω)
kg
m
kg
m
3
this
would
be
a
waste
of
time.
weber
(Wb)
s
are
awkward.
some
In
situations
astronomy,
where
for
the
use
example,
of
the
SI
1
A
2
s
2
There
1
WA
2
V
A
Sometimes,
2
however,
1
A
by
2
converting
s
s
2
also
J
answer
volt
will
1
s
are
coulomb
substituted
kg
3
becomes
distances
tesla
(T)
kg
involved
s
1
A
V
s
1
2
A
Wb
m
1
becquerel
(Bq)
s
PREfIxES
To
avoid
booklet.
the
repeated
These
can
be
use
of
very
scientic
useful
but
notation,
they
can
an
alternative
also
lead
to
is
to
errors
example,
1
kW
=
1000
W
.
1
mW
=
10
one
1W
W
(in
other
words,
)
1000
2
M E A S U R E M E N T
A N D
of
the
calculations.
____
3
For
use
in
U N C E R T A I N T I E S
list
It
is
of
agreed
very
easy
prexes
to
given
forget
to
on
page
include
the
2
in
the
IB
data
conversion
factor.
Estimation
1
ORDERS Of MAgNITUDE
kelvin
1K
is
and
It
is
important
that
a
you
simple
way
of
use.
to
develop
When
mistake
(eg
checking
resorting
to
the
a
using
by
the
‘feeling’
a
entering
answer
calculator.
for
calculator,
is
the
to
The
some
it
is
data
rst
of
the
very
make
an
very
low
at
373
temperature.
K.
Room
Water
freezes
temperature
is
to
A
estimate
paper
make
1
mol
12
good
g
of
carbon
carbon
in
the
12.
About
‘lead’
of
a
not
allow
the
use
of
273
K
300
K
the
number
of
atoms
of
pencil
before
(paper
The
same
1
s
process
can
happen
with
some
of
the
derived
units.
1)
1
does
at
about
numbers
easy
incorrectly).
multiple-choice
a
boils
m
1
Walking
speed.
A
car
moving
at
30
m
s
would
be
fast
calculators.
2
1
Approximate
values
for
each
of
the
fundamental
SI
units
m
s
Quite
a
slow
acceleration.
The
acceleration
of
gravity
are
2
given
1
is
10
m
s
small
below.
kg
A
packet
about
of
50
sugar,
kg
or
1
litre
of
water.
A
person
would
1
N
A
1
V
Batteries
m
Distance
1
s
Duration
between
one’s
hands
with
arms
a
about
the
range
weight
from
a
of
an
few
apple
volts
heart
beat
(when
resting
–
the
mains
is
several
hundred
up
to
20
or
volts
outstretched
1
of
–
generally
more
so,
1
force
be
it
can
Pa
A
very
small
pressure.
small
amount
Atmospheric
pressure
is
about
easily
5
10
double
with
exercise)
1
1
amp
Current
Pa
owing
from
the
mains
electricity
when
J
A
very
lifting
computer
domestic
is
connected.
device
would
The
be
maximum
about
10
A
current
or
of
energy
–
the
work
done
a
to
an
apple
off
the
ground
a
so
POSSIblE RE A SONAblE A SSUMPTIONS
Everyday
these
it
is
The
situations
assumptions
often
possible
table
some
below
quantity
are
are
to
go
lists
is
very
not
complex.
absolutely
back
some
constant
and
work
common
even
if
In
physics
true
they
out
we
allow
what
know
that
us
would
assumptions.
we
often
Be
in
to
simplify
gain
happen
careful
reality
it
Assumption
Object
treated
Friction
No
is
Mass
of
as
point
particle
energy
(“heat”)
connecting
string,
of
ammeter
Resistance
of
voltmeter
resistance
Material
obeys
Machine
Gas
is
Object
of
is
loss
etc.
is
negligible
is
innite
is
is
Many
as
Only
a
perfect
black
body
153.2
=
not
to
Additionally
At
be
we
the
Even
end
if
of
we
the
know
calculation
true.
often
have
to
assume
that
time.
Linear
mechanics
all
motion
and
situations
thermal
mechanics
–
translational
but
you
need
equilibrium
to
be
very
careful
situations
situations
situations
are
too
big
or
number
1.532
×
molecules
have
equilibrium,
perfectly
e.g.
elastic
collisions
planets
SIgNIfIC ANT fIgURES
too
small
for
decimals
are
often
notation:
×
between
10
gas
Thermal
1
Any
experimental
uncertainty.
the
This
quantity
measurement
indicates
being
the
measured.
should
possible
At
the
be
quoted
range
same
of
with
values
time,
the
its
for
number
10
and
10
and
b
is
2
e.g.
the
out
on.
Circuits
efcient
scientic
a
all
assumptions.
going
Circuits
b
a
slightly
turned
much!
is
Circuits
zero
law
a
where
too
simple
what
Circuits
elastic
that
in
varying
of
assumption
assume
Many
SCIENTIfIC NOTATION
written
making
Thermodynamics
radiates
Numbers
our
to
Almost
zero
battery
Ohm’s
100%
is
by
Mechanics:
ideal
Collision
is
if
problem
Many
Resistance
Internal
not
a
understanding
Example
negligible
thermal
an
an
of
signicant
gures
of
uncertainty.
used
will
act
as
a
guide
to
the
amount
integer.
For
example,
a
measurement
of
mass
which
3
;
0.00872
=
8.72
×
10
is
quoted
(it
has
an
A
as
ve
rule
the
of
for
LEAST
For
a
more
implies
±
gures),
0.1
to
precise
g
results,
M E A S U R E M E N T
the
value
complete
calculated
an
(it
that
page
A N D
whereas
three
one
of
of
number
is
±
of
0.001
23.5
signicant
(multiplication
same
analysis
see
uncertainty
has
calculations
answer
the
g
signicant
uncertainty
simple
quote
in
23.456
or
g
g
implies
gures).
division)
signicant
is
digits
to
as
used.
of
how
to
deal
with
uncertainties
5.
U N C E R T A I N T I E S
3
Uncer tainties and error in eperimenta measurement
ERRORS – RANDOM AND SySTEMATIC (PRECISION
Systematic
graph
of
and
the
random
errors
can
often
be
recognized
from
a
results.
AND ACCURACy)
experimental
between
Errors
the
can
be
Repeating
error
recorded
just
categorized
readings
means
value
does
and
as
that
the
random
not
reduce
there
is
‘perfect’
or
a
or
A ytitnauq
An
difference
‘correct’
value.
systematic.
systematic
errors.
perfect results
Sources
of
random
errors
include
random error
•
The
readability
•
The
observer
•
The
effects
of
the
being
instrument.
less
than
systematic error
perfect.
quantity B
Sources
of
of
a
change
systematic
in
errors
the
surroundings.
Perfect
include
results,
proportional
•
An
instrument
value
•
An
•
The
should
with
be
instrument
zero
error.
subtracted
being
from
wrongly
To
correct
every
for
zero
error
random
and
systematic
errors
of
two
quantities.
the
reading.
calibrated
ESTIMATINg ThE UNCERTAINTy RANgE
observer
being
less
than
perfect
in
the
same
way
every
An
uncertainty
range
applies
to
any
measurement.
experimental
value.
The
idea
is
that,
3
An
accurate
experiment
is
one
that
has
a
small
cm
systematic
instead
error,
whereas
a
precise
experiment
is
one
that
has
a
of
giving
one
value
that
100
small
implies
random
just
perfection,
we
give
the
likely
error.
90
range
measured
measured
value
value
for
the
measurement.
80
1.
Estimating
from
rst
principles
value
70
All
measurement
involves
a
readability
probability
error.
If
we
use
a
measuring
cylinder
60
to
that result has a
nd
the
volume
of
a
liquid,
we
might
50
cer tain value
3
think
that
the
best
estimate
is
73
cm
,
40
but
we
know
that
it
is
not
exactly
this
30
3
value
(73.000 000 000 00
cm
).
20
3
Uncertainty
value
=
Normally
73
examples
illustrating
the
nature
of
experimental
(a)
an
(b)
a
accurate
less
experiment
accurate
but
more
of
low
cm
.
We
say
±
5
cm
10
.
the
many
situations
analysing
data
is
representing
graphs
below
to
the
the
best
experiment.
use
a
method
graph.
If
uncertainties
the
explains
line
created
error
of
by
the
the
bar
their
error
should
range
as
due
to
below.
of
this
is
to
presenting
is
the
use
case,
error
Example
Analogue
Rulers,
scale
moving
Digital
scale
a
digital
neat
bars.
way
2.
Estimating
The
meters
with
±
pointers
Top-pan
and
Uncertainty
balances,
±
meters
uncertainty
(half
scale
the
smallest
division)
(the
smallest
scale
division)
range
from
several
repeated
measurements
use.
represents
graph
uncertainty
estimated
Device
If
Since
is
precision
precise
gRAPhIC Al REPRESENTATION Of UNCERTAINTy
t’
5
results:
readability
of
±
(b)
Two
In
is
3
volume
(a)
range
value
the
pass
uncertainty
through
ALL
range,
of
the
the
‘best-
rectangles
bars.
the
time
times,
1.94.
the
A ytitnauq
C ytitnauq
=
the
The
taken
average
largest
0.18;
and
1.98
uncertainty
would
for
readings
also
of
trolley
1.82
these
=
In
to
seconds
smallest
range.
be
a
in
go
ve
0.16).
can
The
to
is
be
quote
the
a
slope
2.01,
1.98
value
time
this
is
as
measured
1.82,
s.
is
2.0
is
1.97,
The
calculated
largest
example
appropriate
be
readings
readings
this
down
might
taken
±
s
0.2
and
deviation
(2.16
1.98
ve
2.16
±
of
1.98
as
the
0.18
s.
It
s.
SIgNIfICANT fIgURES IN UNCERTAINTIES
In
order
from
quantity B
quantity D
gure,
E ytitnauq
be
be
e.g.
4.264
4.3
to
cautious
calculations
±
a
N
0.4
acceptable
when
often
calculation
with
N.
are
an
This
to
quoting
rounded
that
nds
uncertainty
can
express
be
of
uncertainties,
up
the
±
to
one
value
0.362
unnecessarily
uncertainties
to
of
N
a
is
force
values
to
quoted
pessimistic
two
nal
signicant
and
signicant
as
it
is
also
gures.
19
mistake
For
The
best
t
line
example,
the
charge
on
an
electron
is
1.602176565
×10
is
19
assumed
±
included
by
all
the
0.000000035
×10
C.
In
data
booklets
error
19
expressed
bars
in
graphs.
quantity F
4
the
M E A S U R E M E N T
the
upper
This
lower
A N D
is
two
not
true
in
graph.
U N C E R T A I N T I E S
as
1.602176565(35)
×
10
C.
this
is
sometimes
C
Uncer tainties in cacuated resuts
MAThEMATICAl REPRESENTATION Of UNCERTAINTIES
For
example
if
the
mass
of
a
block
was
measured
as
10
±
1
Then
g
the
fractional
uncertainty
is
±∆p
_
,
3
and
the
volume
was
measured
as
5.0
±
0.2
cm
,
then
the
p
full
which
calculations
for
the
density
would
be
as
makes
the
percentage
uncertainty
follows.
±∆p
mass
10
______
Best
value
for
density
=
_
3
=
=
2.0
g
×
cm
In
11
___
The
largest
possible
value
of
100%.
p
5
volume
density
the
example
above,
the
fractional
uncertainty
of
the
density
is
3
=
=
2.292
g
cm
4.8
±0.15
9
___
The
smallest
possible
value
of
density
=
3
=
1.731
g
Thus
cm
or
±15%.
equivalent
ways
of
expressing
this
error
are
5.2
3
density
=
2.0
±
density
=
2.0
g
0.3
g
cm
3
Rounding
these
values
gives
density
=
2.0
±
0.3
g
cm
3
OR
We
can
express
absolute
If
a
be
this
uncertainty
fractional
quantity
p
expressed
is
as
or
in
one
percentage
measured
then
the
of
three
ways
–
Working
uncertainties
absolute
cm
±
15%
using
uncertainty
out
the
uncertainty
There
are
some
These
are
introduced
range
mathematical
is
very
‘short-cuts’
time
that
consuming.
can
be
used.
would
in
the
boxes
below.
±∆p
ab
_
MUlTIPlIC ATION, DIvISION OR POwERS
In
symbols,
if
y
=
c
∆y
Whenever
two
or
more
quantities
are
multiplied
or
∆a
_
_
divided
Then
and
they
each
have
uncertainties,
the
overall
+
approximately
equal
to
the
addition
of
∆c
_
+
a
[note
this
is
ALWAYS
added]
c
b
uncertainty
Power
is
∆b
_
=
y
the
relationships
are
just
a
special
case
of
this
law.
percentage
n
If
(fractional)
y
=
a
uncertainties.
∆y
∆a
_
Then
Using
the
same
numbers
from
For
∆m
=
±
1
|n
=
y
above,
|
(always
positive)
a
example
if
a
cube
is
measured
along
1
∆m
_
each
side,
±
(
=
)
g
±
0.1
=
±
±
±
0.1
cm
in
length
0.1
_
10%
%
Uncertainty
in
length
=
±
0.2
=
±
2.5
%
4.0
Volume
3
=
4.0
g
10
3
∆V
be
then
_
=
m
to
g
=
(length)
3
=
(4.0)
3
=
64
cm
cm
3
%
Uncertainty
in
[volume]
=
%
uncertainty
=
3
×
(%
=
3
×
(±
=
±
in
[(length)
]
3
∆V
_
0.2 cm
_
=
±
(
5
V
The
total
%
=
)
3
±
0.04
=
±
4%
uncertainty
in
[length])
cm
uncertainty
in
the
result
=
=
±
±
(10
14
+
2.5
%)
4)%
7.5
%
%
3
Absolute
3
14%
of
2.0
g
cm
3
=
0.28
g
uncertainty
=
7.5%
of
64
cm
3
cm
≈
0.3
g
cm
3
=
4.8
cm
3
≈
5
cm
3
So
density
=
2.0
±
0.3
g
cm
as
before.
3
Thus
the
calculation
involves
mathematical
operations
other
division
or
raising
to
a
cube
=
64
±
5
cm
than
There
multiplication,
of
Oter unctions
OThER MAThEMATIC Al OPERATIONS
If
volume
power,
then
one
has
are
no
‘short-cuts’
possible.
Find
the
highest
and
lowest
to
values.
nd
the
highest
and
lowest
possible
values.
e.g.
the
each
two
have
addition
or
more
quantities
uncertainties,
of
the
sin
θ
if
θ
=
60°
±
5°
the
absolute
are
added
overall
or
subtracted
uncertainty
uncertainties.
is
and
equal
to
nis
they
of
θ
Addition or sutraction
Whenever
uncertainty
1
0.91
0.87
In
If
symbols
y
∆y
=
=
a
±
∆a
0.82
b
+
∆b
uncertainty
(note
of
ALWAYS
thickness
in
added)
a
pipe
wall
55
60
65
θ
external
=
radius
6.1cm
±
of
if
pipe
0.1cm
(≃
best
internal
=
thickness
of
pipe
radius
5.3cm
wall
=
±
6.1
of
θ
=
60°
sin
θ
=
0.87
max.
sin
θ
=
0.91
min.
sin
θ
=
0.82
∴
sin
θ
=
0.87
value
to
uncertainty
in
thickness
5°
pipe
0.1cm
(≃
2%)
±
0.05
5.3cm
worst
=
±
2%)
value
used
0.8cm
=
±(0.1
=
0.2cm
+
=
±25%
0.1)cm
M E A S U R E M E N T
A N D
U N C E R T A I N T I E S
5
Uncer tainties in raps
ERROR bARS
Plotting
one
a
time.
their
well
allows
Ideally
error
be
UNCERTAINTy IN SlOPES
graph
bars.
In
different
individually
all
of
one
the
to
points
principle,
for
every
worked
visualize
the
should
size
single
all
of
point
the
be
the
and
readings
plotted
error
so
If
with
bar
they
at
the
gradient
quantity,
could
should
to
be
out.
an
the
the
uncertainty
shallowest
with
of
then
the
lines
error
obtained.
This
in
the
has
been
bars)
the
process
of
gradient.
possible
a ytitnauq
a ytitnauq
best t line
graph
uncertainties
(i.e.
the
lines
to
the
that
range
represented
calculate
points
Using
uncertainty
is
used
the
will
give
steepest
are
for
still
the
a
rise
and
the
consistent
gradient
is
below.
steepest gradient
shallowest
gradient
quantity b
quantity b
A
full
analysis
gradient
use
of
of
the
a
in
order
best
to
determine
straight-line
error
bars
for
the
graph
all
of
uncertainties
should
the
data
always
in
the
make
points.
UNCERTAINTy IN INTERCEPTS
In
practice,
it
would
often
take
too
much
time
to
add
all
the
If
correct
error
bars,
so
some
(or
all)
of
the
following
the
intercept
quantity,
could
be
Rather
worst
the
that
value
plot
is
and
out
assume
error
that
bars
all
of
for
the
each
point
–
use
other
error
bars
the
are
within
Only
the
the
the
plot
These
error
limits
gradient
or
of
error
often
considering
the
the
bar
from
limits
the
are
an
the
the
uncertainty
shallowest
with
the
result.
furthest
within
•
working
same.
Only
is
than
the
for
the
of
this
all
the
bars
the
line
bar,
error
the
rst
ranges
(see
point,
t.
If
then
i.e.
the
it
the
line
will
of
error
This
graph
has
uncertainties
point
best
probably
and
the
points
last
in
t
be
points.
the
been
of
intercept.
possible
bars)
process
bars.
important
uncertainty
intercept
best
error
for
most
‘worst’
of
lines
a ytitnauq
•
then
used
the
to
points
calculate
will
give
a
rise
considered.
to
•
of
short-cuts
we
is
(i.e.
can
the
Using
lines
obtain
represented
the
the
that
steepest
are
still
uncertainty
and
the
consistent
in
the
below.
maximum value
of intercept
when
calculated
for
the
right).
best value
minimum value
•
Only
include
the
error
bars
for
the
axis
that
has
the
worst
for intercept
of intercept
uncertainty.
quantity b
6
M E A S U R E M E N T
A N D
U N C E R T A I N T I E S
vectors and scaars
DIffERENCE bETwEEN vECTORS AND SC Al ARS
If
you
unit.
measure
Together
Some
any
they
quantities
quantity,
express
also
quantity
that
quantity
whereas
has
have
a
that
must
the
have
and
has
a
number
magnitude
direction
magnitude
one
it
the
associated
direction
only
of
is
with
is
REPRESENTINg vECTORS
a
In
quantity.
called
magnitude
AND
a
them.
A
vector
called
most
books
whereas
would
a
be
quantity.
For
example,
all
forces
are
used
direction.
bold
The
to
letter
letter
is
represent
list
used
to
represents
below
a
force
shows
a
represent
scalar.
in
some
For
a
vector
example
magnitude
other
F
AND
recognized
methods.
a
F,
scalar
a
normal
F
or
F
vectors.
Vectors
are
best
shown
in
pull
The
in
table
the
lists
table
some
are
common
linked
to
one
quantities.
another
The
by
rst
their
two
quantities
denitions
diagrams
9).
All
the
others
are
in
no
particular
arrows:
(see
•
page
using
the
relative
magnitudes
order.
of
Vectors
Scalars
Displacement
Distance
Velocity
Speed
Acceleration
Mass
Force
Energy
Momentum
Temperature
the
are
vectors
shown
length
of
involved
by
the
the
relative
friction
arrows
normal
•
the
direction
is
shown
strength
(all
Potential
or
strength
of
the
arrows.
ADDITION / SUbTRACTION Of vECTORS
potential
If
we
have
force)
eld
the
forms)
difference
Magnetic
by
weight
direction
eld
the
reaction
vectors
Electric
of
can
a
3
N
and
a
4
N
force,
the
overall
force
(resultant
be
Density
3 N
anything
between
=
Gravitational
eld
strength
1
Although
the
vectors
used
7 N
Area
in
many
of
the
given
examples
N
and
7
4 N
N
are
depending
on
5 N
3 N
forces,
the
techniques
can
be
applied
to
all
vectors.
the
directions
=
involved.
4 N
The
way
to
take
the
It
is
also
possible
to
‘split’
one
vector
into
two
(or
more)
directions
vectors.
=
into
This
process
is
called
resolving
and
the
vectors
that
we
get
the
components
of
the
original
vector.
This
can
be
account
4 N
are
is
called
a
to
do
a
way
of
analysing
a
situation
if
we
choose
to
resolve
all
into
two
directions
that
are
at
right
angles
to
one
and
3 N
use
4 N
=
the
the
vectors
scale
very
diagram
useful
3 N
3 N
COMPONENTS Of vECTORS
1 N
parallelogram
another.
law
of
vectors.
b
This
process
is
adding
vectors
‘tail’
one
the
in
same
turn
–
as
the
F
F
a + b
F
ver tical
of
starting
vector
from
the
is
drawn
head
of
a
the
previous
vector.
Parallelogram
of
vectors
F
horizontal
Splitting
a
vector
into
components
TRIgONOMETRy
These
‘mutually
perpendicular’
directions
are
totally
Vector
independent
of
each
other
and
can
be
analysed
separately.
both
directions
can
then
be
combined
at
the
work
out
the
nal
resultant
this
can
be
always
time
be
solved
consuming.
using
The
scale
diagrams,
mathematics
trigonometry
often
makes
it
much
easier
to
use
the
vector.
mathematical
forces
can
very
end
of
to
problems
If
but
appropriate,
Push
appropriate
calculate
functions
when
the
of
sine
resolving.
values
of
either
or
The
of
cosine.
diagram
these
This
is
below
particularly
shows
how
components.
Surface
A
v
Weight
θ nisA =
force
A
A
v
components
θ
A
P
H
V
A
H
= Acos θ
S
P
H
H
See
page
14
for
an
example.
S
V
W
Pushing
a
block
along
a
rough
surface
M E A S U R E M E N T
A N D
U N C E R T A I N T I E S
7
to
Ib Questions – measurement and uncer tainties
1.
An
object
is
rolled
from
rest
down
an
inclined
plane.
The
3.
A
stone
is
dropped
down
a
well
and
hits
the
1
distance
travelled
by
the
object
was
measured
at
seven
different
it
is
released.
Using
the
equation
d
=
water
2.0
s
after
2
g
t
and
taking
2
2
times.
A
graph
was
then
constructed
of
the
distance
travelled
g
=
9.81
m
s
,
a
calculator
yields
a
value
for
the
depth
d
of
2
against
the
(time
taken)
as
shown
below.
the
well
the
best
as
19.62
m.
estimate
of
If
the
the
time
is
absolute
measured
error
in
d
to
±0.1
s
then
is
)mc( /del levart ecnatsid
9
A.
±0.1
m
B.
±0.2
m
In
order
to
C.
±1.0
m
D.
±2.0
m
8
4.
determine
the
density
of
a
certain
type
of
wood,
7
the
6
following
measurements
were
made
on
a
cube
of
the
wood.
Mass
=
493
g
5
Length
of
each
side
=
9.3
cm
4
The
percentage
±0.5%
3
of
and
length
uncertainty
the
is
percentage
in
the
measurement
uncertainty
in
the
of
mass
is
measurement
±1.0%.
2
The
1
0
0.0
0.1
0.2
0.3
0.4
2
(time taken)
A.
±0.5%
B.
±1.5%
Astronauts
What
a
(ii)
quantity
is
given
by
the
gradient
of
such
data
is
valid
the
but
graph
suggests
includes
a
that
the
systematic
Planet
Using
[2]
why
wish
cliff
a
as
error.
=
Do
these
results
suggest
that
distance
is
to
(time
2.46
taken)
?
Explain
your
answer.
allowance
for
the
systematic
from
s,
the
error,
acceleration
following
graph
of
the
shows
object.
that
have
been
calculated
and
D.
±3.5%
density
is
the
gravitational
tape
measure
m
the
of
a
stones
±
0.01
cliff,
they
m.
timing
They
then
fall
displays
The
an
acceleration
overhanging
measure
each
which
second.
from
the
height
drop
using
three
a
times
the
similar
hand-held
readings
recorded
cliff.
of
to
for
one-
three
s
and
2.40
drops
are
s.
Explain
a
why
tenth
the
of
time
a
readings
second,
vary
although
by
the
more
stopwatch
[2]
same
data
after
the
drawn
as
error
readings
to
one
hundredth
of
a
second.
[1]
uncertainty
b)
ranges
the
±3.0%
calculate
gives
The
determine
stopwatch
2.31
than
b)
in
C.
[2]
a)
Making
uncertainty
proportional
2
(iv)
to
dropping
7.64
electronic
[2]
by
steel
s
stones
collected
X
hundredth
(iii)
the
2
graph?
Explain
for
/ s
on
(i)
estimate
0.5
5.
a)
best
Obtain
the
average
time
t
to
fall,
and
write
it
in
bars.
)mc( /del levart ecnatsid
the
form
(value
±
uncertainty),
to
the
appropriate
9
number
of
signicant
digits.
[1]
8
c)
The
astronauts
then
determine
the
gravitational
2s
acceleration
a
on
the
planet
using
the
formula
a
g
=
Calculate
a
from
the
values
of
s
and
t,
and
.
2
g
t
7
determine
the
g
uncertainty
6
the
in
the
calculated
value.
Express
the
result
in
form
5
a
=
(value
±
uncertainty),
g
to
the
appropriate
number
of
signicant
digits.
[3]
4
3
HL
2
6.
1
This
question
forces
In
an
is
between
about
nding
magnets
experiment,
two
and
the
their
magnets
relationship
between
the
separations.
were
placed
with
their
North-
0
0.0
0.1
0.2
0.3
0.4
0.5
2
(time taken)
seeking
poles
and
separation
two
lines
acceptable
2.
The
the
lengths
of
diagram
to
show
values
the
for
sides
shows
the
the
the
of
a
range
of
gradient
rectangular
measured
the
of
another.
The
force
of
repulsion,
the
are
shown
of
in
the
the
magnets,
table
d,
were
measured
f,
and
the
below.
possible
the
plate
values
one
2
/ s
results
Add
facing
with
graph.
are
[2]
measured,
their
Separation
and
d/m
Force
of
repulsion
0.04
4.00
0.05
1.98
0.07
0.74
0.09
0.32
f/N
uncertainties.
50 ± 0.5 mm
25 ± 0.5 mm
a)
Plot
a
b)
The
law
of
Which
one
of
the
following
would
be
the
best
estimate
of
graph
the
of
log
relating
(force)
the
force
against
to
the
log
(distance).
separation
[3]
is
form
the
n
f
percentage
uncertainty
in
the
calculated
area
A.
±
0.02%
C.
±
3%
B.
±
1%
D.
±
5%
8
I B
Q U E S T I o N S
–
of
the
=
kd
plate?
M E A S U R E M E N T
A N D
(i)
Use
the
graph
(ii)
Calculate
a
to
value
U N C E R T A I N T I E S
nd
for
the
k,
value
giving
of
its
n.
units.
[2]
[3]
2
m e c h a n i c s
m
Definitions
These
technical
terms
•
Vector
quantities
•
Generally,
•
The
units
•
The
denition
should
always
velocity
and
not
be
have
a
speed
confused
direction
are
NOT
with
their
associated
the
same
‘everyday’
with
thing.
use.
This
is
•
acceleration
of
acceleration
Whenever
the
constantly
increasing
A
motion
deceleration
come
of
an
speed
means
is
–
it
Symbol
its
denition.
precise.
object
slowing
Displacement
from
is
It
is
related
changes,
possible
down,
i.e.
(m
it
to
is
s
to
negative
÷
s
=
change
while
at
acceleration
Denition
s
The
m
should
note
that
important
if
the
object
is
not
in
velocity
For
this
constant
if
(not
speed
velocity
is
the
reason
if
same
thing
acceleration
the
direction
as
moved
in
a
The
the
does
is
in
a
straight
line.
change
not
in
speed).
necessarily
mean
changed.
positive.
Example
distance
going
s
acceleration.
accelerate
one
2
)
the
called
particular
particularly
1
of
In
them.
displacement
from
London
SI
Vector
Unit
scalar?
to
m
or
Vector
6
particular
direction.
Rome
is
1.43
×
10
m
southeast.
1
Velocity
v
or
u
The
rate
of
change
of
The
average
velocity
during
a
ight
m
s
m
s
m
s
Vector
1
displacement.
from
change
of
displacement
to
Rome
is
160
m
s
southeast.
________________
velocity
London
=
time
taken
1
Speed
v
or
u
The
rate
of
change
distance
of
distance.
The
speed
average
speed
during
a
ight
Scalar
1
gone
from
__________
London
to
Rome
is
160
m
s
=
time
taken
2
Acceleration
a
The
rate
of
change
of
change
velocity.
of
The
velocity
on
_____________
acceleration
average
the
acceleration
runway
during
of
a
plane
take-off
Vector
is
=
time
taken
2
3.5
m
s
means
in
that
a
forwards
on
direction.
average,
its
This
velocity
1
changes
But
instantaneous vs average
It
should
of
time)
be
is
noticed
very
that
different
the
to
average
the
value
(over
instantaneous
a
(at
during
time.
period
value
This
one
every
At
second
the
the
means
race,
end
that
of
by
her
the
her
3.5
m
s
instantaneous
rst
2.0
average
speed
seconds,
speed
over
she
the
was
had
rst
changing
travelled
2.0
all
the
10.04
seconds
m.
was
1
particular
5.02
time).
m
speed
In
the
example
below,
the
positions
of
a
sprinter
are
shown
s
.
was
During
these
increasing
–
rst
she
two
was
seconds,
her
accelerating.
If
instantaneous
she
started
at
rest
at
1
(speed
different
times
after
the
start
of
a
=
0.00
m
s
)
5.02
m
and
her
average
speed
(over
the
whole
two
race.
1
seconds)
The
average
speed
over
the
whole
race
is
easy
to
work
out.
the
total
distance
(100
m)
divided
by
the
total
time
(11.3
s
then
her
instantaneous
speed
at
2
seconds
It
must
is
was
be
more
than
this.
s)
In
fact
the
instantaneous
speed
for
this
1
sprinter
was
9.23
m
s
,
but
it
would
not
be
possible
to
work
this
1
giving
8.8
m
s
.
out
from
the
information
given.
star t
nish
d = 0.00 m
d = 10.04 m
d = 28.2
1 m
d = 47
.89 m
d = 69.12 m
t = 2.0 s
t = 4.0 s
t = 6.0 s
t = 8.0 s
t = 0.0 s
In
frames of reference
technical
moving
If
two
things
are
moving
in
the
terms
what
we
from
one
frame
of
another.
The
velocities
are
doing
is
reference
to
a
d = 100.00 m
t = 11.3 s
stationary
road.
We
observer
moved
from
on
this
the
side
frame
of
of
the
reference
same
1
into
straight
line
but
are
travelling
at
of
25
m
into
s
the
driver’s
frame
of
reference.
different
1
and
speeds,
then
relative
straight
can
velocities
subtraction
imagine
we
as
two
road
work
by
out
simple
appropriate.
cars
at
different
m
s
were
measured
or
increases
example,
along
30 m s
1
1
by 5 m s
a
speeds.
25 m s
1
If
one
car
(travelling
overtakes
the
other
at
car
according
gap between the cars
addition
For
travelling
30
their
30
m
s
1
)
(travelling
at
1
25
m
the
s
),
slow
then
car,
according
the
relative
to
the
driver
velocity
of
of
the
one
car
overtaking
another,
as
seen
by
an
1
fast
car
is
+5
m
s
observer
on
the
side
of
the
road.
one
car
driver
overtaking
of
the
slow
another,
as
seen
by
the
car.
m e c h a n i c s
9
g   
2.
the use of graphs
Graphs
are
happen
graphs
•
very
when
that
useful
an
can
for
object
is
provide
displacement–time
representing
in
motion.
useful
or
the
changes
There
are
that
three
To
possible
make
in
one
not
graphs
velocity–time
or
speed–time
much
acceleration–time
are
two
common
from
these
methods
of
determining
particular
depends
graphs.
The
particular
physical
Finding
To
be
a
the
little
on
what
gradient
more
is
being
plotted
on
precise,
of
the
one
the
a
straight-line
are
it
nd
either
at
value),
the
clear
object
this
is
the
versions
from
the
section
of
the
graph
(this
the
nds
component
moves
distinguish
gradient
of
positive.
should
Many
an
graphs
the
that
case
scalar
then
versions
(displacement
or
are
are
just
there
moving
is
(distance
velocity)
or
as
situation.
More
complicated
the
graphs
of
a
velocity
in
a
particular
direction.
the
forward
two
then
velocities
backward
by
(or
choosing
up
then
which
down),
direction
be
It
does
clearly
For
matter
labelled
examination
up.
not
on
candidates
example
a
which
the
get
direction
we
choose,
graph.
the
three
speed–time
types
graph
of
graph
might
as
a
distance–time
graph
or
even
an
be
acceleration–
or
tangent
to
the
instantaneous
graph
at
one
point
(this
nds
graph.
Always
look
at
the
axes
of
a
graph
very
carefully.
an
value).
Displ acement–time graphs
velocity–time graphs
acceleration–time graphs
•
•
•
The
gradient
graph
is
the
of
a
displacement–time
The
velocity
gradient
graph
is
the
of
a
velocity–time
The
acceleration
The
area
under
a
displacement–time
•
The
area
under
a
gradient
time
is
•
to
but
average
time
the
If
between
vector
the
interpreted
•
direction.
the
objects
graph.
line.
could
look
muddled
•
beginning,
considering
quantity
call
1.
the
line.
physical
we
determined
at
by
the
graphs.
If
quantities
the
under
graphs
can
There
simple
difference
and
directions
•
area
introduced
particular
speed)
•
the
things
normally
information
distance–time
Finding
graph
actually
of
is
an
not
the
acceleration–
often
rate
of
useful
change
(it
of
velocity–time
acceleration)
graph
does
not
represent
anything
graph
is
the
displacement
useful
•
The
area
time
e
e
speed =
1
rst 4 seconds
acceleration is constantly
object at constant
increasing, rate of change
acceleration of 20 m s
of velocity is increasing
see = 20 m s

velocity still changing all
=
5 m s
the time
acceleration is ero
rate of
object is slowin
own acceleration
=
1
20
1
= 5 m s
2
=
acceleration
is decreasing,
20.0
20
10.0
10.0
speed =
velocity
2
at constant
speed
in
2
1
4
s m / noitarelecca
= 20 m s
acceleration–
change
acceleration =
20.0
1
20.0
an
the
2
m / tnemecalpsid
20
1
speed = 0 m s
object at constant
20
s m / yticolev
1–
faster speed
for 3 seconds
is
e
objects velocity is increasin
object returns at a
object stationary
under
graph
-20 m s
4
but velocity
continues
to increase
10.0
0
0
1.0
2.0
3.0
4.0
5.0
6.0
7
.0
8.0
1.0
2.0
3.0
4.0
5.0
6.0
7
.0
time / s
8.0
time / s
istance travelle in rst 4 secons
Object moves at constant speed, stops then returns.
time / s
= area ner rah
1
1
1
Change in velocity =
=
× 4 × 20 = 40 m s
× 4 × 20 m = 40 m
2
2
Object moves with increasing, then constant,
Object moves with constant acceleration
then decreasing acceleration.
then constant velocity then ecelerates.
initial war
2
velocity is +ve
ma. heiht = area ner rah
s m / noitarelecca
1
4.0
=
+10.0
× 0. × .0 m = 4.05 m
.0
2
1
.0
s m / yticolev
m / tnemecalpsid
highest point at t = 0.9 s
2.0
object returns to
hand at t = 1.8 s
2
acceleration
10 m s
0.
1.0
2.0
time / s
1.0
2.0
time / s
–10.0
change in velocity = area under graph
instantaneos
ownwar
1.0
level of
1
= -10.0 ×1.8 m s
velocity = ero
.0
2.0
velocity is
1
= -18 m s
at hihest oint
time / s
hand as zero
neative
displacement
1
(change from +9.0 to
t = 0. s
Object is thrown ver tically upwards.
Object is thrown ver tically wars.
9.0 m s
)
Object is thrown ver tically upwards.
e x ample of equation of uniform motion
A
car
speed
accelerates
after
8
1
(i)
s
=
ut
uniformly
from
rest.
After
8
s
it
has
travelled
120
m.
Calculate:
s
+
2
2
at
(ii)
v
2
=
u
=
0
+
=
900
=
30
2
as
2
1
∴
120
=
0
×
8
a
=
3.75
+
+
2
×
2
a
×
8
=
32
a
2
–1
–2
10
m
s
m e c h a n i c s
∴
v
m
s
3.75
×
120
(i)
its
average
acceleration
(ii)
its
instantaneous
u d 
practic al
equations of
falling objects
A
c alcul ations
very
order
velocity
of
an
to
the
object
situations,
to
determine
(or
record
its
is
how
the
acceleration)
varies
it
in
often
real
used
is
necessary
motion.
These
when
the
constant
if
–
this
can
the
forget
eld.
If
being
in
Taking
in
of
variables
considered
symbols)
is
as
(and
their
follows
l 
light
senses
gate
when
through
time
is
a
for
broken
a
that
object
cuts
an
beam
which
is
device
of
light.
the
recorded.
of
the
breaks
the
If
object
beam
is
initial
v
nal
a
acceleration
average
speed
t
time
s
distance
be
through
velocity
velocity
(const)
two
a
timer
can
light
be
following
equations
different
link
quantities.
can
v
=
u
s
=
(
+
at
u + v
_
gates
used
)
t
to
2
calculate
the
between
the
average
two
velocity
joined
and
a
gates.
=
u
computer
together
to
calculations
of
+
s
=
ut
+
make
s
velocity
=
The
light
gives
as
positive,
out
the
graphs
ashes
of
time
intervals.
light
at
2
vt
at
rst
from
very
equation
the
is
denition
acceleration.
In
derived
of
xed
these
symbols,
terms
this
a
pointed
at
an
camera
object
is
known
as
of
the
motion
of
any
object
20
5
2.0
3.0
time / s
2.0
3.0
time / s
3.0
time / s
30
20
10
2
acceleration / m s
10
of
would
2.0
the
absence
of
air
resistance,
all
falling
objects
have
the
be
SAME
is
this
denition
In
If
resistance,
displacement / m
1.0
brief
air
are
1.0
2
acceleration.
strobe
of
2
at
can
s 
A
effects
2as
2
or
the
Several
1
direct
is
45
ev+ sdrawnwod
be
gates
motion
gravitational
2
v
1
light
accelerated
uniform
1
the
gate
a
velocity / m s
2
and
ignore
down
travelled
known,
the
in
taken
calculated.
Alternatively,
uniformly
1.0
these
object
of
object
free-fall
the
that
of
an
is
The
the
we
free-fall
ev+ sdrawnwod
length
u
The
beam
of
to
include.
be
A
example
motion
to
case!
Possible
list
vertical
be
acceleration
don’t
is
only
ev+ sdrawnwod
methods
equations
check
The
laboratory
important
uniform motion
the
In
and
(v
acceleration
of
free-fall,
INDEPENDENT
of
their
mass.
u)
_
a
the
only
source
the
strobe
developed
light,
then
picture
an
light
object’s
Air
t
the
will
=
is
have
This
can
give
the
be
rearranged
rst
objects.
to
The
=
u
+
at
second
(1)
equation
comes
t = 0.0 s
from
the
denition
of
t = 0.1 s
will
Typically,
resistance
equation.
motion.
v
resistance
ev+ sdrawnwod
captured
of
(eventually)
the
become
graphs
the
of
shapes
affect
a
the
falling
shown
motion
object
of
all
affected
by
air
below.
displacement / m
straight line as
20
average
velocity.
t = 0.2 s
s
average
velocity
=
t
velocity becomes
constant
5
t = 0.3 s
Since
the
velocity
is
changing
1.0
uniformly
we
know
that
2.0
3.0
time / s
this
1
velocity / m s
given
velocity
must
be
ev+ sdrawnwod
average
t = 0.4 s
23
by
20
(v
+
u)
_
average
velocity
=
2
terminal
10
(u
s
+
v)
_
or
=
t
t = 0.5 s
This
2
can
be
rearranged
velocity of
1
23 m s
to
1.0
t 
2.0
acceleration = zero
timer
can
be
(u
+
v)t
_
s
a
strip
of
to
make
paper
dots
of
at
time
regular
other
ftieth
of
a
second).
piece
of
paper
is
an
object,
and
the
allowed
to
fall,
strip
will
the
the
distance
have
these
two
substituting
dots
by
equations
for
one
of
1.0
variables
(see
As
for
an
recorded
2.0
3.0
time / s
previous
on
page
the
of
derived
at terminal velocity
object
the
is
be
attached
and
to
can
If
using
the
equations
(typically
motion
every
(2)
2
The
intervals
=
on
ev+ sdrawnwod
ticker
arranged
time / s
2
acceleration / m s
A
3.0
give
example
the
graphs
show,
the
velocity
does
not
keep
on
rising.
It
of
eventually
reaches
a
maximum
or
terminal
velocity.
A
theiruse).
object
in
a
moved
known
by
the
time.
piece
of
shorter
falling
time
paper
than
a
will
reach
falling
its
terminal
velocity
in
a
much
book.
m e c h a n i c s
11
p 
hz 
components of projectile motion
If
two
them,
children
are
the
of
known
as
path
throwing
the
ball
projectile
is
and
catching
always
motion
and
the
the
a
tennis
same
ball
shape.
shape
is
This
called
There
between
a
motion
are
no
horizontal
is
must
parabola
be
forces
in
the
acceleration.
horizontal
This
means
direction,
that
the
so
there
horizontal
is
no
velocity
constant.
ball travels at a constant horizontal velocity
v
v
v
3
H
2
v
v
v
v
4
H
H
v
1
H
v
5
path taken by ball
v
v
H
H
is a parabola
d
d
H
d
H
d
H
d
H
H
v
6
v  
The
only
forces
acting
during
its
ight
are
gravity
and
friction.
There
In
many
situations,
air
resistance
can
be
a
It
is
moving
horizontally
and
vertically
is
a
constant
vertical
force
acting
down,
so
there
is
ignored.
at
the
same
constant
vertical
acceleration.
The
value
of
the
vertical
time
2
acceleration
but
the
horizontal
and
vertical
components
of
the
motion
is
10
m
s
,
which
v
of
one
another.
Assuming
the
gravitional
the
acceleration
v
2
independent
is
force
3
is
is
always
gravity.
H
v
v
this
to
v
ver tical
constant,
due
are
v
4
H
true.
H
velocity
v
v
1
H
v
5
v
v
H
H
changes
v
6
mathematics of parabolic motion
The
graphs
of
the
components
of
parabolic
   x-d
e x ample
motion
are
shown
below.
   y-d
A
a
projectile
is
launched
horizontally
from
the
top
of
cliff.
2
2
initial horizontal velocity
s m /
s m /
u
H
a
a
y
x
0
0
t / s
t / s
g
height
of cli
h
1
1
u
y
s m /
s m /
slope =
y
x
v
u
-g
v
x
0
0
t / s
t / s
x
u
H
m / y
m / x
slope = u
x
maximum
height
v
v
vertical
0
u
0
=
motion
horizontal
0
u
=
f
motion
u
H
t / s
v
t / s
=
?
v
=
u
H
2
a
Once
the
components
have
been
worked
out,
the
actual
velocities
s
(or
t
displacements)
at
any
time
can
be
worked
out
by
vector
=
=
=
10
m
s
a
h
?
=
0
s
=
x
t
=
?
addition.
1
The
solution
of
any
problem
involving
projectile
motion
is
as
2
follows:
s
=
ut
+
at
2
•
use
the
angle
•
the
time
of
launch
to
resolve
the
initial
velocity
into
components.
1
2
of
ight
will
be
determined
by
the
vertical
component
of
so
h
=
0
+
×
10
×
t
2
velocity.
2h
2
∴
•
the
range
time
of
will
be
determined
by
the
horizontal
component
(and
t
=
10
the
2h
ight).
t
=
s
10
•
the
velocity
at
any
point
can
be
found
by
vector
addition.
Since
Useful
‘short-cuts’
in
calculations
include
the
following
v
=
u
+
at
v
=
0
+
10
facts:
2h
•
for
•
if
a
given
two
speed,
objects
the
are
greatest
released
range
is
together,
achieved
one
if
with
the
a
launch
angle
horizontal
is
m
45°.
x
=
u
×
t
H
10
velocity
1
=
20h
2h
=
u
×
m
H
and
one
from
rest,
they
will
both
hit
the
ground
together.
10
The
final
velocity
v
f
12
m e c h a n i c s
is
the
vector
addition
of
v
and
u
H
fd  d  -
fluiD resistance
When
An
or
an
object
example
a
of
moves
this
parachutist
Modelling
Physics
moves
the
option
precise
through
•
Viscous
•
The
falling
(see
a
drag
drag
Relative
•
The
shape
•
The
uid
For
example
path.
When
effect
will
is
and
a
12
of
the
of
(and
a
shows
resistance
reduced
uid
the
(a
to
or
a
note
a
gas),
that
page
on
is
11
for
moving
mathematical
are
there
will
reached
how
by
the
objects
a
frictional
free-falling
motion
is
analysis
be
a
graphs
complex
of
the
uid
resistance
object,
but
will
be
simple
frictional
e.g.
drag
a
that
altered
in
predictions
force
affects
spherical
that
mass
these
are
acts
the
motion.
through
a
liquid
situations.
possible.
on
object’s
falling
a
The
perfect
Engineering
sphere
when
it
that:
through
a
uid
on:
object
with
object
property
how,
in
and,
called
its
to
of
account,
the
the
the
uid
object
is
aerodynamic
or
not)
viscosity).
absence
into
in
respect
(whether
the
taken
range
See
resistance
motion
the
is
liquid
velocity
Earth.
uid
points
oppose
size
a
terminal
introduces
dependent
used
uid
be
to
of
167)
Key
velocity
page
the
effect
page
acts
is
towards
uid.
force
•
through
effect
uid
the
extreme,
resistance,
vertical
the
and
horizontal
an
the
object
that
horizontal
velocity
can
is
in
projectile
components
be
reduced
to
of
motion
will
velocity
near
follow
will
both
a
parabolic
be
reduced.
The
zero.
parabolic path (no uid resistance)
path (with uid resistance)
e xperiment to Determine free-fall acceleration
All
experiments
recorded
than
to
determine
measurements
others.
equipment
This
often
of
increased
means
the
free-fall
displacement
use
that,
of
acceleration
and
time.
technology
with
a
limited
for
Some
potentially
time
an
object
are
experimental
brings
available
for
based
on
set-ups
greater
the
will
precision
experimentation,
use
be
but
it
is
of
a
more
can
constant
introduce
easier
acceleration
sophisticated
for
and
more
many
use
an
object
free-falls
a
height,
h,
from
rest
in
a
time,
t,
the
acceleration,
g,
can
be
calculated
using
s
=
ut
to
be
with
equipment
complications.
repetitions
1
If
equation
more
Simple
attempted.
2
+
at
which
rearranges
2
2h
to
give
=
.
Rather
than
just
calculating
a
single
value,
a
more
reliable
value
comes
from
taking
a
series
of
measurement
of
the
2
t
1
different
times
of
fall
for
different
heights
h
2
=
gt
2
.
A
graph
of
h
on
the
y-axis
against
t
on
the
x-axis
will
give
a
straight
line
graph
2
1
that
goes
through
the
origin
with
a
gradient
equal
to
g,
making
g
twice
the
gradient.
2
Possible
set-ups
include:
Set-up
Direct
e.g.
a
Comments
measurement
ball
metre
bearing
of
with
a
a
falling
stop
object,
watch
and
ruler
Very
simple
eliminated.
equipment
air
release
timing
of
version
Motion
of
recording
falling
falling
on
the
and
electronic
above
object
attached
object
Distance
The
to
fall
to
be
and
data
logger
All
to
analysis
of
falling
object
allows
Addition
all
Capturing
a
the
how
often
involves
a
of
of
to
simple
the
just
easily
controlled,
everyday
the
achieved
great
object
laboratory
timing
be
record
so
random
precision
such
whereas
and
the
of
the
of
overall
paper
tape
as
a
the
motion
plot
very
errors
and
taken)
introduces
is
ball
error
possible
bearing,
effect
of
air
fall
a
from
set-up
harder
allows
the
to
graphs.
can
even
the
be
though
effect
resistance
of
on
a
will
to
object’s
to
be
motion,
can
take
be
longer.
identify.
whole
graphically
however.
programmed
Experimenter
needs
to
software.
known
the
the
data
the
Software
associated
against
information
but
are
for
friction
precise.
and
thus
and
appropriate
logger
object’s
Timing
accuracy
systematic
time
and
to
data
improve
that
analysis
automated
taken.
can
mean
the
moving
operate
be
in
can
calculations
to
repetitions
carefully
detailed
(not
can
visual
measurements
For
technology
measurements
perform
is
signicant.
precision
of
many
fall
negligible
be
considered
understand
Video
be
will
record
analysed.
sensor
of
standard.
will
increased
Physical
meaning
height
ball
Introduction
automatically
ticker-tape
is
resistance
Ping-Pong
Electromagnet
set-up
If
scale,
video
allows
recording
detailed
needed,
which
ICT.
m e c h a n i c s
13
f d  -d d
•
forces – what the y are anD what the y Do
In
the
(the
ball
gains
but
of
examples
a
in
a
below,
changes
velocity).
general
shape)
There
terms
deformation
measurement
of
a
or
or
are
one
a
force
a
can
is
kick)
change
many
in
the
can
any
change’.
newton
cause
motion
different
describe
velocity
forces
(the
(the
types
force
The
SI
of
as
can
say
force
cause
the
•
The
(N).
but
causes
is
zero
change
a
NOT
the
one
(a) deformation
is
a
that
fact
force
force
Remember
ball
for
(resultant)
(resultant)
forces,
‘the
unit
A
deformation
force
needed
that
a
force
–
there
CHANGE
in
for
of
was
the
velocity
causes
a
can
deformation
force
a
then
is
an
the
called
is
an
velocity
was,
one
in
the
acceleration,
A
(see
is
fact,
from
If
constant.
deformation
ball
another
velocity.
acceleration.
constant
cause
in
velocity
page
also
not
the
so
we
(resultant)
16).
important,
caused
by
just
wall.
(b) change in velocity
•
One
force
precise
can
the
act
on
only
description
•
its
magnitude
•
its
direction
•
the
object
on
•
the
object
that
•
the
nature
of
one
a
object.
force
To
should
be
absolutely
include
kick
kick
kick causes
deformation of football
of
a
force
on
a
motion of football
following
pulls)
that
exist
in
all
describe
the
Electrostatic
Magnetic
One
way
(the
pushes
or
force
the
Normal
force
reaction
(or
the
part
of
a
large
object)
force
Upthrust
Tension
Lift
these
N
of
the
force
push
the
at
force
20°
to
shown
the
in
the
horizontal
example
acting
would
ON
the
thus
football
boot’.
forces
is
forces
are
resultant
split
into
vectors,
force
its
from
vector
two
components.
mathematics
or
more
See
other
page
7
for
must
forces.
more
be
A
used
force
to
nd
can
also
details.
Compression
Friction
categorizing
50
Since
nature.
force
of
the
acts
the
forces a s vectors
forces
be
Gravitational
‘a
FROM
football
words
description
be
Different types of forces
The
of
it
exerts
kick causes a change in
A
Effect
which
whether
(a) by vector mathematics
example: block being pushed on rough surface
they
force diagram:
result
S, surface force
from
the
contact
between
two
surfaces
or
whether
the
P, push force
force
resultant
exists
even
if
a
distance
separates
the
W
S
objects.
force
The
origin
of
all
these
everyday
forces
is
either
gravitational
W
or
P
electromagnetic.
observe
are
due
The
to
vast
majority
of
electromagnetic
everyday
effects
that
weight
we
forces.
(b) by components
example: block sliding down a smooth slope
me a suring forces
The
simplest
force
is
to
experimental
use
the
method
extension
of
a
for
measuring
spring.
When
a
the
size
spring
of
is
a
in
resultant down
R
tension
it
increases
in
length.
The
difference
between
the
natural
slope = W sin θ
length
and
stretched
length
is
called
the
extension
of
a
spring.
component into slope
resultant into
= W cos θ
θ
slope = W cos θ - R
θ
= zero
original
component down slope
length
= W sin θ
W, weight
ex tension
Vector
= 15.0 cm
addition
ex tension
2 N
= 5.0 cm
free-boDy Diagrams
6 N
In
a
free-body
diagram
mc / noisnet xe
15.0
mathematically,
•
one
•
all
object
(and
ONL
Y
one
object)
is
chosen
F ∝ x
10.0
the
forces
on
that
object
are
shown
and
labelled.
F = kx
For
example,
if
we
considered
the
simple
situation
of
a
book
spring constant
5.0
resting
on
a
table,
we
can
construct
free-body
diagrams
for
1
(units N m
)
either
the
book
force / N
2.0
4.0
6.0
or
the
table.
free-body diagram
8.0
free-body diagram
for book:
situation:
for table:
P, push from
R
, reaction from table
T
Hooke’s
law
Hooke’s
law
spring
is
book
states
of
a
of
proportionality
that
up
proportional
k
is
to
to
called
the
the
the
elastic
limit,
tension
spring
the
force,
F.
extension,
The
constant.
x,
R
constant
The
, reaction
E
from Ear th’s
SI
R
E
W
1
units
the
for
the
spring
extension,
14
we
constant
can
are
calculate
m e c h a n i c s
N
m
the
.
Thus
by
measuring
surface
w, weight of book
weight of table
gravitational pull of Ear th
gravitational pull of Ear th
force.
n’  
ne wton’s first l aw
Newton’s
acts’.
says
On
is
rst
rst
that
a
law
of
motion
reading,
resultant
this
states
can
force
that
sound
causes
‘an
object
complicated
acceleration.
continues
but
No
it
in
does
resultant
uniform
not
really
force
b     
motion
add
means
in
a
straight
anything
no
to
the
acceleration
–
line
or
at
rest
description
i.e.
of
‘uniform
unless
a
a
force
motion
resultant
given
in
a
on
external
page
straight
14.
force
All
it
line’.
l    
R
P, pull from person
R, reaction from ground
W
W, weight of suitcase
If the suitcase is too heavy to lift, it is not moving:
∴ acceleration = zero
since
acceleration
= zero
resultant force
= zero
∴ P + R = W
c     
∴ R
W
= zero
R
R
P
p   
F, air friction
F
W
F is force for wards, due to engine
P is force backwards due to air resistance
At all times force up (2R) = force down (W).
If F > P the car accelerates for wards.
If F = P the car is at constant velocity (zero acceleration).
parachutist
If F < P the car decelerates (i.e. there is negative
free-falling
acceleration and the car slows down).
downwards
p       d
W, weight
If W > F the parachutist accelerates downwards.
W = F
The parachutist is at constant velocity
(the acceleration is zero).
sdrawpu gnivom tfil
As the parachutist gets faster, the air friction increases until
R
R
2
2
W
The total force up from the oor of the lift = R
The total force down due to gravity = W
If R > W the person is accelerating upwards.
If R = W the person is at constant velocity
(acceleration = zero).
If R < W the person is decelerating (acceleration is
negative).
m e c h a n i c s
15
e
equilibrium
If
the
be
in
resultant
force
translational
Mathematically
on
an
object
is
equilibrium
this
is
expressed
zero
(or
as
then
just
in
it
is
said
to
equilibrium).
follows:
Translational
equilibrium
being
For
is
at
rest.
allowed
to
swing
instantaneously
Σ
From
F
=
rst
situations
law,
we
must
know
be
in
1.
An
object
that
is
constantly
2.
An
object
that
is
moving
in
Since
a
at
back
rest
if
and
but
NOT
the
forth,
he
mean
child
is
in
there
never
the
the
in
same
thing
previous
are
times
as
example
when
she
is
equilibrium.
zero
Newton’s
following
does
example
straight
forces
are
that
the
objects
in
the
equilibrium.
at
rest.
with
constant
(uniform)
velocity
line.
vector
quantities,
a
zero
resultant
force
means
T
no
force
IN
ANY
DIRECTION.
T
For
2-dimensional
forces
case
balance
then
the
in
problems
any
object
two
is
in
it
is
sufcient
non-parallel
to
show
directions.
that
If
this
T
the
is
the
equilibrium.
θ
W
W
tension, T
W
At the end of the
Forces are not
swing the forces
balanced in the centre
are not balanced
as the child is in circular
but the child is
motion and is
instantaneously
accelerating (see page 65).
P, pull
at rest.
weight, W
if in equilibrium:
Tsin θ = P (since no resultant horizontal force)
Tcos θ = W (since no resultant ver tical force)
Different types of forces
Name
of
force
Gravitational
force
Description
The
force
object
Electrostatic
Normal
is,
as
objects
force
The
force
between
magnets
reaction
The
force
between
two
this
is
The
force
Tension
Compression
The
Upthrust
be
a
rod
(or
force
is
tension
This
is
that
causes
the
This
force
wing
m e c h a n i c s
a
is
of
result
and/or
as
spring)
the
is
between
force
electric
right
of
force
it
that
it
–
is
page
sometimes
referred
to
as
the
weight
of
the
19.
charges.
angles
to
two
of
has
the
surfaces
technically
has
end
(squashed),
the
This
see
currents.
at
motion
the
–
the
surfaces.
If
two
surfaces
are
smooth
then
them.
stretched,
that
masses.
their
acts
frictional
force
is
of
electric
that
relative
a
their
ambiguous
equal
the
equal
ends
and
string
the
acts
is
to
opposite
rod
along
known
opposite
applies
and
of
and
this
the
as
forces
on
another
forces
applies
to
surfaces.
uid
its
Air
resistance
or
friction
ends
pulling
outwards.
object.
on
its
another
ends
pushing
object.
This
inwards.
is
the
opposite
force.
some
can
a
acts
compressed
upward
of
as
the
of
force
result
surfaces
that
opposes
compression
the
the
force
thought
string
tension
When
of
a
only
that
can
When
The
the
a
unfortunately,
between
drag
16
objects
term
force
Friction
Lift
this
The
Magnetic
force
between
but
be
an
force
that
objects
exerted
aircraft
to
acts
oat
on
an
causes
on
in
an
object
the
object
water
(see
when
a
when
page
uid
aerodynamic
lift
it
is
submerged
in
a
uid.
It
is
the
buoyancy
force
164).
ows
that
over
enables
it
in
the
an
asymmetrical
aircraft
to
y
way.
(see
The
page
shape
166).
of
n’ d 
e x amples of ne wton’s seconD l aw
ne wton’s seconD l aw of motion
Newton’s
an
rst
acceleration.
calculating
way
of
law
of
the
the
stating
states
His
second
value
the
that
of
of
force
provides
a
acceleration.
law
an
resultant
law
this
second
momentum
a
is
use
object.
the
This
causes
means
The
1.
of
Use
of
F
=
ma
in
a
simple
situation
12 N
3 kg
best
concept
concept
no friction between block and surface
is
If
a
mass
of
3
kg
is
accelerated
in
a
4
m
straight
line
by
a
resultant
force
of
2
explained
A
correct
using
on
page
23.
statement
momentum
of
12N,
Newton’s
would
be
second
‘the
law
resultant
force
F
=
a
=
the
acceleration
to
the
rate
of
change
of
F
momentum’.
be
s
.
Since
ma
is
12
_
2
=
=
m
proportional
must
4
m
s
3
If
2
2.
we
use
SI
units
(and
you
always
should)
then
the
Use
a
is
even
easier
to
state
–
‘the
resultant
force
is
of
F
=
ma
equal
slightly
more
to
complicated
the
rate
of
expressed
change
as
of
momentum’.
In
symbols,
acceleration = 1.5 m s
in
law
this
situation
is
12 N
If
follows
a
mass
of
3
accelerated
∆p
kg
in
a
3 kg
friction force
is
straight
line
by
a
force
of
12
N,
and
the
resultant
_
In
SI
units,
F
=
2
acceleration
∆t
dp
have
_
or,
in
full
calculus
notation,
F
been
is
1.5
acting.
m
s
,
then
we
can
work
out
the
friction
that
must
Since
=
dt
F
p
is
the
symbol
for
the
momentum
of
a
resultant
Until
not
you
make
given
have
studied
much
here
for
sense,
what
but
this
this
means
version
this
of
action
of
(but
second
a
force
=
will
the
law
3
is
This
resultant
force
=
×
1.5
4.5
N
forward
force
friction
completeness.
equivalent
Newton’s
ma
=
therefore,
An
=
body.
force
more
law
on
a
common)
applies
single
when
mass.
way
we
If
of
=
forward
force
resultant
force
stating
consider
the
friction
=
12
=
7.5
4.5
N
the
N
amount
friction
normal reaction
of
mass
stays
constant
‘The
resultant
we
can
state
the
law
as
proportional
to
3.
Use
of
F
=
ma
in
a
(max. 8.0 N)
follows.
the
acceleration.’
resultant
and
In
the
force
is
If
force
we
also
equal
to
is
use
the
SI
units
product
then
of
the
2-dimensional
situation
‘the
3 kg
mass
acceleration’.
symbols,
in
SI
units,
30 N
30°
F
=
m
a
A
mass
What
of
3
will
maximum
rl fr
kg
feels
happen
friction
if
a
gravitational
it
is
placed
between
on
the
pull
a
towards
30
block
degree
and
the
the
Earth
slope
slope
given
is
8.0
of
30
that
N.
the
N?
lr
rd 
 rd
rd 
w
 klgr
 
normal reaction
2
friction
Note:
•
The
‘F
=
ma’
version
of
the
law
only
applies
if
we
3 kg
use
SI
must
units
be
in
–
for
the
equation
kilograms
rather
to
work
than
in
the
component
mass
into slope
grams.
30°
•
F
is
the
acting
one
resultant
on
an
needs
force.
object
to
work
If
(and
out
there
this
the
is
are
several
usually
resultant
forces
true)
force
then
before
component down
applying
the
law.
30 N
30°
•
This
is
•
There
an
are
experimental
no
throughout
Einstein’s
exceptions
the
–
Universe.
theory
of
Newton’s
(To
relativity
be
laws
takes
over
into
apply
absolutely
at
slope:
values
of
speed
and
normal
The
precise,
reaction
block
does
the
the
F
=
ma
version
situation
is
of
simple
the
acting
on
acceleration.
changing
If
a
–
law
for
constant
the
force
or
can
be
example,
mass
situation
a
is
changing
accelerate
into
into
slope
the
slope.
mass.)
used
a
giving
more
mass)
down
slope
=
30
N
=
15
N
8
×
sin
30°
whenever
constant
maximum
force
component
slope:
component
The
=
not
very
down
large
the slope
law.
a
friction
force
up
slope
=
N
down
slope
=
15
=
7
F
=
ma
slope
=
constant
difcult
then
(e.g.
one
∴
a
needs
resultant
force
8
to
N
dp
_
use
the
F
=
version.
dt
F
∴
acceleration
down
m
7
=
2
=
2.3
m
s
3
m e c h a n i c s
17
n’ d 
In
statement of the l aw
Newton’s
second
law
is
an
experimental
law
that
allows
us
symbols,
to
F
=
-
F
AB
calculate
the
effect
that
a
force
has.
Newton’s
third
law
Key
the
fact
that
forces
always
come
in
pairs.
It
provides
a
way
to
see
if
we
have
remembered
all
the
forces
points
The
two
is
very
easy
to
state.
‘When
two
bodies
A
and
B
interact,
that
A
exerts
on
B
is
equal
and
opposite
to
the
force
exerts
on
A
’.
Another
way
of
saying
the
same
thing
is
every
action
on
one
object
there
is
an
equal
but
Not
on
another
that
equal
the
and
pair
act
on
opposite
different
forces
objects
that
act
on
–
this
the
are
NOT
Newton’s
third
law
same
pairs.
only
the
are
same
the
forces
type.
In
equal
other
and
words,
opposite,
if
the
but
force
they
that
A
must
exerts
be
on
opposite
B
reaction
include
that
of
‘for
in
that
•
B
forces
the
object
force
notice
involved.
means
It
to
of
•
checking
BA
highlights
is
a
gravitational
force,
then
the
equal
and
opposite
force
object’.
exerted
by
B
on
A
is
also
a
gravitational
force.
e x amples of the l aw
f   -
a -     
push of wall
push of girl
on girl
on wall
If one roller-skater
push of
push of
pushes another, they
B on A
A on B
both feel a force. The
forces must be equal
and opposite, but the
acceleration will be
dierent (since they
A
B
hae dierent


. m s
. m s
2.5 m s
masses.
1
The mass of the
The force on the
The person with the
smaller mass will
ain the reater
girl causes her
to accelerate
backwards.
elocity.
wall (and
Ear th) is so
large that the
force on it does
not eectivel
cause an
acceleration.
A
B
a     – n’ d 
a  
R, reaction from table
These two forces are not third
law pairs. There must be another
force (on a dierent object) that
W, weight
F, push for ward from the ground on the car
pairs with each one:
R
In order to accelerate, there must be a for ward force on the car
The engine makes the wheels turn and the wheels push on the
W
ground.
EART
force from car on ground =
f the table pushes
f the Ear th pulls the boo
upwards on the boo
down with force W, then the
with force R, then the
boo must pull the Ear th up
boo must push down on
with force W
the table with force R
18
m e c h a n i c s
- force from ground on car
m d 
weight
their
Mass
in
and
weight
meanings
is
kg)
the
are
have
amount
whereas
the
two
very
become
of
matter
weight
of
different
muddled
contained
an
things.
in
object
in
is
Unfortunately
everyday
a
an
language.
object
force
in
(measured
(measured
Although
in
N).
these
equilibrium,
situations.
were
put
denitions
If
an
but
object
its
Moon
are
are
often
about
is
taken
weight
less
than
muddled
wanting
worried
to
would
about
to
is
the
be
on
Moon,
less
the
or
gaining
they
lose
or
its
mass
would
gravitational
Earth).
because
gain
(the
On
are
Earth
the
proportional.
weight
losing
the
be
–
what
the
forces
they
on
denitions
are
very
example,
a
lift
would
and
give
if
are
both
the
the
different
lift
the
same
in
object
accelerated
different
object
is
and
the
scale
upwards
then
the
same,
two
terms
talk
actually
R
2M
W
weight, W
the
values.
mass.
M
if
non-equilibrium
the
People
are
For
into
two
they
sdrawpu noitarelecca
Mass
new weight = 2W
If the lift is accelerating
Double
the
mass
means
double
the
upwards:
weight
R >
To
make
even
to
things
worse,
physicists.
the
Some
people
choose
object.
Other
force
reading
supporting
you
weigh
bottom
–
a
less
the
at
pull
‘weight’
on
gravitational
on
term
the
of
an
scale.
top
of
gravity
a
is
can
to
slightly
W
ambiguous
dene
people
Whichever
building
be
weight
dene
denition
compared
it
you
with
as
to
at
the
be
the
use,
the
The
safe
possible!
gravity
thing
to
do
Stick
to
the
and
you
is
to
avoid
phrase
cannot
go
using
the
term
‘gravitational
weight
force’
or
if
at
force
all
of
wrong.
less!
Gravitational
situation:
force
=
m
g
1
On
the
surface
of
the
Earth,
g
is
approximately
10
N
kg
,
1
whereas
on
the
surface
of
the
moon,
g
≈
1.6
N
kg
Weight can be dened as either
(a) the pull of gravity, W or
(b) the force on a suppor ting
scale R
OR
R
W
Two
different
denitions
of
‘weight’
m e c h a n i c s
19
s d 
factors affecting friction – static
push = zero
anD Dynamic
block
friction, F = zero
P = 0 N
Friction
is
surfaces.
smooth
It
force
arises
on
the
relative
that
opposes
because
the
microscopic
motion
(they
the
relative
surfaces
scale.
are
at
If
motion
involved
the
rest)
are
surfaces
then
not
are
this
is
of
stationary
two
F = 0 N
perfectly
prevented
an
example
ecrof hsup gnisaercni
from
the
block
of
static
friction.
If
the
surfaces
are
moving,
then
it
is
called
P = 5 N
stationary
dynamic
friction
or
kinetic
friction
F = 5 N
push
friction
block
P = 10 N
stationary
F = 10 N (= F
)
max
block accelerates
Friction arises from the
unevenness of the surfaces.
P = 15 N
F = 9 N
The
value
of
F
depends
upon
max
push causes
•
the
nature
•
the
normal
motion to
of
maximum
the
two
surfaces
in
contact.
reaction
force
between
frictional
force
and
the
the
two
normal
surfaces.
reaction
The
force
are
RIGHT
proportional.
If
the
two
surfaces
are
kept
in
contact
by
gravity,
the
value
of
friction opposes motion,
F
acting to LEFT
does
key
experimental
fact
is
that
the
value
of
static
depending
the
on
the
applied
force.
Up
to
a
F
,
the
resultant
force
is
zero.
object
slightly
<
k
force,
the
area
of
contact
has
started
reduces.
In
moving,
other
the
maximum
value
of
words,
certain
F
maximum
upon
friction
friction
changes
depend
max
Once
A
NOT
For
example,
F
max
if
max
For
we
try
to
get
a
heavy
block
to
move,
any
value
of
pushing
two
surfaces
F
would
fail
to
get
the
block
to
moving
over
one
another,
the
dynamic
force
frictional
below
force
remains
roughly
constant
even
if
the
speed
accelerate.
max
changes
coefficient of friction
Experimentally,
reaction
force
coefcient
the
are
of
maximum
friction,
e x ample
frictional
proportional.
slightly.
We
use
force
this
to
and
the
dene
normal
the
If
a
block
is
increased
µ
This
turns
coefcient
coecient of friction = µ
placed
until
out
of
the
to
on
be
static
a
slope,
block
an
just
easy
the
angle
begins
to
of
the
slide
experimental
slope
down
way
to
can
the
be
slope.
measure
the
friction.
reaction, R
P
R, reaction
friction F
F
frictional force
component of W down
W
θ
F
= µR
max
gravitational attraction
slope (W sin θ)
component of W into
The
coefcient
of
friction
is
dened
from
the
maximum
value
slope (W cos θ)
W
that
friction
can
take
θ
F
=
µ
R
max
where
It
•
should
since
the
R
be
the
=
normal
noted
force
If
value
value
for
for
static
dynamic
friction,
friction
the
is
values
less
of
friction
will
be
<
for
W
sin
=
W
cos
θ
θ
the
is
increased.
block
just
starts
F
F
moving,
µ
d
•
=
R
different
When
µ
F
than
θ
coefcients
balanced,
that
maximum
maximum
reaction
the
coefcient
has
no
of
friction
is
a
ratio
between
two
forces
–
=
max
it
F
max
units.
static
=
R
•
if
the
zero
surfaces
i.e.
µ
=
are
smooth
then
the
maximum
friction
is
0.
W
sin
θ
W
cos
θ
=
•
the
coefcient
are
stuck
F
≤
f
20
µ R
s
of
friction
is
together.
and
F
f
=
less
than
1
unless
the
surfaces
=
µ
R
d
m e c h a n i c s
tan
θ
w
Definition of work
when is work Done?
Work
is
done
when
a
force
moves
the
direction
of
the
force.
If
the
the
direction
of
the
force,
then
its
force
point
of
moves
application
at
right
in
angles
Work
is
a
scalar
quantity.
Its
denition
is
as
follows.
to
F
no
work
has
been
done.
θ
1) before
after
block now
v
work done = Fs cos θ
at rest
s
moving –
work has
Work
done
=
F
s
cos
θ
been done
If
the
this
distance
force
can
and
be
the
displacement
simplied
‘Work
done
=
are
in
the
same
direction,
to
force
×
distance’
block now
2) before
after
higher up –
From
this
work has
dene
a
denition,
new
unit
the
called
SI
units
the
for
joule:
1
work
done
J
N
=
1
are
N
m.
We
m.
been done
force
e x amples
distance
(1) lifting ver tically
small distance
force
large force
3) before
(2) pushing along a rough slope
force
dis
e
rg
la
a
sm
ce
n
ta
e
rc
fo
r
lle
spring has been
compressed –
after
The
task
in
the
second
case
would
be
easier
to
perform
(it
work has
involves
less
force)
but
overall
it
takes
more
work
since
work
been done
has
force
to
4) before
done
work
is
If
force
the
spring
distance
be
the
is
to
overcome
friction.
In
each
case,
the
useful
same.
doing
work
compressed),
is
not
then
constant
graphical
(for
example,
techniques
can
when
be
a
used.
original length
after
F
A
book suppor ted by shelf –
x
no work is done
F
max
5) before
after
v
constant
x
max
velocity v
the
total
area
work
under
done
is
ecrof
The
the
F = kx
force–displacement
friction-free su rface
friction-free su rface
graph.
F
total work done
max
object continues at constant velocity –
= area under graph
no work is done
1
=
In
the
examples
above
the
work
done
has
had
different
x
0
•
In
•
In
•
In
1)
the
force
has
made
has
been
the
object
move
the
object
lifted
higher
in
the
the
spring
has
been
equations
In
4)
and
object
5),
is
NO
work
moving
in
is
the
work
done
include:
work
done
when
lifting
something
vertically
=
mgh
compressed.
done.
Note
that
even
the
last
example,
there
m
represents
mass
(in
kg)
though
g
the
for
eld.
where
•
ex tension
max
gravitational
•
3)
x
faster.
Useful
2)
2
k x
2
results.
is
no
represents
the
Earth’s
gravitational
eld
strength
force
1
(10
moving
along
its
direction
of
action
so
no
work
is
N
kg
)
h
represents
the
height
change
(in
m)
done.
1
•
work
done
in
compressing
or
extending
a
spring
=
2
k
∆x
2
m e c h a n i c s
21
e  d 
concepts of energy anD work
Energy
and
work
are
linked
together.
When
you
do
work
on
an
object,
it
gains
K ≈ 0 J
energy
and
you
lose
energy.
The
amount
of
energy
transferred
is
equal
to
the
P = 1000 J
work
units
done.
of
Energy
energy
is
must
a
be
measure
the
same
of
the
as
amount
the
units
of
of
work
work
–
done.
This
means
that
the
joules.
energy transformations – conservation of energy
In
by
any
situation,
one
object,
it
we
must
must
conservation
of
•
Overall
total
•
Energy
•
There
be
be
able
gained
energy.
to
by
There
account
another.
are
several
for
the
This
is
ways
changes
known
of
in
as
stating
energy.
the
this
If
it
is
principle
‘lost’
of
principle:
K = 250 J
P = 750 J
K = 250 J
the
is
is
energy
neither
no
of
created
change
in
any
nor
the
closed
system
destroyed,
total
energy
it
in
must
just
the
be
constant.
changes
form.
Universe.
P = 750 J
energy types
Kinetic
energy
Radiant
Nuclear
Gravitational
energy
Electrostatic
energy
Electrical
Equations
Solar
energy
for
the
energy
rst
three
Elastic
of
Light
energy
are
given
energy
energy
Chemical
energy
types
potential
Thermal
energy
Internal
1
Kinetic
potential
potential
energy
energy
below.
2
=
1
mv
where
m
is
the
mass
(in
kg),
v
is
the
velocity
(in
m
s
)
2
2
p
1
=
where
p
is
the
momentum
(see
page
23)
(in
kg
m
s
),
and
m
is
2m
the
Gravitational
mass
potential
(in
kg)
energy
=
mgh
where
m
represents
mass
(in
kg),
g
represents
1
the
Earth’s
gravitational
eld
(10
N
kg
1
K = 500 J
Elastic
potential
energy
),
h
represents
the
height
change
(in
m)
2
=
k
∆x
1
where
k
is
the
spring
constant
(in
N
m
),
∆x
is
2
the
extension
(in
m)
P = 500 J
power anD efficiency
1.
e x amples
Power
Power
is
energy
the
is
rate
1.
dened
as
the
transferred.
at
which
This
work
energy
RATE
is
is
at
the
which
same
grasshopper
hindlegs
as
done.
transferred
to
jumps
(i)
take
its
(mass
push
result
power
__
Power
A
1.8
off
for
m
8
g)
0.1s
high.
speed,
uses
and
its
as
a
Calculate
(ii)
the
developed.
=
time
taken
(i)
PE
gained
=
mgh
work done
__
Power
=
time
1
taken
KE
The
SI
unit
for
power
is
the
joule
at
start
=
2
mv
2
per
K = 750 J
1
1
second
(J
s
).
Another
unit
for
power
2
mv
is
P = 250 J
=
mgh
(conservation
of
2
1
dened
If
the
something
velocity
v
watt
is
(W).
moving
against
a
1
W
at
a
=
1
J
s
energy)
.
constant
constant
v

=
√2gh
√
2 × 10 × 1.8
=
frictional
1
=
force
F,
the
power
P
needed
is
P
=
F
6
m
s
v
mgh
_
(ii)
2.
Power
=
Efciency
t
Depending
on
the
situation,
we
can
0.008 × 10 × 1.8
__
categorize
the
energy
transferred
(work
=
0.1
done)
K = 1000 J
as
useful
useful
energy
or
not.
would
In
be
a
light
light
bulb,
energy
,
the
the
≈
P = 0 J
‘wasted’
energy
would
be
thermal
2.
(and
non-visible
forms
of
radiant
dene
A
60W
efciency
as
the
ratio
energy
to
the
total
equation
Possible
forms
of
total
work
IN
energy
OUT
__
Efciency
=
total
useful
energy
power
IN
OUT
__
Efciency
=
total
Since
units.
22
m e c h a n i c s
this
is
Often
a
it
ratio
is
it
power
does
expressed
efciency
is
wasted
hour?
wasted
Energy
useful work OUT
__
=
useful
an
energy
not
as
=
90%
=
54W
of
60W
the
include:
Efciency
has
much
energy
Power
transferred.
lightbulb
How
of
every
useful
W
energy).
10%.
We
1.4
energy
IN
have
a
any
percentage.
wasted
=
54
=
190
×
60
kJ
×
60J
of
m d 
Definitions – line ar momentum anD impulse
conservation of momentum
Linear
The
the
momentum
product
of
(always
mass
and
given
the
symbol
p)
is
dened
as
velocity.
law
linear
of
constant
Momentum
=
p
=
mass
×
of
provided
a
of
linear
system
there
is
of
no
momentum
interacting
resultant
states
that
particles
‘the
total
remains
external
force’.
velocity
To
m
conservation
momentum
see
why,
we
start
by
imagining
two
isolated
particles
A
and
v
B
that
collide
with
one
another.
1
The
SI
units
for
momentum
must
be
kg
m
s
.
Alternative
•
units
of
Ns
can
also
be
used
(see
below).
Since
velocity
The
force
vector,
momentum
must
be
a
vector.
In
any
if
it
happens
quickly,
the
change
by
of
is
called
the
impulse
(∆p
=
F
onto
certain
B,
F
will
cause
B’s
momentum
to
amount.
If
the
time
taken
was
∆t,
then
the
momentum
change
(the
momentum
impulse)
∆p
a
situation,
•
particularly
A
AB
change
a
from
is
given
to
B
will
be
given
by
∆p
∆t).
=
F
B
•
By
Newton’s
third
law,
the
force
from
B
∆t
AB
onto
A,
F
will
be
BA
equal
and
Since
the
opposite
to
the
force
from
A
onto
B,
F
=
-
F
AB
BA
use of momentum in newton’s seconD law
•
Newton’s
second
law
states
that
the
resultant
to
the
rate
of
change
of
of
contact
for
A
and
B
momentum
change
change
for
for
B,
A
is
∆p
equal
=
-
can
write
this
•
(nal
momentum
initial
momentum)
∆p
_
=
then
opposite
to
the
(momentum
of
A
∆t.
AB
This
means
plus
the
that
the
total
momentum
of
momentum
B)
will
remain
the
same.
Total
=
time
Example
of
same,
as
____
F
and
F
A
we
the
momentum.
momentum
Mathematically
is
forceis
the
proportional
time
taken
∆t
momentum
1
A
jet
is
brought
This
water
leaves
a
hose
and
hits
a
wall
where
its
interacting
velocity
2
to
rest.
If
the
hose
cross-sectional
area
is
25
cm
argument
isolated.
,
is
conserved.
can
be
particles
If
this
is
extended
so
the
long
case,
as
up
the
the
to
any
system
number
of
momentum
of
particles
is
still
is
still
conserved.
1
the
velocity
of
the
water
is
50
m
s
and
the
density
of
the
3
water
is
1000
kg
m
,
what
is
the
force
acting
on
the
wall?
el a stic anD inel a stic collisions
The
law
of
conservation
of
linear
momentum
is
not
enough
to
1
velocity = 50 m s
always
50
predict
the
outcome
after
a
collision
(or
an
explosion).
m
This
depends
example,
a
on
the
moving
nature
of
railway
the
truck,
colliding
m
,
bodies.
velocity
v,
For
collides
with
A
density of
an
cross-sectional
3
stationary
truck
m
.
Possible
one
second,
a
jet
of
water
50
m
long
hits
= 0.0025 m
(a) elastic collision
the
wall.
new velocity = v
So
m
of
water
mass
of
water
hitting
every
wall
=
0.0025
=
0.125
×
50
=
0.125
m
A
3
volume
are:
2
area = 25 cm
at rest
In
outcomes
B
2
water = 1000 kg m
identical
B
m
second
hitting
wall
×
1000
=
125
kg
(b) totally inelastic collision
v
every
new velocity =
second
2
1
momentum
of
water
hitting
wall
=
125
×
50
=
6250
kg
m
s
m
m
A
every
This
water
is
all
brought
to
rest,
(c) inelastic collision
1
∴
change
in
B
second
momentum,
∆p
=
6250
kg
m
s
v
force
=
new velocity =
6250
_
_
∴
3v
new velocity =
∆p
=
=
6250
4
N
4
1
∆t
m
Example
The
a
graph
football
below
of
m
A
2
mass
shows
500
g.
the
variation
Calculate
with
the
time
nal
of
the
velocity
of
force
the
B
on
ball.
In
(a),
were
in
the
the
the
trucks
case
would
then
collision.
A
no
have
to
have
mechanical
collision
in
elastic
energy
which
no
at
bumpers.
If
all
be
would
mechanical
this
lost
energy
The football was given an impulse of approximately
is
lost
is
called
an
elastic
collision.
In
reality,
collisions
N/ecrof
100 × 0.01 = 1 N s during this 0.01 s.
between
everyday
objects
always
lose
some
energy
–
the
100
Area under graph is the total
only
real
example
of
elastic
collisions
is
the
collision
between
90
molecules.
impulse given to the
For
an
elastic
collision,
the
relative
velocity
of
80
ball ≈ 5 N s
approach
p = mv
always
equals
the
relative
velocity
of
separation.
70
p
v
In
(b),
the
railway
trucks
stick
together
during
the
collision
=
60
50
m
(the
5 N s
is
relative
what
is
velocity
known
as
of
a
separation
totally
is
zero).
inelastic
This
collision
collision.
A
large
=
amount
0.5 kg
40
1
the
of
total
mechanical
momentum
energy
is
still
is
lost
(as
heat
and
sound),
but
conserved.
= 10 m s
30
∴ nal
In
energy
terms,
(c)
is
somewhere
between
(a)
and
(b).
Some
20
1
velocity v = 10 m s
energy
is
lost,
but
the
railway
trucks
do
not
join
together.
This
10
is
0.00
0.02
0.04
0.06
0.08
an
example
of
an
inelastic
collision.
Once
again
the
total
0.10
momentum
is
conserved.
time/s
Linear
momentum
is
also
conserved
in
explosions.
m e c h a n i c s
23
ib q – 
1.
Two
identical
objects
A
and
B
fall
from
rest
from
6.
different
A
car
and
a
truck
are
both
travelling
at
the
speed
limit
of
1
heights.
what
is
If
the
Neglect
A.
2.
A
air
takes
ratio
it
twice
of
the
as
long
heights
as
A
to
from
reach
which
the
A
60
ground,
and
B
fell?
km
twice
B.
is
1:2
given
moving.
C.
an
The
initial
trolley
1:4
D.
push
the
A.
slowing.
trolley
There
while
is
a
then
a
travels
horizontal
forward
oor
along
What
it
is
is
true
of
the
horizontal
mass
is
a)
force
and
a
backward
force,
but
vehicles
the
is
a
collide
During
the
the
on
car
as
shown.
The
truck
has
car.
forward
force
only
a
and
the
truck
how
become
does
compare
the
entangled
force
together.
with
the
exerted
force
by
exerted
the
the
force
and
a
backward
force,
but
In
truck
what
is
on
the
direction
car?
will
Explain.
the
[2]
entangled
vehicles
move
after
the
or
will
they
be
stationary?
Support
larger.
your
is
head-on
collision,
larger.
collision,
There
of
directions
on
slowing?
forward
force
backward
C.
opposite
oor,
force(s)
b)
There
the
in
to
the
by
forward
B.
but
1:8
along
The
gradually
h
resistance.
√2
1:
trolley
get
B
forward
force,
which
diminishes
with
answer,
referring
to
a
physics
principle.
[2]
time.
1
c)
D.
There
is
only
a
backward
Determine
A
mass
is
suspended
from
a
ring
How
immediately
does
the
attached
by
two
further
to
the
ceiling
and
wall
as
cord
makes
than
as
from
an
45°
three
the
angle
with
shown.
the
S
and
of
the
The
f)
less
R
vertical
tensions
cords
T
Both
the
do
are
in
the
The
of
T
the
in
of
the
car
[3]
compare
with
the
the
truck
during
the
collision?
Explain.
[2]
car
and
is
truck
likely
drivers
to
be
are
more
wearing
severely
seat
belts.
jolted
in
the
total
the
kinetic
collision.
violated?
in
a)
A
net
R,
[2]
energy
Is
the
of
the
system
principle
of
decreases
conservation
as
of
a
result
energy
Explain.
[1]
force
three
I
of
of
magnitude
the
F
acts
on
a
body.
Dene
the
force.
[1]
S
A
ball
of
mass
0.0750
cords
kg
is
travelling
horizontally
with
a
1
speed
in
Explain.
diagram.
tensions
the
compare
combined
labelled
b)
and
acceleration
of
driver
impulse
How
collision.
the
ceiling
7.
R,
the
of
shown.
collision?
The
after
)
T
Which
the
h
S
e)
cords
km
which
acceleration
is
(in
by
d)
cord
speed
force.
wreck
3.
the
of
2.20
m
s
.
It
strikes
a
vertical
wall
and
rebounds
magnitude?
horizontally.
A.
R
>
T
>
S
C.
R
=
S
=
T
B.
S
>
D.
R
=
R
>
T
S
>
T
ball mass
2
4.
A
a
5.
24
N
force
horizontal
A.
0.0
C.
0.6
An
athlete
horizontal
causes
a
surface.
trains
by
2.0
kg
The
mass
to
coefcient
dragging
a
accelerate
of
dynamic
B.
0.4
D.
0.8
heavy
at
load
8.0
m
friction
s
0.0750 kg
along
is:
1
2.20 ms
across
a
rough
surface.
Due
to
kinetic
(i)
the
collision
energy
is
Show
that
speed
of
Show
that
with
the
wall,
20
%
of
the
ball’s
initial
dissipated.
the
ball
rebounds
from
the
wall
with
a
1
1.97
m
s
.
[2]
F
(ii)
wall
is
the
0.313
impulse
N
given
to
the
ball
by
the
s.
[2]
25°
c)
The
wall
The
athlete
exerts
a
force
of
magnitude
F
on
the
load
at
of
25°
to
the
at
Once
the
load
time
t
the
=
wall
at
time
t
=
0
and
leaves
the
T
sketch
graph
shows
how
the
force
F
that
the
wall
horizontal.
exerts
a)
strikes
an
The
angle
ball
is
moving
at
a
steady
speed,
the
on
the
ball
is
assumed
to
vary
with
time
t
average
F
horizontal
Calculate
load
b)
The
to
frictional
the
move
load
is
force
average
at
value
constant
moved
a
acting
of
F
on
the
that
load
will
is
470
enable
N.
the
speed.
horizontal
[2]
distance
of
2.5
km
in
1.2
hours.
Calculate
0
t
T
(i)
the
work
done
(ii)
the
minimum
the
load.
on
the
average
load
by
power
the
force
required
F.
to
[2]
The
time
[2]
Use
the
value
c)
The
part
athlete
the
load
uphill
at
the
same
speed
as
in
(a).
Explain,
average
24
pulls
in
terms
power
i B
of
energy
required
is
changes,
greater
Q u e s t i o n s
–
why
than
in
the
T
is
measured
electronically
to
equal
0.0894
s.
move
minimum
(b)(ii).
m e c h a n i c s
[2]
of
impulse
F.
given
in
(b)(ii)
to
estimate
the
average
[4]
3
T h e r m a l
p h Y s i C s
T cc t
TemperaTure and he aT flow
Kelvin and Celsius
Hot
Most
and
the
cold
are
direction
(sometimes
in
just
labels
which
known
as
that
thermal
heat)
identify
energy
will
transferred
when
two
placed
order
in
thermal
contact.
to
the
concept
of
the
object.
‘hotness’
of
The
direction
thermal
energy
of
the
is
determined
by
the
use
each
object.
Thermal
ows
from
temperature
of
everyday
in
is
words,
of
if
to
hot
object
between
determine
the
transfer
thermal
of
is.
are
then
is
to
chose
you
do
not
dened,
need
but
you
to
understand
do
need
to
the
know
details
the
of
how
relation
either
between
thermometers
Celsius
are
marked
with
the
Celsius
scale
and
temperature
the
In
easy
the
relationship
corresponding
between
a
temperature
temperature
t
as
T
as
measured
measured
on
the
on
Celsius
the
Kelvin
scale.
relationship
The
is
(K)
of
=
t
means
(°C)
+
273
the
zero
‘size’
of
the
units
used
on
each
scale
is
identical,
but
they
have
points.
in
objects
the
energy.
that
other
temperature
two
direction
is
naturally
temperature
high
will
natural
Thermal
transferred
difference
temperature
to
is
(°C).
Notice the size of the units is
700 K
identical on each scale.
400 °C
mercury boils
630 K
357 °C
600 K
energy
of
them.
a
placed
the
the
scales
elacs suisleC
contact,
difference
it
objects
temperature
scale.
cold.
elacs nivleK
thermal
how
two
been
degrees
an
and
different
measure
sensible
Celsius
energy
hot
an
two
the
‘hotness’
This
The
only
and
two
T
naturally
them,
has
approximate
of
are
scale
natural
between
scale
objects
there
Kelvin
of
There
ow
to
scales
quoted
an
time,
the
This
Most
leads
–
objects
these
are
the
be
In
naturally
of
between
low
–
‘down’
300 °C
from
temperature.
500 K
Eventually,
be
the
expected
to
temperature.
two
objects
reach
When
the
this
would
200 °C
same
happens,
400 K
they
are
said
to
be
in
thermal
100 °C
water boils
373 K
equilibrium
Heat
is
from
not
one
a
substance
object
has
happened
has
been
(heat)
and
its
is
that
ows
another.
What
that
thermal
transferred.
refers
transfer
to
of
to
the
energy
300 K
water freezes
273 K
energy
Thermal
0 °C
mercury freezes
energy
200 K
carbon dioxide freezes
-100 °C
non-mechanical
between
a
system
surroundings.
100 K
oxygen boils
-200 °C
hydrogen boils
-273 °C
0 K
The
this
Kelvin
scale
Zero
is
scale
also
Kelvin
is
is
an
absolute
called
called
the
thermodynamic
absolute
absolute
zero
temperature
scale
and
a
measurement
on
temperature
(see
page
29).
direction of transfer of thermal energy
In
e x amples: Ga ses
For
a
given
temperature
•
The
at
sample
are
all
pressure,
90°
on
the
of
a
gas,
related
P,
is
the
to
the
one
force
container
pressure,
the
volume
and
the
another.
per
unit
•
area
from
the
gas
wall.
order
to
investigate
how
these
quantities
are
interrelated,
we
choose:
one
quantity
alter
acting
•
and
another
thing
to
be
quantity
we
the
independent
variable
(the
thing
we
measure)
to
be
the
dependent
variable
(the
second
measure).
F
p
=
A
•
2
The
SI
units
of
pressure
are
N
m
or
Pa
Pa
=
1
N
third
The
specic
mass
needs
values
that
to
will
be
controlled
be
recorded
(i.e.
also
kept
constant).
depend
of
gas
being
investigated
and
the
type
of
gas
on
being
the
used
m
so
Gas
quantity
(Pascals).
2
1
The
pressure
can
also
be
measured
in
these
need
to
be
controlled
as
well.
atmospheres
5
(1
atm
≈
10
Pa)
3
•
The
volume,
3
(1
•
m
The
V,
6
=
10
of
the
gas
is
measured
in
m
3
or
cm
3
cm
)
temperature,
t,
of
the
gas
is
measured
in
°C
or K
T H E R M A L
P H Y S I C S
25
ht  t g y
KineTiC TheorY
miCrosCopiC vs maCrosCopiC
When
analysing
something
physical,
we
have
a
choice.
Molecules
phase
•
The
macroscopic
whole
and
sees
point
how
it
of
view
interacts
considers
with
its
the
system
as
of
are
the
arranged
in
substance
different
(i.e.
solid,
ways
liquid
depending
or
on
the
gas).
a
surroundings.
solids
•
The
see
microscopic
how
its
point
component
of
view
parts
looks
interact
inside
with
the
each
system
to
other.
Macroscopically,
This
is
because
However
So
far
we
have
looked
at
the
temperature
of
a
system
in
way,
but
all
objects
are
made
up
of
atoms
have
a
molecules
bonds
are
not
xed
are
volume
held
in
absolutely
and
a
position
rigid.
The
xed
by
shape.
bonds.
molecules
a
vibrate
macroscopic
the
solids
the
around
a
mean
(average)
position.
The
higher
the
and
temperature,
the
greater
the
vibrations.
molecules
According
random
to
kinetic
motion
–
theory
hence
the
these
name.
particles
See
are
below
constantly
for
more
Each molecule vibrates
in
around a mean
details.
position.
Although
is
a
atoms
and
combination
this
stage.
mass
The
with
of
molecules
atoms),
particles
velocities
the
can
that
be
are
are
different
difference
thought
is
of
continually
as
things
not
(a
molecule
important
little
‘points’
at
of
changing.
Bonds
between
The molecules in a solid are
molecules
held close together by the
inTernal enerGY
intermolecular bonds.
If
the
temperature
gained
the
(or
lost)
molecules
of
an
energy.
must
object
From
have
changes
the
gained
then
it
microscopic
(or
lost)
this
must
point
have
of
liquids
view,
A
energy.
liquid
The
The
two
possible
forms
are
kinetic
energy
and
potential
xed
speed in a random direction
also
has
molecules
a
xed
are
also
volume
but
vibrating,
its
but
shape
they
can
are
change.
not
completely
energy.
in
position.
There
are
still
strong
forces
between
the
v
molecules.
This
keeps
the
molecules
move
around
close
to
one
another,
but
∴ molecule has KE
they
are
free
to
each
other.
F
equilibrium
resultant force back towards equilibrium
Bonds between
position
position due to neighbouring molecules
neighbouring
molecules; these can
∴ molecule has PE
be made and broken,
•
The
molecules
have
kinetic
energy
because
they
are
allowing a molecule to
Each molecule is free
moving.
To
be
absolutely
precise,
a
molecule
can
have
move.
to move throughout the
either
translational
moving
(the
in
a
kinetic
certain
molecule
is
energy
direction)
rotating
or
about
(the
whole
rotational
one
or
molecule
kinetic
more
is
liquid by moving around
its neighbours.
energy
axes).
Ga ses
•
The
molecules
have
potential
energy
because
of
the
A
intermolecular
forces.
If
we
imagine
pulling
gas
put.
molecules
further
apart,
this
would
require
will
work
The
intermolecular
plus
total
energy
inter
that
molecule
are
to
not
ll
the
xed
in
container
position,
in
which
and
any
it
is
forces
against
the
molecules
are
very
weak.
This
means
that
the
forces.
molecules
The
expand
molecules
between
the
always
two
the
molecules
potential)
is
possess
called
the
(random
internal
they
kinetic
do
are
essentially
occasionally
independent
collide.
More
of
detail
one
is
another,
given
on
but
page
31.
energy
Molecules in random
of
a
substance.
Whenever
we
heat
a
substance,
we
increase
its
motion; no xed bonds
internal
energy.
between molecules so
they are free to move
Temperature
of
If
the
two
is
a
molecules
substances
molecules
have
measure
in
a
have
the
of
the
average
kinetic
energy
substance.
the
same
same
temperature,
average
kinetic
then
their
energy.
he aT and worK
Many
same temperature
people
answers
for
of
When
have
confused
examination
example,
transfer
•
to
that
‘heat
thermal
a
force
ideas
rises’
energy
moves
about
questions
–
is
it
when
is
heat
very
what
and
work.
common
is
meant
to
is
In
read,
that
the
upwards.
through
a
distance,
we
say
that
work
same average
is
v
done.
Work
is
the
energy
that
has
been
transmitted
from
V
kinetic energy
one
system
to
another
from
the
macroscopic
point
of
view.
m
M
•
When
work
individual
is
molecules with large
molecules with small
place.
Heat
mass moving with
mass moving with
either
increase
lower average speed
higher average speed
potential
In
26
T H E R M A L
P H Y S I C S
both
done
is
the
microscopic
is
of
say
that
kinetic
or,
energy
a
we
energy
the
energy
cases
on
molecules),
that
has
energy
course,
being
level
heating
been
of
both.
on
taken
transmitted.
the
transferred.
(i.e.
has
molecules
It
or
can
their
scc t ccty
meThods of measurinG heaT CapaCiTies and
definiTions and miCrosCopiC e xpl anaTion
In
theory,
if
an
object
could
be
heated
up
with
no
energy
loss,
speCifiC heaT CapaCiTies
then
the
increase
in
temperature
∆T
depends
on
three
things:
The
•
the
energy
•
the
mass,
given
to
the
object
1.
m,
the
two
basic
ways
to
measure
heat
capacity.
Electrical
method
and
The
•
are
Q
substance
from
which
the
object
is
experiment
would
be
set
up
as
below:
made.
heater (placed in object)
1000 J
1000 J
mass m
mass m
substance X
substance Y
V
voltmeter
dierent temperature
ammeter
change
A
small temperature
variable power supply
change since
more molecules
I t V
_
•
the
specic
heat
capacity
c
=
m( T
large temperature
T
2
Sources
of
experimental
)
1
error
change since fewer
•
loss
•
the
of
thermal
energy
from
the
apparatus.
molecules
Two
different
blocks
with
the
same
mass
and
same
container
warmed
input
will
have
a
different
temperature
dene
required
the
to
thermal
raise
its
capacity
temperature
C
of
by
1
an
K.
object
as
Different
the
it
different
samples
of
the
same
substance)
values
of
heat
capacity.
Specic
by
energy
1
K.
required
‘Specic’
to
here
raise
just
a
unit
means
mass
‘per
and
the
heater
will
also
be
heat
of
a
unit
take
some
time
through
for
the
the
energy
to
be
shared
substance.
Method
of
mixtures
have
capacity
known
specic
heat
capacity
of
one
substance
can
be
used
is
to
the
substance
objects
will
The
different
will
uniformly
energy
2.
(even
the
up.
change.
•
We
for
energy
nd
the
specic
heat
capacity
of
another
substance.
substance
before
mass’.
temperature T
(hot)
temperature T
A
In
B
symbols,
(cold)
Q
_
Thermal
capacity
C
=
1
(J
1
K
or
J
°C
)
∆T
Specic
Q
_
heat
c
1
=
(J
(m
capacity
Q
=
kg
1
K
1
or
J
1
kg
°C
)
∆T)
mc∆T
mix together
Note
•
A
particular
gas
can
have
many
different
values
of
specic
heat
mass m
mass m
A
capacity
–
it
depends
on
the
conditions
used
–
see
page
B
161.
temperature T
max
•
These
equations
refer
to
the
temperature
difference
after
resulting
other
to
•
from
words,
raise
the
it
does
for
is
only
true
If
an
it
addition
generally
temperature
the
object
the
so
is
same
long
as
a
takes
of
object
raised
of
an
to
the
is
not
402
lost
of
amount
from
from
room
amount
same
object
go
energy
above
certain
25
°C
from
°C
to
of
to
it
In
energy
35
412
the
temperature,
energy.
°C
°C.
as
This
object.
starts
to
Procedure:
lose
energy.
which
it
The
loses
hotter
it
becomes,
the
greater
the
rate
at
•
measure
the
masses
•
measure
the
two
of
the
liquids
m
and
m
A
energy.
starting
B
temperatures
T
and
T
erutarepmet
A
B
increase in
•
mix
the
•
record
two
liquids
together.
temperature if no
the
maximum
temperature
of
the
mixture
T
max
energy is lost
If
no
energy
is
lost
energy
lost
gained
by
from
by
hot
the
system
substance
then,
cooling
down
=
energy
increase in
cold
substance
heating
up
temperature
m
in a real situation
c
A
Again,
(T
A
the
thermal
T
A
)
=
m
max
main
energy
from
c
B
source
of
the
(T
B
T
max
)
B
experimental
apparatus;
error
is
particularly
the
loss
while
of
the
time
liquids
Temperature
constant
rate
change
of
an
object
being
heated
at
a
the
are
being
container
more
transferred.
also
accurate
need
to
be
The
changes
taken
into
of
temperature
consideration
for
of
a
result.
T H E R M A L
P H Y S I C S
27
p (tt)  tt  tt t
definiTions and miCrosCopiC vie w
When
a
substance
constant
even
changes
though
phase,
thermal
the
energy
meThods of me a surinG
temperature
is
still
being
remains
The
transferred.
two
below
possible
are
very
C/
° erutarepmet
measuring
methods
similar
specic
in
heat
for
measuring
principle
capacities
to
latent
the
(see
heats
methods
previous
shown
for
page).
500
1.
A
molten lead
method
for
vaporization
measuring
of
the
specic
latent
heat
of
water
400
set-up
electrical circuit
heater
300
liquid and
solid
to electrical
solid mix
200
circuit
100
V
water
voltmeter
1
2
3
4
6
5
7
8
9
10
11
12
Cooling
curve
for
molten
lead
ammeter
A
13 14
time / min
heater
(idealized)
beaker
The
amount
called
of
the
phase
change
The
of
energy
latent
from
from
energy
associated
heat.
The
technical
solid
to
liquid
liquid
to
gas
given
to
the
with
is
is
the
phase
term
fusion
and
for
change
the
the
change
term
for
The
the
does
amount
point
vaporization
molecules
is
increase
of
thermal
calculated
vaporized
not
variable power supply
is
needs
energy
using
to
be
provided
electrical
energy
so
it
must
be
increasing
their
potential
When
the
releases
It
is
a
•
The
specic
latent
heat
L
up
vapour
are
being
freezes
broken
bonds
are
and
this
created
takes
and
common
during
at
100
a
°C
mistake
phase
must
to
think
change.
be
The
moving
that
the
with
the
in
same
of
as
the
molecules
in
liquid
water
at
100
•
Loss
•
Some
of
specic
amount
change
of
of
latent
energy
heat
per
of
unit
a
substance
mass
is
absorbed
boiling
mass
m
)
2
error
thermal
energy
must
2.
A
method
fusion
water
vapour
from
of
for
will
be
the
lost
apparatus.
measuring
before
the
and
specic
after
timing.
latent
heat
of
water
average
we
as
released
example
the
during
a
the
know
the
specic
specic
below,
temperature
of
ice
latent
heat
(at
the
0
heat
°C)
is
resulting
of
capacity
fusion
added
mix
is
to
for
of
water,
water.
warm
In
we
can
water
the
and
the
measured.
phase.
ice
In
its
°C.
dened
or
experimental
water
calculate
The
at
The
process
Providing
speed
V.
energy.
this
molecules
molecules
t
=
1
energy.
very
speed
bonds
substance
I
recorded.
energy.
Sources
Intermolecular
water
=
I t V
_
their
(m
kinetic
to
energy
water
symbols,
Q
1
Specic
latent
heat
L
=
(J
kg
)
Q
=
ML
M
In
of
the
idealized
energy
constant
point
is
situation
transfer
rate
of
into
a
increase
of
no
solid
in
energy
loss,
substance
a
constant
would
temperature
until
result
the
rate
in
mix together
a
melting
C°/erutarepmet
mass: m
water
temp.: T
graph
with
temperature
vs
no
energy
energy
m
energy
example
above,
the
specic
heat
capacity
of
the
liquid
ice
mix
than
the
line
specic
that
gradient
of
heat
capacity
corresponds
the
line
that
to
the
of
the
liquid
corresponds
solid
as
phase
to
the
the
is
amount
of
energy
will
cause
a
greater
for
the
liquid
when
compared
the
•
Loss
•
If
T H E R M A L
P H Y S I C S
)
=
m
mix
then,
down
L
ice
=
+
energy
m
fusion
c
ice
gained
by
ice
T
water
of
experimental
mix
error
(or
the
gain)
ice
had
of
thermal
not
started
energy
at
from
exactly
the
zero,
apparatus.
then
an
additional
term
in
the
equation
in
there
order
to
would
account
in
the
energy
needed
to
warm
the
ice
up
to
solid.
•
28
system
phase.
increase
with
T
water
the
cooling
than
for
temperature
from
water
gradient
greater
solid
lost
(T
water
be
given
is
by
is
Sources
the
lost
c
water
A
+ m
solid
the
the
water
solid and liquid mix
Phase-change
of
temp.: T °C
liquid
If
less
mass: m
temp.: 0 °C
ice
energy supplied/J
In
mass: m
reached:
Water
clinging
to
the
ice
before
the
transfer.
0
°C.
T g  1
The
Ga s l aws
For
the
outline
experimental
what
might
methods
be
shown
below,
the
graphs
below
trends
presented
can
in
a
be
seen
slightly
more
clearly
different
if
this
information
is
way.
observed.
(1) constant volume
(a) constant volume
pressure / Pa
graph ex trapolates
back to
273 °C
absolute temperature / K
300
200
100
0
100
temp. / °C
(2) constant pressure
(b) constant pressure
3
volume / m
graph ex trapolates
back to
273 °C
300
200
100
m / emulov
100
3
0
absolute temperature / K
temp. / °C
aP / erusserp
(3) constant temperature
(c) constant temperature
3
volume / m
1
3
/ m
volume
Points
•
to
note:
Although
Celsius
pressure
and
temperature,
volume
neither
both
vary
pressure
linearly
nor
From
with
volume
to
Celsius
graphs
for
a
xed
mass
of
gas
we
can
say
that:
p
1.
proportional
these
is
At
constant
V,
p
∝
T
or
=
constant
(the
pressure
=
constant
(Charles’s
law)
T
temperature.
V
•
A
different
sample
of
gas
would
produce
a
different
straight-
2.
At
constant
p,
V
∝
T
or
law)
T
line
variation
for
pressure
(or
volume)
against
temperature
1
3.
but
both
graphs
temperature,
absolute
would
273
extrapolate
°C.
This
back
to
temperature
the
is
same
known
At
As
These
as
increases,
the
volume
decreases.
In
fact
they
or
∝
p
V
=
constant
(Boyle’s
law)
do
not
is
are
always
always
known
as
expressed
apply
to
the
in
ideal
Kelvin
experiments
gas
(see
done
laws.
page
with
The
25).
real
These
gases.
are
A
real
gas
is
said
to
‘deviate’
from
ideal
behaviour
under
certain
proportional.
conditions
•
e xperimenTal invesTiGaTions
1.
p
relationships
temperature
zero
pressure
inversely
T,
V
laws
•
constant
low
Temperature
dependent
t
as
the
variable;
independent
V
as
the
variable;
P
as
(e.g.
Volume
high
of
gas
concentrated
the
control.
•
temperature t measured
pressure).
is
trapped
sulfuric
Concentrated
in
capillary
tube
by
bead
of
acid.
sulfuric
acid
is
used
to
ensure
gas
remainsdry.
pressure gauge
•
Heating
gas
causes
•
Pressure
•
Temperature
it
to
expand
moving
bead.
to measure P
remains
equal
to
atmospheric.
surface of water
of
gas
altered
by
temperature
of
bath;
time
xed volume of air
is
needed
to
ensure
bath
and
gas
at
same
as
the
temperature.
water (or oil) bath
air in
3.
P
as
the
independent
variable;
V
dependent
variable;
ask
tas
•
Fixed
volume
of
gas
is
trapped
in
the
ask.
Pressure
the
control.
zero of scale
is
trapped air
measured
by
a
pressure
pressure gauge
gauge.
scale to
to measure p
•
Temperature
of
gas
altered
by
temperature
of
bath
–
time
is
measure V
needed
to
ensure
bath
and
gas
at
same
temperature.
(length
2.
Temperature
dependent
t
as
the
variable;
independent
P
as
the
variable;
V
as
the
air
and volume)
pump
oil column
surface of oil
control.
oil
temperature t measured
capillary tube
•
Volume
of
gas
•
Increase
•
Temperature
measured
against
calibrated
scale.
scale to measure V
(length and volume)
surface of water
water bath
bead of
of
changed;
pressure
of
time
gas
is
forces
will
be
needed
to
oil
column
altered
ensure
to
when
gas
is
compress
volume
always
gas.
is
at
room
sulfuric acid
temperature.
gas (air)
zero of scale
volume V
T H E R M A L
P H Y S I C S
29
T g  2
equaTion of sTaTe
The
three
ideal
gas
one
mathematical
definiTions
laws
can
be
combined
together
to
produce
The
relationship.
concepts
constant
of
pV
a
gas
of
are
(an
the
all
mole,
molar
introduced
easily
so
measurable
mass
as
to
be
and
able
quantity)
to
the
to
Avogadro
relate
the
the
number
mass
of
_
=
constant
T
This
If
constant
we
of
molecules
will
compare
different
molecules
weuse
ideal
gases,
that
the
depend
the
value
it
are
on
of
turns
in
out
the
denition
of
the
this
gas
the
to
–
mass
and
constant
depend
not
mole
their
to
type
for
on
the
type.
state
of
gas.
different
this
for
n
are
present
gas
An
case
Mole
moles
The
of
pV
of
_
a
universal
is
one
values
that
of
of
follows
P,
V
the
and
T
gas
(see
29).
mole
any
gas.
gas
all
‘amount
gas
=
the
for
page
of
in
ideal
laws
masses
number
In
that
Ideal
that
is
of
the
substance
that
basic
SI
substance’.
is
equal
substance
that
unit
One
to
for
mole
the
of
amount
contains
the
constant.
nT
same
number
of
particles
as
0.012
kg
of
12
The
universal
The
SI
constant
is
called
1
unit
for
R
is
J
=
8.314
J
molar
gas
constant
carbon–12
R.
it
1
mol
1
R
the
C).
(slightly)
When
writing
shortened
to
the
the
unit
mol.
K
Avogadro
1
mol
is
(
This
is
the
number
of
atoms
in
0.012
kg
K
12
constant,
N
of
carbon–12
(
23
C).
It
is
6.02
×
10
.
A
pV
_
Summary:
=
R
Or
p
V
=
n
R
T
Molar
nT
mass
The
is
mass
called
applies.
e x ample
a)
What
number
4)
at
will
be
room
occupied
by
temperature
8
g
(20
of
helium
°C)
and
A
(mass
×
10
an
A,
mole
element
then
of
a
mass.
the
substance
A
has
simple
a
molar
rule
certain
mass
mass
will
be
N
_
atmospheric
n
(1.0
one
molar
grams.
=
N
5
pressure
If
number,
volume
of
the
Pa)
A
number of atoms
__
number
of
moles
=
8
n
=
=
2
Avogadro
moles
constant
4
T
=
20
+
273
nRT
_
V
=
293
K
2 × 8.314 × 293
__
=
ide al Ga ses and re al Ga ses
3
=
=
0.049
m
5
p
1.0
×
10
An
b)
How
many
atoms
are
there
in
8
g
of
helium
(mass
number
of
4)?
ideal
p,
V
gas
and
is
T
microscopic
a
one
and
that
thus
follows
ideal
description
of
the
gases
an
gas
laws
cannot
ideal
gas
be
for
all
values
liqueed.
is
given
to
ideal
on
The
page
31.
8
n
=
=
2
moles
Real
4
providing
23
number
of
atoms
=
2
×
6.02
×
1.2
×
however,
that
the
can
approximate
intermolecular
forces
are
behaviour
small
enough
10
to
24
=
gases,
be
ignored.
For
this
to
apply,
the
pressure/density
of
the
10
gas
must
Equating
linK beTween The maCrosCopiC and
hand
be
the
side
of
low
and
the
right-hand
the
temperature
side
macroscopic
of
this
must
formula
equation
of
be
with
state
for
moderate.
the
an
right-
ideal
gas
miCrosCopiC
shows
The
equation
macroscopic
of
state
for
properties
an
of
ideal
a
gas
gas,
(p,
V
pV
and
=
nRT,
T).
links
Kinetic
the
that:
three
2
theory
nRT
=
N E
K
3
(page
26)
random
describes
motion
a
and
gas
for
as
being
this
composed
theory
to
be
of
molecules
valid,
each
of
in
these
N
_
But
macroscopic
properties
must
be
linked
to
the
n
=
microscopic
,
so
N
A
behaviour
of
A
analysis
molecules.
N
_
2
RT
detailed
of
how
a
large
number
of
=
N E
randomly
K
N
3
A
moving
molecules
interact
beautifully
predicts
another
3
formula
that
allows
the
links
between
the
macroscopic
and
∴
E
R
_
=
T
K
2
N
A
the
microscopic
only
uses
The
the
by
be
Newton’s
assumptions
mean
to
an
detail
of
laws
describe
ideal
this
assumptions
identied.
and
from
a
the
The
derivation
handful
of
of
the
formula
assumptions.
microscopic
perspective
These
what
we
(the
molar
derivation
the
is
not
required
approach
are
by
the
outlined
IB
on
syllabus
the
but
following
gas
constant)
T
∝
is
so
this
proportional
The
result
the
of
this
derivation
idealized
gas
are
is
that
related
the
to
pressure
just
two
E
=
called
the
K
3
•
The
number
•
The
average
of
molecules
random
present,
kinetic
N
energy
per
molecule,
E
K
30
T H E R M A L
P H Y S I C S
average
KE
per
molecule
Boltzmann’s
constant
k
.
k
=
B
N
A
3
T
R
_
=
T
B
2
N
A
N E
the
B
k
2
=
are
absolute
R
is
K
2
pV
constant)
the
R
and
quantities:
to
that
E
ratio
3
of
(Avogadro
shows
K
The
A
volume
N
equation
N
page.
and
A
numbers
temperature
gas.
and
R
xed
mc     g
KineTiC model of an ide al Ga s
A
before
•
Newton’s
single
molecule
laws
apply
to
•
molecular
When
walls
a
molecule
of
a
changes
there
are
no
intermolecular
during
a
the
molecules
•
the
molecules
•
the
collisions
are
treated
as
(due
There
–
are
in
random
between
there
is
(no
no
to
its
the
off
the
momentum
change
momentum
must
is
a
in
vector).
the
energy
time
is
spent
been
the
a
wall
force
on
the
(Newton
II).
There
must
have
been
an
equal
and
molecules
lost)
in
have
from
motion
opposite
•
the
points
•
elastic
bounces
container
molecule
are
of
collision
•
•
walls
forces
direction
except
the
container.
behaviour
•
hitting
wall
Assumptions:
after
wall
force
molecule
on
the
(Newton
wall
from
the
III).
these
•
Each
time
there
is
a
collision
between
collisions.
a
The
pressure
of
a
gas
is
explained
as
molecule
exerted
and
on
the
the
wall,
a
force
is
wall.
follows:
•
The
average
forces
time
on
means
constant
•
This
we
all
the
wall
that
force
force
what
of
the
call
there
on
per
microscopic
over
the
unit
a
is
period
of
effectively
wall
area
from
of
the
a
the
gas.
wall
is
pressure.
result
F
P
=
A
Since
overall force
of
on wall
on molecule
The
pressure
of
a
gas
is
a
result
between
the
the
molecules,
molecules
and
gas
the
of
the
have
any
pressure l aw
Macroscopically,
the
•
(see
can
If
be
the
the
at
Charles’s l aw
a
constant
pressure
proportional
this
to
its
page
of
29).
gas
is
the
its
follows
of
have
–
in
Microscopically
as
temperature
energy
a
Macroscopically,
temperature
analysed
molecules
kinetic
a
gas
more
they
are
•
Fast
on
molecules
change
hit
Thus
the
the
goes
up,
•
average
A
higher
moving
The
of
will
have
momentum
walls
of
the
microscopic
molecule
molecules
hit
Faster
For
the
both
in
is
this
pressure,
proportional
kelvin
can
(see
be
temperature
molecules
moving
a
walls
with
force
(see
page
to
lower
the
will
we
a
measure
the
temperature
slower.
the
on
Thus
the
energy.
we
any
29).
analysed
as
temperature,
inversely
of
At
molecules
We
cannot
cannot
reduce
go
their
further!
(see
means
(see
faster
•
left).
page
seen
The
molecules
hit
at
the
a
constant
pressure
proportional
29).
to
be
to
of
its
a
gas
is
volume
Microscopically
this
can
correct.
constant
means
that
will
temperature
the
molecules
average
of
gas
have
a
speed.
are
walls
these
greater
the
microscopic
•
left).
The
the
If
the
volume
of
the
gas
exerts
force
on
moving
more
goes
the
rate
at
which
will
constant.
increases,
these
take
place
on
a
unit
•
area
Increasing
container
the
volume
decreases
of
the
faster
the
wall
must
go
down.
which
so
the
average
The
average
force
on
a
unit
area
molecules
the
rate
total
force
hit
the
with
wall
–
decreases.
of
often.
the
wall
can
thus
be
the
same.
•
If
the
average
total
force
decreases
total
Thus
the
pressure
remains
the
pressure
decreases.
same.
up.
high pressure
low temperature
each
wall
from
low temperature
pressure
that
the
greater.
reasons,
wall
microscopic
molecule
container.
force
be
a
remain
•
force
because
energy
the
•
is
of
move
imagine
kinetic
Macroscopically,
when
the
•
zero,
zero
constant
•
they
gas
constant
moving
of
•
gas
boYle’s l aw
a
be
collisions
each
a
temperature
then
•
of
at
follows
•
they
a
average.
moving
greater
volume
Microscopically
•
faster
we
molecules
lower
kinetic
kelvin
of
energy
container.
to
volume
as
kinetic
the
absolute
walls
temperature
average
of
a
collisions
the
overall force
goes
low pressure
high temperature
up.
high temperature
constant
constant
temperature
pressure
low volume
high volume
low volume
high volume
constant volume
low pressure
high pressure
Microscopic
Microscopic
pressure
justication
of
the
Charles’s
justication
of
Microscopic
justication
of
Boyle’s
law
law
law
T H E R M A L
P H Y S I C S
31
ib qt – t yc
The
following
information
relates
to
questions
1
and
2
b)
below.
An
electrical
information
A
substance
A
graph
of
is
its
heated
at
a
constant
temperature
against
rate
time
of
is
energy
shown
heater
for
written
swimming
on
its
pools
has
the
following
side:
transfer.
below.
50 Hz
erutarepmet
(i)
2.3 kW
Estimate
how
many
days
it
would
take
this
P
heater
N
O
(ii)
to
Suggest
heat
two
the
water
reasons
in
why
the
this
swimming
can
only
be
pool.
[4]
an
approximation.
6.
L
a)
A
cylinder
tted
[2]
with
a
piston
contains
0.23
mol
of
M
helium
gas.
K
piston
time
helium gas
1.
Which
regions
existing
in
a
of
the
mixture
graph
of
correspond
two
to
the
substance
phases?
The
A.
KL,
MN
and
following
piston
B.
2.
LM
and
All
regions
D.
No
regions
which
in
B.
LM
×
10
Pressure
=
1.0
×
10
Temperature
=
290
region
MN
OP
of
the
graph
is
the
specic
heat
capacity
7.
This
A
When
the
volume
of
a
gas
is
isothermally
compressed
smaller
volume,
the
pressure
exerted
by
the
gas
on
Use
this
the
walls
increases.
The
best
microscopic
of
the
C.
the
pressure
increase
individual
gas
student
at
gas
is
that
molecules
molecules
average
repel
velocity
frequency
walls
4.
A
lead
stops.
is
at
the
smaller
a
a
value
for
the
universal
are
of
each
gas
of
other
with
is
made
in
the
calculation
is
(1)
about
determines
home.
mass
more
determining
the
specic
latent
heat
of
She
the
takes
specic
some
ice
latent
from
heat
the
of
fusion
freezer,
of
measures
and
mixes
jug.
it
She
with
stirs
a
known
until
all
the
mass
ice
of
has
water
in
melted
an
and
the
nal
the
temperature
gas
temperature
in
of
the
the
mixture.
freezer
and
She
the
also
initial
strongly
hitting
the
wall
is
molecules
with
of
the
water.
greater
records
her
measurements
as
follows:
the
Mass
of
ice
used
m
0.12
kg
12
°C
0.40
kg
i
red
result,
assumption
compressed
molecules
collisions
the
(2)
volume
greater
bullet
As
calculate
explanation
She
the
to
(a)(i).
temperature
D.
data
ice.
measured
B.
the
question
measures
A.
Pa
K
constant.
State
insulating
for
3
m
the
its
container
the
to
ice
a
with
of
fusion
3.
helium
greatest?
in
D.
the
3
5.2
(ii)
C.
for
=
gas
KL
available
shown.
5
substance
A.
are
position
Volume
(i)
the
the
NO
C.
In
data
OP
into
the
an
iron
plate,
temperature
of
where
the
it
lead
deforms
increases
Initial
and
by
temperature
of
ice
T
i
an
Initial
mass
Initial
temperature
of
water
m
w
amount
∆T.
For
an
identical
bullet
hitting
the
plate
with
twice
of
water
T
22
°C
15
°C
w
the
speed,
what
is
the
best
estimate
of
the
temperature
increase?
Final
A.
∆T
B.
2
∆T
C.
2
∆T
temperature
of
mixture
f
The
specic
heat
capacities
1
c
=
4.2
kJ
4
In
and
c
=
2.1
kJ
and
ice
are
1
kg
°C
i
winter,
in
some
countries,
the
water
in
a
swimming
Set
up
the
needs
to
be
appropriate
during
the
process
of
representing
coming
to
energy
thermal
heated.
equilibrium,
Estimate
equation,
pool
transfers
a)
water
1
°C
∆T
a)
5.
of
1
kg
w
D.
T
the
cost
of
heating
the
water
in
a
typical
specic
swimming
that
latent
will
heat
L
enable
of
ice.
her
to
Insert
solve
values
for
the
into
the
i
pool
from
may
choose
Clearly
5
°C
to
show
information
to
a
suitable
consider
any
will
temperature
any
estimated
be
reasonable
values.
for
size
The
swimming.
of
heat
Density
of
the
pool.
b)
following
capacity
of
water
4186
J
1000
kg
kg
(i)
(ii)
per
kW
h
Estimated
of
electrical
energy
c)
K
Explain
in
State
do
not
solve
the
[5]
physical
your
meaning
equation
an
assumption
experiment,
in
Why
she
should
the
mixture
has
melted?
(but
of
not
each
each
energy
transfer
symbol).
[4]
you
setting
up
have
made
your
about
equation
in
the
take
the
temperature
(a).
[1]
of
immediately
after
all
the
ice
[1]
[7]
Explain
of
–
T H E R M A L
P H Y S I C S
from
molecular
ice
Q u E S T I o n S
but
[4]
Calculations
I B
above,
$0.10
values
e)
32
data
m
d)
Cost
the
1
3
water
from
equation.
term
useful:
1
Specic
equation
You
does
not
the
microscopic
behaviour,
increase
why
while
it
point
the
is
of
view,
in
temperature
melting.
terms
of
the
[4]
4
w a V e s
Oo
DefinitiOns
Many
an
systems
object
xed
the
continually
average
same
time
simple HarmOnic mOtiOn (sHm)
involve
point
path
between
interchange
vibrations
moves
(the
through
repeats.
of
energy
or
oscillations;
to-and-fro
mean
space
about
position)
taking
Oscillations
between
a
Simple
a
when
retracing
xed
involve
kinetic
the
and
proportional
is
caused
mean
F
Mass
moving
Kinetic
Potential
energy
energy
Moving
Elastic
mass
energy
store
two
springs
a
moving
vertical
an
and
force
also
as
object
displacement
restoring
position
-x
F
=
in
the
is
from
that
motion
always
a
must
proportional
xed
point.
always
to
the
that
takes
directed
be
place
towards,
This
and
acceleration
pointed
towards
displacement
from
the
or
F
=
-
(constant)
×
x
ma
∝
-x
or
a
=
-
(constant)
×
x
the
negative
sign
signies
that
the
acceleration
is
always
pointing
back
springs
towards
Mass
its
dened
of
potential
The
horizontal
a
to,
is
a,
position.
∝
Since
a
between
by
mean
motion
acceleration,
is
the
potential.
harmonic
the
on
spring
Moving
Elastic
mass
energy
the
mean
position.
potential
in
the
2
acceleration a / m s
springs
and
gravitational
potential
A
energy
displacement x / m
Simple
pendulum
Moving
Gravitational
-A
Buoy
up
bouncing
and
down
in
pendulum
potential
bob
energy
Moving
Gravitational
buoy
PE
water
An
of
of
bob
buoy
and
water
oscillating
Moving
Points
Elastic
PE
•
ruler
as
a
result
sections
the
bent
to
note
about
SHM:
of
The
time
period
T
does
not
depend
on
the
amplitude
A.
It
is
ruler
isochronous
of
one
end
being
of
the
•
displaced
while
Not
of
the
other
is
all
oscillations
are
SHM,
but
there
are
many
everyday
examples
ruler
natural
SHM
oscillations.
xed
Denition
e x ample Of sHm: ma ss between twO springs
Displacement,
The
instantaneous
x
measurement:
m)
object
mean
from
its
distance
of
the
(SI
moving
position
(in
a
simple harmonic motion
specied
Amplitude,
A
The
direction)
maximum
displacement
(SI
displacement
velocity
acceleration
against ti me
against ti me
against ti me
large displacement to right
measurement:
m)
from
the
mean
maximum
right
zero
displacement
velocity
position
zero velocity
mass m
Frequency,
f
The
number
of
maximum
left
oscillations
large force to left
completed
per
measurement
unit
is
time.
the
The
number
acceleration
SI
of
small displacement to right
cycles
per
second
or
Hertz
right
(Hz).
small velocity
to left
Period,
T
The
time
taken
(SI
measurement:
s)
mass m
left
1
for
one
complete
oscillation.
T
=
small force to left
f
Phase
This
is
a
measure
of
how
‘in
step’
zero
right
difference,
ø
different
particles
are.
If
moving
displacement
large velocity
to left
maximum
together
they
are
in
phase.
ø
is
mass m
velocity
left
measured
in
either
degrees
(°)
or
zero net force
zero
acceleration
radians
(rad).
complete
360°
cycle
so
or
2π
180°
rad
or
π
is
one
rad
small displacement to left
is
right
completely
out
of
phase
by
half
a
small velocity
to left
cycle.
A
phase
difference
of
90°
or
mass m
left
π/2
rad
is
a
quarter
of
a
cycle.
small force to right
object oscillates betwe en ex tremes
large displacement to left
maximum
right
displacement
zero velocity
zero velocity
mass m
maximum
left
acceleration
large force to right
displacement, x
amplitude, A
mean position
W A V E S
33
gh o  ho oo
acceleratiOn, VelOcity anD Displ acement During sHm
•
acceleration
•
velocity
•
acceleration
leads
leads
velocity
by
displacement
and
90°
by
displacement
90°
are
displacement
180°
velocity
out
of
phase
•
displacement
•
velocity
lags
lags
velocity
acceleration
by
by
90°
90°
time
T
T
3T
4
2
4
T
acceleration
energy cHanges During simple HarmOnic mOtiOn
During
total
SHM,
energy
Energy
in
energy
must
SHM
is
is
interchanged
remain
constant.
proportional
•
the
mass
•
the
(amplitude)
•
the
(frequency)
between
The
KE
and
oscillation
is
PE.
said
Providing
to
be
there
are
no
resistive
forces
which
dissipate
this
energy,
the
undamped
to:
m
2
2
E

tot

p
Graph showing the
variation with distance, x
of the energy during SHM

k
x
x
x
0
0


tot

k
Graph showing the
variation with time, t
of the energy
during SHM

p
t
34
W A V E S
T
T
4
2
3T
4
T
tv v
intrODuctiOn – rays anD waVe frOnts
Light,
sound
examples
of
and
ripples
wave
They
all
transfer
•
They
do
so
•
They
all
surface
of
a
pond
trans Verse waVes
are
all
Suppose
shown
energy
without
they
the
motion.
•
which
on
a
from
net
one
motion
place
of
to
the
another.
medium
a
stone
is
thrown
into
a
pond.
The
continuous
oscillations.
Two
A
out
as
situation
through
oscillations
(vibrations)
of
one
sort
or
(1) wave front diagram
A
spread
travel.
involve
another.
Waves
below.
oscillations
wave
wave
important
are
involves
pulse
categories
a
succession
involves
of
(2) ray diagram
SHM.
wave
just
are
of
one
individual
oscillation.
transverse
and
direction of
longitudinal
(see
below).
The
pages
The
table
gives
some
examples.
energy ow
following
common
to
all
analyse
some
of
the
properties
that
are
waves.
cross-section through water
Example
of
energy
A
object
transfer
wave pattern moves
Water
ripples
oating
(Transverse)
down’
Sound
The
waves
(Longitudinal)
the
Light
The
gains
an
‘up
wave pattern at a given
out from centre
and
instant of time
motion.
sound
received
eardrum
at
an
ear
makes
wave pattern slightly
vibrate.
‘up’ and ‘down’
wave
(Transverse)
back
of
stimulated
the
eye
when
(the
light
is
retina)
is
waves
Buildings
collapse
during
top
T
and
L)
along
of
a
A
‘sideways
pulse’
will
travel
rope
a
rope
that
is
held
taut
between
that
there
These
Compression waves
A
compression
pulse
will
spring
The
down
(Longitudinal)
a
spring
that
is
is
held
between
two
lOngituDinal waVes
longitudinal
The
wave.
parallel
to
The
fronts
This
the
is
the
several
of
are
the
parts
direction
of
of
of
aspects
to
important
wave
the
energy
energy
because
direction
of
oscillations
should
whereas
the
this
to
wave
all
that
can
be
waves.
pattern.
wave
that
The
are
wave
fronts
moving
transfer.
The
rays
together.
highlight
the
transfer.
in
be
of
noted
the
the
that
above
medium.
the
rays
diagrams.
are
This
at
is
right
always
angles
the
to
the
wave
case.
the
This
are
crest,
trough
people.
It
a
the
the
taut
•
is
are
aspects
movement
direction
oscillations
as
as
travel
•
Sound
known
known
people.
highlight
a
is
is
two
•
(Transverse)
down
wave
wave
down
studied.
stretched
the
the
earthquake.
Note
Waves
of
an
bottom
(Both
edge of pond
received.
The
Earthquake
later in time
centre of pond
energy
wave
is
an
example
of
a
transverse
wave
because
the
transfer.
oscillations
are
at
right
angles
to
the
direction
of
energy
o
transfer.
Transverse
uids
A
point
(high
view from above
on
the
pressure)
where
(2) ray diagram
(1) wave front diagram
mechanical
(liquids
or
wave
is
everything
waves
cannot
be
propagated
through
gases).
where
known
is
‘far
as
everything
a
is
‘bunched
compression.
apart’
(low
pressure)
A
is
together’
point
known
as
a
rarefaction.
displacement
loudspeaker
of molecules
to the right
distance
to the left
along wave
rarefaction
rarefaction
rarefaction
v
wave moves
to right
situation
cross-section through wave at one instant of time
rekaepsduol
compression
direction
compression
variation of
pressure
of energy
transfer
average
pressure
motion of air molecules in
wave pattern moves
same direction as energy
out from loudspeaker
transfer
distance
along wave
Relationship
between
displacement
and
pressure
graphs
W A V E S
35
wv h
DefinitiOns
There
are
motion
also
some
in
more
shown
Because
waVe equatiOns
useful
on
the
terms
detail.
the
The
that
table
need
to
below
be
dened
attempts
to
in
order
explain
to
analyse
these
terms
wave
and
There
they
are
graphs.
graphs
seem
The
on
to
be
identical,
you
need
to
look
at
the
axes
of
the
graphs
The
displacement–time
wave.
they
The
will
All
the
not
graph
other
start
on
the
points
their
displacement–position
points
along
left
on
represents
the
oscillations
wave
at
will
exactly
the
oscillations
oscillate
the
same
in
a
for
one
similar
point
manner,
In
this
the
wave
at
graph
one
on
the
instant
of
right
time.
represents
At
a
later
a
The
on
but
graphs
it
can
will
be
retain
used
to
the
same
represent
simple
It
applies
taken
for
‘snapshot’
time,
all
one
the
period
time,
the
wave
moved
on
by
the
of
wave
all
the
will
means
one
must
be
have
that
and
waves.
complete
of
the
wave,
pattern
will
wavelength,
given
the
speed
T
have
transverse
waves
of
λ
the
wave
by
distance
________
λ
=
=
T
time
AND
wavelength
to
is
shape.
longitudinal
relationship
speed,
time.
c
moved
•
very
wave
time
This
•
a
oscillation
the
but
is
links
frequency.
carefully.
•
that
because
1
Since
=
f
T
the
y-axis
records
direction
of
this
direction
of
the
displacement
only
the
value
displacement.
wave
were
at
energy,
right
of
So,
the
the
if
displacement.
this
wave
angles
to
displacement
would
the
It
be
a
direction
does
were
NOT
parallel
longitudinal
of
the
specify
wave
to
wave.
energy,
the
c
the
If
this
the
In
be
a
transverse
λ
words,
=
frequency
×
wavelength
wave.
x / tnemecalpsid
x / tnemecalpsid
+
f
wave
velocity
would
=
T
A
time / s
λ
e x ample
A
stone
is
thrown
onto
a
still
water
A
surface
and
oating
impact
creates
cork
point
1.0
a
m
has
away
the
displacement–time
wave.
A
small
from
the
following
graph
(time
is
position / m
measured
hits
Symbol
Displacement
x
mc/tnemecalpsid
Term
Denition
This
as
a
measures
result
Zero
its
a
change
wave
displacement
position.
the
of
the
For
distance
passing
refers
to
mechanical
(in
undisturbed
that
metres)
a
has
particular
the
mean
waves
that
taken
the
the
place
point.
(or
average)
displacement
particle
moves
is
from
the
from
the
instant
the
stone
water):
2
1
time/s
0
1.4
1.5
1.6
1.7
1.8
1
position.
2
Amplitude
A
This
is
the
position.
its
maximum
If
the
amplitude
wave
is
displacement
does
not
lose
from
any
the
of
its
mean
energy
a)
constant.
the
2
amplitude
of
the
wave:
cm
.........................................................
Period
T
This
is
the
time
taken
(in
seconds)
for
one
complete
b)
oscillation.
It
is
the
time
taken
for
one
complete
the
speed
d
to
pass
any
given
of
the
wave:
wave
c
point.
=
1.0
____
=
t
1
=
0.67
m
s
1.5
.........................................................
Frequency
f
This
is
the
number
of
oscillations
that
take
place
in
one
c)
second.
50
Hz
The
unit
means
used
that
50
is
the
cycles
hertz
are
(Hz).
A
frequency
completed
every
the
frequency
of
the
wave:
of
1
second.
f
=
1
____
=
T
=
3.33
Hz
0.3
.........................................................
Wavelength
This
λ
is
wave
one
the
shortest
between
another.
distance
two
‘In
points
phase’
(in
that
means
metres)
are
in
that
along
phase
the
two
the
with
d)
the
c
λ
are
moving
exactly
in
step
with
one
wavelength
points
another.
=
f
For
of
the
wave:
0.666
______
=
=
0.2
m
3.33
.........................................................
example,
crest
on
the
a
distance
water
compression
to
from
ripple
the
or
next
one
the
one
crest
to
distance
on
a
the
from
sound
next
one
wave.
1
Wave
speed
c
Intensity
I
This
is
pass
a
The
the
speed
stationary
intensity
of
(in
m
s
)
at
which
the
wave
fronts
observer.
a
wave
is
the
power
per
unit
area
2
that
The
is
received
intensity
by
of
a
the
observer.
wave
is
The
unit
proportional
to
is
W
the
m
.
square
2
of
The
period
and
the
frequency
of
its
amplitude:
any
wave
are
I
∝
A
inversely
related.
For
example,
if
1
___
frequency
of
a
wave
is
100
Hz,
then
its
period
must
be
exactly
of
100
In
symbols,
1
T
=
f
36
W A V E S
a
second.
the
eo 
electrOmagnetic waVes
Visible
light
part
a
is
one
possible
of
much
f / H
larger
λ / m
3
source
spectrum
of
similar
22
waves
that
are
10
all
13
10
electromagnetic.
Charges
that
are
2
1
10
accelerating
radium
generate
12
10
frequency
electromagnetic
elds.
If
an
wavelength
20
electric
10
15
3 × 10
charge
7
Hz
m
10
γ-rays
oscillates,
11
10
it
will
produce
a
19
varying
electric
magnetic
eld
and
10
10
at
10
X-rays
right
angles
to
one
18
10
another.
X-ray tube
9
10
These
oscillating
propagate
elds
(move)
1
7
10
as
a
transverse
wave
8
10
through
space.
Since
the sun
no
physical
UV
matter
16
10
is
involved
in
this
7
10
propagation,
can
travel
vacuum.
they
through
The
15
a
10
V I S I B L E
speed
6
10
UV
wave
can
light bulb
be
14
calculated
electric
from
and
constants
10
basic
magnetic
and
it
IR
is
the
same
electromagnetic
8
3.0
×
13
10
forall
waves,
1
10
m
s
.
15
1 × 10
Although
H
5
10
4
10
12
z
10
all
3
10
electromagnetic
waves
are
11
identical
m / λ htgnelevaw
this
zH / f ycneuqerf
of
electric heater
10
in
their
nature,
they
2
10
have
very
different
microwaves
10
properties.
This
is
10
Violet
because
range
of
of
the
1
huge
10
Indigo
frequencies
microwave oven
9
(and
thus
in
ELBISIV
involved
10
energies)
the
electromagnetic
spectrum.
1
Blue
8
shor t radio waves
Green
10
1
Yellow
See
page
10
132
Orange
(option
A)
for
more
7
10
Red
details.
2
10
standard broadcast
TV broadcast aerial
6
10
3
10
5
10
4
10
long radio
waves
4
10
6
14
3 × 10
Hz
10
m
5
10
3
10
radio broadcast aerial
W A V E S
37
iv d o od 
1.
Direct metHODs
The
most
d
direct
method
to
measure
the
speed
of
sound
is
to
record
the
time
taken
t
for
sound
to
cover
a
known
distance
d:
speed
c
=
.
In
t
1
air
a
at
normal
possible
race
or
can
side
hands)
the
experiment
seeing
Echoes
the
pressures
of
to
claps
two
be
a
wooden
used
to
heard.
with
temperatures,
would
building
be
and
their
With
to
use
planks
put
that
be
the
is
stop
being
source
not
hit
it
When
observer
by
is
for
at
and
of
the
difference
hearing
the
the
sound
buildings)
an
achieved,
approximately
time
other
possible
this
to
together)
and
is
travels
watch
surrounded
practice,
echoes.
a
sound
in
the
can
frequency
of
s
.
event
same
allow
to
m
between
same
experimenter
the
330
8
Given
seeing
some
place.
the
adjust
clapping
f
an
much
event
distance
Standing
echo
the
the
from
a
can
be
(e.g.
away
a
the
of
of
of
light
ring
of
a
d
or
in
sound
clapping
recorded
speed
(100m
distance
pulse
frequency
larger
to
(3×
starting
m
pistol
s
),
for
a
more).
front
(e.g.
a
of
a
tall
single
synchronize
(counting
1
10
the
wall
clap
the
number
of
(e.g.
of
the
sound
claps
of
in
1
a
given
time)
and
the
time
period
T
between
claps
is
just
T
=
.
In
this
time,
the
sound
travels
to
the
wall
and
back.
The
speed
of
f
sound
In
A
is
thus
either
graph
should
go
Timing
the
above
through
pulses
a
of
A
or
c
=
fλ
,
the
reliable
allow
the
result
speed
will
of
be
achieved
sound
to
be
if
a
range
calculated
of
distances,
from
the
rather
gradient
of
than
the
one
single
best-t
value
straight
is
line
used.
(which
over
smaller
timers
through
distances
and
water
to
/
or
be
requires
data
small
loggers.
time
This
intervals
equipment
to
be
would
recorded
allow,
for
with
precision.
example,
the
It
is
possible
speed
of
a
to
sound
automate
wave
investigated.
a
sound
electronic
a
can
cathode
of
techniques
fork)
that
sound
be
can
ray
from
(e.g.
is
can
frequency
Comparisons
Wavelength
and
measurements
tuning
Frequency
The
of
calculated
if
we
measure
a
sound’s
frequency
and
wavelength.
measurement
microphone
(e.g.
a)
more
will
origin).
speed
Stroboscopic
d)
the
a
time
electronic
rod
Appropriate
c)
situations
against
sound
using
metal
Frequency
b)
2df.
inDirect metHODs
Since
a)
the
=
distance
process
along
2.
of
of
c
the
be
oscilloscope
the
graph
ashing
source
of
controlled
light
the
at
(CRO)
allow
of
[page
the
time
known
116]
can
period
display
and
frequency)
a
hence
can
be
graph
the
used
of
the
frequency
to
measure
oscillations
to
be
the
of
a
sound
wave.
calculated.
frequency
of
the
vibrating
object
sound.
source
using
a
known
frequency
source
(e.g.
a
standard
tuning
fork)
or
a
calibrated
generator.
also
be
made
between
the
unknown
frequency
and
a
known
frequency.
measurement
interference
destructive
of
waves
interference.
(see
The
page
path
40)
can
be
difference
employed
between
to
these
nd
two
the
path
situations
difference
will
be
between
consecutive
positions
of
λ
source of
S
frequency
f
*
path A
path B
1
λ
detector (microphone
2
D
and cathode ray oscilloscope (CRO))
b)
Standing
can
•
in
•
c)
A
be
waves
revealed
the
(see
period
pattern
electronically
resonance
page
48)
in
a
gas
can
be
employed
using
tube
(see
made
a
small
page
by
dust
in
movable
49)
allows
the
maxima
will
be
column
length
.
factOrs tHat affect tHe speeD Of sOunD
Factors
include:
•
Nature
•
Density
•
Temperature
•
Humidity
38
of
material
(for
(for
the
location
of
adjacent
nodes.
The
positions
in
an
enclosed
tube
microphone.
the
2
3.
nd
tube
λ
adjacent
to
either:
an
air).
W A V E S
ideal
gas,
c
∝ √T )
for
different
maxima
to
be
recorded.
The
length
distance
between
i
intensity
The
sound
intensity,
I,
is
the
amount
of
energy
that
a
sound
wave
brings
to
a
unit
area
every
second.
The
units
of
sound
2
intensity
It
are
depends
W
on
m
the
.
amplitude
of
the
sound.
A
more
intense
sound
(one
that
is
louder)
must
have
a
larger
amplitude.
2
Intensity
This
∝
(amplitude)
relationship
between
intensity
and
amplitude
is
true
for
all
waves.
elcit r ap a fo tnemecalpsid
sessap evaw dnuos a nehw
louder sound
of same pitch
time
2
I ∝ A
The
inVerse square l aw Of raDiatiOn
As
the
distance
increases,
the
of
an
power
observer
received
from
by
a
the
point
source
observer
will
of
A
the
energy
spreads
out
over
a
larger
area.
A
doubling
will
result
in
the
reduction
of
the
power
the
point
received
to
the
of
the
original
A
source
power
of
a
sphere
of
radius
r
is
calculated
using:
radiates
received
a
total
per
unit
power
area
P
in
(the
all
directions,
intensity
I)
at
a
a
distance
quarter
area
4πr
of
then
distance
=
decrease
If
as
surface
2
light
r
away
from
the
point
source
is:
value.
P
_____
I
=
2
4πr
area 4A
For
area A
a
given
radiation
from
the
inverse
is
area
of
receiver,
inversely
point
source
square
law
the
intensity
proportional
to
the
and
to
the
receiver.
applies
to
of
This
all
the
received
square
is
of
the
known
as
distance
the
waves.
2
I
A
waVefrOnts anD rays
As
introduced
the
motion
of
on
a
page
35,
wavefront
waves
can
and/or
in
be
described
terms
of
rays.
in
terms
of
∝
ray
from
A
is
the
the
path
taken
by
the
wave
energy
as
it
travels
out
source.
wavefront
where
rays spreading out
x
the
dimensions,
wavefront
is
a
surface
oscillations
is
the
a
are
joining
in
wavefront
neighbouring
phase
is
a
line
with
and
one
in
points
another.
one
In
two
dimension,
the
point.
wavefront
point source
of wave energy
W A V E S
39
soo
If
interference Of waVes
When
and
two
we
waves
can
of
work
of
superposition.
at
any
time
the
out
the
The
where
same
meet,
resulting
overall
the
type
they
wave
using
disturbance
waves
meet
is
at
the
the
any
vector
point
and
of
waves
have
then
constructive
principle
sum
the
frequency
interfere
the
the
or
same
amplitude
interference
at
a
and
the
same
particular
point
can
be
destructive.
graphs
the
wave 1 displacement (at P)
disturbances
individual
that
would
waves.
This
have
is
been
shown
produced
by
each
of
the
below.
A
(a) wave 1
tinu /
time
time
wave 2 displacement (at P)
1
y
0
t / s
A
time
time
(b) wave 2
tinu /
resultant displacement (at P)
2
y
0
t / s
zero result
2A
(c) wave 1 + wave 2 = wave 3
tinu / y
time
time
0
constructive
t / s
Wave
superposition
tecHnic al l anguage
Constructive
are
‘in
There
step’
is
a
–
interference
with
zero
interference
step’
destructive
they
one
another
phase
takes
are
place
–
they
difference
place
said
e x amples Of interference
takes
to
when
be
out
are
the
said
between
the
of
when
waves
phase.
two
to
be
them.
are
waves
in
A
Destructive
exactly
There
Water
phase.
are
‘out
waves
ripple
waves.
of
tank
can
Regions
interference.
be
of
used
to
view
the
large-amplitude
Regions
of
still
water
interference
waves
are
are
of
water
constructive
destructive
interference.
several
Sound
different
ways
of
saying
this.
One
could
say
that
the
phase
It
difference
is
equal
to
‘half
a
cycle’
or
‘180
degrees’
or
‘π
is
possible
frequencies
Interference
can
take
place
if
there
are
two
possible
routes
ray
to
travel
between
the
constructive
from
two
source
rays
is
a
interference
to
observer.
whole
will
If
number
take
the
of
path
the
difference
=
n
λ
path
difference
=
(n
→
make
any
it
noise
up.
A
in
terms
computer
of
the
can
component
then
same
frequencies
but
of
different
generate
phase.
This
difference
wavelengths,
‘antisound’
will
interfere
a
position
with
the
original
sound.
An
observer
particular
in
space
could
have
the
overall
noise
level
if
the
waves
superimposed
destructively
at
that
position.
constructive
Light
1
+
)
λ
→
destructive
The
colours
seen
on
the
surface
of
a
soap
bubble
are
a
result
2
of
n
=
0,
1,
2,
3
.
.
constructive
constructive
or
destructive
interference
to
take
place,
of
the
waves
must
be
destructive
interference
of
two
light
One
phase
linked
or
ray
is
reected
off
the
outer
surface
of
the
bubble
the
whereas
sources
and
.
rays.
For
the
other
is
reected
off
the
inner
surface.
coherent.
superpOsitiOn Of waVe pulses
Whenever
different
each
wave
waves
individual
pulses
meet,
meeting
wave.
at
y
=
overall
a)
the
the
principle
same
y
+
1
point
y
+
2
y
of
in
superposition
space
is
just
applies:
the
At
vector
any
sum
of
instant
the
in
time,
the
displacements
net
that
displacement
would
have
that
been
results
3
b)
i)
A
A
pulse Q
A
pulse P
A
pulse Q
ii)
P + Q = 2A
P + Q = A
ii)
A = 0
iii)
iii)
A
A
A
A
pulse Q
pulse P
pulse Q
40
W A V E S
from
produced
etc.
i)
pulse P
in
then
place.
reduced
path
analyse
that
for
exactly
a
to
radians’.
pulse P
by
po o
pOl arizeD ligHt
Light
is
part
oscillating
to
one
the
electric
another
transverse
of
of
waves
and
is
the
and
(for
waves;
propagation.
dened
magnetic
more
both
The
to
direction
bre wster’s l aw
electromagnetic
be
of
elds
details
elds
plane
the
see
are
of
spectrum.
at
that
page
right
vibration
plane
that
It
are
at
132).
angles
of
is
made
right
They
to
up
of
A
angles
are
the
is
direction
the
electric
of
in
light
for
this
be
partially
refracted
reected
eld
incident
general,
always
the
electromagnetic
contains
ray
will,
ray
ray
is
the
boundary
and
are
at
is
right
angles
If
the
to
as
the
The
two
ray
another,
The
angle
polarizing
media
reected
reected
one
plane-polarized.
known
between
refracted.
plane-polarized.
totally
condition
on
reected
ray
and
then
of
the
incidence
angle
propagation.
reected ray is totally
plane-polarized
plane vibration of
EM wave containing
direction of electric
incident ray
electric oscillations
direction
θ
is unpolarized
eld oscillation
medium 1 (vacuum)
i
medium 2 (water)
of
magnetic
θr
distance
eld
transmitted ray
along
is par tially polarized
oscillation
the wave
represents electric eld oscillation into the paper
There
are
an
innite
number
of
ways
for
the
elds
to
be
oriented.
represents electric eld oscillation in the plane of the paper
Light
of
(or
any
vibration
EM
wave)
varies
is
said
randomly
to
be
unpolarized
whereas
if
the
plane-polarized
plane
light
θ
+
θ
i
has
a
xed
plane
of
vibration.
The
diagrams
below
represent
=
Brewster’s
electric
elds
of
light
when
being
viewed
‘head
90°
r
the
law
relates
the
refractive
index
of
medium
2,
n,
to
on’.
the
incident
angle
θ :
i
sin
θ
sin
i
=
θ
i
_____
n
_____
=
=
tan
θ
i
sin
θ
cos
θ
r
A
unpolarized light: over a
period of time, the electric
period of time, the electric
eld only oscillates in
eld oscillates in
one direction
random directions
A
mixture
of
polarized
plane-polarized.
If
light
the
and
plane
unpolarized
of
light
polarization
polarizer
from
polarized light: over a
an
detect
is
partially
rotates
in
is
a
device
plane
at
that
beam.
plane
90°
which
of
light
Most
and
the
is
light
said
to
sources
laboratory
processes
be
circularly
emit
that
produce
are
the
is
a
polarizer
light
used
to
to
preferentially
absorbs
allowing
any
light
in
transmission
only
this.
uniformly
b)
polarized.
unpolarized
microwaves
plane-polarized
analyser
polarization
a)
the
produces
An
light.
material
particular
the
any
unpolarized
polarized
Polaroid
one
is
i
light
often
whereas
radio
plane-polarized
waves.
Light
can
be
waves,
as
a
radar
result
polarized
as
of
trans-
a
parent
result
of
exhibit
ray
reection
double
that
enters
or
selective
refraction
a
crystal
is
or
split
absorption.
In
addition,
birefringence
into
two
where
some
an
plane-polarized
crystals
unpolarized
beams
that
indicates the
have
mutually
perpendicular
planes
of
polarization.
zero transmission
preferred directions
malus’s l aw
When
Optic ally actiVe substances
plane-polarized
preferred
direction
light
will
is
allow
incident
a
on
an
component
analyser,
of
the
light
its
to
An
be
optically
plane
transmitted:
of
active
polarization
solutions
(e.g.
sugar
substance
of
light
is
that
solutions
of
one
that
passes
rotates
through
different
the
it.
Many
concentrations)
are
opticallyactive.

0
analyser
analyser ’s
θ

plane of
θ
θ
preferred
vibration
original
direction
rotated
plane of

0
through
vibration
cos θ
angle θ
transmitted component of
plane-polarized light
optically active
electric eld after analyser
seen head-on with
substance
 = 
cos θ
0
electric eld amplitude, 
0
2
The
intensity
of
light
is
proportional
to
the
(amplitude)
.
2
I
is
transmitted
intensity
of
light
in
W
W
m
m
2
Transmitted
intensity
I
∝
E
2
I
is
incident
intensity
of
light
in
0
∴
2
I
∝
E
2
cos
θ
as
expressed
by
Malus’s
law:
θ
0
is
the
angle
between
the
plane
of
vibration
and
the
analyser’s
2
I
=
I
cos
θ
preferred
direction
0
W A V E S
41
u o o o
pOl arOiD sungl a sses
Polaroid
is
molecules
aligned
wires
a
material
selectively
with
will
the
absorb
molecules
selectively
cOncentratiOn Of sOlutiOns
containing
light
in
absorb
long
chain
that
the
molecules.
have
same
electric
way
that
a
The
For
elds
grid
a
given
optically
which
the
plane
•
length
of
active
solution,
polarization
is
the
angle
rotated
is
θ
through
proportional
to:
of
The
of
the
solution
through
which
the
plane-
microwaves.
polarized
•
The
light
passes.
concentration
A
polarimeter
It
consists
is
a
of
the
device
solution.
that
measures
θ
for
a
given
solution.
grid
electric
of
two
polarizers
(a
polarizer
and
an
analyser)
that
absorption
viewed
are
eld
initially
aligned.
The
optically
active
solution
is
introduced
head on
between
the
maximum
two
and
the
transmitted
analyser
is
rotated
to
nd
the
light.
stress analysis
Glass
grid
electric
When
be
light
transmitted
electric
•
normally
allow
The
by
with
and
a
person
vertically
absorb
light
standing
up,
with
Polaroid
electric
plastics
under
through
coloured
oscillating
some
placed
passed
head on
worn
glasses
transmission
viewed
eld
and
when
lines
become
stress.
stressed
are
birefringent
When
plastics
observed
in
polarized
and
the
then
(see
page
white
analysed,
regions
of
41)
light
is
bright
maximum
stress.
dark
elds
to
horizontally
oscillating
overall
intensity
elds.
absorption
will
mean
that
the
light
is
reduced.
•
Light
that
has
horizontally
•
Polaroid
reducing
reýected
from
plane-polarized
sunglasses
‘glare’
will
from
horizontal
to
some
preferentially
horizontal
surfaces
will
be
extent.
absorb
reýected
light,
surfaces.
•
liquiD-crystal Displ ays (lcDs)
LCDs
are
include
crystal
used
in
a
calculator
is
wide
displays
sandwiched
birefringent.
One
surrounding
the
variety
and
between
possible
liquid
of
different
computer
two
glass
arrangement
crystal
is
shown
applications
monitors.
electrodes
with
The
and
crossed
With
no
liquid
polarizer
that
rst
liquid
is
•
polarizers
below:
polarizer.
The
liquid
of
potential
a
rotate
•
This
The
all
screen
has
a
the
the
electrodes,
light
would
twisted
difference,
through
the
between
absorb
crystal
means
reach
reector
crystal
would
that
appear
the
second
through
the
black.
structure
causes
the
passed
and,
plane
of
in
the
absence
polarization
to
90°.
that
light
reecting
can
pass
surface
through
and
be
the
second
transmitted
polarizer,
back
along
its
polarizers
original
•
With
•
A
no
direction.
pd
between
the
electrodes,
the
LCD
the
molecules
appears
light.
liquid
pd
across
the
liquid
crystal
causes
to
crystal
align
with
the
transmitted
•
electrodes
The
extent
controlled
electric
and
to
by
this
which
the
eld.
This
section
the
of
means
the
screen
LCD
appears
less
will
light
will
appear
grey
or
be
darker.
black
can
be
pd
etched into glass
light enters the
•
Coloured
ålters
•
A
can
picture
can
be
be
built
used
up
to
from
create
a
colour
individual
image.
picture
elements.
LCD from the front
•
furtHer pOl ariz atiOn e x amples
Only
transverse
concentrated
are
waves
on
transverse
the
are
can
be
polarized.
polarization
able,
in
of
principle,
light
to
Page
but
be
41
all
EM
Microwave
cm)
has
waves
that
can
laboratory.
conducting
polarized.
plane
•
Sound
waves,
being
longitudinal
waves,
cannot
be
The
nature
of
radio
and
TV
broadcasts
means
that
of
radiation
used
are
42
to
polarized
receive
and
the
aerials
need
maximum
W A V E S
to
be
possible
properly
signal
the
signal
aligned
strength.
(with
wires.
vibration
a
typical
demonstrate
Polarization
absorbed.
If
of
can
the
be
grid
the
wavelength
wave
electric
are
eld
of
a
characteristics
demonstrated
wires
if
Rotation
of
the
grid
is
microwaves
often
to
using
aligned
the
a
few
in
the
grid
parallel
microwaves
to
of
the
will
polarized.
be
•
be
they
to
be
transmitted.
through
90°
will
allow
the
wv hvo – o
reflectiOn Of twO-DimensiOnal pl ane waVes
reflectiOn anD transmissiOn
In
general,
different
when
media
any
it
is
wave
meets
partially
the
reected
boundary
and
between
partially
two
The
transmitted.
are
diagram
reected
convention
below
at
we
a
shows
what
boundary.
always
happens
When
measure
when
working
the
angles
plane
with
waves
rays,
between
by
the
rays
normal
and
incident ray
the
drawn
normal.
at
right
The
angles
normal
to
incident
incident
ray
angle i
the
is
a
construction
line
that
is
surface.
normal
reected ray
reected
reected
angle r
ray
medium (1)
medium (2)
transmitted ray
surface
medium (2) is optically
Law of reection: i = r
denser than medium (1)
types Of reflectiOn
When
ray.
a
single
This
place
type
from
single
ray
of
an
light
‘perfect’
uneven
incident
diffuse
of
ray
is
l aw Of reflectiOn
strikes
a
reection
surface
smooth
is
such
generally
as
very
the
scattered
mirror
it
different
walls
in
all
of
produces
to
a
the
single
reection
room.
directions.
a
In
this
This
is
reected
that
takes
situation,
an
can
a
example
The
the
of
a
reection.
a
location
be
principles
line
at
and
one direction
of
the
angles
to
all directions
The
the
of
in
ray
A
which
must
ignores
and
ray
is
light
always
wavefront.
optics
nature
images
diagrams
optics.
direction
geometric
particle
optical
ray
geometric
noitceer esuid
noitceer rorrim
light leaves in
of
of
using
propagated.
right
study
light leaves in
is
nature
out
showing
energy
be
and
worked
The
the
wave
light.
rays spreading out
wavefront
point source
of light
We
see
objects
not
give
become
out
by
light
visible
receiving
by
with
light
themselves
a
source
of
that
so
has
we
light
come
cannot
(e.g.
the
from
see
Sun
them.
them
or
a
in
Most
the
light
objects
dark.
bulb)
do
Objects
because
diffuse
When
reections
have
taken
place
that
scatter
light
from
the
source
towards
our
a
mirror
reection
takes
place,
the
eyes.
direction
using
the
specify
The surfaces of the
to
picture scatter the
of
the
laws
the
ray
measure
imaginary
all
reected
of
ray
reection.
directions
angles
In
be
respect
line
predicted
order
involved,
with
construction
can
it
to
called
to
is
usual
an
the
normal.
light in all directions.
For
as
example,
the
angle
normal.
Light from the
right
incident
between
The
angles
the
normal
to
the
the
to
a
angle
is
always
incident
surface
surface
as
ray
is
the
shown
taken
and
the
line
at
below
.
central bulb sets
normal
o in all directions.
incident ray
r
i
reected ray
The
•
laws
the
of
reection
incident
angle
are
is
that:
equal
to
the
reýected
angle
•
An obser ver ‘sees’ the painting by
the
incident
the
normal
ray,
all
the
lie
in
reýected
the
same
ray
and
plane
(as
receiving this scattered light.
shown
Our
brains
travel
in
are
able
straight
to
lines.
work
out
the
location
of
the
object
by
assuming
that
rays
The
second
order
to
should
mirror
is
not
(e.g.
in
be
be
the
statement
precise
obvious
(such
as
suddenly
out
diagram).
of
the
is
and
that
the
only
is
a
one
of
ray
in
arriving
the
an
in
omitted.
represented
reected
plane
included
often
odd
at
It
a
above)
direction
page).
W A V E S
43
s ’  d  v d
e x amples
refractiVe inDe x anD snell’s l aw
Refraction
In
takes
general,
change
the
of
a
place
wave
that
direction.
change
in
at
wave
the
boundary
crosses
The
the
reason
speed
that
boundary
for
has
between
this
will
change
taken
two
media.
undergo
in
1.
a
direction
A
is
place.
Parallel-sided
ray
will
parallel
The
always
direction
overall
sideways.
As
with
reection,
considering
ray
travels
water),
If
the
ray
into
then
ray
of
the
an
the
is
ray
directions
between
optically
ray
travels
light
the
angles
of
into
refracted
denser
light
an
the
is
always
and
less
from
from
towards
the
medium
a
the
of
the
example
parallel-sided
one
with
block
of
this
has
is
block
which
been
shown
it
to
travelling
entered
move
the
the
in
a
block.
ray
below.
by
normal.
(e.g.
dense
the
specied
the
medium
refracted
optically
away
are
ray
to
effect
An
block
leave
air
If
incident ray
a
into
normal.
then
the
glass
normal.
normal
ray leaves block
parallel to incident ray
2.
Ray
travelling
between
two
media
refracted
If
incident ray
a
ray
goes
between
two
different
media,
the
two
individual
ray
less dense medium
refractive
indices
using
following
the
can
be
used
to
calculate
the
overall
refraction
equation
n
more dense medium
sin
θ
2
_____
n
sin
θ
1
=
n
1
sin
θ
2
or
=
n
2
sin
2
n
refractive
index
of
θ
medium
1
medium
2
1
θ
angle
in
medium
1
1
n
refractive
index
of
2
ray refracted
θ
angle
in
medium
2
2
towards normal
Suppose
water.
shown
normal
a
ray
The
of
light
refraction
is
shone
that
into
takes
a
sh
place
tank
would
that
be
contains
calculated
1st
air
refraction:
water
sin a
_____
n
(n
refracted
as
below:
= 1.0)
(n
glass
air
(n
=
glass
= 1.3)
sin
water
b
= 1.6)
glass
ray
incident ray
more dense medium
2nd
refraction:
n
×
c
less dense medium
sin
b
=
glass
n
b
×
sin
c
water
a
n
ray refracted away
glass
sin c
_____
_____
=
n
from the normal
Snell’s
law
allows
us
to
work
out
the
angles
a
ray
is
refracted
sin(angle
of
between
two
b
involved.
Overall
When
sin
water
different
the
refraction
is
from
incident
angle
a
to
refracted
media,
angle
incidence)
c.
________________
the
ratio
is
sin(angle
of
a
constant.
sin a
_____
refraction)
i.e.
n
=
sin a
_____
=
sin b
_____
×
overall
sin
The
constant
is
called
the
refractive
index
n
between
the
=
media.
waves
This
in
ratio
the
is
two
equal
to
the
ratio
of
the
speeds
of
c
sin
sin
b
c
two
n
the
water
media.
sin i
_____
=
sin
If
the
refractiOn Of pl ane waVes
refractive
particular
then
n
r
you
index
number
can
for
and
assume
a
the
that
particular
other
the
substance
medium
other
is
is
not
medium
is
given
as
a
mentioned
air
(or
to
The
in
reason
speed
of
for
the
the
change
in
direction
in
refraction
is
the
change
wave.
be
normal
absolutely
correct,
a
vacuum).
Another
way
of
expressing
this
medium 1
is
to
say
that
the
refractive
index
of
air
can
be
taken
to
(e.g. air)
be
1.0.
i
For
example
given
the
refractive
index
for
a
type
of
glass
might
boundary
be
as
n
r
medium 2 (e.g. glass)
=
wavelength smaller
1.34
glass
since speed reduced
This
means
incident
that
angle
a
of
ray
40°
entering
would
the
have
a
glass
from
refracted
air
with
angle
refracted ray
an
given
by
Snell’s
law
(an
experimental
law
of
refraction)
states
that
sin 40°
_______
sin
r
=
=
0.4797
sin
i
sin
r
____
1.34
∴
r
=
the
ratio
The
n
sin
θ
glass
glass
=
V
air
air
_______
=
n
air
_____
=
sin
ratio
n
is
equal
sin
1
2
for
a
given
frequency.
ratio
of
the
speeds
in
the
different
←
speed
of
wave
in
medium
2
←
speed
of
wave
in
medium
1
2
______
___
=
sin
V
θ
1
W A V E S
the
V
2
___
n
glass
to
θ
=
V
θ
glass
44
constant,
28.7°
____
n
=
1
media
ro d  
tOtal internal reflectiOn anD critic al angle
e x amples
In
1.
general,
boundary
It
is,
both
between
under
complete
This
can
1
certain
(total)
in
the
two
and
refraction
can
happen
at
the
What
a
media.
when
a
denser
with
ray
sh
sees
under
water
entire world outside water is
circumstances,
reection
happen
travelling
n
reection
possible
no
to
meets
the
visible in an angle of twice the
guarantee
transmission
at
boundary
critical angle
all.
and
it
is
medium.
< n
par tial
2
transmission
θ
grazing
1
emergence
n
1
critical angle c
n
2
θ
θ
At greater than the critical angle, the surface
total
c
of the water acts like a mirror. Objects inside
2
reection
1
the water are seen by reection.
2
3
2.
Prismatic
reectors
O
source
A
prism
surface
Ray1
This
ray
is
partially
Ray2
This
ray
has
reected
and
partially
be
a
refracted
angle
of
nearly
90°.
ray
refracted
of
is
the
angle
incidence
θ
name
of
90°.
for
the
given
The
to
the
critical
critical
ray
that
angle
be
the
used
prism
in
at
place
of
greater
a
mirror.
than
the
If
the
critical
light
strikes
angle,
it
the
must
totally
internally
reected.
The
Prisms
critical
can
of
refracted.
is
has
the
are
used
in
many
optical
devices.
Examples
include:
a
angle
•
periscopes
over
ray.
a
–
the
double
reection
allows
the
reection
means
user
to
see
crowd.
c
Ray3
This
ray
angle.
has
The
critical
is
angle
Refraction
reected
ray
an
at
said
angle
the
to
can
of
incidence
cannot
occur
boundary
be
be
totally
and
so
greater
the
stay
inside
internally
worked
out
as
ray
than
must
the
be
medium
critical
•
totally
2.
The
•
reected
follows.
For
the
binoculars
–
binoculars
do
SLR
the
critical
the
cameras
double
not
–
have
the
to
view
be
too
that
the
long
through
the
lens
is
reected
up
to
eyepiece.
ray
,
binoculars
periscope
n
sin
θ
1
=
n
1
2
sin
The prism arrangement delivers the
θ
2
image to the eyepiece the right way
θ
=
90°
=
θ
up. By sending the light along the
1
instrument three times, it also allows
θ
2
c
the binoculars to
1
___
∴
sin
θ
be shor ter.
=
c
n
2
eyepiece
lens
metHODs fOr Determining refractiVe inDe x
e xperimentally
Par t of ray identied
in several positions
objective lens
glass
block
1.
Locate
paths
through
a
taken
solid
by
and
different
measuring
rays
its
either
by
position
or
sending
aligning
a
ray
objects
Par t of
Par t of ray in
by
eye.
Uncertainties
in
angle
measurement
are
dependent
on
ray identied
block can be inferred
protractor
measurements.
(See
diagrams
on
left)
from measurements
2.
Use
a
travelling
depth
and
microscope
apply
following
to
measure
real
and
apparent
formula:
ray heading towards
real
centre will not be
depth
of
object
_______________________
n
=
apparent
refracted entering the block
3.
Very
accurate
depth
of
object
measurements
of
angles
of
refraction
can
be
semi-circular
achieved
using
a
prism
of
the
substance
and
a
custom
piece
glass block
of
equipment
call
a
spectrometer
centre
W A V E S
45
Do
There
DiffractiOn
When
Waves
waves
also
pass
through
spread
around
apertures
obstacles.
they
This
tend
to
wave
spread
property
out.
is
•
called
d >
some
(or
•
the
The
important
becomes
wavelength
diffraction
geometric
are
Diffraction
is
large
points
relatively
in
to
note
more
comparison
from
these
important
to
the
size
diagrams.
when
of
the
the
aperture
object).
wavelength
needs
to
be
of
the
same
order
of
magnitude
λ
as
shadow region
the
aperture
for
diffraction
to
be
noticeable.
practic al signific ance Of DiffractiOn
Whenever
d
of
an
observer
electromagnetic
receives
waves,
information
diffraction
from
causes
the
a
source
energy
to
λ
spread
out.
This
spreading
takes
place
as
a
result
of
any
geometric
obstacle
d ≈
in
the
way
and
the
width
of
the
device
receiving
the
λ
shadow region
electromagnetic
radiation.
Two
sources
of
electromagnetic
diraction more
waves
impor tant with
smaller
that
out
and
not
they
are
angularly
interfere
can
be
with
close
one
resolved
to
one
another.
(see
page
another
This
can
will
both
affect
spread
whether
or
101).
d
obstacles
Diffraction
λ
d >
because
λ
effects
they
meaning
are
that
mean
that
smaller
light
will
it
than
is
impossible
the
diffract
ever
wavelength
around
the
to
of
see
atoms
visible
atoms.
It
is,
light,
however,
d
geometric
geometric
possible
shadow
shadow
devices
region
region
•
CDs
can
to
image
where
and
be
DVDs
stored
recording
λ
•
The
–
the
smaller
needs
to
be
maximum
depends
on
the
wavelengths.
considered
amount
size
and
of
the
Practical
include:
information
method
that
used
for
microscope
using
a
light
–
resolves
microscope.
items
The
that
cannot
electrons
be
have
λ
geometric
geometric
d
an
shadow
effective
wavelength
region
of
visible
light
is
much
(see
page
smaller
than
the
127).
region
•
Radio
telescopes
resolution
together
a
λ
in
greater
geometric
–
the
possible.
an
size
of
Several
array
diameter
astronomical
λ
that
shadow
wavelength
d <
using
information.
electron
resolved
d ≈
atoms
diffraction
to
and
objects.
the
create
with
(See
dish
radio
a
a
virtual
greater
page
limits
telescopes
radio
the
can
maximum
be
linked
telescope
ability
to
with
resolve
181)
d
geometric
shadow region
shadow region
e x amples Of DiffractiOn
Diffraction
even
if
provides
we
can
not
the
see
reason
why
we
can
hear
something
it.
λ
d
=
width
of
Diffraction
obstacle/gap
wave
energy
is
received
in
geometric
shadow
region.
ba sic ObserVatiOns
Diffraction
etc.)
have
is
a
a
wave
size
that
effect.
is
of
The
the
objects
same
involved
order
of
(slits,
apertures,
magnitude
as
the
wavelength.
If
There is a central maximum
intensity.
halfway between the minima.
As the angle increases,
the
intensity of the
maxima decreases.
angle
1st minimum
46
W A V E S
look
eyes
diffraction
say,
Other maxima occur roughly
Diraction of a single slit.
you
your
intensity
this
at
the
a
distant
light
taking
street
spreads
place
explanation
is
light
around
a
at
sideways
your
night
and
–
is
this
as
eyelashes!
simplication.)
then
a
squint
result
of
(Needless
to
to -o     o v
principles Of tHe twO-sOurce interference
matHematics
The
location
of
the
light
and
dark
fringes
can
be
mathematically
pattern
derived
T
wo-source
interference
is
simply
another
application
in
one
of
two
ways.
The
derivations
do
not
need
to
be
of
recalled.
the
principle
roughly
the
of
superposition,
same
for
two
coherent
sources
having
amplitude.
Method
T
wo
sources
•
they
•
there
are
have
is
a
coherent
the
same
constant
The
if:
the
frequency
phase
relationship
between
the
two
1
simplest
slits
as
way
is
shown
to
consider
two
parallel
rays
setting
off
from
below.
sources.
parallel rays
regions where waves are in phase:
constructive interference
destructive
S
1
interference
path dierence
d
θ
p = dsin θ
θ
S
2
If
these
must
S
S
but
2
1
two
arrive
the
light
distance
T
wo
dippers
in
water
moving
together
are
coherent
sources.
rays
in
is
from
called
regions
of
water
ripples
and
other
regions
with
no
loudspeakers
both
connected
to
the
same
signal
coherent
sources.
This
forms
regions
of
loud
and
is
soft
set-up
is
shown
for
viewing
below.
experiment.
A
It
is
two-source
known
as
interference
Young’s
monochromatic
source
with
double
out
only
one
frequency.
Light
of
from
light
the
called
interferes
fringes,
can
and
be
patterns
seen
on
of
the
light
and
From
is
one
that
twin
slits
(the
dark
of
an
then
light
the
two
started
extra
rays
out
distance.
in
phase
This
extra
difference.
whole
can
number
only
of
happen
if
the
wavelengths.
path
Mathematically,
=
n
λ
n
is
an
integer
–
e.g.
1,
2,
3
etc.]
light
slit
In
sources)
travels
path
interference
a
difference
the
Path
gives
2
patch,
rays
sound.
[where
A
bright
two
generator
Path
are
source
the
a
waves.
difference
T
wo
in
The
This
Constructive
forms
result
phase.
geometry
difference
other
words
of
=
n
the
d
λ
situation
sin
=
θ
d
sin
θ
regions,
screen.
Method
If
a
2
screen
is
used
to
make
the
fringes
visible,
then
the
rays
region in which
Se-p 1
from
the
two
slits
cannot
be
absolutely
parallel,
but
the
superposition occurs
physical
set-up
means
that
this
is
effectively
true.
p
separation
sin
monochromatic
θ
=
s
of slits
light source
X
tan
P
θ
=
D
If
θ
is
small
sin
θ
≃
tan
θ
S
1
p
S
X
1
S
so
X
0
=
s
D
θ
S
2
Xs
___
θ
∴
p
=
D
s
N
For
2
constructive
source
interference:
twin source
slit
p
possible
p
=
nλ
=
nλ
slits (less than 5 mm)
screen
0.1 m
S
2
X
1 m
D
s
n
____
∴
positions
D
nλD
_____
∴
Se-p 2
The
use
of
X
=
n
a
laser
makes
the
set-up
s
λD
___
easier.
fringe
separation
d
=
X
X
n
double
+
1
=
n
s
screen
λD
___
laser
∴
s
=
d
slit
This
equation
only
applies
when
the
angle
is
small.
Example
The
experiment
strips
across
the
results
screen
in
as
a
regular
pattern
represented
of
light
and
dark
below.
Laser
are
light
0.1
placed
intensity distribution
of
mm
5.0
wavelength
apart.
m
How
450
far
nm
apart
is
are
shone
the
on
two
fringes
on
slits
a
that
screen
away?
view seen
7
fringe width, d
d
=
=
s
intensity
dark
bright
dark
4.5 × 10
× 5
______________
λ D
_____
=
0.0225
m
=
2.25
cm
4
1.0
×
10
bright
W A V E S
47
n d odo o d (o) v
stanDing waVes
A
special
that
case
of
interference
occurs
when
two
waves
meet
There
are
some
These
are
called
movement
are:
standing
•
of
the
same
of
the
takes
wave
same
in
space
travelling
In
these
on
place
is
so
the
nodes.
are
called
rope
The
that
points
called
are
always
where
antinodes.
because
the
wave
the
The
at
rest.
maximum
resulting
pattern
–
it
is
its
amplitude
that
changes
remains
over
time.
A
frequency
comparison
•
the
amplitude
xed
•
points
in
opposite
conditions
a
with
a
normal
(travelling)
wave
is
given
below.
directions.
standing
wave
will
be
Stationary
formed.
wave
Normal
(travelling)
wave
The
conditions
needed
to
form
standing
waves
seem
quite
Amplitude
specialized,
but
standing
waves
are
in
fact
quite
common.
All
points
wave
often
occur
when
a
wave
reects
back
from
a
boundary
on
have
route
are
that
interfere
Perhaps
of
it
came.
(nearly)
Since
and
the
equal
the
reected
produce
simplest
a
amplitude,
wave
standing
way
of
these
and
two
the
waves
to
consider
two
along
a
waves
rope.
The
is
2A
It
is
at
amplitude
the
a
standing
wave
travelling
in
zero
The
series
of
at
antinodes.
the
nodes.
would
All
points
oscillate
All
what
points
oscillate
opposite
diagrams
the
same
with
the
same
below
frequency.
shows
same
amplitude.
wave.
picturing
transverse
stretched
the
the
incident
with
directions
on
have
can
Frequency
be
points
wave
along
maximum
wave
All
different
amplitudes.
the
the
They
frequency.
happens.
Wavelength
This
is
twice
the
This
is
the
shortest
resultant wave
distance
from
one
distance
(in
metres)
wave 1 moves →
node
(or
antinode)
along
the
wave
a)
wave 2 moves ←
to
the
next
node
(or
between
tnemecalpsid
antinode).
that
are
two
in
points
phase
with
total
one
Phase
distance
All
points
one
next
b)
tnemecalpsid
total
in
between
node
and
node
are
All
another.
points
along
the
wavelength
moving
different
a
have
phases.
phase.
wave 1
Energy
Energy
is
not
transmitted
distance
wave,
an
c)
but
it
energy
Energy
by
the
does
by
the
is
transmitted
wave.
have
associated
t
with
it.
total
wave 2
Although
the
example
left
involved
transverse
waves
on
a
rope,
distance
sid
a
All
d)
can
also
instruments
wave
inside
the
be
created
involve
the
instrument.
using
sound
creation
The
of
a
or
light
waves.
standing
production
of
laser
light
tnemecalpsid
total
involves
a
standing
light
wave.
Even
electrons
in
hydrogen
wave 2
atoms
distance
A
can
be
standing
explained
longitudinal
imagine.
The
example
–
a
diagram
standing
in
terms
wave
below
sound
of
can
standing
be
waves.
particularly
attempts
to
hard
represent
to
one
wave.
tnemecalpsid
total
wave 2
zero
wave 1
Production
antinode
distance
of
standing
waves
antinode
tnemecalpsid latot
etc.
etc.
node
max.
distance
movement
A
node
standing
antinode
movement
antinode
48
wave
musical
sound
e)
A
standing
wave
node
–
the
W A V E S
pattern
node
remains
xed
node
longitudinal
standing
wave
node
bod odo
As
bOunDary cOnDitiOns
The
boundary
that
must
when
met
standing
meets
mode
be
conditions
these
of
at
the
waves
the
edges
are
boundary
the
of
system
(the
taking
the
boundaries)
place.
conditions
specify
will
Any
be
a
of
the
standing
possible
the
that
that
standing
pipe
resonant
boundary
can
displacement
system
wave
before,
waves
conditions
exist
node.
waves
closed
at
are
one
in
conditions
the
An
tube.
open
shown
A
end
for
a
determine
closed
must
pipe
end
be
an
open
the
standing
must
be
a
antinode.
at
both
Possible
ends
and
a
end.
system.
N = node
l
1.
If
Transverse
the
string
oscillate.
is
waves
xed
Both
at
ends
on
a
each
of
the
A = antinode
string
end,
string
the
ends
would
of
the
reect
string
a
1st harmonic
cannot
travelling
wave
frequency = f
0
and
thus
a
standing
wave
is
possible.
The
only
standing
waves
λ
0
that
t
these
boundary
conditions
are
ones
that
have
nodes
at
A
each
end.
The
diagrams
below
show
the
possible
resonant
= 2l
A
N
modes.
λ' = l
N = node
l
A = antinode
f ' = 2f
0
λ
0
= 2l
st
1
harmonic =
A
f
N
0
A
N
A
2l
λ" =
A
N
N
3
f " = 3f
0
A
N
A
N
A
N
A
f ' = 2f
λ' = l,
0
Harmonic
modes
for
a
pipe
open
at
both
ends
A
N = node
l
A = antinode
1st harmonic
2
λ" =
f " = 3f
l
frequency = f
0
3
0
λ
0
A
N
A
N
= 4l
A
4l
λ' =
3
l
f ' = 3f
f "' = 4f
λ"' =
0
0
2
A
N
A
N
A
N
A
N
A
4l
λ" =
3
Harmonic
modes
for
a
string
f " = 5f
0
The
resonant
fundamental
are
called
mode
or
the
that
has
rst
harmonics.
the
lowest
harmonic.
Many
frequency
Higher
musical
is
called
resonant
instruments
the
N
modes
(e.g.
A
guitar
etc.)
involve
similar
oscillations
of
metal
Longitudinal
sound
waves
in
a
longitudinal
enclosed
the
in
a
standing
pipe.
reections
As
that
wave
in
take
the
can
be
set
example
place
at
both
up
in
above,
organ
the
this
modes
for
a
pipe
closed
at
one
end
instruments
column
results
of
include
the
ute,
that
the
involve
a
trumpet,
standing
the
wave
recorder
in
and
a
column
organ
of
pipes.
air
from
ends.
e x ample
An
A
pipe
air
A
N
‘strings’.
Musical
2.
A
piano,
Harmonic
violin,
N
resOnance tube
pipe
(open
at
one
end)
is
1.2
m
long.
Tuning fork of
Calculate
its
fundamental
A
frequency.
known frequency
1
The
speed
of
sound
is
330
m
s
λ
l
=
1.2
m
∴
=
1.2
m
=
4.8
m
(rst
harmonic)
x
4
N
∴
λ
v
=
fλ
330
____
f
=
≃
69
Hz
4.8
Resonance will occur
at dierent values of x
The distance between
λ
adjacent resonance lengths =
2
W A V E S
49
ib
1.
A
qo – v
surfer
region
is
out
where
beyond
the
the
ocean
breaking
waves
are
surf
in
a
deep-water
sinusoidal
in
shape.
b)
On
the
(i)
crests
are
20
m
apart
and
the
surfer
rises
a
vertical
diagram
distance
draw
m
from
the
speed
wave
trough
to
crest,
in
a
time
of
2.0
s.
What
the
marker
1
1.0
m
s
C.
5.0
m
s
is
to
indicate
the
direction
in
which
moving.
[1]
indicate,
with
the
letter
A,
the
amplitude
of
waves?
the
A.
arrow
is
(ii)
of
an
of
the
4.0
above
The
wave.
[1]
1
B.
2.0
m
D.
10.0
s
(iii)
1
indicate,
with
the
letter
λ,
the
wavelength
of
the
1
m
s
wave.
2.
A
standing
wave
is
established
in
air
in
a
pipe
with
one
[1]
T
closed
(iv)
draw
the
displacement
of
the
string
a
time
later,
4
and
one
open
end.
where
T
is
Indicate,
the
period
with
the
of
oscillation
letter
N,
the
of
new
the
wave.
position
of
the
marker.
X
[2]
1
The
c)
The
air
molecules
near
X
wavelength
of
the
wave
is
5.0
cm
and
its
speed
is
10
cm
s
Determine
are
(i)
the
(ii)
how
frequency
of
the
wave.
[1]
T
A.
always
at
the
centre
of
a
compression.
far
the
wave
has
moved
in
s.
[2]
4
B.
always
C.
sometimes
at
D.
the
at
the
centre
never
at
centre
at
the
the
of
of
a
centre
a
Interference
rarefaction.
of
a
compression
and
d)
sometimes
By
what
rarefaction.
centre
of
a
compression
or
a
The
rarefaction.
of
is
This
question
is
about
sound
to
meant
diagram
the
by
below
arrangement
3.
waves
reference
for
principle
of
constructive
(not
drawn
observing
the
superposition,
explain
interference.
to
scale)
[4]
shows
interference
an
pattern
waves.
produced
by
the
light
from
two
narrow
slits
S
and
S
1
A
sound
wave
variation
wave
at
of
of
frequency
particle
one
660
Hz
displacement
instant
of
time
is
passes
through
withdistance
shown
air.
along
2
The
P
the
below.
y
0.5
n
S
1
θ
ϕ
monochromatic
displacement /mm
d
distance / m
0
M
light source
O
0
x
S
2
0.5
D
a)
State
a
whether
transverse
this
wave
is
an
example
of
a
longitudinal
wave.
single slit
or
screen
[1]
double slit
b)
Using
data
from
the
above
graph,
deduce
for
this
sound
The
distance
and
screen
S
S
1
is
d,
the
distance
between
the
double
slit
2
wave,
on
(i)
the
wavelength.
the
is
D
and
diagram
are
D
≫
d
small.
such
M
is
that
the
the
angles
mid-point
θ
of
and
S
[1]
observed
(ii)
the
amplitude.
[1]
(iii)
the
speed.
[2]
S
1
that
a
distance
y
a
distance
S
there
from
is
a
bright
point
O
on
fringe
the
at
point
screen.
P
on
Light
ϕ
the
from
diagram
below
represents
the
direction
of
oscillation
of
e)
a
(i)
is
travels
2
X
further
to
point
P
than
light
from
S
2
The
it
screen,
S
n
4.
shown
and
2
1
State
the
condition
in
terms
of
the
distance
S
X
2
disturbance
that
gives
rise
to
a
and
wave.
a
a)
By
redrawing
direction
of
the
wave
diagram,
energy
add
arrows
transfer
to
to
show
illustrate
the
wavelength
bright
fringe
at
of
the
light
λ,
for
there
to
be
P
.
[2]
(ii)
Deduce
an
expression
for
θ
(iii)
Deduce
an
expression
for
ϕ
in
terms
of
S
X
and
d.
[2]
2
the
the
in
terms
of
D
and
y
.
[1]
n
difference
(i)
(ii)
a
a
between
transverse
For
wave
longitudinal
and
[1]
wave.
a
particular
1.40
mm
1.50
m.
and
The
arrangement,
the
distance
distance
y
is
the
from
the
separation
the
slits
distance
to
of
of
the
the
the
slits
screen
eighth
is
is
bright
n
[1]
3
fringe
A
wave
travels
along
a
stretched
string.
The
diagram
below
string
the
f)
shows
the
variation
displacement
of
small
is
M.
marker
The
the
with
string
attached
undisturbed
distance
at
to
a
the
particular
the
position
along
of
string
the
at
instant
the
string
is
of
in
point
time.
Using
(i)
A
labelled
shown
as
O
your
the
(ii)
and
the
answers
wavelength
the
angle
separation
to
of
of
θ
(e)
the
the
=
to
2.70
×
10
rad.
determine
light.
fringes
[2]
on
the
screen.
[3]
a
5.
dotted
from
A
bright
source
of
light
is
viewed
through
two
polarisers
line.
whose
preferred
directions
are
initially
parallel.
Calculate
the
direction of wave travel
angle
the
M
50
I B
Q u E S t I o n S
–
W A V E S
through
which
transmitted
one
intensity
sheet
to
half
should
its
be
turned
original
to
value.
reduce
5
E l E C t r i C i t y
a n d
m a g n E t i s m
Eecc ce  C ' 
ConsErvation of ChargE
T
wo
of
types
of
positive
contains
charge
and
no
exist
negative
charge,
or
–
Coulomb’s l aw
positive
charge
matter
and
cancel
that
negative.
each
contains
Equal
other.
equal
amounts
Matter
that
amounts
The
are
diagram
far
shows
away
from
the
the
force
between
inuence
of
two
any
point
other
charges
that
charges.
of
distance
positive
and
Charges
are
negative
known
charge,
to
exist
is
said
to
because
be
of
electrically
the
forces
neutral.
that
r
F
exist
F
q
between
all
charges,
called
the
electrostatic
force:
force
like
q
1
charge
charges
repel,
F
unlike
charges
+
The
F
directions
charges.
-
-
+
F
F
-
-
F
F
+
are
important
always
experimental
other
Each
observation
is
that
charge
In
order
are
for
be
created
physically
the
they
each
by
charge
moved
friction.
In
this
to
from
remain
on
repel.
other
must
feel
and
one
object
to
another.
the
the
object,
it
normally
between
kq
an
along
they
they
force
force
the
q
q
1
be
If
–
a
inversely
2
the
the
line
forces
joining
are
away
the
from
are
unlike
charges,
the
forces
attract.
of
the
same
size
as
the
force
on
is
proportional
proportional
to
to
the
the
size
square
of
of
both
the
=
k
2
2
r
This
is
2
_
=
insulator.
charges.
q
1
_
to
are
charges,
process
F
needs
forces
like
one.
Experimentally,
can
distance
electrons
–
charge
other
charges
objects
the
are
is
conserved.
Charged
of
they
towards
the
very
If
F
each
A
charge
attract.
+
F
force
2
r
known
as
Coulomb’s
law
and
the
constant
k
is
called
before
the
neutral hair
a
neutral
Coulomb
slightly
medium
constant.
different
called
In
form
the
fact,
using
the
a
permittivity,
law
is
different
often
quoted
constant
for
in
the
ε
comb
value
of
first
charge
value
force
between
q
F
two
point
second
charge
2
=
charges
of
q
1
distance
between
2
4πε
0
the
charges
constants
permittivity
of
free
after
space
(a
constant)
attraction
positive hair
negative
1
_
-
k
=
+
-
4πε
+
comb
0
+
-
If
+
-
there
overall
are
two
force
or
can
more
be
charges
worked
out
near
another
using
vector
charge,
the
addition.
+
force on q
(due to q
A
)
C
overall force on q
A
electrons have been transferred
(due to q
and q
B
q
from hair to comb
B
q
A
The
total
charge
before
any
process
must
be
equal
to
the
force on q
(due to q
A
total
charge
charge
afterwards.
without
an
It
equal
is
impossible
negative
to
create
charge.
This
is
a
)
B
positive
the
law
of
q
C
conservation
of
charge.
Ve  f ee fe
Electrical
ConduCtors and insul ators
A
material
electrical
it
is
charge
to
atom.
allows
conductor.
called
of
that
an
is
the
If
electrical
always
as
a
ow
charge
of
cannot
insulator.
result
charge
of
In
the
through
ow
solid
ow
it
is
through
a
conductors
of
called
electrons
conductors
Electrical
insulators
an
material
the
ow
from
atom
brass
acetate
dry
wood
glass
ceramics
E l E c t r i c i t y
a n d
M a g n E t i s M
)
C
51
Eecc e
In
ElECtriC fiElds – dEfinition
A
charge,
electric
in
the
or
combination
eld
eld,
around
the
value
it.
of
of
If
charges,
we
the
place
force
is
a
said
test
that
it
to
produce
charge
feels
at
at
any
any
point
point
practical
doesn’t
an
The
denition
will
on
the
value
of
the
test
charge
the
of
the
test
charge
electric
or
charge
charges
eld,
E,
needs
that
to
are
be
small
being
so
that
=
is
=
force
per
unit
positive
point
test
charge.
q
only.
2
Coulomb’s
law
can
be
used
to
relate
the
electric
eld
around
a
A
point
charge
to
the
charge
producing
the
eld.
q
A test charge placed at A
1
_
E
=
2
4πε
would feel this force.
r
0
When
A test charge placed at B
•
not
using
to
charge
would feel this force.
•
q
not
to
these
muddle
sitting
use
the
equations
up
in
the
the
you
charge
eld
(and
mathematical
have
to
be
producing
thus
very
the
feeling
equation
for
careful:
eld
a
and
A
test
charge
around
a
would
charge
point
feel
a
different
force
at
different
charge
for
other
situations
(e.g.
the
eld
parallel
around
plates).
points
q
1
rEprEsEntation of ElECtriC fiElds
This
At
is
any
done
using
point
in
a
eld
lines.
eld:
two opposite
•
the
direction
of
eld
is
represented
by
the
direction
of
the
charges
eld
•
the
lines
closest
magnitude
eld
lines
to
of
that
the
passing
point
eld
near
is
that
represented
by
the
number
of
point.
+
The eld here
+
The eld here
must be strong
must be
as the eld lines
weak as the
two like
are close
eld lines
charges
together.
are far apar t.
The direction of the
force here must be
a negatively
F
as shown.
charged
Field
The
around
resultant
point
The
a
charges
parallel
eld
is
positive
electric
is
eld
shown
eld
point
lines
to
at
conducting
charge
any
the
position
due
to
a
collection
of
right.
between
two
plates
mean
that
the
two oppositely charged
electric
parallel metal plates
uniform.
Electric
eld
•
begin
•
never
•
are
on
+
lines:
positive
sphere
(radial eld)
charges
and
end
on
negative
+
+
+
+
+
+
+
+
charges
cross
close
together
when
the
eld
is
strong.
–
–
–
–
–
–
–
–
–
parallel eld lines
in the centre
Patterns
52
+
E l E c t r i c i t y
a n d
M a g n E t i s M
of
electric
elds
–
the
force)
1
B
it
considered.
F
E
depend
situations,
disturb
a
Eecc e ee  eecc e eece
EnErgy diffErEnCE in an ElECtriC fiEld
When
placed
means
done.
that
As
potential
a
charge
is
the
mass
a
if
it
an
energy.
has
as
the
charge
result
as
a
eld,
around
Electric
a
idea
its
electric
moves
result,
same
up,
in
a
in
charge
an
will
its
mass
gravitational
electric
either
potential
of
a
in
is
an
energy
or
This
work
lose
the
ElECtriC potEntial diffErEnCE
force.
will
eld.
we
increases.
If
example
we
A
a
is
the
falls,
potential
its
gravitational
below
a
positive
potential
charge
is
energy
moved
decreases.
from
the
left,
the
depended
the
charge
quantity
difference
difference,
we
the
energy
charge
would
that
per
or
actual
on
double
remains
unit
pd,
was
the
xed
charge.
between
difference
that
In
difference
position
energy
between
This
the
is
B.
This
results
in
an
increase
in
electric
Since
the
eld
is
uniform,
the
force
is
=
the
two
points
per
unit
charge
moved
difference
work done
__
=
potential
constant.
=
This
it
very
easy
to
calculate
the
work
charge
W
_
V
makes
B
the
A
charge
energy.
and
points.
__
position
A
called
difference
energy
to
moved.
energy
the
between
example
B
The
energy
Potential
mass
on
and
doubled
difference.
This
lift
the
between
If
that
eld.
If
In
be
electric
energy
electric
gravitational
potential
a
eld
gain
energy
position
in
feels
=
q
done.
The
basic
unit
for
potential
difference
is
the
joule/coulomb,
1
J
C
.
A
electric
very
important
eld,
the
point
potential
to
note
difference
is
that
for
between
a
given
any
two
points
force needed to move charge
is
single
two
+
+
q
a
xed
points
scalar
does
not
quantity.
depend
on
The
the
work
path
done
taken
between
by
the
these
test
q
charge.
A
technical
way
of
saying
this
is
‘the
electric
eld
is
conservative’.
distance d
units
The
smallest
amount
of
negative
charge
available
is
the
charge
position of lower electric
on
an
electron;
the
smallest
amount
of
positive
charge
is
the
potential energy
potential energy
charge
small
Charge
moving
in
an
electric
on
so
a
we
proton.
use
the
In
everyday
coulomb,
situations
C.
One
this
unit
coulomb
of
is
far
too
negative
eld
18
charge
Change
in
electric
potential
energy
=
force
=
E
×
From
q
×
d
page
52
for
a
denition
of
electric
eld,
the
example
above
the
electric
potential
its
.
than
the
electric
potential
energy
at
B
volt
A.
We
B.
to
If
put
we
in
let
this
go
amount
of
the
of
work
charge
at
B
to
it
push
would
eld.
This
push
potential
would
energy
accelerate
would
be
the
it
the
be
charge
pushed
from
by
so
same
of
6.25
×
10
electrons.
given
the
a
unit
new
of
potential
name,
the
difference
volt,
V
.
(pd)
is
Thus:
that
as
=
1
J
C
potential
difference
are
Potential
difference
is
different
the
the
loss
gain
probably
words
the
for
better
the
name
A
use
as
it
reminds
you
that
it
is
measuring
the
difference
the
two
points.
in
working
at
the
atomic
scale,
the
joule
is
far
too
big
to
in
use
kinetic
is
and
When
electrical
total
thing.
between
electric
a
is
to
to
by
would
same
have
carried
1
energy
at
denition,
This
Voltage
greater
charge
E
1
In
the
1
JC
See
is
distance
for
a
unit
for
energy.
The
everyday
unit
used
by
physicists
energy.
for
its
this
gained
of
situation
name,
1
the
by
is
the
electronvolt.
electronvolt
an
electron
is
simply
moving
As
the
through
could
be
energy
a
guessed
that
potential
from
would
be
difference
volt.
19
1
B
electronvolt
=
1
volt
×
1.6
A
+
×
10
C
19
+
=
velocity v
The
in
normal
SI
prexes
kiloelectronvolts
latter
unit
is
1.6
very
×
10
also
(keV)
apply
or
common
J
so
one
can
measure
megaelectronvolts
in
particle
energies
(MeV).
The
physics.
Exmpe
Calculate
the
speed
of
an
electron
accelerated
in
a
vacuum
by
A positive charge released at B will be
a
pd
of
1000
V
(energy
=
1
KeV).
accelerated as it travels to point A
19
KE
gain
in
kinetic
energy
1
=
loss
=
Eqd
in
electric
2
mv
potential
of
electron
=
V
×
e
=
1000
=
1.6
×
1.6
energy
×
10
16
1
2
mv
2
×
10
J
16
=
1.6
×
10
J
2
2
mv
=
2Eqd
7
v
=
1.87
×
10
1
m
s

2Eqd
_
∴
v
=
√
m
E l E c t r i c i t y
a n d
M a g n E t i s M
53
Eecc ce
ElECtriC al ConduCtion in a mEtal
Whenever
for
charges
moving
complete
move
charges
circuit,
a
and
we
the
current
say
path
that
a
that
cannot
be
CurrEnt
current
they
is
follow
maintained
owing.
is
called
for
any
A
current
the
is
circuit.
length
of
the
name
Without
Current
a
of
time.
is
the
symbol,
denition
Current
ows
THROUGH
an
object
when
there
is
a
potential
difference
ACROSS
dened
electrical
as
the
charge.
I.
for
It
rate
is
Mathematically
current
is
of
ow
always
given
the
expressed
as
the
follows:
object.
A
battery
(or
power
supply)
is
the
device
that
creates
the
potential
difference.
charge
owed
__
By
convention,
currents
are
always
represented
as
the
ow
of
positive
charge.
Current
Thus
=
time
conventional
current,
as
it
is
known,
ows
from
positive
to
negative.
taken
Although
∆Q
_
currents
can
ow
in
solids,
liquids
and
gases,
in
most
everyday
electrical
I
circuits
=
dQ
_
or
(in
calculus
notation)
I
=
dt
∆t
the
currents
ow
through
wires.
In
this
case
the
things
that
actually
move
are
the
1 coulomb
__
1
negative
electrons
opposite
to
the
–
the
conduction
direction
of
the
electrons.
representation
The
of
direction
conventional
in
which
current.
they
As
move
they
ampere
=
is
1
second
move
1
1
the
interactions
work
needs
speed
of
to
the
between
be
done.
electrons
the
conduction
Therefore,
due
to
the
electrons
when
current
a
and
current
is
called
the
ows,
their
lattice
the
drift
ions
metal
means
heats
up.
A
=
1
C
s
that
If
a
it
is
current
ows
in
just
one
direction
The
known
as
a
direct
current.
velocity
A
current
that
constantly
changes
conventional current, I
direction
is
positive ions
conduction electrons
drift velocity
is
conduction
possible
speed
these
could
electrons.
unit
a
the
so
be
time
All
a
is
the
or
that
are
metal
drift
currents
positive
that
speed
in
estimate
Suppose
volume
average
In
to
equation.
velocity
are
negative;
the
not
number
available
to
of
electrons
comprised
all
the
currents
density
move)
of
is
n,
of
the
the
using
the
movement
involve
generalized
of
just
the
charge-carriers
charge
on
each
drift
charge-carriers
movement
(the
carrier
is
q
and
v
average
so
number
so
charge
distance
of
charge
of
moved
moved
by
charge-carriers
moved
past
a
a
past
charge-carrier
a
point
moved
point,
ΔQ
=
=
past
A
a
nAvΔt
×
=
point
×
v
×
Δt
vΔt
=
n×AvΔt
q
∆Q
_
current
I
=
I
=
∆t
It
is
interesting
nAvq
to
compare:
4
•
A
typical
drift
speed
of
an
electron:
10
1
m
s
2
(5A
current
•
The
speeds
•
The
speed
in
metal
conductor
of
cross
section
1
to
their
random
mm
)
6
of
the
electrons
due
motion:
10
1
m
s
8
54
of
an
electrical
signal
E l E c t r i c i t y
down
a n d
a
conductor:
approx.
M a g n E t i s M
3
×
10
1
m
s
and
of
number
Δt,
volume
ac.
In
SI
per
their
as
units,
the
1
held in place
It
or
and
metal wire
Electrical
known
(rst
C
one
an
the
1
A
s
then
alternating
ampere
coulomb
=
way
is
a
is
the
other)
current
the
derived
base
unit
unit
Eecc cc
ohm’s l aw – ohmiC and non-ohmiC bEhaviour
The
graphs
typical
below
show
how
the
current
varies
with
rEsistanCE
potential
difference
for
some
Resistance
devices.
between
current.
(a) metal at constant
(b) lament lamp
is
the
mathematical
potential
If
difference
something
has
a
ratio
and
high
(c) diode
resistance,
it
means
that
you
would
temperature
tnerruc
tnerruc
tnerruc
need
it
in
a
large
order
potential
to
get
a
difference
current
potential
to
across
ow.
difference
__
Resistance
=
current
V
In
symbols,
R
=
I
potential
potential
potential
dierence
dierence
dierence
We
be
dene
equal
a
to
new
one
unit,
volt
the
per
ohm,
Ω,
to
amp.
1
1
If
current
the
and
device
is
potential
said
to
proportional
(like
Ohm’s
states
law
temperature
In
ohmic.
lament
that
remains
the
are
proportional
Devices
where
lamp
the
or
current
current
diode)
owing
(like
are
the
and
said
through
a
metal
at
potential
to
be
piece
constant
ohm
=
1
V
A
temperature)
difference
are
not
non-ohmic
of
metal
is
proportional
to
the
potential
difference
across
it
providing
the
constant.
symbols,
V
A
be
the
difference
∝
I
[if
device
temperature
with
constant
is
constant]
resistance
(in
other
words
an
ohmic
device)
is
called
a
resistor
resistor.
powEr dissipation
All
this
energy
is
going
into
heating
up
the
resistor.
In
symbols:
energy
difference
__
Since
potential
difference
=
charge
charge
P
owed
And
current
=
V
×
Sometimes
is
more
useful
to
use
this
equation
in
a
slightly
taken
different
means
it
=
time
This
I
owed
__
that
(energy
potential
difference
difference)
×
(charge
__
owed)
×
(charge
e.g.
current
energy
__
=
form,
difference
__
P
=
V
×
P
=
(I
P
=
I
I
×
but
R)
×
V
=
I
×
R
so
I
=
owed)
(time
time
taken)
2
R
2
V
_
Similarly
This
energy
difference
per
time
is
the
power
dissipated
by
P
=
the
R
CirCuits – KirChoff ’s CirCuit l aws
An
electric
circuit
can
contain
E x amplE
many
A
1.2
kW
electric
kettle
is
plugged
into
3.4A
different
devices
or
components.
The
the
250
V
mains
supply.
Calculate
5.5A
mathematical
applied
to
relationship
any
components
V
component
in
a
=
or
IR
can
groups
be
(i)
the
(ii)
its
(i)
I
current
drawn
2.7A
of
resistance
circuit.
1200
_
When
to
analysing
look
at
the
a
circuit
circuit
as
a
it
is
important
whole.
=
=
x
4.8
A
250
The
250
_
power
supply
providing
the
is
the
device
energy,
but
that
it
is
The
second
any
loop,
that
determines
what
through
the
the
fundamental
the
total
must
potential
when
conservation
be
sum
circuits:
electric
of
to
completely
per
and
zero.
Any
within
energy.
These
laws
dissipated
loop
The
across
in
the
loop
(potential
known
as
and
can
be
Kirchoff’s
stated
example
larger
circuit.
below
across
the
component).Care
betaken
to
get
the
sign
of
any
of
loop
the
12.0
-
means
5.3
x
potential
mathematically
=
0
loop
in
a
consideration
+
that:
2.7
3.2
difference
=
0.
across
the
bulb,
circuit
If
the
chosen
loop
direction
is
from
x
=
6.2
V
as:
negative
side
of
a
battery
to
its
pd = +12v
∑I
one
correct.
the
law:
shows
Anti-clockwise
are
•
First
Ω
source
the
The
laws
52
the
pd
collectively
=
unit
the
charge
needsto
conservation
=
laws
drop
of
R
circuit.
analysing
conservation
energy
difference
thecomponents
apply
(ii)
around
whole
must
Two
that
current
of
ows
states
4.8
charge
circuit
law
is
pd = -3.2v
(junction)
positive
side,
this
is
an
increase
in
+
Second
The
rst
law:
law
∑
states
V
=
that
0
(loop)
the
potential
when
algebraic
and
the
calculating
value
the
is
positive
+
sum.
+
pd = -5.3v
sum
of
the
currents
at
any
junction
in
•
If
the
direction
around
the
loop
pd = +2.7v
is
+
the
circuit
into
a
is
zero.
junction
current
The
must
owing
out
current
be
of
equal
a
owing
to
the
junction.
In
in
the
same
owing
is
a
direction
through
potential
the
drop
as
the
component,
and
the
value
this
pd =
example
(right)
the
unknown
negative
when
calculating
the
example
x
=
5.5
+
2.7
3.4
=
4.8
of
the
use
of
Kirchoff’s
sum.
circuit
current
-x
is
An
the
-
current
laws
is
shown
on
page
59.
A
E l E c t r i c i t y
a n d
M a g n E t i s M
55
re  ee  e
ElECtriC al mEtErs
rEsistors in sEriEs
A
series
circuit
The
current
The
total
has
must
be
potential
components
the
same
difference
connected
everywhere
is
shared
one
in
after
the
among
another
circuit
the
since
in
a
continuous
charge
is
chain.
conserved.
A
current-measuring
ammeter.
series
components.
needs
at
It
the
to
be
should
point
meter
be
where
measured.
is
called
connected
A
the
an
in
current
perfect
power supply
ammeter
would
have
zero
resistance.
(24V)
A
meter
that
difference
should
I
is
be
measures
called
placed
component
or
considered.
A
a
in
potential
voltmeter.
parallel
components
perfect
It
with
the
being
voltmeter
has
I
innite
(2A)
resistance.
(2A)
R
R
1
R
2
(3Ω)
3
(4Ω)
(5Ω)
Example
of
a
series
circuit
M
We
resistor
ul
by
eletrial
thermal enery
can
work
out
what
share
they
take
motor
liht enery
mehanial
and thermal
enery
enery
and thermal
looking
at
each
component
in
turn,
e.g.
enery is
The
oner ted
potential
resistor
=
I
×
difference
across
the
across
the
R
1
into
enery
The
potential
bulb
potential
=
I
×
difference
R
2
dierene
R
=
R
total
+
R
1
+
2
R
3
(6 + 8 + 10 = 24 V)
This
always
applies
to
a
series
circuit.
pd of power supply
Note
that
potential
Total
resistance
=
3Ω
+
4
Ω
+
5
Ω
=
12
V
rEsistors in parallEl
parallel
the
circuit
branches
and
allows
the
charges
more
than
one
possible
route
circuit.
V
total
I
I
total
total
V
I
I
R
1
1
1
I
+ I
2
I
3
+ I
2
3
V
I
I
2
2
R
2
V
I
I
R
3
3
3
M
Example
Since
the
potential
each
of
a
parallel
power
circuit
supply
difference
xes
across
it.
the
potential
The
total
difference,
current
is
just
each
the
component
addition
of
has
the
1
_
1
=
=
total
I
+
1
I
+
2
V
_
=
I
R
3
V
_
+
R
=
3
47
_
1
_
=
R
R
total
1
_
+
R
1
1
_
1
Ω
60
+
R
2
60
_
3
∴
R
=
Ω
total
47
=
56
E l E c t r i c i t y
a n d
1
Ω
60
R
=
1
_
1
Ω
5
20 +
15 +
12
__
V
_
2
1
+
4
total
+
R
1
1
+
3
1.28
M a g n E t i s M
Ω
the
same
currents
branch.
I
correctly
across
calculates
each
the
individual
Ω
the
around
IR
difference
component
A
=
in
total.
as
well
as
calculating
it
across
pe  e cc  e
potEntial dividEr CirCuit
The
a
example
potential
‘divide
up’
calculate
the
the
common
circuits
constant
way
to
to
the
this
needs
after
way
a
called
by
one
also
to
be
solving
to
the
divider
a
divider,
the
does
(a
a
circuit
the
from
not
One
After
to
start
potentiometer)
power
smallest
supply.
resistor
is
is
(a)
the
pd
(b)
the
reading
resistance
needs
to
should
be
be
taken
into
signicantly
account:
the
across
has
a
resistance
of
20
kΩ.
the
on
20
the
kΩ
resistor
voltmeter
with
with
the
the
switch
switch
open
closed.
6.0 V
internal
10 kΩ
remains
the
20 kΩ
only
again.
V
the
designing
going
to
20 kΩ
be
20
_________
(a)
connected
voltmeter
of
often
When
that
the
electrical
change,
is
below
the
difference
a
correct
circuit
most
involving
potential
are
unless
the
the
Calculate:
can
ratio
ammeter’s
of
In
resistors
You
the
work
An
involving
two
battery.
problems
or
circuit.
variable
the
of
resistor
considered.
current
of
because
considered.
calculations
produce
potential
so
E x amplE
example
approach
is
change
your
an
difference
the
potential
to
is
when
assume
ensure
variable
best
but
is
taken
resistance
also
It
potential
mistakes
is
right
‘share’
resistances
resistance
the
divider.
the
voltmeter’s
A
on
pd
=
×
(20
potentiometer’s
smaller.
(b)
+
resistance
given
A potentiometer has
6.0
=
4.0
V
10)
of
20
kΩ
resistor
and
voltmeter
combination,
R,
by:
1
1
_
1
_
=
R
1
+
kΩ
20
20
3 terminals – the 2 ends
∴
and the central connection
R
=
10
kΩ
10
_
∴
pd
=
×
(10
+
6.0
=
3.0
V
10)
sEnsors
A
light-dependent
resistor
(LDR),
is
a
device
whose
output voltage
resistance
surface.
In
order
resistor
Both
to
R,
will
two
both
difference,
the
measure
the
V,
R
>>
is
preferred
the
V
I
circuits
provide
across
resistance
a
and
range
and
of
characteristics
(A
of
current,
the
B)
of
below
readings
I,
for
through
potentiometer,
an
are
R.
this
depends
An
on
increase
the
in
amount
light
causes
LDR
circuit
decrease
in
on
its
resistance.
LDR, there will be a
the LDR.
that
(circuit
shining
decrease in pd across
potential
Providing
a
light
When light shines on the
unknown
constructed.
the
of
B)
pd V
total
•
Circuit
because
B
allows
the
the
range
of
readings
potential
is
greater.
difference
across
R
(and
When light shines on the
hence
the
current
through
R)
to
be
reduced
down
to
zero.
10 k
Circuit
A
will
not
go
below
the
minimum
value
LDR, there will be an
achieved
increase in pd across
when
the
variable
resistor
is
at
its
maximum
value.
the xed resistor.
•
Circuit
the
B
allows
current
value
V
the
potential
through
R)
to
that
be
supplied
can
be
difference
increased
by
across
up
the
to
R
the
power
(and
hence
maximum
supply
in
A
supply
regular
intervals.
depends
on
a
The
range
maximum
of
of
values
resistance
obtainable
of
the
by
Circuit
variable
A
resistor.
thermistor
on
its
have
is
a
resistor
temperature.
a
means
negative
that
an
Most
whose
are
temperature
increase
in
value
of
resistance
semi-conducting
coefcient
temperature
(NTC).
causes
depends
devices
a
that
This
decrease
in
Circuit A – variable resistor
resistance.
variable resistor
divider
Both
circuits
difference
of
a
of
to
these
create
sensor
devices
sensor
circuit
can
be
circuits.
depends
used
The
on
in
potential
output
an
potential
external
factor.
A
V
R
supply
When the temperature
V
of the thermistor increases,
10 kΩ
there will be an increase in
pd across the xed
resistor.
Circuit B – potentiometer
pd V
total
A
of the thermistor
NTC
increases, there will be a
thermistor
decrease in pd across
V
R
supply
V
the thermistor.
potentiometer
E l E c t r i c i t y
a n d
M a g n E t i s M
57
re 
rEsistivity
The
resistivity,
ρ,
of
a
material
is
dened
in
terms
of
Exmpe
its
7
resistance,
R,
its
length
l
and
its
cross-sectional
area
A
The
resistivity
100
m
of
copper
is
3.3
×
10
Ω
m;
the
resistance
of
a
2
length
of
wire
of
cross-sectional
area
1.0
mm
is:
l
R
=
ρ
A
100
_
7
R
=
3.3
×
10
×
=
0.3
Ω
4
The
units
this
is
of
the
resistivity
ohm
must
multiplied
be
by
ohm
the
metres
metre,
(Ω
not
m).
Note
‘ohms
per
10
that
metre’.
invEstigating rEsistanCE
The
resistivity
equation
a)
Proportional
b)
Inversely
These
a)
relationships
with
R
l
has
through
the
predicts
length
proportional
Increasing
b)
to
is
an
the
Increasing
can
like
be
to
l
the
the
the
resistance
by
another
resistance
R
of
a
substance
A
of
will
be:
substance
cross-sectional
predicted
putting
overall
that
of
of
area
considering
resistor
2R.
in
the
resistors
series.
Doubling
l
substance.
in
series
Doubling
means
l
is
and
the
doubling
R.
in
parallel:
same
So
R
as
∝
putting
l.
A
is
with
like
R
has
putting
an
another
overall
resistor
resistance
in
of
To
through
practically
Control
the
vs
I
resistor
will
be
a
in
series.
straight
R
in
line
series
going
parallel.
Doubling
Doubling
A
means
A
is
the
same
halving
R.
So
as
R
putting
∝
an
identical
resistor
in
parallel.
R
in
1
.
A
graph
of
R
vs
will
be
a
straight
line
A
A
origin.
investigate
Independent
R
1
.
2
going
identical
of
origin.
A
R
parallel
an
graph
these
relationships,
variable:
we
Either
variables:
A
or
l
have:
l
or
A
(depending
on
above
choice);
Temperature;
Substance.
Data
Data
collection:
For
analysis
Possible
sources
•
Temperature
•
The
each
•
a
•
R
range
can
Values
of
error/uncertainty
variation
of
the
value
of
be
of
R
of
independent
values
for
calculated
and
the
V
and
from
variable:
I
the
should
be
gradient
independent
recorded
of
variable
a
V
vs
I
analysed
graph.
graphically.
include:
substance
(particularly
if
currents
are
high).
Circuits
should
not
be
left
connected.
2
πd
___
2
cross-sectional
area
of
the
wire
is
calculated
by
measuring
the
wire’s
diameter,
d,
and
using
A
=
πr
=
.
Several
sets
of
4
measurements
•
The
small
using
58
a
should
value
vernier
of
the
be
taken
wire’s
calliper
or
diameter
a
E l E c t r i c i t y
along
the
will
length
mean
of
the
that
the
micrometer
a n d
wire
M a g n E t i s M
and
the
readings
uncertainties
in
generated
a
set
should
using
a
be
ruler
mutually
will
be
perpendicular.
large.
This
will
be
improved
Ee  e  K c' 
KirChoff CirCuit l aws E x amplE
Great
care
positive
an
needs
or
alternative
equal
to
the
∑(emf)
Process
to
=
of
the
whe n
co ncep t
of
the
a pp lyi ng
of
em f
se cond
pr od ucts
K ir c h off ’s
( se e
l aw
pa g e
wh ic h
of
c urr e nt
in
symbols
a nd
6 0)
ma y
l aws
as
to
e n su r e
s ou rc es
he lp
of
a voi d
that
e ver y
e l e c t r ic a l
c on fu s io n :
term
e n e rg y
in
can
‘R ou n d
t he
be
an y
eq u at i on
used
c l os e d
is
a l on g
c o rr e c t ly
wit h
c ir c u it ,
t he
V
=
su m
id en t i  ed
IR
of
to
as
provide
t he
em fs
is
r e si s t an c e ’.
follow
Draw
It
helps
to
set
•
It
helps
to
be
a
points
full
are
the
•
Use
•
Identify
circuit
up
as
being
diagram.
the
for
a
Kirchoff’s
a
and
×
as
rst
to
R
possible.
considered
law
the
to
apply
in
(use
currents
current,
loop
I
equations
precise
unknown
direction
emfs
ta ke n
T he
∑(IR)
•
Give
be
state me nt
sum
•
•
to
negativ e .
symbols
solution
identify
shown
substituting
difference
V
is
a
numbers
and
difference
units.
between
two
points
in
the
circuit
so
specify
which
two
labels).
Kirchoff’s
senses
before
Potential
and
to
mark
the
their
equations
directions
will
be
appropriate
relationships
second
Go
law.
all
on
the
diagram.
If
you
make
a
mistake
and
choose
the
wrong
negative.
between
around
the
currents.
loop
in
one
direction
(clockwise
or
anticlockwise)
adding
the
below:
emf ε
With chosen direction around
I
I
loop in the direction shown, ε and
IR are both positive in the
Kircho equation:
chosen direction around loop
∑
R
I
(emf) =
∑
(IR)
I
(If chosen direction opposite to that
shown, values are negative)
•
The
total
problem
•
Use
•
A
number
to
be
simultaneous
new
loop
of
able
can
different
to
be
equations
be
equations
generated
by
Kirchoff’s
laws
needs
to
be
the
same
as
the
number
of
unknowns
for
the
solved.
to
identied
substitute
to
check
and
that
solve
for
calculated
the
unknown
values
are
values.
correct.
Exmpe
Sub
6v
(1)
into
(4)
20Ω
30i
+
10(i
2
1
i
)
=
5
=
5
(5)
=
24
(6)
=
19
=
0.1727
=
172.7
=
6
=
0.8182
=
0.08182
2
+
1
i
i
+
1
10i
40i
2
1
(3)
×
4
120i
+
1
40i
2
10Ω
(6)
-
(5)
110i
1
i
i
3
∴
3
i
A
1
i
i
2
mA
2
(3)
⇒
10i
30i
2
1
30Ω
5v
∴
i
A
2
Kirchoff
1st
law
junction
i
+
i
1
C(or
=
2
i
D)
=
81.8
=
172.7
mA
+
=
254.5
mA
(1)
3
i
81.8
mA
3
Kirchoff
2nd
law
10i
and
+
ACDB
20i
3
Sub
(1)
into
10
(i
∴
6
=
6
=
6
(2)
(2)
+
1
i
)
+
20i
2
1
30i
+
10i
1
Kirchoff
=
1
2nd
law
30i
CEFD
10i
2
(3)
2
and
=
-5
(4)
3
E l E c t r i c i t y
a n d
M a g n E t i s M
59
ie ece  c e
ElECtromotivE forCE and intErnal rEsistanCE
When
used
a
6V
up
some
battery
inside
the
internal
charge
is
connected
battery
itself.
resistance.
around
the
circuit
The
is
still
in
a
circuit
In
other
some
words,
TOTAL
energy
6
but
volts,
energy
the
will
battery
difference
some
of
this
‘perfect battery’
be
internal resistance
ε (e m f) = 6 V
r
has
per
unit
energy
is
terminals of battery
used
from
up
one
made
For
inside
the
terminal
available
historical
charge
battery.
of
by
the
the
a
energy
battery
chemical
reasons,
around
The
the
circuit
is
to
difference
the
other
reaction
TOTAL
called
in
energy
the
is
the
per
less
unit
than
charge
the
R
total
battery.
difference
ex ternal resistance
per
electromotive
unit
force
e
(emf).
m
f
=
I
×
R
=
I(r
=
Ir
=
emf
total
However,
but
In
an
remember
energy
practical
that
difference
terms,
emf
it
is
per
is
not
a
force
charge
exactly
(measured
(measured
the
same
as
in
in
newtons)
if
no
current
=
I
(R
+
Ir
p
d,
V
‘lost’
volts
r)
CElls and battEriEs
electric
IR
potential
V
An
+
ows.
terminal
ε
R)
volts).
IR
difference
+
battery
is
a
=
ε
-
Ir
dEtErmining intErnal rEsistanCE
device
consisting
of
one
or
more
cells
E xpErimEntally
joined
together.
which
converts
In
a
cell,
a
chemical
reaction
takes
place,
To
stored
chemical
energy
into
electrical
experimentally
(and
There
are
two
different
types
of
cell:
primary
and
determine
the
internal
resistance
r
of
a
cell
energy.
its
emf
ε),
the
circuit
below
can
be
used:
secondary.
terminal pd, V
A
primary
cell
cannot
be
recharged.
During
the
lifetime
of
battery
battery
V
the
cell,
the
chemicals
in
the
cell
get
used
in
a
non-reversible
terminal
terminal
reaction.
Once
electrical
energy,
a
primary
cell
is
no
longer
able
to
internal
provide
emf, ε
resistance, r
zinc–carbon
secondary
reaction
reverse
reused
car
that
cell
many
and
designed
the
current
times.
away.
batteries
is
produces
electrical
battery,
thrown
to
alkaline
be
Common
nickel–cadmium
batteries.
energy
the
cell
examples
and
examples
recharged.
electrical
charges
Common
is
The
include
lithium-ion
chemical
reversible.
allowing
a
tnerruc
A
is
I
include
it
it
to
A
ex ternal resistance, R
be
lead–acid
I
batteries.
Procedure:
The
charge
capacity
of
a
cell
is
how
much
charge
can
ow
before
•
the
cells
stops
working.
T
ypical
batteries
have
charge
capacities
Vary
external
more)
are
measured
in
Amp-hours
(A
h).
1
A
h
is
the
charge
that
a
current
of
1
A
ows
for
one
hour
i.e.
1
A
h
=
3600
current
terminal
(and
thus
potential
electrical
difference
energy)
varies
is
with
drawn
time.
A
from
Repeat
readings.
•
Do
leave
a
maintain
however,
•
loses
do
its
its
not.
terminal
The
initial
pd
terminal
value
throughout
potential
its
of
a
values
get
of
a
V
number
and
I
(ideally
over
as
10
wide
a
or
range
Take
care
real
current
of
running
for
too
long
(especially
at
I).
that
nothing
overheats.
cell
analysis:
cells,
typical
•
The
•
A
relevant
equation,
V
=
ε
Ir
was
introduced
above.
cell:
plot
of
V
on
the
y-axis
and
I
on
the
x-axis
gives
a
straight
quickly,
line
•
to
cell,
perfect
lifetime;
difference
not
high
Data
would
R
readings
possible.
•
•
the
matching
C.
disChargE CharaCtEristiCs
When
of
ows
as
when
resistance
that
graph
with
has a stable and reasonably constant value for most of its lifetime.
This
The
is
followed
graph
particular
by
below
type
of
a
rapid
shows
decrease
the
lead–acid
to
discharge
car
zero
(cell
•
gradient
•
y-intercept
=
- r
discharges).
characteristics
for
=
ε
one
battery.
rECharging sECondary CElls
In
discharge characteristics
order
DC
to
power
recharge
source.
a
secondary
The
negative
cell,
it
is
terminal
connected
of
the
to
an
external
secondary
cell
is
ambient temperature: 25 °C
connected
13
positive
to
the
terminal
negative
of
the
terminal
power
of
source
the
power
with
the
source
positive
and
the
terminal
12
)V( egatlov lanimret
of
11
the
the
secondary
voltage
cell.
output
of
In
order
the
for
power
a
charging
source
current,
must
be
10.5
9.5A
14.3A
5.6A
3.0A
than
that
source
9.6
33A
55A
9
charging
8
of
and
the
the
battery
.
cell's
process
A
large
terminal
will
take
less
difference
potential
time
but
between
difference
risks
ow
,
higher
secondary cell being charged
7
.6
I
7
2
3
5 10
20
30 60
2
3
5
10
I
20
power source
discharge time
(slightly higher pd)
E l E c t r i c i t y
a n d
M a g n E t i s M
the
power
means
damaging
+
60
to
slightly
10.8
10
0
I,
that
the
the
cell.
mec ce  e
magnEtiC fiEld linEs
There
the
are
many
similarities
electrostatic
force.
In
between
fact,
both
the
magnetic
forces
have
force
been
geographic Nor th Pole
and
shown
to
ar th
be
two
(see
aspects
page
78).
of
It
one
is,
force
–
the
however,
electromagnetic
much
easier
to
interaction
consider
them
as
A magnet free to
completely
separate
forces
to
avoid
confusion.
move in all
Page
52
introduced
the
idea
of
electric
elds.
A
similar
S
concept
directions would
is
used
for
magnetic
elds.
A
table
of
the
comparisons
between
line up pointing
these
two
elds
is
shown
below.
along the eld
lines. A compass is
Electric
eld
Magnetic
eld
normally only free to
Symbol
E
N
B
move horizontally, so it
Caused
by
…
Charges
Magnets
(or
electric
ends up pointing along the
horizontal component of the eld.
currents)
geographic
The magnetic Nor th pole of the
Affects
…
Charges
Magnets
(or
electric
South Pole
compass points towards the geographic
currents)
Nor th Pole − hence its name.
Two
types
Charge:
positive
and
Pole:
North
and
An
of
…
negative
electric
current
mathematical
Simple
force
Like
charges
repel,
Like
poles
unlike
charges
attract
unlike
poles
given
order
the
to
help
concept
of
visualize
eld
a
lines.
magnetic
This
time
eld
the
we,
eld
once
lines
again,
are
on
page
be
seen
in
pole
•
is
eld
placed
The
–
also
in
a
direction
eld
called
ux
magnetic
of
the
lines.
eld,
force
is
it
If
will
shown
a
‘test’
feel
by
a
the
63.
the
also
the
cause
a
magnetic
magnetic
elds
eld.
The
produced
The
eld
patterns
due
to
in
different
magnetic
diagrams
this
way
currents
below.
thumb
use
lines
(current direction)
of
I
magnetic
of
attract
can
In
value
repel,
is
rule:
can
South
current
North
force.
direction
of
the
lines.
curl of ngers
•
The
strength
of
the
force
is
shown
by
how
close
the
lines
are
gives direction of
I
to
one
eld lines
another.
A ‘test’ South pole
The
eld
lines
are
circular
around
the
current.
here would feel a
Force here
The
direction
of
the
eld
lines
can
be
remembered
with
the
right-
force in the
strong since
hand
grip
rule.
If
the
thumb
of
the
right
hand
is
arranged
to
point
opposite direction.
eld lines are
along
the
direction
of
a
current,
the
way
the
ngers
of
the
right
close together.
hand
naturally
Field
pattern
curl
will
give
the
direction
of
the
eld
lines.
Overall force is in
N
S
direction shown
Force here
of
a
straight
wire
because a Nor th
weak since
carrying
cross-section
pole would feel a
eld lines are
current into
repulsion and an
far apar t.
current
page
attraction as shown.
rotate
N
N
S
S
current out
A
small
with
of
magnet
the
iron
eld
(iron
placed
lines.
in
This
lings)
will
the
is
eld
how
also
a
line
would
rotate
compass
up
with
until
works.
the
lined
Small
eld
lines
pieces
–
of page
up
Field
Field
induced
pattern
Despite
all
to
of
the
become
an
little
isolated
similarities
bar
of
a
at
circular
coil
they
A
willbe
pattern
long
current-carrying
coil
is
called
a
solenoid.
magnets.
magnet
between
electric
elds
and
magnetic
eld pattern of
elds,
For
•
it
should
be
remembered
that
they
are
very
different.
A
magnet
as a bar magnet
does
not
feel
a
force
when
placed
in
an
electric
cross-section
eld.
•
solenoid is the same
example:
A
positive
stationary
•
Isolated
•
The
charge
in
a
does
not
magnetic
charges
exist
feel
a
force
when
placed
N
eld.
whereas
isolated
poles
do
S
poles of solenoid can
not.
be predicted using
Earth
itself
has
a
magnetic
eld.
It
turns
out
to
be
right-hand grip rule
similar
to
that
of
a
bar
magnet
with
a
magnetic
South
pole
Field
near
the
geographic
North
Pole
as
shown
pattern
for
a
solenoid
below.
E l E c t r i c i t y
a n d
M a g n E t i s M
61
mec ce
magnEtiC forCE on a CurrEnt
force on current
S
When
a
current-carrying
magnetic
interaction
is
as
wire
is
between
placed
the
two
in
a
magnetic
results
in
a
eld
force.
the
This
N
known
the
motor
effect.
The
direction
of
this
force
is
at
I
I
right
as
angles
shown
to
the
plane
that
contains
the
eld
and
the
current
below.
thumb
force (F)
rst nger
(force)
(eld) B
eld (B)
I
F
zero force
second nger
current (I)
F
(current) I
force at right
I
θ
angles to plane of
N
Fleming’s
left-hand
Experiments
show
rule
that
the
force
is
proportional
to:
S
current and eld
•
the
magnitude
of
the
magnetic
•
the
magnitude
of
the
current,
•
the
length
•
the
sine
eld,
B
lines
of
the
current,
L,
I
that
is
in
the
magnetic
eld
F
force maximum
I
of
the
angle,
θ,
between
the
eld
and
current.
when current and
The
magnetic
eld
strength,
B
is
dened
as
follows:
eld are at right
F
= BIL
B
=
sin
θ
or
angles
F
_
IL
A
new
sin
unit,
1
1
N
A
θ
the
tesla,
is
introduced.
1
T
is
dened
to
be
equal
to
1
m
.
Another
possible
unit
for
magnetic
eld
strength
is
2
Wb
Since
magnEtiC forCE on a moving ChargE
A
single
force
in
charge
moving
exactly
the
through
same
way
a
magnetic
that
a
eld
current
feels
m
also
a
feels
a
force.
this
case
the
force
on
a
moving
charge
is
proportional
the
the
magnitude
of
the
magnetic
•
the
magnitude
of
the
charge,
•
the
velocity
•
the
sine
eld,
force
velocity
An
example
the
of
of
possible
on
the
a
moving
charge
this
eld
be
is
is
magnetic
charge
the
would
magnetic
term
is
always
resultant
when
at
right
ux
an
at
motion
to
right
can
electron
angles
density.
its
angles
be
enters
below.
B
S
of
of
the
the
charge,
angle,
θ,
q
v
between
the
velocity
of
the
charge
electron
and
We
of
can
the
the
use
these
magnetic
equivalent
F
eld.
to
relationships
eld
the
strength,
previous
to
B.
give
This
an
alternative
denition
is
denition
F
F
exactly
denition.
N
F
F
_
F
=
Bqv
sin
θ
or
B
=
qv
sin
θ
An
62
E l E c t r i c i t y
a n d
M a g n E t i s M
electron
moving
at
right
angles
to
a
magnetic
eld
to
circular.
a
velocity
to:
shown
•
Another
the
where
In
.
region
as
Ee  e ec e e  c e
The
formulae
used
on
this
page
do
not
need
to
be
two parallEl wirEs – dEfinition
remembered.
of thE ampErE
straight wirE
The
eld
one
moves
weaker.
•
the
pattern
Two
around
away
from
the
Experimentally
value
of
the
a
long
straight
wire,
the
the
eld
current,
is
wire
shows
strength
of
the
proportional
that
as
eld
of
gets
parallel
the
there
to:
is
a
it
feels
a
the
inverse
distance
of
the
away
is
distance
doubled,
away
from
the
the
magnetic
the
medium
wire,
eld
r.
will
If
a
magnetic
owing
magnetic
force.
Newton’s
•
of
current
producing
I
current-carrying
concepts
The
third
law
down
eld.
forces
pair
wires
eld
of
the
The
on
provide
and
wire,
other
the
a
good
magnetic
each
wire
wires
are
is
wire
in
an
example
force.
Because
is
this
eld
example
the
r
l
length
l
1
The
eld
These
also
depends
factors
are
on
summarized
in
the
around
the
2
wire.
I
equation:
I
1
2
B
µI
1
B
_
B
a
forces.
halve.
length
•
so
of
1
=
B
1
2πr
F
r
force
I
felt
by
I
2
B
1
=
B
×
I
1
B
=
field
produced
by
×
l
2
2
I
1
1
r
 I
∴
1
force
per
unit
=
2πr
length
of
I
2
I
B
I
1
l
2
2
=
l
Magnetic
eld
of
a
straight
2
current
=
The
constant
medium
µ
is
around
called
the
the
wire
permeability
changes.
and
Most
of
changes
the
time
if
B
we
I
1
the

consider
2
I
I
1
2
=
r
the
eld
around
a
wire
when
there
is
nothing
there
–
so
we
2πr
length
l
1
use
the
value
for
the
permeability
of
a
vacuum,
µ
.
There
length
is
l
2
0
almost
no
difference
between
the
permeability
of
air
and
the
I
I
1
2
permeability
of
a
vacuum.
There
are
many
possible
units
for
this
B
2
2
constant,
but
it
is
common
to
use
N
A
or
T
m
2
A
force
Permeability
and
permittivity
are
related
constants.
In
felt
you
know
one
constant
you
can
calculate
I
the
F
B
1
2
B
×
I
2
if
by
other
=
words,
B
1
×
r
l
1
1
other.
B
2
In
the
SI
system
of
units,
the
permeability
of
a
vacuum
is
∴
force
per
unit
B
=
field
produced
by
I
2
7
dened
to
have
a
value
of
exactly
4
π
×
2
2
10
N
A
.
See
the
length
of
I
 I
1
2
=
denition
of
the
ampere
(right)
for
more
detail.
2πr
B
I
2
l
1
1
=
l
1
magnEtiC fiEld in a solEnoid
=
B
I
2
The
magnetic
eld
of
a
solenoid
is
very
similar
to
1
the
I
I
1
magnetic
eld
of
a
bar
magnet.
As
shown
by
the
parallel
2
=
eld
2πr
lines,
the
magnetic
eld
inside
the
solenoid
is
constant.
It
µI
I
2
____
might
seem
surprising
that
the
eld
does
not
vary
at
all
inside
Magnitude
of
force
per
unit
length
on
either
wire
=
2πr
the
solenoid,
but
this
can
be
experimentally
veried
near
the
This
centre
of
a
long
solenoid.
It
does
tend
to
decrease
near
The
ends
of
the
solenoid
as
shown
in
the
graph
equation
is
experimentally
used
to
dene
the
ampere.
the
coulomb
is
then
dened
to
be
one
ampere
second.
If
we
below.
imagine
amp
two
innitely
separated
by
a
long
wires
distance
of
carrying
one
a
metre,
current
the
of
one
equation
would
7
predict
not
up
the
force
possible
can
be
to
per
have
arranged
unit
length
innitely
with
very
to
be
2
×
long
wires,
long
wires
10
an
N.
Although
experimental
indeed.
This
it
is
set-
allows
the
axis
forces
to
be
measured
The
mathematical
of
long
and
equation
ammeters
for
this
to
be
properly
constant
eld
at
calibrated.
the
centre
I
I
a
solenoid
is
n
magnetic
field
along
B
axis
constant
=
µ
(
)
I
l
field
B
in
centre
Thus
the
eld
only
•
the
current,
I
•
the
number
of
•
the
nature
depends
on:
n
turns
per
unit
length,
distance
l
(n
=
number
of
turns,
l
=
length)
It
Variation
of
magnetic
eld
in
a
is
of
independent
the
of
solenoid
the
core,
µ
cross-sectional
area
of
the
solenoid.
solenoid
E l E c t r i c i t y
a n d
M a g n E t i s M
63
ib Qe – eecc  e
1.
Which
one
of
the
eld
patterns
below
could
be
produced
by
12 V battery
two
point
charges?
A.
C.
B.
D.
a)
On
the
correct
circuit
voltmeter
2.
Two
long,
vertical
wires
X
and
Y
carry
currents
in
the
and
pass
through
a
horizontal
X
sheet
of
The
Y
b)
An
are
scattered
on
the
card.
Which
one
of
diagrams
best
shows
the
pattern
be
(The
dots
show
where
the
wires
formed
X
and
Y
voltmeter
allow
the
V-I
showing
and
an
the
ideal
characteristics
of
this
[2]
and
why
(i)
cannot
(ii)
cannot
alternative
a
the
ammeter
are
connected
correctly
in
above.
Explain
the
potential
be
increased
be
reduced
circuit
potential
by
the
(i)
Draw
enter
for
difference
to
to
12
across
the
lamp
V
.
[2]
zero.
[2]
measuring
the
V-I
characteristic
divider.
a
circuit
[3]
that
uses
a
potential
divider
to
iron
the
the
V-I
characteristics
of
the
lament
to
card.)
be
A.
symbols
ammeter
measured.
enable
lings?
would
circuit
ideal
the
c)
following
to
circuit
uses
lings
add
an
card.
the
Iron
that
of
same
lamp
direction
above,
positions
found.
[3]
C.
(ii)
Explain
why
difference
zero
The
graph
lament
B.
this
circuit
across
the
enables
lamp
to
be
the
potential
reduced
to
volts.
below
lamps
[2]
shows
A
and
the
V-I
characteristic
for
two
12
V
B.
D.
potential
dierence / V
lamp A
lamp B
12
3.
This
question
sphere
The
and
diagram
vacuum
is
the
about
below
that
the
motion
of
electric
shows
carries
a
eld
electrons
an
in
isolated
negative
due
that
to
metal
electric
a
charged
eld.
sphere
charge
of
in
9.0
a
nC.
0
0
0.5
1.0
current / A
a)
On
the
diagram
pattern
due
to
draw
the
arrows
charged
to
represent
the
electric
eld
sphere.
[3]
d)
State
and
explain
dissipation
b)
The
at
electric
points
that
9.0
the
nC
eld
outside
sphere
is
strength
the
acts
situated
at
at
sphere
as
the
can
though
its
a
centre.
surface
be
the
sphere
determined
point
The
of
charge
radius
of
by
of
a
which
lamp
potential
has
the
difference
greater
of
12
power
V
.
[3]
and
assuming
magnitude
the
for
The
two
battery
lamps
as
are
shown
now
connected
in
series
with
a
12
V
below.
12 V battery
sphere
2
is
4.5
×
10
m.
Deduce
that
the
magnitude
of
the
eld
4
strength
An
c)
at
electron
(i)
(ii)
is
surface
initially
Describe
the
(iii)
the
the
surface
Calculate
State
at
the
the
rest
path
of
the
and
of
sphere
on
the
followed
is
4.0
×
surface
by
the
10
of
1
V
the
electron
m
.
[1]
sphere.
as
it
leaves
sphere.
initial
explain
[1]
acceleration
whether
the
of
the
electron.
acceleration
of
[3]
the
lamp A
electron
remains
decreases
as
it
constant,
moves
increases
away
from
the
sphere.
the
electron
[2]
e)
(i)
State
that
(iv)
At
a
certain
point
6
6.0
×
10
between
4.
In
of
order
a
64
to
lamp,
the
student
i B
the
speed
of
how
in
the
lamp
current
in
lamp
A
compares
with
B.
[1]
is
1
m
measure
a
P
,
lamp B
or
s
.
Determine
point
the
sets
P
and
the
the
potential
surface
voltage-current
up
the
Q u E s t i o n s
following
–
of
(V-I)
(ii)
difference
the
sphere.
electrical
the
[2]
characteristics
Use
(iii)
the
total
Compare
circuit.
E l E c t r i c i t y
a n d
V-I
M a g n E t i s M
characteristics
current
the
from
power
the
of
the
lamps
to
deduce
battery.
dissipated
by
the
[4]
two
lamps.
[2]
6
c i r c U l a r
M o t i o n
a n d
g r a v i t a t i o n
Um u m
Mechanics of circUlar Motion
MatheMatics of circUl ar Motion
The
The
is
phrase
used
going
Most
to
‘uniform
describe
around
of
the
a
circular
an
object
circle
time
this
at
motion’
that
constant
also
means
is
diagram
acceleration
speed.
is
below
–
constantly
allows
which
us
must
to
also
work
be
out
the
the
direction
direction
of
the
of
the
centripetal
centripetal
force.
This
direction
changing.
that
situation diagram
vector diagram
v
B
the
circle
is
horizontal.
An
example
of
change in velocity directed in
uniform
circular
motion
would
be
the
towards centre of circle
B
motion
of
a
small
mass
on
the
end
of
a
v
A
string
as
shown
below.
v
v
B
A
A
v
+ change = v
A
The
mass moves at
object
has
changed
has
changed.
work
of
uniform
circular
is
important
to
remember
that
the
speed
direction
is
of
the
changing
object
all
the
out
is
this
the
to
v
Since
the
.
The
two
points
magnitude
of
A
and
velocity
B
is
B
on
a
always
horizontal
the
same,
circle.
but
Its
the
velocity
direction
B
velocities
average
are
change
in
vector
quantities
velocity.
This
we
vector
need
to
diagram
use
is
vector
also
mathematics
shown
to
above.
example,
circle.
the
This
is
direction
always
of
the
the
case
average
and
change
thus
true
in
for
velocity
the
is
towards
instantaneous
the
centre
acceleration.
constant,
For
its
v
between
even
of
though
from
moving
motion
In
It
shown
A
constant speed
Example
is
a
mass
m
moving
at
a
speed
v
in
uniform
circular
motion
of
radius
r
time.
2
v
Centripetal
acceleration
a
=
[in
centripetal
towards
the
centre
of
the
circle]
r
1
v m s
A
1
force
must
have
caused
this
acceleration.
The
value
of
the
force
is
worked
out
v m s
using
Newton’s
second
law:
1
v m s
Centripetal
force
(CPF)
F
=
m
a
centripetal
centripetal
2
m v
____
1
v m s
=
1
[in
towards
the
centre
of
the
circle]
r
v m s
1
For
example,
if
a
car
of
mass
1500
kg
is
travelling
at
a
constant
speed
of
20
m
s
speed is constant but the direction is
around
a
circular
track
of
radius
50
m,
the
resultant
force
that
must
be
acting
on
it
constantly changing
works
Circular
motion
–
the
direction
out
to
be
of
2
1500(20)
__________
motion
is
changing
all
the
time
F
=
=
12
000
N
50
This
constantly
changing
direction
means
It
that
the
velocity
of
the
object
is
is
really
starts
changing.
The
word
‘acceleration’
is
an
object’s
velocity
means
that
an
object
in
total
motion
MUST
be
the
The
if
the
speed
acceleration
circular
motion
acceleration.
the
centripetal
centripetal
is
a
when
it
that
goes
centripetal
in
a
circle.
force
It
is
a
is
NOT
way
of
a
new
force
working
that
out
must
have
been.
This
total
force
must
result
from
all
the
other
what
forces
object.
See
point
to
the
examples
note
is
that
below
the
for
more
centripetal
details.
force
does
NOT
do
any
work.
(Work
accelerating
particle
called
The
understand
=
force
×
distance
in
the
direction
of
the
force.)
constant.
of
is
force
nal
done
even
to
something
uniform
One
circular
on
changes.
on
This
acting
used
the
whenever
important
constantly
force
the
travelling
needed
acceleration
in
centripetal
is
to
cause
called
the
force
e x aMples
Ear th's gravitational
attraction on
Moon
Ear th
Moon
R
R cos θ
θ
F
T cos θ
T
friction forces
T sin θ
R sin θ
between
tyres and road
θ
mg
W
A conical pendulum – centripetal force
provided by horizontal component
of tension.
At a par ticular speed, the horizontal component
of the normal reaction can provide all the
centripetal force (without needing friction).
C i r C u l a r
m o t i o n
a n d
g r a v i t a t i o n
65
au      u m
radians
Angles
been
angUl ar velocity, ω, and tiMe period, T
measure
achieved.
(symbol:
°)
(symbol:
rad)
the
fraction
They
but
in
is
can,
of
studying
a
more
of
a
complete
course,
circular
useful
be
circle
that
measured
motion,
the
in
has
An
degrees
radian
measure.
object
travelling
changing
direction.
changing
even
We
dene
the
if
in
As
its
motion
a
its
result
speed
average
angle
circular
is
constant
angular
turned
is
be
constantly
constantly
(uniform
velocity,
circular
symbol
ω
motion).
(omega)
as:
∆θ
____
____________
ω
must
velocity
=
=
average
radius
time
r
taken
∆t
1
The
s
θ
distance along
units
The
of
angular
instantaneous
velocity
angular
are
radians
velocity
is
the
per
second
rate
of
(rad
change
s
of
).
angle:
circular arc
dθ
___
ω
=
rate
of
change
of
angle
=
dt
y
1.
Link
between
ω
and
v
v
The
arc
fraction
length
s
of
to
the
the
circle
that
has
been
achieved
is
the
ratio
of
In
circumference:
a
an
time
angle
s
____
fraction
of
circle
Δt,
the
object
rotates
Δθ
s
=
θ
=
∴
s
=
rΔθ
r
2πr
θ
In
degrees,
the
whole
circle
is
divided
up
into
360°
v
denes
the
angle
θ
degrees)
=
=
Δt
v
s
____
θ(in
=
as:
=
×
=
x
rΔθ
_
s
___
which
rω
Δt
rω
360
2πr
2.
In
radians,
the
whole
circle
is
divided
up
into
2π
Link
The
which
denes
the
angle
θ
Angle/°
radians)
×
period
ω
T
2π
In
this
time,
the
total
r
0
0.00
5
0.09
2π
___
small
angles
(less
ω
than
0.1
rad
or
5°),
the
arc
45
two
radii
form
a
3.
shape
to
a
or
are
just
a
triangle.
ratio,
angle
turned
is
complete
2π
one
radians,
full
circle.
so:
T
=
ω
0.74
Circular
motion
equations
=
60
Since
Substitution
of
the
above
equations
into
the
formulae
for
1.05
centripetal
radians
T
to
that
4
approximates
period
taken
and
π
the
time
time
2π
___
=
T
about
the
=
2πr
For
and
is
Angle/radian
s
=
time
as:
s
____
θ(in
between
radians
force
and
centripetal
acceleration
(page
65)
provide
π
the
90
1.57
=
versions
that
are
sometime
more
useful:
2
following
relationship
applies
if
2
180
working
in
3.14
=
π
2
v
centripetal
radians:
acceleration,
a
4π
r
_____
2
=
=
rω
=
2
r
T
3π
___
sinθ
≈
tanθ
≈
270
4.71
=
360
6.28
=
2
θ
2
2
mv
____
centripetal
2π
force,
F
4π
mr
_______
2
=
=
mrω
=
2
r
T
1.
circUl ar Motion in a vertic al pl ane
Uniform
circular
motion
of
a
mass
on
the
end
of
a
string
At
The
a
horizontal
plane
requires
a
constant
centripetal
force
the
to
tension
the
magnitude
of
the
tension
in
the
string
will
not
of
in
the
the
circle:,
string,
T,
and
the
weight,
mg,
are
in
the
same
act
direction
and
top
in
and
add
together
to
provide
the
CPF:
change.
2
mv
Circular
motion
in
the
vertical
plane
is
more
complicated
as
top
the
______
T
+
mg
=
top
weight
of
the
object
always
acts
in
the
same
vertical
to
object
the
will
speed
component
the
circle.
the
bottom
tension
in
The
its
and
the
slow
weight
maximum
and
the
of
up
string
will
that
speed
minimum
also
down
acts
will
speed
change
be
during
along
when
will
the
the
occur
during
its
at
one
r
direction.
To
The
motion
tangent
object
the
is
top.
remain
in
the
vertical
circle,
the
object
must
be
moving
with
due
a
to
certain
minimum
speed.
At
this
minimum
top
speed,
v
top
at
the
The
tension
object’s
is
zero
and
the
centripetal
force
is
provided
by
min,
the
weight:
revolution.
2
m(v
)
top
In
a
vertical
circle,
the
tension
of
the
string
will
always
act
at
min
__________
90°
mg
=
r
to
the
up
or
object’s
slowing
velocity
it
down.
so
this
The
force
does
no
conservation
of
work
in
energy
speeding
means
it
that:
v
=
top
1
mgy
+
rg
√
min
2
mv
=
constant
2.
At
the
bottom
of
the
circle:,
2
a) SITUATION DIAGRAM
The
H
tension
directions
in
the
and
the
string,
T,
resultant
and
the
force
weight,
provides
mg,
the
are
in
opposite
CPF:
2
mv
bottom
________
T
mg
=
bottom
r
r
In
small mass
order
the
O
m
to
circle
complete
must
top
of
the
circle.
the
be
circle
the
large
with
vertical
enough
sufcient
circle,
for
the
the
speed(v
the
object
gains
min
PE
at
to
=
top
Energetically
KE
object
the
rg )
√
(=
bottom
arrive
mg
to
×
at
of
the
complete
2r)
so
it
path
must
y
lose
the
same
1
)
bottom
of
KE:
1
2
m(v
L
amount
mg2r
=
1
2
m(v
min
)
top
2
=
mrg
min
2
2
2
∴
b) FREE-BODY DIAGRAM
(v
)
bottom
∴
v
=
bottom
The
4gr
=
rg
min
5rg
√
min
mathematics
in
the
above
example
(a
mass
on
the
end
of
a
F
instantaneous
string)
acceleration
In
can
place
of
also
T,
apply
the
for
tension
any
in
mg
reaction
66
C i r C u l a r
m o t i o n
a n d
g r a v i t a t i o n
from
the
surface.
vehicle
the
rope,
that
is
there
‘looping
is
N,
the
the
loop’.
normal
n’   
11
Universal
ne wton’s l aw of Univers al gravitation
If
you
trip
over,
you
Newton’s
theory
going
It
of
on.
this
is
of
called
theory
is
will
fall
down
universal
the
‘universal’
statement
towards
the
gravitation
gravitation
that
every
ground.
explains
because
mass
in
The
what
at
the
the
is
gravitational
following
•
The
•
There
law
all
attraction
the
other
between
masses
two
in
point
the
Universe.
masses
is
The
given
by
deals
with
be
=
6.67
×
2
10
N
m
2
kg
noticed:
is
a
force
acting
point
on
masses.
each
of
the
masses.
These
forces
are
Universe
value
an
only
should
G
core
EQUAL
attracts
points
constant
of
and
OPPOSITE
(even
if
the
masses
are
not
equal).
the
•
The
forces
are
•
Gravitation
always
attractive.
equation.
m
forces
act
between
ALL
objects
in
the
Universe.
m
1
2
The
forces
objects
only
become
involved
are
signicant
massive,
but
if
one
they
(or
are
both)
there
of
the
nonetheless.
r
The
m
m
1
Gm
2
r
below
should
same
the
spheres.
In
gravitational field strength
table
the
as
if
the
two
masses
be
compared
with
the
one
on
page
61.
the
example
gravitational
on
eld
the
can
left
be
F
the
eld
g
=
strength
the
out
to
centres
be
of
The
be
value
using
for
the
Newton’s
law:
2
r
gravitational
the
numerical
=
r
g
by…
turns
at
GM
____
2
Gravitational
Caused
masses
concentrated
calculated
GMm
_____
Symbol
spherical
were
2
r
The
between
2
F =
2
interaction
m
1
F ∝
same
as
eld
the
strength
at
acceleration
the
due
surface
to
of
gravity
a
on
planet
the
must
surface.
Masses
force
_____
Field
Affects…
One
strength
is
dened
to
be
Masses
type
of…
mass
force
_____
Mass
Acceleration
=
(from
F
=
ma)
mass
Simple
force
rule:
All
masses
attract
For
The
gravitational
eld
is
therefore
dened
as
the
the
Earth
force
24
M
=
6.0
×
10
kg
F
per
unit
mass.
g
m
=
=
small
point
test
mass
6
m
r
=
g
=
6.4
×
10
m
11
test
24
6.67 × 10
× 6.0 × 10
________________________
2
=
6
mass m
(6.4
×
10
9.8
m
s
2
)
2
F
F
value of g =
F
m
2
mass M
e x aMple
producing
1
In
order
to
calculate
the
overall
gravitational
eld
strength
gravitational eld g
at
any
point
we
gravitational
1
The
SI
units
for
g
are
N
kg
These
are
the
same
as
m
s
strength
use
of
is
eld
a
vector
vector
strength
addition.
at
any
of
both
point
The
overall
between
the
Earth
and
the
Earth
and
.
quantity
and
can
be
represented
Moon
must
be
a
result
pulls.
by
There
the
eld
use
2
.
the
Field
must
will
be
a
single
point
somewhere
between
lines.
the
Moon
gravitational
masses
eld lines
Earth,
is
where
zero.
after
the
Up
this
total
to
this
point
the
gravitational
point
the
overall
eld
overall
pull
is
due
pull
to
is
towards
these
back
the
to
two
the
Moon.
beyond X overall
pull is towards
Ear th
Moon
up to X overall pull
is back to Ear th
Moon
sphere
point mass
r
X
1
Field
strength
around
masses
(sphere
and
r
2
point)
gravitational
eld
distance
between
Earth
and
Moon
=
(r
+
1
If
resultant
gravitational
GM
EARTH
Earth
_______
near
surface
of
the
Earth
=
zero,
_______
2
r
1
C i r C u l a r
X
Moon
=
2
eld
at
)
2
GM
r
Gravitational
eld
r
2
m o t i o n
a n d
g r a v i t a t i o n
67
iB Qu – u m  
1.
A
a
ball
is
tied
vertical
position.
acting
to
a
plane.
Which
on
the
string
The
and
rotated
diagram
arrow
at
shows
shows
the
a
the
uniform
ball
direction
at
of
speed
its
the
in
a)
(i)
lowest
net
On
to
force
the
diagram
represent
position
ball?
above,
the
forces
draw
on
and
the
label
ball
in
arrows
the
shown.
[2]
[1]
(ii)
A
b)
5.
State
and
Determine
This
the
question
a)
Dene
b)
The
explain
is
whether
speed
about
gravitational
of
the
rotation
of
gravitational
eld
ball
is
the
in
equilibrium.
ball.
[2]
[3]
elds.
strength.
[2]
B
C
gravitational
eld
strength
at
the
surface
of
1
Jupiter
is
25
N
kg
and
the
radius
of
Jupiter
is
7
7.1
(i)
×
10
m.
Derive
an
expression
for
the
gravitational
D
eld
2.
A
particle
uniform
of
mass
circular
thecentripetal
m
is
moving
motion.
force
with
What
during
is
one
constant
the
total
speed
work
v
of
in
done
its
strength
mass
constant
by
revolution?
[1]
(ii)
Use
at
M,
the
its
surface
radius
R
of
and
a
planet
the
in
terms
gravitational
G.
your
[2]
expression
in
(b)(i)
above
to
estimate
2
mv
____
A.
Zero
the
mass
of
Jupiter.
[2]
B.
2
6.
2
C.
Gravitational
D.
A
particle
P
is
moving
anti-clockwise
with
Derive
an
in
a
horizontal
diagram
correctly
shows
the
direction
of
v
and
a
acceleration
a
of
the
particle
P
in
mass
The
radius
B.
a
of
a
c)
v
C.
D.
[3]
the
Earth
On
the
A
eld
a
value
diagram
satellite
eld
the
P
r
=
0.33
angle
mass
with
of
is
about
0.25
kg
constant
m.
30°
for
The
eld
at
km
its
mass
draw
outside
of
that
the
orbits
Earth.
satellite
feels
circular
is
and
surface
of
the
lines
the
the
is
9.8
N
kg
.
to
Earth.
[2]
represent
the
Earth.
the
is
v
to
along
attached
a
to
a
string
and
horizontal
the
ceiling
is
made
circle
and
of
to
radius
makes
an
vertical.
ver tical
r = 0.33 m
Q u e s t i o n s
–
C i r C u l a r
m o t i o n
a n d
Earth
why
weightless.
motion.
attached
the
Discuss
P
speed
string
with
i B
the
below
30°
68
6400
strength
a
of
is
[2]
a and v
d)
ball
a
P
v
rotate
eld
from
v
P
A
away
1
gravitational
question
gravitational
[1]
A.
This
the
distance
the
shown?
Calculate
4.
of
M.
gravitational
position
for
function
the
b)
velocity
as
circle.
point
Which
potential
expression
constant
strength
speed
and
2πmv
a)
3.
elds
2
mv
g r a v i t a t i o n
is
in
an
the
gravitational
astronaut
inside
[3]
7
ato mi c ,
nuclE ar
and
pa r t ic l E
p h ys i cs
E   e
Emission spEctra and absorption spEctra
E xpl anation of atomic spEctra
When
In
light
(or
an
can
be
called
a
are
a
few
a
of
prism
light
is
element
the
light
element
sodium
is
frequencies
a
in
diffraction
present,
The
various
an
lamp
the
are
absent
an
element
if
is
For
a
all
a
emits,
Exactly
continuous
the
input
the
is
levels
the
to
spectrum
topic
This
is
is
shone
called
through
an
absorption
when
it
is
in
gaseous
If
be
of
enough
a
the
electron
for
the
reason
can
is
of
put
is
to
in,
an
77,
without
electron
and
can
overall
given
energy
quantized.
these
quantum
page
positive
occupy
be
elements
only
See
‘escape’
now
only
said
why
part
is
cannot
atom
particular
signicant
nucleus.
they
energy
Electrons
the
The
to
that
happens,
xed
obitals.
bound
means
ionized.
are
forms
If
this
energy
levels
are
This
energy.
to
‘allowed’
These
correspond
energies
theory
are
(see
HL
12).
of
When
light
of
the
‘allowed’
same
electrons
model.
atom.
said
–
energy
that
the
atom,
atomic
and
emitted
sign
an
the
leave
contains
example,
often
lamp.
If
be
element
but
This
colours
would
frequencies
question.
light.
grating.
this
light
The
in
emits
a
street
present
its
continuous,
colours.
from
it
into
or
not
energy
it
were
spectrum.
characteristic
to
enough
splitting
spectrum,
particular
particular
by
using
continuous
yellow-orange
the
given
frequencies
emission
only
is
analysed
frequencies)
possible
its
element
an
electron
moves
between
energy
levels
it
must
emit
form.
or
absorb
to
the
energy.
The
energy
emitted
or
absorbed
corresponds
spectrum.
difference
prism
This
(or diraction
photons.
grating)
frequency
energy
is
between
emitted
or
the
two
allowed
absorbed
as
energy
levels.
‘packets’
of
light
corresponds
to
a
of
light
called
spectra: emission set-up
slit
sample
A
higher
energy
(shorter
photon
wavelength)
of
higher
light.
of gas
The
energy
of
a
photon
is
given
by
spectral
energy
light emitted
in
joules
frequency
of
1
Speed
lines
light
in
in
m
s
Hz
from gas
E
=
hf
prism
spectra: absorption set-up
Since
c
=
λ
=
fλ
slit
(or diraction
light
hc
___
Planck’s
constant
E
grating)
34
6.63
source
×
10
J
s
spectral
Wavelength
in
m
lines
sample of gas
all frequencies
Thus
the
frequency
of
the
light,
emitted
or
absorbed,
is
xed
(continuous spectrum)
by
sodium
330
415
420
569
the
levels
590 6
15
energy
are
emission
334
365 366
398 405
389
helium
319
36
1
37
1
382
546
to
the
a
given
the
element,
absorption)
levels.
this
spectrum
Since
means
will
also
the
that
be
energy
the
unique.
579
403
396
384
436
unique
(and
between
ygrene
mercury
313
difference
412
439 447
47
1
492502
588
668
397
hydrogen
380
389
410
434
486
656
allowed
energy
wavelength, λ
300
310
/ nm
320
330
levels
340
350
360
370
380 390 400
450
500
550
600
650
700
yellow
approximate colour
blue
orange
invisible
(IR)
Emission
spectra
sodium
330
415
420
569
365 366
398 405
389
helium
319
36
1
37
1
382
436
546
frequency absorbed
ygrene
334
photon of pa r ticular
energy level to higher energy level
590 6
15
mercury
313
electron ‘promoted’ from low
579
403
396
412
439 447
47
1
492 502
588
668
allowed
384
397
hydrogen
380
389
410
434
486
656
energy
levels
wavelength,
300
310
λ / nm
320
330
340
350
360
370
380
390 400
450
500
550
600
650
700
yellow
approximate colour
blue
orange
invisible
(IR)
Absorption
electron ‘falls’ from high
photon of pa r ticular
energy level to lower energy level
frequency emitted
spectra
A t o m i c ,
n u c l e A r
A n d
p A r t i c l e
p h y s i c s
69
ne 
isotopEs
When
a
electrons
different
outer
nuclE ar stability
chemical
of
the
properties
of
a
of
takes
properties
varies
that
protons.
from
because
element
in
the
general,
it
involves
Different
element
exists
In
place,
concerned.
particular
charge
number
atoms
chemical
electrons
positive
reaction
the
to
are
elements
xed
nucleus
outer
have
arrangement
element.
different
the
–
by
in
The
the
atomic
decay
is
words,
of
the
The
numbers
imply
name
given
contains
of
different
a
to
a
particular
specied
neutrons).
chemical
Some
properties.
species
number
of
nuclides
of
atom
protons
are
the
A
nuclide
(one
and
same
a
alpha
of
of
same
chemical
properties
and
is
the
nuclides
same
These
number
nuclides
of
are
protons
called
but
isotopes
different
–
whose
–
same
they
numbers
(α),
decay
beta
particular
that
For
small
the
number
(β)
The
(see
or
gamma
nuclide
present.
process
page
The
(γ)
depends
graph
by
72).
It
which
they
involves
radiation.
greatly
below
on
shows
the
the
exist.
nuclei,
•
For
large
•
Nuclides
of
of
the
number
of
neutrons
tends
to
equal
protons.
nucleus
nuclei
there
are
more
neutrons
than
protons.
number
they
above
the
band
of
stability
have
‘too
many
have
number
contain
and
will
tend
to
decay
with
either
alpha
or
beta
of
decay
protons.
unstable.
the
specied
element
contain
a
neutrons
neutrons’
the
are
radioactive
structures
•
will
of
stability
stable
nuclei
called
emission
chemical
amount
other
nuclear
of
Many
(see
page
72).
the
neutrons.
•
Nuclides
below
neutrons’
N ,rebmun nortuen
notation
mass number – equal to number
of nucleons
A
chemical symbol
and
the
will
band
tend
of
to
stability
emit
have
‘too
few
positrons
(see
page
73).
160
150
140
130
120
Z
110
atomic number – equal to number of
100
protons in the nucleus
Nuclide
notation
90
80
E x amplEs
70
Notation
Description
Comment
C
carbon-12
isotope
of
2
C
carbon-13
isotope
of
1
60
12
1
6
13
2
50
6
238
3
U
uranium-238
Pt
platinum-198
40
92
198
4
same
mass
number
78
30
as
5
198
5
Hg
mercury-198
same
mass
20
number
80
as
4
10
Each
element
atomic
number.
whereas
In
has
No.4
general,
concerned
Chemists
a
No.1
and
use
chemical
No.2
are
use
nucleus
same
are
symbol
examples
and
of
its
two
own
isotopes,
0
not.
physicists
the
the
and
No.5
when
with
unique
this
rather
notation
notation
than
but
the
tend
to
they
10
20
30
40
50
60
80
atom.
include
the
represent
the
Key
12
overall
charge
on
the
atom.
Thus
C
can
6
carbon
nucleus
to
a
physicist
or
the
N
carbon
atom
to
a
on
the
context.
If
the
charge
is
present
becomes
unambiguous.
Cl
must
refer
number
of
of
neutrons
protons
the
35
situation
number
chemist
Z
depending
to
a
■
naturally
occurring
stable
●
naturally
occurring
α-emitting
nuclide
○
articially
▲
naturally
▵
articially
produced
▿
articially
▼
articially
▼
articial
chlorine
17
ion
–
an
atom
that
has
gained
one
extra
electron.
produced
occurring
nuclide
α-emitting
β
-emitting
nuclide
nuclide
+
70
A t o m i c ,
n u c l e A r
A n d
90
100
atomic number, Z
are
whole
70
p A r t i c l e
p h y s i c s
β
-emitting
nuclide
produced
β
-emitting
nuclide
produced
electron-capturing
nuclide
decaying
by
nuclide
spontaneous
ssion
fe e
strong nuclE ar forcE
The
protons
repel,
they
means
a
nucleus
must
there
together.
few
in
be
must
things
about
repelling
be
Without
are
it
this
positive.
one
another
the
WE ak nuclE ar forcE
all
another
force
nucleus
Since
all
keeping
would
‘y
like
the
the
charges
time.
The
This
do
nucleus
apart’.
We
It
must
be
a
force.
strong.
If
the
proton
repulsions
are
calculated
it
that
is
far
too
It
must
the
gravitational
small
to
be
able
attraction
to
keep
between
the
nucleus
the
force
be
very
short-ranged
anywhere
other
than
as
we
inside
do
the
not
It
is
likely
tend
to
Large
to
The
to
involve
have
nuclei
keep
name
the
equal
need
to
neutrons
of
force
aspects
as
together.
observe
well.
more
•
force
is
things
It
strong
neutrons
nuclear
in
•
order
standard
that
we
that
have
observe
(above).
been
As
fundament
These
detail
model
are
a
on
information
daily
in
(or
Gravity,
all
a
particle
identied
result
forces
about
of
along
being
two
is
with
that
force
on
around
the
in
two
It
must
are
and
interactions
to
Weak.
page
78.
is
have
is
the
force
of
attraction
between
all
be
•
forces
is
attractive
–
masses
are
we
alpha
wish
beta
to
but
be
able
emission.
and
another
to
We
explain
know
a
force:
Many
nuclei
are
stable
and
beta
emission
short-ranged
other
than
as
we
inside
do
the
not
observe
this
nucleus.
name
ones
given
nuclear
electrons,
to
(e.g.
this
force,
it
positrons
protons
force
is
involves
and
and
the
the
lighter
neutrinos)
as
well
as
neutrons).
weak
nuclear
force
This
single
force
as
includes
either
all
the
electrostatic
forces
or
that
we
normally
magnetic.
•
Electromagnetic
forces
involve
•
Electromagnetic
forces
can
•
The
range
•
The
electromagnetic
charged
matter.
be
attractive
or
repulsive.
exist.
More
below:
objects
pulled
if
including
explain
identied
four
Outline
listed
heavier
of
imbalances
•
that
At
the
end
the
of
electromagnetic
of
force
charges
forces
electrostatic
always
involved
to
be
nuclei
nuclei,
occur.
strong
(e.g.
categorize
mass.
Gravity
very
the
different
•
weak.
always
signicant
Gravity
can
why
Most
Electromagnetic
forces
Gravity
•
72)
stable.
stability
only
known
this
anywhere
Unlike
The
the
‘new’
nuclear
there
are
Strong
discussed
‘everyday’
based
model,
interactions)
forces
is
involved
standard
Electromagnetic,
these
about
basis
as
the
physics
be
not
particles
othEr fundamEntal forcEs/intEractions
The
be
nucleus
about
must
the
the
Mechanisms
page
explains
are
nuclei
neutrons.
together.
this
must
the
left)
they
this
•
Small
and
of
(see
box
why
nucleons
nucleus.
protons
proportionately
nucleus
given
the
numbers
unstable.
nuclear
(see
thus
is
force
•
are
force
and
emission
does
•
nuclear
apart
gamma
few
clear
y
however,
know
all
•
strong
not
on
the
force
aspects
on
the
19th
and
of
is
an
relatively
atomic
laboratory
century,
the
the
force
innite.
strong
level
–
give
tiny
rise
to
scale.
Maxwell
magnetic
more
is
showed
force
were
fundamental
that
just
the
two
electromagnetic
together.
force.
•
The
•
Despite
range
of
the
gravity
force
is
innite.
•
the
above,
the
gravity
force
is
relatively
quite
The
mathematics
Maxwell’s
At
least
one
of
the
masses
involved
needs
to
be
large
for
to
be
noticeable.
For
example,
the
gravitational
attraction
between
you
and
this
book
is
negligible,
but
(drop
between
this
book
and
the
Earth
is
easily
Friction
(and
of
The
Newton’s
law
governing
of
this
gravitation
describes
the
is
described
by
the
many
force
other
‘everyday’
between
atoms
forces)
and
this
is
is
simply
the
governed
by
the
interaction.
demonstrable
it).
electromagnetic
considered
•
force
the
electromagnetic
force
electromagnetic
force
result
of
the
equations.
the
•
effects
of
weak.
to
be
force
aspects
and
of
the
the
weak
single
nuclear
force
electroweak
are
now
force.
mathematics
force.
particlEs that E xpEriEncE and mEdiatE thE fundamEntal forcEs.
See
page
78
summarizes
onwards
which
for
more
particles
details
about
experience
the
these
standard
forces
and
Gravitational
Particles
experience
All
model
how
mediate
Graviton
W
the
are
Weak
Quark,
fundamental
structure
of
matter.
The
following
table
mediated.
Electromagnetic
Gluon
+
Particles
for
they
Charged
Strong
Quark,
Gluon
0
,
W
,
Z
A t o m i c ,
Gluon
γ
n u c l e A r
A n d
p A r t i c l e
p h y s i c s
71
r  1
ioniZing propErtiEs
Many
atomic
process
by
which
radioactive
involves
are
they
decay.
the
different
nuclei
possible
decay
Every
emission
EffEcts of radiation
unstable.
of
is
The
At
called
decay
one
radiations
of
molecular
the
complex
alpha
(α),
beta
(β)
ionization
as
DNA
ionization
chemical
in
or
the
reactions
could
RNA.
cause
This
surrounding
(called
damage
could
medium
metabolic
directly
cause
is
it
to
to
enough
pathways)
a
cease
to
biologically
functioning.
interfere
taking
with
place.
damage
can
result
in
a
disruption
to
the
functions
that
are
taking
place
or
within
gamma
an
an
such
the
Molecular
nucleus:
level,
molecule
Alternatively,
three
from
the
important
the
cells
that
make
up
the
organism.
As
well
as
potentially
causing
the
cell
(γ).
to
die,
could
this
be
could
the
just
cause
prevent
of
the
cells
from
dividing
transformation
of
the
and
cell
multiplying.
into
a
On
malignant
top
of
this,
it
form.
α
As
all
body
systems
tissues
that
have
are
built
been
up
of
affected.
cells,
The
damage
to
these
non-functioning
can
of
result
these
in
damage
systems
can
to
the
result
body
in
death
β
for
the
animal.
If
malignant
cells
continue
to
grow
then
this
is
called
cancer
γ
radiation s afEty
Alpha,
the
All
beta
and
gamma
all
come
from
There
nucleus
three
means
that
radiations
that
substance,
electrons
as
they
are
go
collisions
to
be
ionizing.
through
occur
removed
from
should
This
a
which
There
cause
that
have
lost
or
be
called
ions.
This
the
radiations
to
be
as
explains
Run
their
ionizations
detected.
dangerous
occur
in
molecules,
can
be
dose
terms
ways
of
of
an
of
ionizing
extra
the
dose
radiation.
(for
information
protecting
oneself
Any
example
received
from
or
too
hospital
having
the
large
an
procedures
X-ray
benet
a
dose.
it
scan)
gives.
These
can
be
follows:
away!
simplest
method
you
and
and
of
the
reducing
source.
the
Only
dose
received
is
electromagnetic
to
increase
radiation
the
can
distance
travel
this
follows
an
inverse
square
relationship
with
large
distance.
nature.
Don’t
waste
time!
biologically
such
as
you
have
to
receive
a
dose,
then
it
is
important
to
keep
the
time
of
this
DNA,
exposure
function
safe
It
If
important
in
main
a
property
•
When
as
receiving
electrons
distances
also
thing
patient
three
between
allows
a
atoms.
gained
ionizing
such
in
justiable
are
The
are
no
summarized
•
Atoms
is
result
to
a
minimum.
affected.
•
If
you
can’t
Shielding
also
be
run
can
used
away,
always
to
limit
hide
be
the
used
behind
to
something!
reduce
exposure
for
the
both
dose
received.
patient
and
Lead-lined
aprons
can
operator.
propErtiEs of alpha , bEta and gamma radiations
Property
Effect
Alpha,
on
photographic
lm
α
Beta,
Yes
4
Approximate
ion
pairs
number
produced
in
of
β
Gamma,
Yes
γ
Yes
2
10
per
mm
travelled
10
per
mm
travelled
1
per
mm
travelled
air
2
Typical
absorb
material
needed
to
10
mm
aluminium;
piece
of
A
paper
few
mm
aluminium
10
cm
lead
it
Penetration
Typical
ability
path
Deection
Low
length
by
E
in
and
B
air
A
elds
few
Medium
cm
Behaves
Less
like
a
7
Speed
About
10
positive
charge
High
than
Behaves
like
1
m
one
a
8
s
About
10
m
Effectively
negative
charge
s
atomic
numbers
and
the
very
variable
mass
balance
on
each
side
of
10
Y
(Z
the
+
e.g.
1
m
s
0
→
+
β
1)
+
ν
1
90
gamma dEc ay
×
A
X
Z
numbers
deected
3
8
,
A
The
naturE of alpha , bEta and
Not
1
m
innite
90
Sr →
0
Y
38
+
β
39
+
ν
1
equation.
When
a
nucleus
decays
the
mass
•
(95
numbers
and
the
atomic
numbers
=
93
+
2
and
241
=
237
+
Gamma
rays
are
unlike
the
other
two
4)
must
radiations
in
that
they
are
part
of
the
0
balance
on
each
side
of
the
nuclear
•
Beta
particles
are
electrons,
β
electromagnetic
spectrum.
After
their
1
0
equation.
or
e
,
emitted
from
the
nucleus.
emission,
the
nucleus
has
less
energy
1
4
•
Alpha
particles
are
helium
nuclei,
The
explanation
is
that
the
electron
is
α
but
its
mass
number
and
its
atomic
2
4
or
formed
2+
He
.
In
alpha
decay,
a
when
a
neutron
decays.
At
the
‘chunk’
number
have
not
changed.
It
is
said
to
2
same
of
the
nucleus
is
emitted.
The
time,
called
that
remains
will
be
a
an
0
1
1
4)
→
+
2)
241
e.g.
α
Since
2
237
Am →
95
+
4
Np
93
+
an
virtually
+
mass
it
A t o m i c ,
does
no
not
charge
affect
α
A n d
p A r t i c l e
p h y s i c s
energy
→
Z
has
an
+
Z
Excited
Lower
state
state
excited
state.
0
X
and
the
from
A
X
2
n u c l e A r
lower
ν
1
antineutrino
no
a
changed
A
β
1
have
to
0
p
equation.
72
emitted
4
Y
(Z
is
antineutrino.
n →
(A
X
Z
particle
different
nuclide.
A
another
portion
γ
0
energy
state
r  2
antimattEr
The
nuclear
One
mode l
important
existence
of
given
thi ng
on
tha t
antima tte r.
is
p ag e
not
Eve r y
77
is
s ome wha t
me n ti on e d
fo rm
of
t he r e
matter
si m p li  ed .
is
has
the
and
the
antimatte r
it s
1
1
p →
form
of
a nti ma tter.
If
ma t t e r
an d
a n t im at t er
0
n
1
equivalent
+
a nnihi l ate
e ac h
oth e r.
N ot
e le c t r on ,
a
p os it r on ,
is
+
ν
+1
19
→
0
F
10
woul d
an
+
β
0
Ne
they
of
c am e
19
together
v e r s io n
emitted.
+
+
β
9
+
ν
+1
s ur pr is i ng l y,
+
antimatter
form
of
is
rare
b ut
radioactiv e
positron
decay.
In
it
do e s
d e ca y
thi s
ex is t.
tha t
de ca y
Fo r
ca n
a
e xa m p le ,
t a ke
pr oton
pla ce
is
de c a ys
The
positron,
The
antineutrino
into
a
pl u s
or
decay
all
you
measured
that
take
in
in
(Bq)
a
natural
time.
terms
place
becquerels
is
the
of
a
The
the
unit
with
1
phenomenon
activity
number
of
time.
Bq
=
more
Some
background radiation
around
1
of
This
of
any
and
is
given
individual
nuclear
decay
going
is
per
α,
on
source
nuclear
information
,
emission
is
the
are
is
cosmic
and
γ
taking
identies
decays
quoted
β
value
in
details
see
gamma
radiation
place
in
typical
varies
page
counter,
this
which
would
detects
be
and
measured
counts
the
using
a
number
accompanied
form
of
by
a
the
neutrino.
neutrino.
rays
will
received
the
be
as
a
responsible,
result
surrounding
sources
from
78.
of
country
of
to
country
but
The
radiation,
and
there
radioactive
materials.
background
from
pie
but
place
will
also
decays
chart
the
to
be
that
below
actual
place.
second.
food
Experimentally
is
antimatter
ne ut r on ,
For
Radioactive
β
a no t h er
be t a
medicine
nuclear industry
Geiger
of
ionizations
buildings/soil
radon
taking
will
place
always
inside
detect
the
GM
some
tube.
A
radioactive
working
Geiger
ionizations
counter
taking
place
cosmic
even
when
there
is
no
identied
radioactive
source:
there
is
a
medicine – 14%
background
A
reading
detector
of
count
30
as
counts
registering
a
result
per
30
of
the
minute,
ionizing
background
which
events,
radiation.
corresponds
would
not
be
to
nuclear industry – 1%
the
unusual.
buildings/soil – 18%
To
analyse
necessary
the
to
activity
correct
of
for
a
given
the
radioactive
background
source,
radiation
it
is
cosmic – 14%
taking
place.
natural
It
radiation
radon – 42%
would
be
necessary
to
record
the
background
count
without
the
85%
food/
radioactive
source
present
and
this
value
can
then
be
subtracted
drinking water – 11%
from
all
readings
with
the
source
present.
start
random dEc ay
Radioactive
external
a
sample
This
of
means
particular
time.
All
decay
is
conditions.
a
random
For
radioactive
that
is
nucleus
we
material
there
is
know
no
going
is
process
example,
the
way
to
does
of
of
not
is
within
a
not
the
affect
knowing
decay
chances
and
increasing
decay
a
affected
by
temperature
the
rate
whether
certain
of
or
period
happening
in
of
decay.
not
a
to
sample,
we
On
average
the
of
time.
atoms
of
a
the
process
is
random,
the
large
numbers
of
rate
in
decay
certain
Mathematically
Although
number
in
the
would
atoms
radioactive
of
given
decay
the
number
a
that
with
number
is
of
next
expect
of
decay
the
an
is
the
of
a
sample.
then
N,
If
sample
This
can
expect
were
decaying
is
to
be
proportional
process.
The
a
more
proportionality
decreases
expressed
we
there
number
exponential
element,
this
atoms
minute.
certain
atoms
to
the
means
number
exponentially
in
larger.
that
of
over
time.
as:
atoms
dN
___
involved
allows
us
to
make
some
accurate
predictions.
If
∝
we
- N
dt
A t o m i c ,
n u c l e A r
A n d
p A r t i c l e
p h y s i c s
73
h- e
half-lifE
There
is
a
decreases
with
exponential
In
of
the
we
decay
choose.
is
an
is
in
a
that
every
exponential
a
particular
below,
the
always
This
happening
think
have
shown
to
to
allows
time
the
us
property
decrease,
that
taken
for
half
whatever
express
called
the
the
In
but
mathematical
same,
to
quantity
property.
the
several
number
starting
chances
half-life,
T1.
life
six
of
of
of
nuclides
statement
decay
(or
nuclide
present
is
that
substance
can
vary
with
with
from
in
the
activity)
substance
is
a
a
time
sample
half-life
of
a
the
a
taken
to
is
short
fractions
of
a
half
An
time
taken
of
takes
long
will
a
decay
second
to
the
number
for
the
nuclides
time
quickly.
millions
of
rate
to
to
times.
a
A
is
days.
In
will
working
a
matter
common
radioactive
reality,
remain,
i.e.
a
how
six
is
much
applying
mistake
material
after
out
of
3
is
to
days
days
the
think
then
(two
radioactive
half-life
it
that
will
half-lives)
if
all
a
property
the
half-
decay
‘half
of
in
a
quarter.
increase of stable
1
equivalent
sample
half-life
half-life
decay.
the
particular
large
for
situations,
remains
decay of radioactive
of
‘daughter ’ nuclei
‘parent’ nuclei
halve.
decay.
nuoma
A
a
of
half’
The
2
half-life
simple
material
ecna
decay
time
curves
graph
nuclides
value
E x amplE
temptation
A
Half-lives
after 2 half-lives
years.
3
yaced ot elbaliava sedilcun fo rebmun
4
1
number = x
N
after 2 half-lives
2
0
1
of the original
The time taken to halve from
parent’ nuclei will remain
4
T1
any point is always T1
2
2
(x)
x
number =
N
3
1/2
6
time / days
12
9
2
x
The
decay
of
parent
into
daughter
2
14
e.g.
The
half-life
of
C
is
5570
years.
6
N
1/4
Approximately
N
how
long
is
needed
before
less
than
1%
of
a
14
1/8
sample
of
C
remains?
6
N
1/16
T
1
T1
T1
2
Fraction
T1
50%
left
T
1
time
2
Time
2
2
2
half-life
25%
2T 1
2
Half-life
of
an
exponential
decay
12.5%
3T 1
2
4T 1
~
6.3%
~
3.1%
~
1.6%
~
0.8%
2
invEstigating
5T 1
2
half-lifE E xpErimEntally
When
measuring
the
activity
of
a
source,
the
background
rate
6T 1
2
should
be
subtracted.
7T 1
2
•
If
the
half-life
activity
→
A
the
against
simple
normal
could
be
method
→
A
is
of
from
of
readings
activity
exponential
simple
graph
then
can
be
taken
of
6
half
lives
7
half
lives
ln
the
and
against
shape.
graph
quick
(activity)
time
Several
and
but
against
values
then
not
the
time
would
should
of
half-life
averaged.
most
∴
approximately
give
calculated
If
the
half-life
a
from
is
straight
line
and
the
could
decay
be
produced.
constant
over
a
way
to
the
gradient.
See
page
result
used
long,
then
the
activity
will
period
of
time.
In
this
case
calculate
the
number
of
nuclei
effectively
one
needs
present,
N,
available
to
throw
of
a
die
radioactive
and
A n d
six
to
decay.
Each
represents
nd
no
longer
then
-λN.
n u c l e A r
the
simulate
Every
dt
A t o m i c ,
of
is
a
random
decay.
The
process
dice
and
can
represent
throw
represents
a
unit
of
be
dN
___
74
to
217.
use
=
years
needed
simul ation
is
a
38990
years
can
time.
constant
=
37000
This
nuclei
•
years
accurate.
be
be
33420
produce
The
This
=
time.
graph
read
is
short,
p A r t i c l e
p h y s i c s
available.
a
nucleus
decaying
meaning
this
die
ne e
artificial transmutations
There
is
nothing
that
we
can
do
to
units
change
the
likelihood
Using
Einstein’s
equation,
1
kg
of
mass
is
equivalent
to
16
of
a
certain
conditions
be
done
particle
we
by
or
articial
rst
radioactive
can
decay
make
nuclear
bombarding
another
a
small
the
nucleus
In
incoming
but
reactions
with
nucleus.
transmutations.
‘captures’
happening,
a
Such
and
happen.
nucleon,
reactions
general,
object
under
the
an
This
an
are
target
then
certain
9
can
×
the
alpha
10
J
atomic
useful.
called
of
energy.
scale
The
This
other
are
nucleus
a
units
electronvolt
megaelectronvolt
is
of
(see
often
huge
amount
energy
page
tend
53),
or
of
to
energy.
be
more
At
more
usually,
the
used.
19
1
eV
=
1
MeV
1
u
1.6
×
10
J
emission
13
takes
out
place.
by
alpha
The
rst
Rutherford
particles
ever
in
and
articial
1919.
the
transmutation
Nitrogen
presence
of
was
was
bombarded
oxygen
was
=
1.6
×
10
J
carried
of
mass
converts
into
931.5
MeV
by
detected
Since
spectroscopically.
work
mass
in
and
units
energy
that
are
avoid
equivalent
having
to
it
do
is
sometimes
repeated
useful
to
multiplications
2
4
14
2+
He
+
17
N
2
→
1
O
7
+
by
the
(speed
of
light)
.
A
new
possible
unit
for
mass
is
thus
p
8
1
2
MeV
The
mass
numbers
(4
+
14
=
17
+
1)
and
the
atomic
c
.
It
works
like
this:
numbers
2
If
(2
+
7
=
8
+
1)
on
both
sides
of
the
equation
must
1
MeV
c
worth
of
mass
is
converted
you
get
1
MeV
worth
balance.
of
energy.
unifiEd ma ss units
WorkEd E x amplEs
The
individual
masses
involved
in
nuclear
reactions
are
Question:
tiny.
In
order
unied
mass
common
in
the
to
compare
units,
isotope
u.
of
carbon-12
atomic
These
carbon,
atom
(6
are
masses
dened
carbon-12.
protons
and
physicists
in
terms
There
6
often
of
are
the
12
neutons)
use
most
How
nucleons
and
one
much
decayed
14
mass
unit
is
dened
as
exactly
one
twelfth
the
energy
shown
14
C
→
would
in
the
be
released
equation
if
14
g
of
carbon-14
below?
0
N
6
unied
as
+
β
7
+
ν
1
mass
Answer:
of
a
carbon-12
atom.
Essentially,
the
mass
of
a
proton
and
the
Information
mass
of
a
neutron
are
both
1
u
as
shown
in
the
table
1
___
1
u
27
=
mass
of
a
(carbon-12)
atom
=
1.66
×
given
below.
10
atomic
mass
of
carbon-14
atomic
mass
of
nitrogen-14
=
14.003242
u;
kg
12
mass*
of
1
proton
=
1.007
276
=
14.003074
u;
u
mass
mass*
of
1
neutron
=
1.008
665
u
mass*
of
1
electron
=
0.000
549
u
of
electron
=
0.000549
u
14
mass
of
left-hand
side
=
nuclear
mass
=
14.003242
=
13.999948
=
14.003074
of
C
6
*
=
Technically
these
are
all
‘rest
masses’
–
see
option
6(0.000549)
u
7(0.000549)
u
A
u
14
nuclear
mass
of
N
7
ma ss dEfEct and binding EnErgy
The
table
above
shows
the
masses
of
=
neutrons
and
mass
It
should
be
obvious
that
if
we
add
together
the
masses
of
right-hand
side
=
protons,
bigger
gone
what
6
than
neutrons
12
wrong?
keeps
u,
the
The
the
and
6
mass
answer
nucleus
electrons
of
a
becomes
bound
we
will
carbon-12
clear
get
atom.
when
a
difference
=
mass
has
of
difference
its
between
component
the
nucleons
assembling
a
mass
is
u
13.999780
LHS
u
RHS
investigate
of
called
a
nucleus
the
mass
and
the
nucleus,
the
protons
defect.
and
released
per
decay
=
0.000168
u
=
0.000168
×
=
0.156492
MeV
931.5
MeV
masses
If
one
14g
imagined
0.000549
together.
energy
The
+
number
What
we
u
13.999231
of
=
6
13.999231
protons.
of
C-14
is
1
mol
neutrons
23
would
initially
need
to
be
brought
together.
Doing
this
∴
takes
Total
number
of
decays
=
N
=
6.022
×
10
A
work
because
the
protons
repel
one
another.
Creating
the
bonds
23
∴
between
the
protons
and
neutrons
releases
a
greater
Total
energy
release
=
6.022
=
9.424
×
×
10
10
=
9.424
×
10
=
1.51
≈
15
×
0.156492
MeV
amount
22
of
energy
than
the
work
done
in
bringing
them
together.
MeV
This
22
energy
released
must
come
from
somewhere.
The
answer
lies
13
×
1.6
×
10
J
in
10
Einstein’s
famous
mass–energy
equivalence
relationship.
2
∆E
=
×
10
J
GJ
∆mc
NB
Many
examination
the
masses
calculations
avoid
the
need
to
consider
1
energy
in
joules
mass
in
kg
speed
of
light
in
m
s
mass
In
is
Einstein’s
possible
The
binding
when
It
a
be
convert
energy
nucleus
comes
also
equation,
to
from
the
a
nucleus
a
measure
a
is
of
is
its
the
is
that
the
in
individual
from
mass.
to
of
its
of
energy
energy
energy
and
that
component
The
be
form
into
amount
needs
binding
another
directly
assembled
decrease
energy
into
mass
mass
binding
added
nucleons.
in
The
is
the
opposed
electrons
to
the
by
atomic
providing
you
with
the
nuclear
mass
it
versa.
released
nucleons.
energy
order
mass
and
vice
as
of
to
would
separate
defect
is
thus
energy.
A t o m i c ,
n u c l e A r
A n d
p A r t i c l e
p h y s i c s
75
f  
fission
Fission
nuclei
is
energy
in
reactors
involve
can
the
are
the
and
A
1
atomic
the
+
is
a
the
nuclear
into
A
typical
to
might
+
single
break
is
and
used
in
reaction
with
up
whereby
nuclei
a
Since
in
nuclear
two
the
the
chain
might
neutron.
into
large
release
the
This
one
original
production
reaction
neutrons
reactions,
to
but
of
is
causing
neutrons,
occurring.
lose
it
neutron
three
It
enough
is
the
there
reaction
is
technically
energy
to
go
the
quite
on
has
resulted
possibility
difcult
and
initiate
of
to
a
get
further
achievable.
smaller
be:
92
Ba
1
Kr
56
that
nucleus
nucleus
reaction
smaller
reaction
uranium
141
92
the
up
bombs.
reaction
U →
0
It
uranium
typical
to
break
process.
235
n
given
to
bombarding
cause
nuclei.
name
induced
+
3
36
n
+
energy
0
Ba141
n
U-235
Kr-92
A
ssion
reaction
A
fusion
Fusion
is
nuclear
are
name
reaction
to
nuclei
process.
reaction
binding EnErgy pEr nuclEon
the
induced
larger
chain
It
join
the
to
whereby
and
is
given
the
small
together
release
reaction
Whenever
nuclei
into
energy
that
in
reaction
of
the
‘fuels’
this
a
are
nuclear
in
energy.
calculate
nucleus
the
a
reaction
lower
In
order
binding
divided
by
to
state
compare
energy
the
(ssion
energy
total
per
or
fusion)
than
the
the
energy
nucleon.
number
of
releases
reactants.
This
states
is
the
nucleons.
of
energy,
Mass
different
total
One
of
the
loss
be
nuclei,
binding
the
products
must
the
source
physicists
energy
nuclei
of
the
with
for
the
the
largest
56
all
stars
including
the
Sun.
A
typical
binding
energy
per
nucleon
is
iron-56,
Fe.
26
reaction
that
is
taking
place
in
the
Sun
A
is
the
fusion
of
two
different
reaction
hydrogen
3
H
+
1
to
produce
4
H →
1
1
He
+
2
n
+
feasible
if
the
products
energy
per
nucleon
when
compared
of
with
the
the
reaction
have
a
greater
reactants.
helium.
VeM / noelcun rep ygrene gnidnib
2
energetically
isotopes
binding
of
is
energy
0
hydrogen-2
helium-4
fusion
hydrogen-3
iron-56
10
ssion energetically
possible
8
6
4
fusion energetically
2
possible
neutron
nucleon
One
of
the
fusion
reactions
happening
20
40
60
80
100
120
140
160
180
200
number
in
the
Sun
Graph
76
A t o m i c ,
n u c l e A r
A n d
of
binding
p A r t i c l e
energy
per
nucleon
p h y s i c s
se  e
introduction
All
or
matter
that
otherwise,
is
combinations
hundred,
present
type
in
form
elements
symbol;
the
or
made
of
so,
atoms.
a
name
has
There
are
types
of
and
the
living
a
the
The
different
a
Each
hydrogen,
elements,
us,
of
Atoms
element.
has
e.g.
up
different
nature.
an
atomic modEl
surrounds
of
only
a
chemical
chemical
of
and
in
all
is
the
atom
the
negative
called
in
at
must
good
all
the
–
known
a
a
energy
nucleus.
vacuum.
to
the
All
an
The
nuclear
small
levels.
The
Overall
evidence
as
very
nucleons).
charge.
nothing
be
model,
describes
different
(collectively
these
simplest
atomic
century
arranged
atoms
single
of
basic
last
central
The
of
Oxygen
The
has
the
combination
with
one
oxygen
molecule
H
O.
chemical
of
two
atom
nucleus
The
called
full
list
itself
contains
positive
charge
electrons
provide
only
atom
is
nuclear
support
neutral.
model
The
of
the
developed
surrounded
a
and
tiny
vast
bit
of
seems
of
so
and
all
the
the
electrons
protons
almost
majority
atom
during
by
the
neutrons
mass
mass
the
but
of
all
volume
strange
that
of
is
there
it.
symbol
symbol
hydrogen
is
was
the
Protons
H.
model,
nucleus
a
of
Neutrons
Electrons
1
Negligible
O.
Relative
atoms
water
mass
1
Charge
Neutral
+1
1
elements
2
is
shown
consist
in
of
protons,
a
a
periodic
table.
combination
neutrons
and
of
Electron ‘clouds’. The positions of the 6 electrons
Atoms
three
are not exactly known but they are most likely to
things:
nucleus
be found in these orbitals. The dierent orbitals
electrons.
correspond to dierent energy levels.


m
cell


m
DNA
protons
nucleus
Atomic model of carbon
atom
This
so
In
the
basic
atomic
model,
we
are
simple
orbital
protons,
nothing
neutrons,
and
has
limitations.
should
Accelerated
constantly
lose
charges
energy
(the
are
known
changing
to
radiate
direction
energy
means
the
made
electrons
upof
model
electrons
electrons
are
accelerating).
–
more.
E vidEncE
One
of
the
most
experiment.
most
of
them
detectable
huge
law
existence
were
The
of
of
alpha
pieces
within
to
the
mathematics
repulsion
isotopes
of
particles
expected
structure
angles.
square
convincing
Positive
from
travel
gold
of
the
provides
evidence
were
straight
atoms.
the
for
‘red’
at
evidence
a
The
gold
the
amazing
for
model
leaf.
gold
that
of
The
leaf.
discovery
showed
Evidence
for
nuclear
thin
through
experiment
nucleus.
the
The
was
numbers
electron
the
energy
atom
relative
idea
that
comes
size
and
behind
some
of
from
this
the
being
deected
levels
comes
the
velocity
Rutherford–Geiger–Marsden
of
the
experiment
alpha
at
from
any
alpha
was
particles
given
emission
to
were
angle
and
particles
see
if
meant
there
deected
agreed
with
absorption
was
that
any
through
an
inverse
spectra.
The
neutrons.
positive nucleus
vacuum
1 in 8000
par ticles
gold foil target
‘rebound’
gold
screens
8
about 10
m thick
from the foil.
atom
beam of
most pass
α-particles
source of
straight
θ
α-particles
stream of
through
α-particles
positive
α-particle
detector
about 1 in 8000
some are deviated
is repelled back
through a large
deected by
nucleus
NB not to scale. Only a minute percentage
angle θ
of α-particles are scattered or rebound.
Rutherford–Geiger–Marsden
experiment
Atomic
explanation
of
Rutherford–Geiger–Marsden
experiment
A t o m i c ,
n u c l e A r
A n d
p A r t i c l e
p h y s i c s
77
de      e
cl a ssific ation of particlEs
consErvation l aWs
Particle
Not
many,
accelerator
many
classes
of
leptons
whereas
were
of
of
the
These
particles.
were
‘light’)
Protons
and
the
into
mesons
are
in
between
the
the
the
(=
hadrons
(linear
no
baryons.
of
Another
are
out
called
elementary
structure,
of
smaller
elementary
mediation
of
the
be
baryons
that
is,
reactions
if
they
are
have
of
not
are
The
quarks,
not
two
strong
boson,
Combinations
particles.
always,
after
a
a
On
were
collision
to
of
these
broken
a
number
the
reactions
–
1)
that
conrmation
charge,
the
‘baryon
the
laws
law
of
the
same.
A
laws
there
conservation
number’
then
of
momentum
fundamental
e.g.
of
always
of
simply
physicists
assigned
was
study
conservation
were
top
never
baryon
number
suggested
of
1
total
(and
number
similar
law
of
applies.
new
such
is
also
an
of
Another
elementry
elementary
composite
particles.
properties.
composed
of
particles,
and
different
combinations
there
are
three
particle
properties
of
inside
all
quarks.
mesons
of
to
one
whether
their
that
always
is
in
conserved
weak
in
all
‘charm’
were
are
often,
examples
electromagnetic
and
interactions.
they
and
are
the
elementary
various
or
composite,
quantum
numbers
can
that
be
specied
are
related
in
to
laws
that
have
been
discovered.
The
quantum
numbers
the
that
particles
are
include:
charge,
property
strangeness,
is
not
the
charm,
same
as
an
lepton
object’s
number,
actual
baryon
colour
–
number
see
page
and
colour
79).
Inside
(or
there
mass
identify
electric
particle
has
its
own
antiparticle.
An
antiparticle
has
the
same
mass
as
three
is
particle
but
all
its
quantum
numbers
(including
charge,
etc.)
are
opposite.
one
There
and
not
and
are
its
antiquarks);
Strangeness
but
‘Strangeness’
are
hadrons
quarks
reactions.
particle,
Every
baryons
in
particle.
particles
All
conserved
interactions,
(this
quark
and
were
baryons
laws
known
mass-energy.
assigned
lepton
these
already
that
The
classes
•
all
all
of
possible.
experimental
leptons
used
called
were
are
some
or
they
constituents.
particles
exchange
Higgs
of
to
Some
rules
If
were
before
particles
rise
and
other
number.
conservation
the
that
angular)
to
conservation
terms
and
and
gave
physics.
laws
antibaryons
All
of
particle
baryon
all
between
place
bosons’.
internal
made
to
appeared
but
Particles
take
conservation
Other
‘exchange
reactions
did
applied
particles.
bosons
all
that
hadrons
and
involved
gauge
are
baryons.
is
called
–
The
interactions
were
original
hadrons
leptons.
neutrons
particles
Two
neutrons
are
identify
identied
and
and
electrons
subdivided
Protons
class
particles
(=
‘heavy’).
‘new’
experiments
are
some
particles
(e.g.
the
photon)
that
are
their
own
antiparticle.
antiquark.
thE standard modEl – lEptons
There
are
six
different
leptons
and
six
different
antileptons.
The
Electric
six
leptons
are
considered
to
be
in
three
different
generations
‘Generation’
or
charge
1
families
three
The
in
exactly
different
electron
the
same
generations
and
the
way
of
electron
that
quarks
there
(see
neutrino
are
considered
page
have
to
ν
79).
a
number
Similar
1
to
a
muon
family
example,
The
lepton
principles
the
Lepton
For
have
+1.
are
and
antielectron
(electron
used
the
number
whenever
to
tau
is
a
family)
assign
family
also
and
lepton
is
antimuon
number
of
neutrino
leptons
in
must
the
also
muon
of
1.
neutrino)
neutrino)
M
M
M
almost
numbers
of
+1
or
=
0
are
only
four
in
created,
be
all
an
created
family
almost
0
=
0
or
=
0
or
almost
0
0
e
μ
(electron)
(muon)
(tau)
M
M
M
τ
1
reactions.
antimuon
=
0.511
or
is
so
that
always
the
interactions
=
105
2
=
1784
2
c
MeV
MeV
c
total
that
exist:
Gravity,
for
greater
which
the
it
mass
can
of
exist.
the
The
exchange
range
of
particle,
the
weak
the
smaller
interaction
+
Electromagnetic,
All
four
interactions
exchange
particle
the
Strong
of
or
can
particles.
particles.
smaller
the
and
be
Each
The
range
of
as
Weak.
thought
of
as
interaction
bigger
the
the
force
being
has
mass
its
of
mediated
own
the
MeV
2
c
conserved.
The
fundamental
or
members.
E xchangE particlEs
There
τ
(tau-neutrino)
MeV
an
ν
μ
(muon-
antielectron
number
conserved
muon
the
ν
(electron(electron
notpeL
neutrino
of
3
e
lepton
0
family)
2
be
by
an
exchange
exchange
boson,
In
the
masses
particle
disappear
virtual
by
out
is
between
its
exchange
physics,
surrounded
particles
concerned.
of
a
of
all
the
inversely
two
real
cloud
in
Interaction
particles
virtual
proportional
takes
one
place
cloud
is
(W
can
W
be
their
when
Z
thought
The
by
)
are
of
as
appear
or
The
being
of
these
interaction
more
the
large.
and
lifetime
mass.
one
absorbed
and
that
vacuum.
to
of
other
the
particle.
Relative
Range
Exchange
Particles
strength
(m)
particle
experience
8
Quarks,
different
15
Strong
1
~10
gluons
gluons
photon
Charged
2
The
exchange
results
in
repulsion
between
the
two
Electromagnetic
10
Weak
10
innite
particles
+
W
13
From
the
needed
as
the
than
to
of
create
energy
is
78
point
of
view
these
the
proscribed
of
quantum
virtual
particle
by
the
A t o m i c ,
mechanics,
particles,
does
not
uncertainty
∆E
exist
is
for
a
longer
(see
A n d
0
,
W
,
Z
so
lepton
long
time
page
Quarks,
18
~10
energy
available
principle
n u c l e A r
the
39
Gravity
10
innite
graviton
All
∆t
126).
p A r t i c l e
Leptons
and
p h y s i c s
time
small
0
,
particles
surrounding
particles
particles
of
particles
the
is
bosons
are
unaffected
by
the
strong
force.
Q
Isolated
standard modEl – Quarks
The
standard
that
six
all
model
matter
types
of
accepted
is
particle
considered
quark
theory.
fundamental,
of
and
six
Each
which
of
to
be
types
these
means
physics
the
composed
of
lepton.
particles
they
is
do
not
of
This
is
theory
is
the
any
threes.
says
combinations
to
quarks
three
Gravity
is
not
explained
by
the
are
exist.
made
whereas
quarks
all
hadrons
are
made
up
particles
from
called
different
quarks.
standard
are
quark
and
six
types
of
antiquark.
This
very
all
up
of
in
twos
quark
a
and
or
an
combination
of
antiquarks).
be
Name
of
particle
proton
(p)
Quark
u
u
d
u
d
d
six
u
d
s
u
u
d
(pi-minus)
d
u
(pi-plus)
u
d
(K
d
s
(n)
structure
of
different
Λ
types
antiproton
of
made
or
only
(a
model.
combinations
There
are
exist
quarks
deeper
lambda
fundamental
can
two
quarks
neutron
All
They
from
baryons
(either
Baryons
structure.
cannot
Mesons
antiquark)
of
currently
considered
have
that
neatly
matches
(p)
the
-
Mesons
six
leptons
that
strong
force
are
also
known
to
exist.
Quarks
are
affected
π
by
+
π
the
(see
below),
whereas
leptons
are
not.
The
weak
0
K
)
zero
interaction
Electric
can
change
one
type
of
quark
into
another.
The
‘Generation’
the
charge
1
2
force
full
between
description
u
c
t
(up)
(charm)
(top)
the
2
e
3
2
=
5
MeV
2
c
this
is
still
the
strong
interaction
M
=
1500
MeV
chromodynamics.
quarks
green
M
of
is
interaction
termed
QCD
but
theory
–
3
quantum
+
quarks
(g)
is
or
a
property
blue
(b).
The
called
quantum
colour.
Antiquarks
can
All
be
difference
quarks
antired
between
can
( r),
be
red
(r),
antigreen
( g)
2
c
M = 174
MeV
c
or
d
s
b
(down)
(strange)
(bottom)
antiblue
because
( b).
they
The
have
two
up
quarks
different
in
a
proton
are
not
identical
colours.
1
-
e
Only
3
skrauQ
2
M
=
10
MeV
2
c
M
=
200
MeV
white
(colour
neutral)
combinations
are
possible.
2
c
M = 4700
MeV
c
Baryons
must
contain
r,
g
and
b
quarks
(or
r,
g,
b)
whereas
1
All
quarks
have
a
baryon
number
of
+
mesons
contain
a
colour
and
the
anticolour
(e.g.
r
and
r
or
b
3
and
1
All
antiquarks
All
quarks
have
a
baryon
number
of
a
strangeness
number
of
0
colour
except
quark
The
all
c
that
quark
other
has
is
a
strangeness
the
quarks
only
have
quark
charm
number
with
a
of
of
interaction
colour
between
objects
with
interaction
and
is
chromodynamics.
The
force-carrying
There
are
gluon
carries
eight
different
explained
types
of
colour
by
a
is
number
gluon
each
force.
details
=
quarks
is
sometimes
Eight
different
types
of
gluon
of
QCD
do
not
need
to
be
combination
of
colour
and
the
mediate
it.
recalled.
called
is
the
zero
rather
gluon.
mass.
Each
Isolated
sufcient
isolate
quantum
called
with
If
the
The
a
energy
quark,
than
six
quarks
is
then
isolated
and
supplied
more
gluons
a
This
is
gluons
cannot
hadron
hadrons
quarks.
colour-changing
to
or
in
known
anticolour
and
are:
G
be
order
mesons
as
will
, G
r g
observed.
to
be
quark
, G
r b
a
called
+1,
increases.
called
theory
particle
between
0.
Quantum chromodynamics (Qcd)
The
force
1.
charm
number
The
the
The
s
etc.)
-
3
have
b,
attempt
to
produced
connement.
, G
b g
, G
br
, G
g b
gr
their
For
example
when
a
blue
up
quark
emits
the
gluon G
it
loses
br
emission
and
absorption
by
different
quarks
causes
the
colour
force.
its
As
the
gluons
interaction
The
themselves
between
overall
effect
is
are
gluons
that
coloured,
themselves
they
bind
there
as
will
well
quarks
as
be
a
colour
between
together.
The
blue
colour
antired,
quarks.
so
absorbing
and
red
this
becomes
colour
gluon
must
will
a
red
be
up
left
become
quark
behind).
a
blue
(the
A
gluon
red
down
contains
down
force
There
are
two
additional
colour-neutral
gluons:
G
and
0
between
quarks
increases
as
the
separation
between
b
g
G
,
0
quarks
making
r
quark
quark.
g
b
a
total
of
r
eight
gluons.
b
g
r
u
u
G
G
+
π
G
G
rb
G
gb
G
gb
rb
G
0
bg
gr
b
p
u
d
r
b
g
g
G
b
G
rg
rb
d
g
strong intEraction
The
colour
the
same
interaction
thing.
fundamental
and
Properly,
force
mesons.
It
is
interaction
is
the
as
the
effect
is
a
proton
of
the
particles
considered
that
and
that
quarks
binds
together
all
between
strong
The
are
is
in
a
the
into
The
in
the
of
can
composite
In
addition
in
the
(such
boson.
which
and
an
This
the
with
nucleons
detection
the
predictions
a
generations
there
additional
Large
of
three
model
was
particles
overall
nucleons.
to
standard
boson
strong
particles
colour-neutral
interaction
exchange
baryons
nucleus.
quarks
essentially
the
residual
colour-neutral
together
the
the
interaction
interaction
gluons.
between
interaction
involve
strong
colour
by
neutron)
mediating
to
the
binds
force
b
higgs boson
the
mediated
interactions
short-range
The
and
r
Hadron
the
in
acquire
particle
for
highly
proposed
can
are
mass.
Higgs
that
of
leptons
four
massive
1964
Collider
that
the
to
In
boson,
explain
2013
of
quarks
gauge
the
the
the
the
Higgs
process
scientists
announced
matched
and
classes
by
working
experimental
standard
model’s
boson.
be
particles
+
(π
mesons:
interaction
π
is
,
π
or
always
π°)
whereas
seen
as
the
the
fundamental
exchange
of
colour
gluons.
A t o m i c ,
n u c l e A r
A n d
p A r t i c l e
p h y s i c s
79
fe 
Some
rulEs for draWing fE ynman diagrams
Feynman
diagrams
interactions.
probability
of
mechanics,
which
an
in
interaction,
in
The
is
be
used
diagrams
are
interaction
order
it
an
can
to
nd
necessary
interaction
represent
used
taking
out
to
can
to
take
calculate
place.
the
add
to
possible
In
overall
together
place.
the
particle
•
overall
in
quantum
Used
the
of
an
possible
properly
and
as
a
tool
simple
for
calculations
pictorial
model
but,
of
at
this
possible
level,
•
Exchange
are
a
the
going
Feynman
from
books
way,
left
reverse
turn
diagrams
to
right
these
the
page
below
and
two
the
axes).
the
x-axis
y-axis
To
anti-clockwise
by
construction
out.
These
will
quark–quark
leptons
are
solid
(vertex)
of
correct
has
an
represent
a
diagrams:
arrow
going
lepton–lepton
transition.
straight
can
particles
are
either
lines.
wavy
or
or
Z°)
or
curly
broken
(photons,
(gluons).
be
Time
ows
from
left
to
right.
Arrows
from
left
to
right
interactions.
represents
represents
view
the
diagram
±
represent
In
a
in
the
ways
they
they
or
help
in
going
or
Quarks
•
seen
one
•
W
mathematical
rules
junction
transition
probability
all
simple
Each
them
in
time
space
the
right
(some
•
The
alternative
the
to
particles
left
labels
travelling
represent
for
the
forward
antiparticles
different
in
time.
travelling
particles
are
Arrows
forward
shown
at
from
in
the
time.
end
of
line.
90°.
•
The
junctions
exchange
will
particle
be
linked
by
a
line
representing
the
involved.
E x amplEs
An
e
electron
emits
a
photon.
An
electron
absorbs
a
photon.
e
e
e
γ
γ
A
positron
emits
a
photon.
A
+
+
e
positron
absorbs
a
photon.
+
e
e
+
e
γ
γ
A
photon
produces
an
electron
and
positron
pair).
a
γ
An
electron
and
a
positron
meet
and
e
e
positron
(an
electron
annihilate
(disappear),
producing
a
γ
+
e
+
photon.
e
u
Beta
decay.
A
down
d
quark
ν

+
d
changes
into
an
up
W
quark
+
π
with
the
emission
of
a
W
W
before
+
particle.
after
This
decays
into
u
an

ν
e
electron
and
an
antineutrino.
Pion
The
top
vertex
decay.
The
quark
and
antiquark
annihilate
to
produce
involves
+
a
quarks,
the
bottom
This
into
an
antimuon
and
a
muon
electron
leptons.
and
positron
annihilate
γ
An
produce
two
up
quark
(in
a
proton)
emits
u
u
to
decays
neutrino.
involves
An
particle.
vertex
e
e
W
photons.
a
+
gluon
which
in
turn
transforms
π
g
into
a
This
reaction
down/antidown
quark
pair.
d
γ
result
of
p
→
a
could
take
place
proton–proton
as
a
collision:
d
+
+
e
Simple
n
diagrams
can
also
p
be
A
p
+
π°
p
p
+
mediates
n
+
the
π
strong
nuclear
p
drawn
with
exchanges
between
force
between
a
proton
and
a
hadrons.
e
neutron
in
a
nucleus.
π°
W
Beta
decay
(hadron
version)
ν
e
n
probability
usEs of fE ynman diagrams
Once
a
possible
diagram,
for
it
certain
is
interaction
possible
vertex
corresponds
all
terms,
the
the
the
More
complicated
80
to
to
a
use
it
been
to
identied
calculate
processes
to
take
mathematical
probability
using
need
to
fundamental
has
of
the
the
with
be
considered
included
Feynman
probabilities
place.
term.
a
By
interaction
Each
line
adding
can
be
In
and
a
together
in
A t o m i c ,
order
to
the
same
calculate
n u c l e A r
overall
the
A n d
outcome
overall
p A r t i c l e
real
momentum
diagram
time
a
chosen
the
Feynman
energy
with
of
in
represent
calculated
diagram.
diagrams
n
diagram,
particles
particles
the
virtual
p h y s i c s
be
The
Lines
or
obey
in
diagrams
accurate
entering
particles
the
leaving
mass,
they
do
exist
to
not
for
a
apply.
that
are
answer.
the
energy
intermediate
and
relationship
detected.
more
more
must
providing
uncertainty
cannot
the
lines
and
relationships.
represent
conservation
for
outcome.
calculation,
diagram
and
stages
have
to
short
Such
in
the
obey
enough
virtual
ib Qe –  , e   e 
1.
A
sample
The
of
radioactive
half-life
of
Ra
226
material
can
be
contains
dened
as
the
the
element
time
it
Ra
takes
(ii)
226.
Explain
a
for
can
A.
the
mass
of
the
sample
to
fall
to
half
its
original
half
the
number
of
atoms
of
Ra
C.
half
the
number
of
atoms
in
the
226
in
the
sample
D.
2.
the
volume
Oxygen-15
of
the
decays
to
sample
to
fall
nitrogen-15
to
half
with
a
to
2
minutes.
A
pure
its
to
original
half-life
sample
of
of
100
masses
g,
of
is
placed
oxygen
in
an
and
airtight
nitrogen
the
but
of
oxygen
Mass
After
container
of
which
is
with
0
g
100
B.
25
g
25
g
C.
50
g
50
g
D.
25
g
75
g
in
4
is
X
undergoes
is
mostly
also
radioactive.
in
the
wood
[2]
a
carbon-12,
small
which
proportion
of
is
carbon-14,
When
at
a
that
tree
time
is
cut
down,
the
with
half-life
decays
a
carbon-14
of
Carbon-14
decays
by
beta-minus
emission
to
nitrogen-14.
be
the
equation
for
this
decay.
[2]
nitrogen
For
an
old
wooden
bowl
from
an
archaeological
site,
the
g
carbon
count-rate
is
13
from
newly
Explain
a
sequence
of
of
counts
beta
per
cut
why
diminishes
nuclide
they
nuclei.
minutes,
will
(i)
radioactive
magnesium
deuterons
years.
rate
A
the
before
dating
trees
there
average
3.
give
a
b)
A.
to
energy
value.
Write
Mass
the
kinetic
of
oxygen-15,
container.
in
with
carbon
carbon
stable,
a)
the
necessary
decay.
5,800
mass
is
decay.
present
approximately
react
Radioactive
The
sample
it
minimum
value.
9.
B.
why
certain
particles
minute.
wood
the
with
beta
is
52
even
per
kg
corresponding
counts
activity
time,
detected
The
per
from
minute.
the
though
of
count
bowl
the
probability
of
radioactive
Z
decay
decays
to
form
a
new
nuclide
Y.
Z
+
could
The
sequence
of
β,
β
α,
B.
α,
β,
In
α
the
D.
Rutherford
constant.
Calculate
α,
β,
scattering
experiment,
a
stream
of
B.
are
is
red
scattered
at
a
thin
gold
foil.
Most
This
question
of
the
α
D.
go
scattered
a
nuclear
of
in
the
time
the
ssion
reactor
for
providing
power
splitting
of
The
is
to
be
of
absorption
the
neutrons
nucleus
and
the
of
generated
by
a
by
into
neutron
the
ssion
two
release
of
U
results
smaller
nuclei
energy.
The
plus
a
splitting
occur
in
activity.
many
ways;
for
example
foil.
radioactive
between
nuclear
uniformly.
+
material
now
has
about
1/16
of
90
U
→
If
the
half-life
measurements
is
is
4
hours
the
143
Sr
+
Xe
38
+
neutrons
+
energy
54
its
a)
previous
a
235
92
piece
the
[3]
about
reactor,
235
A
is
uranium-235.
n
5.
of
power.
of
can
through
age
particles
number
are
approximate
randomly.
rebound.
C.
the
bowl.
α
In
A.
[3]
γ
electrical
particles
nucleus
β
10.
4.
carbon-14
emitted
wooden
C.
individual
be
(ii)
A.
any
2
is
radiations
of
difference
The
nuclear
ssion
reaction
in
(i)
approximately
How
many
neutrons
are
produced
in
this
reaction?
A.
8
[1]
hours.
(ii)
B.
16
hours.
C.
32
hours.
Explain
each
why
the
reaction
is
release
crucial
of
for
several
the
neutrons
operation
of
in
a
ssion
reactor.
D.
60
(iii)
6.
a)
Use
the
standard
fundamental
(i)
A
model
particles,
to
the
describe,
internal
in
terms
structure
of
of:
An
(iii)
Baryons
The
sum
(iv)
electron
(iv)
before
of
Show
nuclear
the
that
the
of
of
masses
the
is
of
uranium
0.22
the
u
this
‘missing
mass’?
released
in
is
power
about
200
plus
greater
ssion
energy
reaction
ssion
masses
reaction
rest
becomes
Draw
Feynman
diagram
for
(i)
β
the
than
products.
[1]
above
MeV
.
[2]
station
Suppose
a
proton
undergoes
a
strong
interaction
with
a
ϕ
content:
ud)
to
produce
a
neutron
and
Use
conservation
produced
in
this
laws
to
deduce
the
of
ssion
structure
of
reactor,
ssion
550
power
MW
.
station
Estimate
generates
the
reactions
occurring
minimum
each
stating
any
assumption
you
second
have
in
made
the
about
particle
at
another
the
particle.
power
particle
number
(quark
nuclear
decay.
electrical
A
the
rest
Mesons
+
7.
the
thesum
ssion
A
of
neutron
What
proton
(ii)
b)
b)
[2]
hours.
efciency.
[4]
reaction.
11.
Which
of
the
following
is
a
correct
list
of
particles
upon
which
238
8.
a)
Two
properties
of
the
isotope
of
uranium,
U
are:
92
the
strong
nuclear
force
may
act?
234
(i)
it
decays
radioactively
(to
Th)
90
(ii)
it
reacts
chemically
(e.g.
with
uorine
to
form
UF
features
responsible
for
of
the
these
structure
two
of
widely
uranium
different
atoms
protons
and
C.
neutrons
neutrons
B.
protons
and
electrons
electrons
D.
protons,
neutrons
).
6
What
A.
and
and
electrons
are
properties?
[2]
2
b)
A
beam
of
deuterons
(deuterium
nuclei,
H)
are
1
accelerated
through
a
potential
difference
and
are
then
26
incident
on
a
magnesium
target
(
Mg).
A
nuclear
reaction
12
occurs
an
(i)
resulting
alpha
the
production
of
a
sodium
nucleus
and
particle.
Write
this
in
a
balanced
nuclear
equation
for
reaction.
[2]
i B
Q u e s t i o n s
–
A t o m i c ,
n u c l e A r
A n d
p A r t i c l e
p h y s i c s
81
8
E N E R G Y
P R O D U C T I O N
Energy nd poer genertion – sney digr
ENERGY CONvERsIONs
The
production
achieved
with
the
thermal
single
into
using
a
release
energy
process,
work
repeating
must
of
variety
of
but
their
be
the
the
ElECTRIC al POwER PRODUCTION
power
energy
continuous
use
in
of
a
transfer
around
different
completely
actions
the
of
thermal
can
implies
involve
electrical
systems,
from
fuel.
that
In
starting
this
in
electrical
A
energy
a
to
energy
fuel
is
turn
is
used
power
used
to
turbines
generate
continuously
the
all
same.
principle,
cyclical
from
is
work
of
are
Any
energy
In
to
conversion
cycle.
some
a
world
often
converted
machines
xed
of
the
boil
and
electrical
difference
(see
stations
to
water
the
to
process
thermal
make
motion
energy.
page
the
release
of
is
steam.
the
essentially
energy.
This
The
turbines
Transformers
alter
steam
is
the
the
thermal
used
is
used
to
potential
114).
process
system
useful
to
the
surroundings
that
is
no
longer
available
to
perform
electrical output
useful
work.
energy,
in
This
accordance
thermodynamics
Energy
An
unavailable
(see
conversions
arrow
(drawn
with
page
are
energy
the
left
known
principle
of
as
the
degraded
second
law
of
162).
energy in from fuel
represented
from
is
to
right)
using
Sankey
represents
diagrams.
the
energy
heating and
changes
taking
place.
The
width
of
the
arrow
represents
the
sound in
power
or
energy
involved
at
a
given
stage.
Created
or
degraded
transformers
energy
is
shown
with
an
arrow
up
or
down.
friction and heating
losses
Note
that
Sankey
diagrams
are
to
scale.
The
width
of
the
is
2.0
cooling tower
useful
electrical
output
in
the
diagram
on
the
right
mm
losses (condenser)
compared
with
12.0
mm
for
the
width
of
the
total
energy
from
Sankey
the
fuel.
This
represents
an
overall
efciency
of
diagram
representing
the
energy
ow
in
a
typical
16.7%.
power
station
steam
fuel (coal)
turbine
to transformer
generator
boiler
water
condenser
Electrical
energy
generation
POwER
1
Power
is
dened
as
the
rate
at
which
energy
is
energy
_
Power
=
time
82
E N E R G Y
P R O D U C T I O N
converted.
The
units
of
power
are
J
S
or
W
.
Priry energ y ourc e
On
RENE wablE / NON-RENE wablE ENERGY sOURCEs
The
law
created
of
nor
societies
the
conservation
destroyed,
are
input
it
concerned,
of
energy,
Renewable
whereas
of
of
just
if
we
sources
run
Renewable
states
changes
we
need
wish
to
energy
non-renewable
eventually
energy
that
form.
to
use
identify
are
sources
As
far
devices
sources
those
of
energy
that
energy
as
of
•
require
be
up
up,
is
and
Of
sources
solar
for
a
as
a
these
fuel
supply
of
It
they
are
possible
in
the
managed
example,
to
burn
however,
cut
renewable
somewhere
be
For
wood
is,
available
can
make
the
source
(fusion).
to
way.
source
be
the
source
rst
then
If
of
this
renewable
are
this
to
is
is
or
cut
clearly
replant
properly
trees
at
the
managed,
energy.
must
place.
a
trees
possible
down.
sources
in
if
have
Most
got
of
their
the
energy
energy
used
by
coal
cells
can
be
traced
back
to
energy
radiated
from
the
Sun,
oil
but
active
a
rate
could
humans
photovoltaic
as
course
from
hydroelectric
possible
same
it
Non-renewable
hand,
renewable
non-renewable.
out.
sources
other
non-renewable
down
used
used
It
a
energy.
be
the
effectively
neither
human
that
cannot
can
is
heaters
natural
wind
not
quite
all
of
it.
Possible
sources
are:
gas
•
the
•
gravitational
Sun’s
radiated
•
nuclear
•
the
energy
nuclear
energy
of
the
Sun
and
the
Moon
biofuels
Sometimes
taken
the
when
sources
deciding
are
hard
whether
to
a
classify
source
is
so
care
needs
renewable
or
to
be
point
that
sometimes
worries
students
is
that
the
no
eventually
source
sources
is
are
run
out
perfectly
as
a
source
renewable!
considered
from
the
of
energy
This
point
is
of
for
true,
view
the
but
of
Earth,
all
life
the
keep
Sun
in
runs
mind
out,
then
so
will
life
on
Earth.
Nuclear
as
their
might
of
above
energy
(such
think
list
is
that
there
complete.
are
Many
other
sources
everyday
of
sources
energy
from
as
coal
or
oil)
can
be
shown
to
have
derived
of
their
these
on
the
Sun’s
radiated
energy.
On
the
industrial
scale,
Earth.
Other
energy
needs
to
be
generated
from
another
source.
things
you
plug
anything
electrical
into
the
mains
electricity
you
include:
have
•
atoms
energy.
so
When
to
you
the
electrical
When
within
heat
Sun
energy,
will
stored
internal
not.
Although
One
energy
Earth’s
sources
source
(both
so
ssion
they
must
and
be
fusion)
consume
a
material
sPECIfIC ENERGY aND ENERGY
DENsITY
you
pay
use.
must
non-renewable.
to
be
In
the
electricity-generating
order
using
to
one
provide
(or
you
more)
of
company
with
the
this
for
energy,
original
list
of
the
the
energy
company
sources.
COmPaRIsON Of ENERGY sOURCEs
Fuel
Renewable?
CO
Specic
Energy
2
1
Two
quantities
are
useful
to
emission
consider
energy(MJ
kg
)
density
3
when
making
different
energy
and
Specic
the
of
fuel
sources
the
energy
comparison
as
comparisons
energy
energy
provides
between
energy
–
per
Specic
vary
depending
specic
(MJ
m
)
on
type)
density
a
fuels
liberated
consumed.
(values
between
the
useful
and
is
unit
Coal
No
Yes
22–33
23,000
Oil
No
Yes
42
36,500
Gas
No
Yes
Nuclear
No
No
Waste
No
Yes
10
variable
Solar
Yes
No
n/a
n/a
Wind
Yes
No
n/a
n/a
Yes
No
n/a
n/a
dened
mass
energy
is
54
37
1
measured
in
J
kg
7
8.3
×
10
12
1.5
×
10
(uranium)
specic
energy
energy
released
from
fuel
___
=
mass
Fuel
choice
inuenced
the
fuel
greater
of
can
by
needs
the
fuel
be
consumed
particularly
specic
to
mass
be
of
energy
transported:
fuel
that
Hydro
when
the
needs
stored
–
water
in
dams
to
Tidal
be
transported,
the
greater
the
Yes
No
n/a
n/a
n/a
No
n/a
n/a
Wave
Yes
No
n/a
n/a
Geothermal
Yes
No
n/a
n/a
Yes
30
21,000
cost.
Pumped
Energy
density
liberated
per
is
unit
dened
volume
as
of
the
energy
fuel
3
consumed.
The
unit
is
J
storage
m
Biofuels
energy
e.g.
Some
density
types
ethanol
energy
release
from
fuel
___
=
volume
of
fuel
consumed
E N E R G Y
P R O D U C T I O N
83
foi ue poer production
As
ORIGIN Of fOssIl fUEl
Coal,
fuels
or
oil
and
have
beneath
Coal
is
of
This
exposure
in
been
hundreds
matter.
natural
to
the
swamps.
very
been
high
a
years
as
fossil
timescale
from
fuels.
that
involves
accumulations
converted
into
temperatures
fossil
and
of
fuels
pressure
the
tens
Oil
by
upon
plant
layer
of
matter
decaying
that
used
matter
to
buried
exist
is
this
formed
in
a
gas,
be
obtained
possible
to
as
as
into
life.
well
a
plant
more
similar
sea.
Natural
more
turned
marine
also
grow
by
became
microscopic
can
dead
was
material
timescale
dead
that
it
the
These
surface.
from
Layer
known
over
of
has
Earth’s
formed
are
produced
millions
matter
the
gas
matter
other
Over
substances,
the
geological
coal.
manner
The
as
and
compressed.
from
the
compression
occurring
by-product
manufacture
in
during
gas
from
remains
took
of
place
under
underground
the
the
pockets,
production
of
oil.
It
is
coal.
decomposed.
ENERGY TRaNsfORmaTIONs
Fossil
fuel
power
once
again
Sun.
For
Some
of
can
stations
be
used
example,
this
release
to
turn
millions
matter
has
of
energy
in
turbines.
years
eventually
ago
fuel
Since
energy
been
by
all
burning
fossil
radiated
converted
it.
fuels
into
The
were
from
the
thermal
Sun
was
storage
in
fossil
the
the
supplied
on
rate
to
used
converted
(by
the
to
convert
original
water
source
photosynthesis)
of
into
into
this
steam
that
energy
living
plant
was
the
matter.
chemical
compression
energy in
in plants
fossil fuels
fuels
EffICIENCY Of fOssIl fUEl POwER sTaTIONs
data
typical
then
matter,
energy
E x amPlE
Use
is
living
coal.
chemical
photosynthesis
solar energy
Energy
energy
originally
a
this
(in
500
page
and
tonnes
MW
per
coal
the
previous
hour)
red
at
page
which
power
to
coal
calculate
must
be
The
efciency
design.
At
of
the
different
time
of
power
stations
publishing,
the
depends
following
on
the
gures
apply.
station.
Fossil
fuel
Typical
efciency
Current
Answer
maximum
8
Electrical
power
supply
=
500
=
5
MW
=
5
×
1
10
J
s
efciency
8
Power
released
from
fuel
×
10
/
efciency
Coal
35%
42%
8
=
5
=
1.43
×
10
/
0.35
9
×
Natural
1
10
J
Rate
of
consumption
of
coal
=
1.43
×
=
43.3
kg
=
43.3
×
60
=
1.56
×
10
7
10
/
gas
45%
52%
38%
45%
s
9
3.3
×
10
Oil
1
kg
s
1
s
Note
that
thermodynamic
considerations
limit
the
maximum
1
×
60
kg
5
hr
achievable
efciency
(see
page
163).
1
kg
hr
1
≈
160
tonnes
hr
aDvaNTaGEs aND DIs aDvaNTaGEs
Advantages
•
Very
of
high
energy
•
Fossil
•
Still
•
Power
Can
84
is
fuels
cheap
links
•
energy’
released
are
and
a
be
‘energy
small
easy
compared
can
water
used
from
relatively
when
stations
and
be
Disadvantages
‘specic
to
built
to
mass
density’
of
fossil
–
a
great
deal
transport.
other
sources
anywhere
with
of
•
fuel.
energy.
good
transport
Combustion
acid
products
can
produce
products
contain
pollution,
notably
rain.
•
Combustion
•
Extraction
•
Non-renewable.
•
Coal-red
of
fossil
fuels
can
‘greenhouse’
damage
the
gases.
environment.
availability.
directly
in
E N E R G Y
the
home
to
provide
P R O D U C T I O N
heating.
power
stations
need
large
amounts
of
fuel.
Nucer poer – proc e
The
PRINCIPlEs Of ENERGY PRODUCTION
Many
This
a
nuclear
fuel
ssion
page
a
moving
Among
to
not
In
each
fast.
In
An
the
nucleus
other
further
are
uranium-235
release
of
of
this
reaction,
to
words
fragments
initiate
–
use
overview
individual
uranium
the
stations
burned
reaction.
76.
causes
on
is
power
split
the
more
reactions
‘fuel’.
energy
is
achieved
is
described
an
incoming
The
If
chain
is
using
on
very
these
of
go
created.
of
run
of
of
that
the
number
the
speed
a
general
the
is
of
surface
reached
so-called
the
the
There
likely
are
the
likely.
have
given
of
of
the
when
mass
be
ensure
reaction
took
time
and
reactions
that,
goes
place
decreasing
as
the
chain
of
then
the
to
the
chain
took
and
on
on
reaction
place,
the
initiate
number
then
ssion
the
process
process
they
of
a
a
has
the
is
can
to
cause
a
important
the
ssion
ones
are:
way’
neutrons.
block
fuel
reaction
on
‘in
the
further
the
neutron
nuclear
slowed
of
on
Two
of
fuel
exact
(before
assembled
occur.
been
increases
reaction
This
of
is
together
happens
assembled.
nature
so
it
The
the
do
a
from
stage
when
exact
fuel
the
lost
a
value
being
of
used
assembly
.
particular
Before
size
fuel
depends
the
nuclei
causing
As
goes
factors.
energy)
block).
a
neutron
potential
mass
of
cause
be
the
fewer
several
the
trend,
ssion
to
would
on
neutron
shape
to
a
(or
critical
critical
and
by
a
all
If
to
each
reactions
increase
control.
depends
•
chances
more
needs
from
stop.
•
As
If
reactions
chance
reactor
neutron
would
out
soon
reaction
nuclear
one
reaction.
reactions
The
a
only
further
would
high.
is
a
number
are
neutrons
reaction
design
average,
would
neutron
fragments
temperature
neutrons.
a
the
process
apart.
then
as
are
can
energies
ssion.
In
moving
cause
that
general,
too
further
fast
make
the
to
them
make
reactions
more
neutrons
the
created
reactions
neutrons
down.
mODERaTOR, CONTROl RODs aND hE aT
distributors
E xChaNGER
to electricity
kinetic energy
Three
important
components
in
the
design
of
all
consumers
nuclear
rods
reactors
and
the
are
heat
the
moderator,
the
control
exchanger.
nuclear
thermal
energy
energy
electrical
energy
thermal
•
Collisions
between
the
neutrons
and
the
nuclei
of
energy
the
moderator
slow
them
down
and
allow
further
losses
reactions
•
The
to
take
control
place.
rods
are
movable
rods
that
readily
to environment
absorb
neutrons.
removed
control
•
The
from
the
heat
to
occur
of
the
They
the
chain
a
be
reaction
introduced
chamber
in
or
order
to
concrete shields
reaction.
exchanger
in
can
place
allows
that
is
the
nuclear
sealed
off
reactions
from
the
rest
control rods
environment.
The
reactions
increase
the
(moveable)
temperature
in
the
core.
This
thermal
energy
is
moderator
steam to
pressurizer
transferred
to
heat
water
and
the
steam
that
is
turbines
produced
A
general
(PWR
It
uses
or
turns
design
the
for
pressurized
water
as
the
turbines.
one
type
water
of
nuclear
reactor)
moderator
and
is
as
reactor
shown
a
here.
HOT
coolant.
secondary
WATER
coolant
circuit
aDvaNTaGEs aND DIs aDvaNTaGEs
Advantages
•
Extremely
high
‘specic
energy’
–
a
great
deal
pump
heat
of
energy
is
released
of
uranium.
from
a
very
small
mass
exchange
steel pressure vessel
•
Reserves
of
uranium
large
compared
to
fuel rods
oil.
Disadvantages
•
Process
primary
produces
radioactive
nuclear
waste
pump
that
is
currently
just
coolant
stored.
circuit
•
Larger
possible
risk
if
anything
should
go
wrong.
Pressurized
•
Non-renewable
(but
should
last
a
long
water
nuclear
reactor
(PWR)
time).
E N E R G Y
P R O D U C T I O N
85
Nucer poer –  ety nd ri
ENRIChmENT aND REPROCEssING
Naturally
occurring
uranium-235.
percentage
more
uranium
Enrichment
composition
is
contains
is
the
less
process
increased
to
NUClE aR wE aPONs
than
by
1%
which
make
of
A
this
nuclear
nuclear
whereas
ssion
likely.
amount
been
of
addition
to
sustaining
by-product
nucleus
uranium-235,
ssion
of
can
a
which
conventional
capture
uranium-239.
This
undergoes
238
1
U
+
nuclide
nuclear
fast-moving
undergoes
further
neutrons
to
to
also
to
A
energy
as
a
•
associated
Moral
uranium-238
is
form
issues
associated
destructive
neptunium-239
threat
plutonium-239:
of
prevent
•
Np
+
β
93
239
239
with
and
weapons
with
warfare.
capability
that
any
huge
Weapons
plutonium
as
the
have
fuel.
include:
weapon
Nuclear
since
deployment
non-nuclear
has
the
Second
been
aggressive
of
aggression
weapons
acts
used
have
World
as
against
a
that
such
War
the
deterrent
the
to
possessors
capability.
+
unimaginable
consequences
of
a
nuclear
war
have
υ
many
+
countries
to
agree
to
non-proliferation
β
+
υ
1
treaties,
Reprocessing
involves
treating
used
fuel
waste
from
recover
uranium
and
plutonium
and
to
deal
A
fast
breeder
reactor
is
one
a
small
attempt
number
of
to
limit
nuclear
power
technologies
nations.
with
•
products.
which
nuclear
to
design
A
by-product
of
the
peaceful
use
of
uranium
for
energy
that
production
utilizes
weapons.
ssion
the
0
Pu
94
to
nuclear
nuclear
produces
1
→
93
waste
with
nuclear
uranium
associated
their
nuclear
The
forced
other
both
in
ssion
U
0
→
92
reactors
released
using
controlled
nuclear
92
239
Np
involves
capable
formed
reactor.
β-decay
β-decay
is
is
of
→
0
U
This
239
n
92
239
plutonium-239
reactions.
of
station
uncontrolled
designed
Issues
In
power
an
is
the
creation
of
plutonium-239
which
plutonium-239.
could
it
be
right
have
used
for
for
the
nuclear
the
small
production
number
capability
to
of
of
nuclear
countries
prevent
other
weapons.
that
Is
already
countries
from
hE alTh, s afETY aND RIsk
acquiring
Issues
associated
generation
•
If
the
of
with
control
rapidly
the
electrical
rods
increase
use
were
its
of
energy
rate
all
of
nuclear
power
stations
nuclear
for
removed,
the
production.
reaction
would
fUsION RE aCTORs
Completely
ssion
would
cause
an
explosion
meltdown
of
the
core.
The
radioactive
the
reactor
could
be
distributed
around
the
causing
many
fatalities.
Some
argue
that
current
the
in
of
such
a
disaster
means
that
the
use
of
is
be
a
risk
not
targets
worth
for
taking.
terrorist
Nuclear
The
reaction
produces
not
much
of
this
radioactive
waste
is
power
decay
within
page
76)
nuclear
ionize
material
of
a
low
level
risk
and
is
produced
for
millions
decades,
which
of
•
The
this
waste
uranium
mining
the
The
will
years.
is
to
mined
involves
extra
a
signicant
remain
The
the
and
of
of
in
from
the
are
is
waste
and
The
necessary
from
from
plant
risk.
ore
to
any
is
also
protect
the
needs
the
mine
nuclear
to
be
to
a
power
secure
safe.
used
to
of
the
produce
civilian
nuclear
E N E R G Y
and
the
reaction
the
signicant
same
as
The
(if
amounts
takes
and
requires
place
creating
use
of
nuclear
hydrogen
into
a
state
power
weapons.
P R O D U C T I O N
of
matter’,
in
which
in
atoms
but
move
fuel
associated
used,
it
could
of
in
be
hydrogen,
sustained)
radioactive
the
Sun
temperatures
plasma
state
(as
high
(this
electrons
and
independently).
design
challenges
are
associated
conning
can
be
the
plasma
at
sufciently
to
sites.
mines.
uranium
reprocessing
reactors.
waste.
outlined
is
enough
the
protons
are
Currently
the
amount
solution
underground
signicant
uranium
the
ssion
with
maintaining
dangerously
current
secure
precautions
involved
station
By-products
86
geologically
transportation
station
•
so
workers
power
and
fuel
operation
radioactive
•
in
is
atomic
bound
density
bury
signicant
will
and
radioactive
of
problems
waste.
principal
of
potential
the
stations
not
radioactively
supply
produce
reaction
‘fourth
While
of
attacks.
to
•
many
terrible
on
could
theoretical
nuclear
The
energy
nuclear
plentiful
would
scale
the
without
surrounding
is
area
offer
generation
material
with
in
reactors
and
power
thermal
knowledge?
include:
Fusion
uncontrolled
that
for
fusion
to
take
place.
high
temperature
and
s or poer nd ydroeectric poer
sOl aR POwER (TwO TYPEs)
There
are
arrives
two
at
the
ways
of
Earth’s
harnessing
surface
from
solar radiation
the
radiated
the
energy
that
glass/plastic
Sun.
cover
A
photovoltaic
photocell)
into
a
cell
converts
potential
a
difference
semiconductor
to
cell
very
produces
much
not
a
current.
require
a
(otherwise
portion
do
are
deal
known
the
as
It
uses
Unfortunately,
voltage
used
of
to
and
run
energy.
a
radiated
(‘voltage’).
this.
small
They
great
of
it
a
solar
a
of
not
electrical
Using
or
directly
piece
typical
is
cell
energy
able
to
in
water out
provide
devices
them
warmer
photovoltaic
that
series
do
would
cold water in
generate
higher
higher
voltages
and
several
in
parallel
can
provide
a
current.
solar radiation
reective insulator
slices of
copper pipe (black)
behind pipe
semiconductor
active solar heater
thermal energy
solar energy
in water
metal layer
aDvaNTaGEs aND DIs aDvaNTaGEs
Advantage
•
Very
‘clean’
•
Renewable
•
Source
production
source
of
–
no
harmful
chemical
by-products.
energy.
photocell
of
energy
is
free.
Diadvantage
An
active
designed
hot
and
solar
to
water
capture
that
would
heater
it
save
as
(otherwise
much
typically
on
the
thermal
produces
use
known
of
as
energy
can
electrical
be
a
as
solar
panel)
possible.
used
source
of
energy
in
a
Can
only
•
Source
•
A
be
utilized
during
the
day.
of
energy
is
unreliable
–
could
be
a
cloudy
day.
The
domestically
energy.
of
hYDROElECTRIC POwER
The
•
is
very
large
area
would
be
needed
for
a
signicant
amount
energy.
aDvaNTaGEs aND DIs aDvaNTaGEs
hydroelectric
power
station
is
the
Advantage
gravitational
potential
energy
of
water.
If
water
is
allowed
to
move
•
downhill,
the
owing
water
can
be
used
to
generate
electrical
Very
‘clean’
production
–
no
harmful
chemical
energy
.
by-products.
The
•
water
As
can
part
stored
of
in
gain
the
its
gravitational
‘water
large
cycle’,
reservoirs
as
potential
water
high
can
up
energy
fall
as
is
as
in
several
rain.
It
ways.
can
•
Renewable
•
Source
source
of
energy.
be
of
energy
is
free.
feasible.
Diadvantage
•
Tidal
power
during
•
Water
a
low
can
reservoir.
be
more
back
of
pumped
Although
than
few
trap
water
at
high
tides
and
release
it
•
Can
•
Construction
only
be
utilized
in
particular
areas.
tide.
be
down
the
schemes
the
hill,
from
the
low
energy
energy
this
a
used
regained
‘pumped
large-scale
reservoir
methods
to
do
when
storage’
of
to
this
the
a
pumping
water
system
storing
high
under
of
dams
will
involve
land
being
submerged
water.
must
ows
provides
one
energy.
energy lost due to friction throughout
gravitational
KE of water
+
PE of water
E N E R G Y
P R O D U C T I O N
87
wind poer nd oter tecnoogie
ENERGY TRaNsfORmaTIONs
There
is
winds
of
great
that
this
the
a
deal
blow
energy
temperature
due
to
hot
as
a
air
of
the
course,
are
The
ows
kinetic
around
is,
atmosphere
of
Earth.
the
heated
to
differences
rising
or
cold
maThEmaTICs
energy
involved
The
Sun.
cause
air
original
Different
different
in
density of air ρ
the
source
parts
of
r
temperatures.
pressure
sinking,
differences,
and
thus
wind speed ν
air
result.
blades turn
wind
2
The
In
So
area
one
‘swept
second
mass
of
air
out’
the
by
the
volume
that
passes
blades
of
air
the
of
the
that
turbine
passes
turbine
in
the
one
energy
m
available
per
second
A
=
πr
turbine
second
1
Kinetic
=
=
=
v A
v Aρ
2
=
mv
2
1
2
=
(vAρ)v
2
heating
1
Ear th
3
=
Aρv
2
1
In
other
words,
power
available
3
=
Aρv
2
In
practice,
calculate,
the
but
kinetic
it
energy
cannot
all
be
of
the
incoming
harnessed
as
the
wind
air
is
easy
must
to
continue
to
energy lost
move
in
other
words
the
wind
turbine
cannot
be
one
hundred
per
KE of turbine
electric energy
due to friction
–
cent
efcient.
A
doubling
of
the
wind
speed
would
mean
that
the
throughout
available
power
would
increase
•
Diadvantage
aDvaNTaGEs aND DIsaDvaNTaGEs
•
Source
of
energy
is
unreliable
–
by
a
factor
Some
to
could
of
consider
spoil
the
eight.
large
wind
generators
countryside.
Advantage
be
•
Very
‘clean’
chemical
production
–
no
a
•
by-products.
A
very
for
•
Renewable
•
Source
of
source
energy
of
is
day
without
a
large
area
signicant
are
far
the
the
three
inclusion
for
90%
include
With
most
amount
of
of
the
is
reduce
that
the
energy
primary
fuels:
at
world’s
the
solar,
demand
developments
dependence
on
coal
of
sources
and
natural
writing
this
consumption.
wind,
tidal,
expected
with
guide,
biomass
to
rise
use
gas.
Other
renewable
fossil
in
in
worldwide
With
this
accounts
primary
and
the
energy
the
fuels
geothermal.
future,
can
the
help
to
The
energy
typically
a
The
sources
are
not
conversion
storage
convenient
for
individual
impossible
going
to
take
however,
fuels
to
for
predict
place
predict
many
decrease
88
to
a
takes
over
this
technological
the
coming
continuing
years
be
Best
positions
noisy.
far
from
for
wind
generators
centres
of
widely
used
are
population.
capability
Power
place
that
results
to
storing
in
to
come.
dependency
E N E R G Y
years.
Current
dependence
The
over
hope
time.
when
is
It
on
that
is
the
we
are
models,
will
important
P R O D U C T I O N
that
use
of
or
is
consumer
are
electrical
with
vary
a
having
typical
the
only
energy
electrical
fuels
challenge,
a
with
everyday
of
Currently
viable
in
for
(a
petrol).
everyday
limited
demands.
electrical
pumped
large-scale
capacity
society.
energy
(e.g.
very
generation
demand.
the
are
rened
capacitors)
to
be
future
storage
method
of
a
typical
system
is
approximately
75%
use.
The
meaning
one
those
of
quarter
the
of
the
energy
development
associated
of
supplied
new
is
wasted.
technologies
particularly
with:
•
renewable
energy
•
improving
the
sources
fossil
be
to
need
systems
spare
energy
or
can
sources
source)
compared
match
that
a
aware
developments
electrical
batteries
companies
energy
source
secondary
users
NE w aND DE vElOPING TEChNOlOGIEs
is
Can
•
secondary
of
(e.g.
hydroelectric
fuels.
process
energy
versatile
devices
that
It
•
often
common
efciency
and
covered
energy.
most
very
of
Primary
of
be
secondary
energy
oil,
time
energy
renewables:
global
hope
fossil
uranium,
the
need
free.
common
main
would
energy.
sECONDaRY ENERGY sOURCEs
By
wind.
harmful
able
be
efciency
of
our
energy
conversion
process.
Ter energ y trn er
PROCEssEs Of ThERmal ENERGY TRaNsfER
CONvECTION
There
In
are
energy
very
and
several
from
a
hot
important
than
There
faster
is
a
one
Any
of
fourth
moving
object
a
are
given
these
by
to
processes
radiation.
more
processes
which
cold
process
molecules
object
called
called
transfer
can
be
situation
happening
the
surface
thermal
the
This
of
a
Three
convection
probably
at
evaporation.
leaving
of
achieved.
conduction,
practical
processes
the
involves
same
time.
involves
liquid
that
the
convection,
of
a
bulk
(a
liquid
expand
the
up
thermal
movement
or
a
and
hotter
gas).
thus
uid
because
a
energy
of
When
its
part
density
rises
moves
matter.
up.
This
of
is
the
current
is
its
boiling
point.
Evaporation
causes
is
set
two
take
colder
causes
up
as
a
points
place
heated
The
heating
it
because
in
a
tends
uid
room
shown
uid
to
sinks
to
and
warm
below.
is
Cool air is denser and
below
only
uid
reduced.
Central
convection
between
can
Hot air is less dense
cooling.
sinks downwards.
CONDUCTION
In
a
thermal
conduction,
substance
substance.
hot
if
the
without
For
from
example,
other
Conduction
is
thermal
any
end
the
molecule
to
is
bulk
one
placed
process
by
energy
(overall)
end
in
of
a
a
hot
which
is
transferred
movement
metal
cup
spoon
of
kinetic
along
of
the
soon
feels
tea.
energy
is
passed
molecule.
The ow of air around a room
Air is warmed
is called a convection current.
by the heater.
macroscopic view
Convection
HOT
COLD
thermal energy
thermal energy
Points
•
to
in
a
room
note:
Convection
cannot
take
place
in
(and
many
a
solid.
Examples:
RESERVOIR
RESERVOIR
•
The
pilots
of
convection
gliders
currents
in
order
birds)
to
stay
use
naturally
above
the
occurring
ground.
Thermal energy ows along the material as a result
•
of the temperature dierence across its ends.
Sea
breezes
the
microscopic view
day
will
rise
onto
•
the
the
(winds)
land
from
Lighting
in
HOT
the
re
room
are
the
During
in
often
hotter
above
shore.
a
is
a
land
the
the
to
the
and
will
convection.
sea.
This
there
night,
chimney
towards
due
than
the
will
be
a
situation
mean
that
During
means
a
hot
air
breeze
is
reversed.
breeze
ows
re.
COLD
The faster-moving molecules at the hot end pass
RaDIaTION
on their kinetic energy to the slower-moving
molecules as a result of intermolecular collisions.
Matter
is
not
radiation.
Points
to
Poor
conductors
are
called
thermal
Metals
tend
because
allows
a
to
be
very
different
quick
good
thermal
mechanism
transfer
of
conductors.
(involving
thermal
All
gases
(and
most
liquids)
radiate
the
This
up
the
transfer
have
a
of
thermal
temperature
energy
above
by
zero
to
a
electromagnetic
re
to
‘feel
the
waves.
heat’,
your
If
you
hands
hold
are
your
receiving
is
the
radiation.
For
most
electrons)
everyday
energy.
objects
•
in
(that
insulators
hand
•
objects
note:
kelvin)
•
involved
All
tend
to
be
poor
this
radiation
is
conductors.
in
the
infra-red
part
of
HOT
Examples:
the
electromagnetic
OBJECT
•
Most
clothes
keep
us
warm
by
trapping
layers
of
air
–
a
spectrum.
poor
more
•
If
For
conductor.
one
walks
around
a
house
in
bare
feet,
the
oors
that
details
of
the
are
electromagnetic
better
conductors
(e.g.
tiles)
will
feel
colder
than
the
see
that
the
a
are
good
same
piece
insulators
temperature.
of
metal
feels
(e.g.
(For
colder
carpets)
the
same
than
a
even
if
reason,
piece
of
they
on
are
a
page
When
used
for
cooking
food,
saucepans
from
the
source
of
heat
to
the
given
o from
at
cold
Points
to
•
object
all surfaces.
note:
day
An
at
room
temperature
absorbs
and
radiates
wood.)
conduct
If
it
is
at
constant
temperature
(and
not
changing
thermal
state)
energy
radiation is
37.
energy.
•
spectrum,
oors
then
the
rates
are
the
same.
food.
E x amPlE
•
A
•
Surfaces
cork – a poor conductor
surface
poor
•
that
that
radiators
Surfaces
that
is
a
are
good
light
(and
are
radiator
in
poor
dark
colour
is
also
and
a
good
smooth
absorber.
(shiny)
are
absorbers).
and
rough
are
good
radiators
(and
outer plastic cover
good
absorbers).
hot liquid
•
If
the
temperature
of
an
object
is
increased
then
the
surfaces silvered
frequency
of
the
radiation
increases.
The
total
rate
at
par tial vacuum between
so as to reduce
which
energy
is
radiated
will
also
increase.
glass walls to prevent
radiation
convection and
•
Radiation
can
travel
through
a
vacuum
(space).
air gap
conduction
(poor conductor)
Examples:
insulating space
A
thermos
ask
prevents
heat
loss
•
The
•
Clothes
the
Sun
warms
in
the
summer
radiation
from
Earth’s
tend
the
to
surface
be
by
white
–
radiation.
so
as
not
to
absorb
Sun.
E N E R G Y
P R O D U C T I O N
89
Rdition: wien’  nd te sten–botnn 
bl aCk-bODY RaDIaTION: sTEfaN-bOlT zmaNN l aw
wIEN’s l aw
In
Wien’s
general,
on
many
the
radiation
things.
It
is
given
possible
out
to
from
come
a
up
hot
object
with
a
depends
theoretical
the
displacement
intensity
of
the
law
relates
radiation
is
the
a
wavelength
maximum
at
λ
to
which
the
max
model
for
emitter
the
will
‘perfect’
also
be
a
emitter
perfect
of
radiation.
absorber
of
The
‘perfect’
radiation
–
a
temperature
of
the
black
body
T.
This
states
that
black
λ
T
=
constant
max
object
absorbs
all
of
the
light
energy
falling
on
it.
For
this
The
reason
the
radiation
from
a
theoretical
‘perfect’
emitter
value
of
the
constant
can
be
found
by
experiment.
It
is
is
–3
2.9
known
as
black-body
×
10
m
radiation
does
not
depend
on
the
nature
of
surface,
but
should
be
noted
that
in
order
to
use
this
the
it
does
depend
upon
its
wavelength
should
be
substituted
into
the
the
equation
emitting
It
radiation
constant,
Black-body
K.
in
metres
and
the
temperature
in
kelvin.
temperature.
3
2.90 × 10
_
At
any
given
temperature
there
will
be
a
range
of
different
λ
(metres)
=
max
emitted.
This
(and
Some
variation
hence
frequencies)
wavelengths
is
shown
in
will
the
be
of
radiation
more
graph
intense
that
than
ytisnetni
wavelengths
T(kelvin)
are
others.
below.
The
is
peak
wavelength
approximately
stinu yrartibra / ytisnetni
λ
500
=
500
nm
=
5
10
from
the
Sun
nm.
max
7
5
×
m
3
2.9 × 10
_
so
T
=
K
7
5
4
=
×
10
5800
K
3
2
wavelength / nm
λ
max
= 500 nm
1
We
its
can
analyse
surface
light
from
temperature.
a
star
This
and
will
be
calculate
much
less
a
value
than
for
the
0
temperature
400
1200
800
egnaro
der
wolley
eulb
neerg
teloiv
wavelength / nm
of
visible
light
be
the
absolutely
above
actually
graph
precise,
as
the
something
it
is
not
correct
intensity,
that
could
but
be
to
this
called
label
is
the
the
the
often
y-axis
done.
intensity
It
and
core.
might
so
Cooler
stars
(lower
frequencies)
Radiation
To
in
Hot
stars
will
give
out
all
frequencies
1600
emitted
will
well
of
tend
only
give
visible
from
to
appear
out
light
planets
–
will
white
the
in
higher
they
will
peak
in
colour.
wavelengths
appear
the
red.
infra-red.
on
is
function.
INTENsITY, I
This
is
dened
so
that
the
area
under
the
graph
(between
two
The
wavelengths)
gives
the
intensity
emitted
in
that
intensity
of
radiation
is
the
power
per
unit
area
that
is
wavelength
2
received
range.
The
total
total
power
page
195)
area
radiated.
under
The
the
graph
power
is
thus
radiated
by
a
a
measure
of
Black-body
(See
given
the
object.
The
unit
is
W
m
per
unit
Power
_
I
is
by
the
=
A
by:
2
Surface
area
in
m
4
P
Total
power
Although
spectrum
=
radiated
stars
is
and
in
σAT
Stefan-Boltzmann
W
planets
approximately
are
the
not
perfect
same
as
emitters,
black-body
constant
their
radiation
radiation.
power
radiated
by
object
area
____________________________________________
e
EqUIlIbRIUm aND EmIssIvITY
=
power
If
the
temperature
of
a
planet
is
constant,
then
the
power
radiated
per
unit
area
by
black
body
at
same
temperature
being
thus
absorbed
by
the
planet
must
equal
the
rate
at
which
energy
is
4
being
If
it
absorbs
must
of
radiated
go
up
into
more
and
absorption
if
space.
energy
the
then
its
The
planet
than
rate
of
it
is
in
radiates,
loss
of
temperature
thermal
then
energy
must
go
is
the
equilibrium.
p
order
to
estimate
the
power
absorbed
than
its
rate
down.
or
eσ A T
temperature
greater
albEDO
Some
back
In
=
emitted,
of
concepts
are
radiation
space.
The
received
fraction
by
that
a
is
planet
is
reected
reected
back
is
straight
called
the
the
albedo,
following
the
into
α.
useful.
The
Earth’s
albedo
varies
daily
and
is
dependent
on
season
Emissivity
(cloud
The
Earth
and
its
atmosphere
are
not
a
perfect
black
e,
is
dened
as
the
ratio
of
power
radiated
per
has
by
an
object
to
the
power
radiated
per
unit
area
by
a
a
high
value.
at
the
same
temperature.
It
is
a
ratio
and
so
on
has
no
P R O D U C T I O N
global
scattered
power
__
=
total
E N E R G Y
The
Earth.
units.
albedo
90
Oceans
black
total
body
latitude.
unit
(30%)
area
and
have
a
low
value
but
body.
snow
Emissivity,
formations)
incident
power
annual
mean
albedo
is
0.3
s or poer
sOl aR CONsTaNT
The
amount
of
power
that
arrives
from
the
Sun
is
measured
by
the
solar
constant.
It
is
properly
dened
as
the
amount
of
solar
2
energy
that
falls
per
second
on
an
area
of
1
m
above
the
Earth’s
atmosphere
that
is
at
right
angles
to
the
Sun’s
rays.
Its
average
2
value
is
about
1400
W
m
.
2
This
is
not
that
often
the
less
same
than
as
the
half
of
power
this
that
arrives
arrives
at
the
on
1
m
Earth’s
of
the
surface.
Earth’s
The
surface.
amount
Scattering
that
arrives
and
absorption
depends
greatly
in
on
the
the
atmosphere
weather
means
conditions.
incoming solar
radiation
100%
1% absorbed
NB These gures are only guidelines
in stratosphere
stratosphere
because gures vary with cloud cover,
water vapour, etc.
troposphere
clouds reect 23%
24% absorbed
in troposphere
clouds absorb 3%
4% reected from
the Ear th’s surface
surface of
the Ear th
24% direct
2
1% diuse
45% reaches Ear th’s surface
Fate
Different
received
parts
will
of
also
the
of
incoming
Earth’s
vary
with
radiation
surface
the
(regions
seasons
at
since
different
this
will
latitudes)
affect
how
will
receive
spread
out
different
the
rays
amounts
have
of
solar
radiation.
The
amount
become.
atmosphere is a near-uniform
23.5
˚
thickness all around the Ear th
MN > PQ
RS
R
S
Nor th Pole
N
o
m
t
a
e
c
a
f
r
u
s
e
r
e
h
p
s
M
> TU
T
r
o
60
˚
incoming solar
p
ic
s
eg
de
h
tr
a
E
fo
o
f
C
radiation
a
n
c
e
r
travelling in
30
˚
E
q
T
r
o
p
i
u
a
to
near parallel
r
lines
c
o
f
P
C
a
p
ri
0
˚
c
o
T
U
rn
Q
South Pole
30
˚
Radiation has to travel through a
60
˚
90
˚
greater depth of atmosphere (RS as
compared with TU) in high latitudes.
When it reaches the surface the radiation
is also spread out over a greater area (MN as
compared with PQ) than in lower latitudes.
The
effect
of
latitude
on
incoming
solar
radiation
Tropic of
Tropic of
Capricorn
Cancer
SUN
Summer
Summer
in nor thern hemisphere
The
Earth’s
orbit
and
in southern hemisphere
the
seasons
E N E R G Y
P R O D U C T I O N
91
Te greenoue eect
PhYsIC al PROCEssEs



O
S


E

Some solar radiation is
Short
wavelength
radiation
is
received
reected by the atmosphere
the
to
Sun
and
warm
causes
up.
The
the
surface
Earth
will
of
emit
the
E
Some of the infrared
from
Earth
radiation passes through
and Ear th’s surface
the atmosphere and is
lost in space
Outgoing solar radiation:
infra-red
2
radiation
(longer
wavelengths
radiation
coming
from
than
Solar radiation passes
the
103 W m
S  
through the clear atmosphere
the
Sun)
Net outgoing infra-red
because
2
ncoming solar radiation:
the
Earth
is
cooler
than
the
Sun.
Some
radiation: 240 W m
of
2
33 W m
this
in
infra-red
the
radiation
atmosphere
is
and
absorbed
by
re-radiated
in
gases
all
directions.
G
This
and
is
known
the
gases
as
in
the
the
greenhouse
atmosphere
effect
that
R
E
E
N
Net incoming
H
radiation
are
called
U
S
E
G
A
S
E
S
absorbed and reemitted by the
solar radiation:
absorb
greenhouse gas molecules he
2
infra-red
O
Some of the infrared radiation is
240 W m
greenhouse
direct eect is the arming of the
gases.
The
net
atmosphere
effect
and
the
is
that
the
surface
of
Solar energy absorbed
upper
the
Earth
Ear th’s surface and the troposphere
by atmosphere:
are
Surface gains more heat and
2
warmed.
as
real
The
name
is
greenhouses
potentially
are
warm
as
2 W m
confusing,
a
result
infrared radiation is emitted again
of
Solar energy is absorbed by the
a
different
mechanism.
Ear th’s surface and arms it
and is coner ted into heat causing
2
The
be
temperature
constant
energy
The
process
the
the
equals
energy.
of
if
of
the
rate
the
Earth’s
at
rate
without
Earth
would
it
temperature
than
°C
effect
the emission of longae infrared
rediation bac to the atmosphere
radiates
it
is
1 W m
will
absorbs
a
E
natural




temperature
much
of
it
which
the
be
average
which
at
greenhouse
and
surface
the
lower;
Moon
the
is
Sources: Oanagan niersity ollege in anada epartment of eography niersity of Oford
nited States Enironmental rotection gency E Washington limate change 1 he
more
science of climate change contribution of oring group 1 to the second assessment report of the
30
colder
than
the
Earth.
nte
rgoernmental anel on limate hange E and WO ambridge ress 1
•
GREENhOUsE Ga sEs
The
main
balance
release
•
greenhouse
in
the
due
to
gases
atmosphere
industry
Methane,
CH
.
can
and
This
is
are
naturally
be
altered
technology.
the
occurring
principal
as
a
They
but
result
of
Chlorouorocarbons
propellants
the
of
their
are:
Each
component
depleting
of
these
resonance
of
and
the
gases
(see
ozone
absorbs
page
(CFCs).
cleaning
168).
Used
solvents.
as
They
refrigerants,
also
have
the
effect
layer.
infra-red
The
radiation
natural
as
a
frequency
result
of
of
oscillation
of
4
natural
gas
and
fermentation.
amounts
of
the
product
Livestock
of
and
decay,
plants
decomposition
produce
the
or
signicant
methane.
bonds
If
the
is
equal
driving
occur.
•
Water,
H
O.
The
small
amounts
of
water
vapour
in
the
within
to
The
the
molecules
frequency
the
natural
amplitude
(from
of
the
frequency
of
the
the
gas
is
radiation
of
the
molecules’
in
the
infra-red
emitted
molecule,
vibrations
from
region.
the
resonance
increases
Earth)
will
and
the
upper
2
temperature
atmosphere
(as
opposed
to
clouds
which
are
condensed
levels
in
greatly
•
the
as
a
a
signicant
atmosphere
result
of
effect.
do
not
industry,
The
average
appear
but
local
to
water
alter
dioxide,
CO
.
Combustion
dioxide
signicantly
Overall,
into
the
increase
plants
atmosphere
releases
the
which
greenhouse
(providing
they
are
will
take
place
at
specic
depending
on
the
molecular
energy
levels.
greenhouse gases in the Ear th’s atmosphere
2
carbon
absorption
Absorption spectra for major natural
levels
vary.
Carbon
The
vapour
can
effect.
growing)
ytiv it prosb A
can
have
increase.
water
frequencies
vapour)
will
1
Methane
CH
4
0
1
Nitrous oxide
N
O
2
remove
carbon
dioxide
from
the
0
atmosphere
1
during
photosynthesis.
This
is
known
as
Oxygen, O
2
carbon
& Ozone, O
xation
3
0
•
Nitrous
oxide,
N
O.
Livestock
and
industries
1
2
(e.g.
of
the
production
nitrous
remain
in
oxide.
the
Its
of
Nylon)
effect
upper
is
are
major
signicant
atmosphere
for
Carbon
sources
as
long
it
can
periods.
dioxide
CO
0
2
1
Water vapour
In
addition
the
following
gases
also
contribute
to
the
H
O
2
0
greenhouse
effect:
1
•
Ozone,
O
.
The
ozone
layer
is
an
Total
important
3
atmosphere
region
of
the
atmosphere
that
absorbs
high
0
energy
UV
photons
which
would
otherwise
be
0.1
harmful
to
living
organisms.
Ozone
also
adds
0.2
0.30.4
0.60.8 1
1.5
2
3
4
5
6
8 10
20
30
to
Wavelength (µm)
the
greenhouse
effect.
[After J.N. Howard, 1959: Proc. I.R.E
47, 1459: and R.M. Goody
and G.D. Robinson, 1951: Quart. J. Roy Meteorol. Soc. 77, 153]
92
E N E R G Y
P R O D U C T I O N
Go ring
•
POssIblE C aUsEs Of GlObal waRmING
Records
show
increasing
in
that
the
recent
mean
temperature
of
the
Earth
has
Changes
linked
been
years.
•
Cyclical
The
rst
in
to,
the
for
intensity
example,
changes
suggestion
in
of
the
radiation
increased
the
Earth’s
could
be
solar
orbit
caused
and
by
emitted
are
by
the
Sun
activity.
volcanic
natural
activity.
effects
or
could
0.6
be
caused
by
human
activities
(e.g.
the
increased
burning
of
annual mean
fossil
fuels).
An
enhanced
greenhouse
effect
is
an
increase
5-year mean
0.4
0.2
in
the
greenhouse
In
2013,
Change)
0
the
IPCC
report
inuence
warming
has
effect
by
human
(Intergovernmental
stated
been
since
caused
the
that
the
‘It
is
mid–20th
Panel
extremely
dominant
activities.
cause
of
on
likely
the
Climate
that
human
observed
century’.
0.2
Although
is
that
it
that
is
still
the
being
debated,
increased
the
generally
combustion
of
fossil
accepted
fuels
has
view
released
0.4
extra
1880
All
1900
atmospheric
suggestions
•
for
Changes
in
1920
models
this
the
1940
are
highly
increase
1960
1980
complicated.
2000
Some
the
carbon
dioxide
greenhouse
into
the
atmosphere,
which
has
enhanced
effect.
possible
include.
composition
of
greenhouse
gases
in
the
atmosphere.
E vIDENCE fOR GlObal waRmING
One
piece
been
of
drilled
Isotopic
evidence
in
the
analysis
atmospheric
variations
that
links
Russian
allows
the
temperature
warming
base
temperature
concentrations
of
global
Antarctic
of
and
at
to
greenhouse
carbon
to
increased
Vostok.
be
estimated
gases.
dioxide
Each
are
The
and
air
record
very
levels
year’s
of
new
bubbles
provides
closely
greenhouse
snow
fall
trapped
data
gases
adds
from
in
comes
another
the
over
ice
from
layer
cores
400,000
ice
to
can
years
core
the
be
used
ago
data.
The
ice
core
has
ice.
to
to
the
measure
present.
the
The
correlated.
Antarctic Ice C ore
Concentration
C°
vmpp /
2
4
380
2
340
0
OC
2
300
-2
-4
260
-6
220
-8
180
-10
400,000
350,000
300,000
250,000
200,000
150,000
100,000
50,000
0
Years before present
ppmv = par ts per million by volume
•
mEChaNIsms
Predicting
deal
in
of
the
There
global
•
the
future
uncertainty,
Earth
are
and
many
its
effects
as
the
of
global
warming
interactions
atmosphere
mechanisms
are
that
involves
between
extremely
may
a
different
the
warming
reduces
overall
the
rate
only
reduces
albedo.
of
heat
This
ice/snow
will
cover,
result
in
an
which
in
increase
The
rst
small
the
Temperature
increase
absorption.
increase
reduces
the
solubility
of
CO
in
the
the
of
exist
increase
trapped
deforestation
atmosphere,
carbon
four
initial
in
feedback.
•
subsoil
An
(called
in
tundra)
temperature
that
may
support
cause
a
CO
result
the
in
the
reduction
release
in
of
further
number
of
trees
2
turn
in
release
does
into
reduces
warming.
Global
Not
CO
of
frozen
2
•
rate
with
vegetation.
signicant
systems
complex.
increase
Regions
simple
great
xation.
mechanisms
temperature
are
temperature.
This
Some
have
people
examples
increase
has
process
of
gone
is
processes
on
to
known
suggested
that
as
the
whereby
cause
a
a
further
positive
current
sea
2
temperature
and
thus
increases
atmospheric
involves
•
Continued
and
the
vapour
global
warming
atmosphere’s
is
a
will
ability
greenhouse
to
increases
may
be
‘corrected’
by
a
process
which
concentrations.
increase
hold
both
water
negative
feedback,
and
temperatures
may
fall
in
the
future.
evaporation
vapour.
Water
gas.
E N E R G Y
P R O D U C T I O N
93
Ib quetion – energ y production
1.
A
wind
The
generator
source
of
this
converts
wind
wind
energy
energy
can
be
into
electric
traced
back
energy.
to
Calculate
solar
a)
energy
arriving
at
the
Earth’s
the
in
a)
Outline
solar
b)
the
energy
energy
List
one
use
of
transformations
converts
advantage
wind
into
and
wind
one
involved
energy.
from
1
P
=
for
a
the
wind
per
cooling
second
carried
away
by
the
water
tower;
[2]
b)
the
energy
per
c)
the
overall
efciency
second
d)
the
mass
produced
by
burning
the
coal;
[2]
[2]
disadvantage
of
of
the
power
station;
[2]
the
generators.
expression
available
the
as
of
coal
burnt
each
second.
[1]
[2]
5.
The
energy
surface.
maximum
generator
theoretical
power,
This
question
is
about
tidal
power
systems.
P,
a)
Describe
b)
Outline
c)
A
the
principle
of
operation
of
such
a
system.
[2]
is
one
advantage
and
one
disadvantage
of
3
Aρv
using
such
a
system.
[2]
2
where
A
is
the
area
swept
ρ
is
the
density
out
by
the
small
air
v
is
the
table
wind
a
wind
the
is
proposed.
Use
the
data
in
to
calculate
the
useful
the
total
output
energy
power
of
available
this
and
system.
speed.
maximum
generator
below
estimate
Height
Calculate
system
and
hence
c)
power
blades,
the
of
tidal
theoretical
whose
blades
are
power,
30
m
P,
for
low
between
high
tide
and
tide
4
m
long
Trapped
water
would
cover
an
1
when
a
20
m
s
wind
blows.
The
density
of
air
6
area
of
1.0
×
10
1.0
×
10
2
m
3
is
1.3
kg
m
.
[2]
3
Density
d)
In
practice,
under
these
conditions,
the
of
Number
only
provides
3
MW
of
electrical
Calculate
(ii)
Give
the
efciency
of
this
generator.
Solar
reasons
of
explaining
why
the
power
output
theoretical
2.
This
a)
question
Give
one
one
is
is
about
example
example
less
power
of
a
than
the
a
[2]
b)
sources.
renewable
energy
non-renewable
energy
are
such.
source
source
and
why
they
classied
day
2
[4]
as
models.
Distinguish,
a
in
terms
solar
of
the
heating
energy
panel
and
changes
a
involved,
photovoltaic
State
an
appropriate
(i)
solar
(ii)
photovoltaic
heating
domestic
use
for
cell.
[2]
a
panel.
[1]
cell.
[1]
and
26
c)
explain
per
climate
maximum
output.
energy
of
tides
and
actual
between
power
m
[2]
a)
two
–3
kg
power.
6.
(i)
water
generator
The
radiant
power
of
the
Sun
is
3.90
×
10
W
.
The
[4]
average
radius
of
the
Earth’s
orbit
about
the
Sun
is
11
b)
A
wind
year.
If
show
farm
there
that
turbine
c)
State
produces
is
are
the
wind
average
about
two
ten
400
35,000
MWh
turbines
power
of
on
output
energy
the
of
in
1.50
a
and
farm
the
one
kW
.
disadvantages
×
it
10
m.
may
be
The
albedo
assumed
of
that
the
no
atmosphere
energy
is
is
0.300
absorbed
by
atmosphere.
[3]
of
using
wind
power
to
generate
Show
that
at
Earth’s
the
the
intensity
surface
incident
when
the
on
Sun
a
solar
is
heating
directly
panel
overhead
2
electrical
3.
This
power.
question
is
is
[2]
about
energy
d)
transformations.
966
Show,
W
m
.
using
[3]
your
answer
to
(c),
that
the
average
intensity
2
incident
Wind
power
can
be
used
to
generate
electrical
an
energy
ow
diagram
which
shows
the
Assuming
starting
with
solar
energy
and
ending
energy,
generated
by
windmills.
Your
indicate
where
energy
is
degraded.
This
question
that
is
about
a
coal-red
power
station
which
Electrical
power
output
from
station
Temperature
at
cooling
Temperature
leaves
which
=
200
MW
=
288
K
=
348
K
=
4000
water
tower
at
cooling
which
water
tower
1
Rate
of
water
ow
through
tower
kg
s
7
Energy
content
of
coal
=
2.8
×
=
4200
10
1
J
kg
–1
Specic
94
.
the
Earth’s
surface
behaves
[3]
as
a
black-body
no
energy
to
(d)
is
to
absorbed
show
by
that
the
the
atmosphere,
average
use
temperature
Earth’s
surface
is
predicted
to
be
256
K.
[2]
Outline,
with
reference
to
the
greenhouse
effect,
why
is
average
than
enters
Wm
surface
temperature
of
the
Earth
is
higher
cooled.
Data:
the
that
answer
the
the
water
242
[7]
f)
4.
is
diagram
of
should
surface
with
your
electrical
Earth’s
energy
and
transformations,
the
energy.
e)
Construct
on
heat
of
I B
water
Q U E s T I O N s
–
J
kg
E N E R G Y
1
K
P R O D U C T I O N
256
K.
[4]
9
W a v e
p h e n o m e n a
Sil i i
hL
SImpLe harmonIc motIon (Shm) equatIon
tWo e x ampLeS of Shm
SHM
Two
the
occurs
when
resultant
the
forces
acceleration,
a,
on
is
an
object
directed
are
such
towards,
that
and
1.
proportional
to,
its
displacement,
x,
from
a
xed
common
Mass,
m,
on
∝
-x
or
a
=
-(constant)
×
mathematics
of
SHM
is
simplied
if
the
constant
the
of
another
Thus
the
between
constant
general
ω
a
and
which
form
for
x
is
the
approximate
to
SHM
are:
spring
that:
mass
is
identied
called
the
equation
as
the
angular
that
of
the
spring
is
negligible
compared
to
the
of
mass
proportionality
that
vertical
x
•
The
a
point.
Provided
a
situations
is
of
the
load
square
frequency.
denes
SHM
•
friction
•
the
(air
friction)
is
negligible
is:
spring
obeys
Hooke’s
law
with
spring
constant,
k
at
2
a
=
-ω
x
all
The
solutions
for
this
equation
follow
below.
The
times
(i.e.
elastic
limit
is
not
exceeded)
angular
•
the
gravitational
•
the
xed
eld
strength
g
is
constant
1
frequency
period,
T,
ω
has
the
units
of
of
the
oscillation
rad
by
s
and
the
is
related
following
to
the
time
ω
end
of
the
spring
cannot
move.
equation.
2π
_
Then
it
can
be
shown
that:
=
T
k
2
ω
=
m
m
Or
T
=
2π
√
IdentIfIc atIon of Shm
In
order
the
to
analyse
following
a
situation
procedure
to
should
decide
be
if
SHM
is
taking
place,
2.
The
simple
Identify
all
the
forces
acting
on
an
object
when
it
an
arbitrary
distance
x
from
its
rest
position
a
free-body
Calculate
this
points
the
•
is
back
towards
of
SHM
must
be
resultant
the
has
in
force
proportional
motion
Once
and
mass
m
the
mass
of
of
the
the
string
is
negligible
compared
with
the
load
diagram.
the
force
l
using
mass
•
length
that:
is
•
displaced
of
followed.
Provided
•
k
pendulum
the
the
the
mean
object
been
to
using
must
second
displacement
position
be
identied,
following
Newton’s
(i.e.
and
F
law.
equation
∝
of
friction
•
the
maximum
(air
friction)
•
the
gravitational
•
the
length
is
angle
negligible
of
swing
is
small
(≤
5°
or
0.1
rad)
always
x)
then
SHM.
the
•
If
motion
Then
it
can
of
be
eld
the
strength
pendulum
shown
g
is
is
constant
constant.
that:
form:
g
2
ω
restoring
force
per
unit
displacement,
=
k
l
____
a
=
-
(
oscillating
mass,
×
)
m
x
l
Or
•
This
identies
the
angular
frequency
ω
as
ω
(
=
m
)
or
Note
√( m )
=
equations
=
that
equation

k
ω
T
.
Identication
of
ω
allows
2π
the
be
g
mass
and
thus
of
the
does
pendulum
not
affect
bob,
the
m,
time
is
not
period
in
of
this
the
quantitative
pendulum,
to
√
k
_
2
T
applied.
e x ampLe
acceLeratIon, veLocIty and dISpL acement
A
600
g
mass
is
attached
to
a
light
spring
with
spring
constant
–1
30
N
m
.
durIng Shm
The
variation
and
displacement,
angular
The
frequency
precise
object
hand
with
is
of
the
of
of
an
the
acceleration,
object
doing
a,
SHM
velocity,
depends
v,
on
the
(a)
Show
(b)
Calculate
that
(a)
Weight
the
the
mass
does
frequency
SHM.
of
its
oscillation.
ω.
format
when
set
time
x,
of
the
clock
equations
relationships
is
started
correspond
depends
(time
to
an
t
=
on
zero).
where
The
oscillation
of
mass
=
mg
=
6.0
N
the
Additional
displacement
on
k x
x
down
means
when
mass
=
upwards.
Since
F
∝
x,
in
the
mean
position
when
t
=
0.
The
right
hand
m
equations
maximum
correspond
displacement
to
an
oscillation
when
t
=
when
the
object
is
Since
SHM,
T
=
2π
at
√(
1
=
x
sin
ωt
x
=
x
0
cos
=
ωt
=
ωx
a
=
-ω
cos
k
0.6
_
)
=
2π
√(
30
)
=
0.889s
ωt
v
=
-ωx
a
=
-ω
0
sin
1
_
=
T
0
v
=
1.1
Hz
0.889
ωt
0
2
2
x
sin
ωt
x
0
cos
ωt
0
displacement
x
The
rst
force
oscillate
0.
f
x
will
set
(b)
of
resultant
mass
_____
____
is
the
the
withSHM.
object
that
left
two
equations
can
be
rearranged
to
produce
0
the
ωx
0
following
relationship:
_______
velocity
2
v
=
±
ω
√
(x
x
)
0
x
is
the
amplitude
of
the
oscillation
measured
in
m
0
t
is
the
time
taken
measured
in
T
T
3T
4
2
4
time
T
s
2
ω
–1
ω
is
ω t
the
is
full
an
angular
angle
=
time
s
measured
in
radians.
x
0
acceleration
A
velocity
equation.
•
acceleration
m
_
•
displacement
k
•
velocity
=
ω
with
rad
acceleration
is
completed
frequency
2π
_
T
increases
in
•
angular
following
that
measured
•
oscillation
The
frequency
2π
is
when
related
(ω t)
to
the
=
2π
time
rad.
period
T
by
the
leads
leads
velocity
by
displacement
and
90°
by
displacement
lags
velocity
by
90°
are
180°
out
of
phase
90°
√
lags
acceleration
by
W a v e
90°
p h e n o m e n a
95
e s i sil i i
hL
During
SHM,
Providing
energy
there
are
is
no
interchanged
resistive
between
forces
which
KE
and
dissipate
PE.
The
total
energy
is
this
1
E =
energy,
the
total
energy
must
remain
constant.
The
oscillation
E
is
+
k
E
2
=
m
ω
p
1
2
(x
x
) +
said
to
be
The
kinetic
1
2
x
=
2
m
ω
x
0
2
2
undamped
Energy
energy
1
E
2
m ω
0
2
=
can
1
2
mv
be
calculated
2
=
m
ω
in
SHM
is
proportional
•
the
mass
m
•
the
(amplitude)
2
(x
k
x
to:
from
2
)
0
2
2
2
•
The
potential
1
E
=
energy
2
m
ω
can
be
calculated
the
(frequency)
from
2
x
p
2
E

total

p
Graph showing the
variation with distance, x
of the energy,  during SHM

k
x
x
x
0
0


total

k
Graph showing the
variation with time, t
of the energy, 
during SHM

p
t
96
W a v e
T
T
3T
T
4
2
4
2
p h e n o m e n a
dii
hL
The
intensity
plot
for
a
single
slit
is:
Ba SIc oBServatIonS
Diffraction
is
a
wave
effect.
The
objects
involved
intensity
(slits,
apertures,
etc.)
have
a
size
There is a central maximum intensity.
that
is
of
the
same
order
of
magnitude
Other maxima occur roughly halfay
10
as
the
wavelength
of
visible
light.
beteen the minima.
n 
gil
dii
bsl
sw

s the angle increases,
the intensity of the
() straight
maxima decreases.
edge
1.1
0.4
angle
b = slit idth
(b) single
1st minimum
λ
θ
long slit
=
λ
θ
=
θ
λ
=
b
b
b
b
~
3λ
λ
The angle of the rst minimum is given by sin θ
b
() circular
λ
For small angles, this can be simplied to θ =
.
b
() single
long slit
b
~
5λ
1
have
e xpL anatIon
The
shape
derived
of
by
1
of
the
maximum
amplitude
and
thus
be
5
the
relative
applying
an
intensity
idea
called
versus
angle
Huygens’
plot
can
maximum
be
principle.
of
the
central
25
intensity.
We
For 1st minimum:
can
treat
the
slit
as
a
series
of
secondary
wave
sources.
In
the
b sin θ
forward
up
to
direction
give
a
(θ
=
zero)
maximum
these
intensity.
are
At
all
any
in
phase
other
so
they
angle,
there
is
a
difference
between
the
rays
that
depends
on
the
λ
λ
∴sin θ
path
=
add
=
angle.
b
The
for
overall
the
rst
sources
The
result
is
the
minimum
across
the
condition
for
slit
addition
is
that
cancel
the
rst
of
the
all
the
angle
sources.
must
The
make
all
condition
of
the
Since angle is small,
b
out.
maximum
out
from
the
centre
θ
sin θ
≈
λ
3λ
when
the
path
difference
across
the
whole
slit
is
θ
is
.
At
this
∴
2
θ
=
b
angle
the
slit
can
be
analysed
as
being
three
equivalent
sections
λ
each
having
a
path
difference
of
across
its
length.
Together,
for 1st minimum
2
two
of
these
sections
will
destructively
interfere
leaving
the
1
resulting
amplitude
to
be
of
the
maximum.
Since
intensity
path dierence across slit = b sin θ
3
2
∝
(amplitude)
,
the
rst
maximum
intensity
out
from
the
1
centre
will
be
of
the
central
maximum
intensity.
By
a
similar
9
argument,
the
second
maximum
intensity
out
from
the
centre
will
SIngLe-SLIt dIffractIon WIth WhIte LIght
When
a
single
component
slit
is
colour
illuminated
has
a
specic
with
white
light,
wavelength
and
each
so
Red
the
rst order
associated
maxima
and
minima
for
each
wavelength
will
be
Violet
located
at
a
different
angle.
For
a
given
slit
width,
colours
incident white light
with
longer
than
colours
wavelengths
with
short
(red,
orange,
wavelengths
etc.)
(blue,
will
diffract
violet,
etc.).
more
zero order
The
Violet
maxima
for
the
resulting
diffraction
pattern
will
show
all
the
rst order
colours
central
of
the
rainbow
position
and
with
red
blue
and
appearing
at
violet
nearer
greater
to
the
Red
angles.
W a v e
p h e n o m e n a
97
tw-s i  ws: y ’s bl-
hL
sli i
douBLe-SLIt Interference
The
double-slit
This
to
can
be
only
taken
Decreasing
that
the
interference
take
into
the
total
place
if
account
slit
slits
when
width
intensity
pattern
the
will
of
will
out
that
be
on
page
innitely
working
mean
light
shown
are
the
the
47
small.
was
In
overall
observed
decreased.
The
derived
practice
double
pattern
assuming
they
slit
have
more
pattern
(a) Young ’s fringes for innitely narrow slits
each
nite
interference
becomes
interference
that
a
was
behaving
The
diffraction
pattern
and
will
slit
width.
as
more
become
shown
to
a
perfect
pattern
of
point
each
source.
slit
needs
below.
‘idealized’.
harder
like
Unfortunately,
it
will
also
mean
observe.
(c) Young ’s fringes for slits of nite width
relative intensity
intensity
angle θ
bright
fringes
(b) diraction pattern for a nite-width slit
intensity
angle θ
λD
s
=
still applies but dierent fringes
d
will have dierent intensities with
it being possible for some fringes
angle θ
to be missing.
InveStIgatIng young’S douBLe-SLIt e xperImentaLLy
Possible
page
set-ups
for
the
double-slit
experiment
are
shown
on
47.
The
most
width
accurate
are
microscope
Set-up 1
measurements
achieved
that
is
using
a
for
slit
travelling
mounted
on
a
separation
and
microscope.
frame
so
that
it
fringe
This
can
be
is
a
moved
region in which
perpendicular
to
the
direction
in
which
it
is
pointing.
The
superposition occurs
microscope
separation
moved
by
of slits
precision
is
moved
the
across
microscope
ten
can
or
more
be
read
is
often
fringes
off
from
and
the
the
distance
scale.
The
monochromatic
of
this
measurement
improved
by
utilizing
a
light source
vernier
In
the
scale.
simplied
version
(set-up
2)
of
the
experiment,
fringes
S
1
can
still
be
bright
enough
to
be
viewed
several
metres
away
S
0
from
the
slits
and
thus
they
can
be
projected
onto
an
opaque
S
2
screen
(it
is
separation
dangerous
can
be
then
to
look
be
into
directly
a
laser
beam).
measured
with
Their
a
source
Set-up 2
twin source
slit
possible
slits (less than 5 mm)
double
screen
0.1 m
laser
1 m
positions
In
the
original
source
is
set-up
diffracted
at
(set-up
S
so
1)
as
to
light
from
ensure
the
that
coherent
light.
monochromatic
S
0
receiving
and
S
1
Diffraction
takes
slit
are
2
place
providing
S
1
and
S
are
narrow
enough.
The
slit
separations
need
to
be
2
approximately
order
were
of
0.1
separated
transparent
will
need
they
1
mm
can
98
to
be
or
be
mm
(or
by
(or
less)
less).
thus
This
approximately
translucent)
darkened
viewed
using
W a v e
to
a
the
would
0.5mm
situated
allow
slit
1m
the
widths
provide
on
a
away.
fringes
microscope.
p h e n o m e n a
are
fringes
screen
The
to
be
of
the
that
(semi-
laboratory
visible
and
screen
ruler.
mlil-sli ii
hL
the dIffractIon gratIng
The
the
slit
diffraction
overall
that
appearance
experiment
considers
further
takes
the
slits.
(see
effect
A
place
of
page
on
series
the
98
the
of
(a) 2 slits
at
an
individual
fringes
for
nal
parallel
in
more
Young’s
details).
interference
slits
slit
(at
a
affects
double-
This
section
pattern
regular
of
adding
separation)
is
(b) 4 slits
called
a
diffraction
Additional
condition
angle
be
at
slits
for
at
grating
the
same
constructive
which
unaffected
the
by
light
the
separation
will
interference.
from
number
slits
of
adds
slits.
In
not
affect
other
the
words,
constructively
The
situation
the
will
is
(c) 50 slits
shown
below.
θ
path dierence
Grating
patterns
between slits = d sin θ
uSeS
θ
One
of
the
main
uses
of
a
diffraction
grating
is
the
accurate
d
experimental
For constructive
interference:
measurement
light
contained
on
diffraction
a
interference
in
a
given
grating,
takes
place
of
the
different
spectrum.
the
angle
depends
at
on
If
wavelengths
white
which
light
is
of
incident
constructive
wavelength.
Different
path dierence = nλ
wavelengths
can
thus
be
observed
at
different
angles.
The
between slits [λ, 2λ, 3λ]
accurate
nλ
measurement
with
an
(and
thus
accurate
formula
also
applies
The
to
the
difference
Young’s
between
angle
of
the
of
provides
the
colour
of
exact
light
the
experimenter
wavelength
that
is
being
= d sin θ
The
measurement
arrangement.
the
measurement
frequency)
considered.
This
of
is
apparatus
called
a
that
is
used
to
achieve
this
accurate
spectrometer.
double-slit
the
patterns
is
R
most
third (and part of the four th)
noticeable
at
the
angles
where
perfect
constructive
interference
order spectrum not shown
does
not
have
a
take
place.
signicant
If
there
angular
are
only
width.
two
T
wo
slits,
sources
the
that
maxima
are
just
2nd order
will
out
V
of
phase
interfere
to
give
a
resultant
that
is
nearly
the
same
R
amplitude
as
two
sources
that
are
exactly
in
phase.
1st order
resultant interference
V
pattern
white
light
white central
source A
maximum
time
V
diraction
grating
source B
R
V
The
addition
just
out
pattern
of
of
more
phase
will
be
slits
with
totally
its
will
mean
neighbour.
that
The
each
new
overall
slit
is
interference
destructive.
R
overall interference pattern is totally destructive
time
The
addition
following
of
further
slits
at
the
•
the
principal
maxima
maintain
•
the
principal
maxima
become
•
the
overall
so
the
same
slit
separation
has
the
effects:
amount
pattern
of
light
increases
in
the
same
much
being
let
separation
sharper
through
is
increased,
intensity.
W a v e
p h e n o m e n a
99
ti lll ls
hL
pha Se changeS
There
that
in
are
also
detail
many
involve
the
condItIonS for Interference patternS
situations
the
reection
conditions
interference,
one
consideration.
A
when
for
needs
phase
to
interference
of
light.
When
constructive
take
change
any
is
or
take
place
analysing
A
parallel-sided
reections
that
lm
are
can
produce
taking
place
interference
at
both
as
surfaces
a
of
result
the
of
the
lm.
destructive
phase
the
can
changes
inversion
of
into
the
wave
E
that
can
take
place
at
a
reection
interface,
but
it
does
not
air
D
always
happen.
It
depends
on
the
two
media
involved.
A
C
The
technical
‘undergone
term
a
for
phase
the
inversion
change
of
of
a
wave
is
that
it
has
π’.
thickness d
•
When
light
is
reected
back
from
an
optically
denser
lm
medium
there
is
a
phase
change
of
π
(refractive index = n)
ϕ
•
When
light
medium
is
reected
there
is
no
back
phase
from
an
optically
less
dense
change.
air
B
n
< n
1
ϕ
transmitted wave
2
(no phase change)
ϕ
=
zero
when viewed
n
1
along the normal
n
2
F
reected wave
From
point
A,
there
are
two
possible
paths:
incident wave
(no phase change)
n
1
1.
along
path
AE
2.
along
ABCD
in
in
air
the
film
of
thickness
d
< n
2
reected wave
These
(π phase change)
incident wave
the
rays
then
interfere
optical
path
difference.
The
path
AE
in
and
air
is
we
need
to
equivalent
calculate
to
CD
in
the
film
n
1
So
path
difference
=
(AB
+
BC)
in
the
film.
λ
n
2
In
addition,
the
phase
change
at
A
is
equivalent
to
path
2
difference.
λ
transmitted wave
So
total
path
difference =
(AB
+
BC)
in
film
2
(no phase change)
λ
=
n(AB
+
BC)
+
2
By
geometry:
(AB
+
BC) =
FC
e x ampLe
=
The
equations
in
the
box
on
the
right
work
out
the
angles
2d
cos
ϕ
for
λ
∴
which
given
eye
If
constructive
wavelength.
receives
white
rays
light
is
and
If
destructive
the
leaving
used
source
the
then
of
lm
the
interference
light
over
is
a
an
place
extended
range
situation
take
of
source,
values
becomes
for
for
path
difference =
2dn
cos
ϕ
+
a
2
if
the
θ
2dn
or
cos
ϕ
=
mλ
when
ϕ
=
0,
:
destructive
2dn
=
mλ:
destructive
λ
more
if
2dn
cos
ϕ
=
m
+
λ:
constructive
2
complex.
or
two
Provided
colours
cancel.
The
as
can
be
•
an
•
soap
oil
the
may
thickness
reinforce
appearance
seen
lm
when
on
the
of
surface
the
along
the
looking
of
lm
a
lm
is
small,
direction
will
be
in
bright
then
which
one
or
others
colours,
when
such
ϕ
=
0,
2dn
=
mλ:
m
=
0,1,2,3,4
constructive
at
of
water
or
appLIc atIonS
bubbles.
Applications
•
The
of
design
aircraft.
If
parallel
of
thin
lms
non-reecting
the
thickness
of
include:
radar
the
coatings
extra
for
coating
is
military
designed
so
rays from an
that
eye focused
radar
signals
destructively
interfere
when
they
reect
ex tended source
from
both
surfaces,
then
no
signal
will
be
reected
and
an
at innity
aircraft
•
of
thickness
of
the
give
Design
constructive
allow
of
the
and
would
reduce
A
surface
thin
place
solar
for
transmittance
p h e n o m e n a
and
the
a
A
can
typical
takes
of
slicks
the
surfaces
strong
amount
lm
oil
be
of
destructive
thickness
cells.
of
wavelengths
non-reecting
panels
takes
W a v e
undetected.
Measurements
angles)
100
go
Measurements
that
•
could
of
oil
for
energy
added
so
at
this
by
spillage.
interference
to
be
that
and
signals
known
(blooming),
at
being
(at
calculated.
lenses
reection
wavelength
place
caused
electromagnetic
any
of
usefully
destructive
thus
wavelength.
solar
these
surfaces
transmitted.
interference
maximum
rsli
hL
ytisnetni evitaler
dIffractIon and reSoLutIon
If
two
sources
another,
the
eye
said
at
to
of
then
can
be
light
they
tell
the
apertures
affects
to
the
the
close
as
one
sources
The
right
very
seen
two
resolved.
examples
are
are
apart,
diffraction
eye’s
show
in
ability
how
angle
single
then
the
pattern
to
the
to
of
light.
sources
that
resolve
(a) resolved
one
source
takes
sources.
appearance
of
If
are
place
The
two
line
angle θ
sources
a
slit.
will
The
depend
resulting
on
the
diffraction
appearance
is
the
that
takes
addition
place
of
the
at
two
appearance
overlapping
relative
These
and
diffraction
intensity
examples
the
of
light
look
diffraction
patterns.
at
that
at
the
The
different
situation
takes
place
graph
of
angles
is
of
at
a
a
line
slit.
A
the
resultant
alsoshown.
source
more
of
two sources clearly separate
light
common
resultant intensity
(b) just resolved
situation
would
be
a
point
source
of
light,
and
the
diffraction
diraction pattern
that
takes
place
at
a
circular
aperture.
The
situation
is
exactly
of source B
the
same,
aperture.
point
but
As
diffraction
seen
source
is
on
thus
takes
page
97,
place
the
concentric
all
the
way
diffraction
circles
around
pattern
around
the
of
the
the
of source A
central
angle θ
position.
The
geometry
of
the
situation
results
in
a
slightly
slightly dimmer
different
value
for
the
rst
minimum
of
the
diffraction
pattern.
appearance
For
a
slit,
the
rst
minimum
was
at
the
angle
λ
θ
=
two maxima visible
b
(c) not resolved
resultant intensity
For
a
circular
aperture,
the
rst
minimum
is
at
the
angle
1.22 λ
_
θ
diraction pattern
=
b
of source B
of source A
If
two
one
sources
are
diffraction
just
resolved,
pattern
is
then
located
on
the
top
rst
of
minimum
the
of
maximum
of
angle θ
the
other
diffraction
pattern.
This
is
known
as
the
Rayleigh
criterion.
appearance
appears as one source
how
e x ampLe
Late
one
night,
a
student
was
observing
rst
a
far
away
the
distinguish
car
two
was
when
points
of
she
light.
could
T
ake
Since
θ
small
1.8
_
the
θ
=
[x
is
distance
to
car]
x
car
approaching
from
a
long
distance
distance
away
.
between
the
headlights
to
be
1.8
m.
1.8
__
She
noticed
that
when
she
rst
observed
the
When
just
⇒
resolved
x
=
4
1.525
headlights
of
the
car,
they
appeared
to
1.22 × λ
_
θ
one
point
of
light.
Later,
when
the
car
×
10
be
=
=
11.803
≃
12
was
b
closer,
she
became
able
to
see
two
km
separate
7
1.22 × 5 × 10
__
points
of
light.
If
the
wavelength
of
the
=
light
0.004
can
be
taken
her
pupil
as
500
nm
and
the
diameter
of
4
=
is
approximately
4
mm,
1.525
×
10
calculate
Example:
reSoLvance of dIffractIon gratIngS
As
a
result
of
on
a
grating’s
resolvance,
between
possible
a
Rayleigh’s
ability
R,
of
a
to
resolve
diffraction
wavelength
resolvable
criterion,
being
there
is
different
grating
is
limit
placed
wavelengths.
dened
investigated,
wavelength
a
difference,
λ,
as
and
the
the
In
The
the
that
ratio
sodium
are
589.00
smallest
a
close
nm
and
diffraction
one
spectrum
another
589.59
grating,
nm.
the
(the
In
there
Na
order
resolvance
are
two
D-lines).
for
these
must
wavelengths
These
to
be
are
resolved
by
be
Δλ.
589.00
_
λ
_
R
=
=
=
1000
0.59
Δλ
λ
_
R
emission
to
=
Δλ
In
For
m,
any
given
being
and
the
grating,
observed
total
R
(rst
number
of
is
dependent
order:
slits,
m
N,
=
on
1;
on
the
second
the
diffraction
order:
grating
that
m
order,
=
are
2,
etc.)
the
rst
order
illuminated
spectrum,
whereas
requirement
drops
in
to
the
only
at
least
second
500
1000
order
slits
must
be
spectrum,
the
slits.
being
illuminated.
λ
_
R
=
=
mN
Δλ
W a v e
p h e n o m e n a
101
t dl 
hL
analysis
doppLer effect
The
Doppler
effect
of
wave
a
a
as
movement
When
a
of
is
result
the
source
the
of
name
the
given
to
movement
the
of
change
the
of
source
the
frequency
or
A
the
observer.
of
sound
is
moving:
change
Sound
waves
are
emitted
at
a
particular
frequency
from
of
shows
sound
but
•
police
When
a
the
of
that
a
frequency
stationary,
hear
•
quickly
observer,
the
pitch
if
can
car
or
the
is
source
be
moving
will
detected
be
if
away
from
received.
the
source
is
moving.
ambulance
sound
is
frequency
also
observer
of
the
lower
passes
change
you
from
on
high
the
to
road,
low
you
can
frequency.
It
the
is
high
when
it
is
approaching
and
low
when
it
is
going
away.
source.
•
•
The
speed
of
the
sound
wave
in
air
does
not
change,
Radar
detectors
moving
the
motion
of
the
source
means
that
the
wave
fronts
are
object.
‘bunched
up’
ahead
of
the
means
that
the
stationary
observer
receives
of
reduced
For
Reduced
the
wavelength
corresponds
to
an
increased
of
speed
the
of
a
change
reected
in
the
wave.
the
effect
to
observer)
be
noticeable
needs
to
be
with
light
moving
at
waves,
high
the
speed.
a
source
an
of
light
observer,
of
a
the
particular
observer
frequency
will
receive
is
moving
light
of
a
away
lower
sound.
overall
higher
effect
is
frequency
that
than
it
the
observer
was
emitted
will
by
hear
the
sound
source.
at
a
This
frequency
the
source
is
moving
towards
the
observer.
A
If
the
source
be
from
red
the
part
other
of
the
spectrum
colours,
this
is
has
lower
called
a
of
light
is
moving
towards
the
red
observer,
shift.
there
a
blue
shift
mathematIcS of the doppLer effect
movIng Source
moves
the
all
similar
will
Source
Since
than
applies
•
when
the
Doppler
(or
frequency.
The
the
measuring
frequency
from
of
measure
by
wavelength.
If
•
to
this
sound
source
waves
used
do
source.
•
This
be
They
all
frequency
•
can
but
A
to
D
with
velocity,
u
,
speed
of
waves
is
v
Mathematical
stated
on
this
equations
that
apply
to
sound
are
page.
Unfortunately
the
same
analysis
does
not
apply
to
light
λ
o
–
the
velocities
medium.
It
is,
can
not
be
however,
worked
possible
out
to
relative
derive
an
to
the
equation
moving source
for
u
s
that
turns
out
to
be
in
exactly
the
same
form
as
t
the
A
light
•
BC D
stationary
equation
the
for
relative
sound
as
velocity
long
of
as
two
source
conditions
and
are
detector
is
met:
used
stationary
in
obser ver
obser ver
receives sound
receives sound
at lower frequency
at higher frequency
•
the
this
equations.
relative
velocity
Providing
change
in
frequency
v
<<
change
due
to
is
a
lot
less
than
the
speed
of
light.
c
wavelength
relative
relative
motion
Δλ
_
v
source
speed
and
of
observer
Received
v
Δf
f' = f
_
frequency
=
v
≈
u
s
v
c
λ
f
at P
speed
f' = f
v ± u
s
of
light
source
Received
v
relative
f' = f
motion
frequency
v + u
s
at Q
movIng oBServer
e x ampLe
The
frequency
measured
200
Hz
by
when
frequency
of
a
the
will
approaching
a
car’s
horn
stationary
car
be
the
is
at
heard
if
observer
is
observer
rest.
the
car
is
at
1
30
m
s
?
(Speed
of
sound
1
330
m
s
.)
u
o
f
S
=
200
f’
=
?
u
=
30
v
=
330
f
=
Hz
O
1
m
s
s
1
in a time
 t, observer has moved
u
t
m
300
_
o
200
(
300
=
200
×
=
220
Hz
1.1
If observer is moving away from source:
v
u
o
f' = f
v ± u
v
o
f' = f
v
If observer is moving towards source:
v + u
o
f' = f
v
102
W a v e
p h e n o m e n a
s
30
)
in
as
What
air
is
els  liis   dl 
hL
1.
Train
The
going
sound
frequency,
on
the
the
through
emitted
but
the
platform
resolved
passenger
a
station
by
a
sound
will
is
used
of
to
train’s
received
change.
component
that
4.
moving
At
the
by
any
a
of
velocity
the
is
of
passenger
instant
train’s
calculate
whistle
constant
•
standing
time,
it
The
galaxies
relative
light
is
towards
frequency
Receding
–
red
shift
intensities
received
from
of
the
the
stars
different
in
wavelengths
distant
galaxies
can
of
be
analysed.
the
•
The
light
•
The
measured
shows
a
characteristic
absorption
spectrum.
received.
received frequency
associated
wavelengths
with
particular
are
not
elements
the
as
same
as
those
measured
in
the
laboratory.
•
For
the
have
vast
been
spectrum
train passing
majority
shifted
(i.e.
to
red
shift
(see
The
magnitude
of
stars,
towards
lower
page
all
the
the
red
received
end
frequencies).
of
The
the
frequencies
visible
light
shows
a
202).
time
•
of
the
red
shift
is
used
to
calculate
the
through station
recessional
2.
Radars
–
speed
many
countries
the
police
use
radar
to
measure
speed
to
see
provides
evidence
for
the
Big
if
they
are
breaking
the
speed
model
of
the
creation
of
the
Universe.
of
5.
vehicles
and
measurement
Bang
In
velocity
Rotating
object
limit.
The
reected
transmitted
wave
RADAR wave
rotation
measured
side
of
of
by
the
luminous
looking
object
for
objects
a
(e.g.
different
compared
with
the
Sun)
Doppler
the
can
shift
be
on
one
other.
λ
RADAR
light red shifted
rotating star
moving car
transmitter
stationary
V
λ'
Police
light blue shifted
•
Pulse
of
microwave
radiation
of
known
frequency
emitted.
•
Pulse
is
reected
off
moving
car
and
received
back
at
view above pole
source.
6.
•
Difference
in
emitted
and
received
frequencies
is
used
Broadening
•
calculate
speed
of
Absorption
Double
Doppler
effect
taking
Moving
car
receives
a
frequency
that
is
higher
as
Moving
car
it
is
lines
emission
a
moving
atomic
energy
spectra
levels
provide
(see
page
evidence
for
69).
Precise
measurements
show
that
each
individual
level
than
is
emitted
and
place:
•
◊
spectral
car.
discrete
•
of
to
actually
equivalent
to
a
small
but
dened
wavelength
observer.
range.
◊
acts
as
a
moving
source
when
sending
•
signal
3.
Medical
physics
–
blood
ow
red
is
can
blood
used
to
gas
will
be
molecules
use
cells
a
in
pulse
an
measure
of
ultrasound
analogous
the
speed
of
way
a
to
that
measure
a
moving
pulse
car
the
of
speed
of
Different
will
microwaves
are
to
moving
Doppler
so
light
from
molecules
shift.
be
molecules
general
have
a
Doppler
range
of
speeds
broadening
of
so
there
the
discrete
(above).
A
higher
kinetic
receiver
a
wavelengths.
•
transmitter
subjected
measurements
•
Doctors
The
back.
temperature
energies
spectral
and
means
hence
a
wider
more
distribution
broadening
to
of
the
line.
skin
incident
reected
sound
sound
ν
s
red blood cell
W a v e
p h e n o m e n a
103
IB Questions – wave phenomena
HL
1.
When
pitch
is
at
a
train
of
the
rest.
travels
sound
This
is
towards
you
hear
you
is
sounding
different
its
from
whistle,
when
the
the
much
train
displacement
the
sound
waves
are
travelling
faster
toward
the
wave
closer
graph
fronts
of
the
sound
reaching
you
are
below
atom
the
wave
further
the
carbon
carbon
atom
atom
may
such
be
that
any
ignored.
shows
x
in
the
from
a
its
variation
with
equilibrium
molecule
of
time
t
position
of
of
a
methane.
spaced
together.
10
x / ×10
C.
than
the
thedisplacement
you.
hydrogen
B.
of
because
The
A.
lessmassive
fronts
of
the
sound
reaching
you
are
spaced
m
2.0
apart.
1.5
D.
the
sound
the
speed
frequency
emitted
by
the
whistle
changes
with
1.0
of
the
train.
0.5
2.
A
car
is
travelling
at
constant
speed
towards
a
stationary
13
t / ×10
0.0
observer
emitted
note
a)
of
whilst
by
the
its
horn
horn
frequency
With
the
aid
frequency
is
is
a
sounded.
660
720
of
is
Hz.
The
The
frequency
observer,
of
the
however,
note
hears
a
Hz.
0.5
1.0
diagram,
explain
why
a
higher
1.5
heard.
[2]
2.0
1
b)
If
the
speed
of
sound
is
330
m
s
,
calculate
the
27
The
speed
of
the
car.
mass
from
3.
This
the
question
emission
Light
from
upon
a
is
about
using
spectrum
a
of
hydrogen
atom
is
1.7
×
10
kg.
Use
data
[2]
sodium
of
a
diffraction
grating
to
the
graph
above
view
(i)
to
determine
(ii)
to
show
its
amplitude
of
oscillation.
[1]
sodium.
discharge
tube
is
incident
that
the
frequency
of
its
oscillation
is
normally
13
9.1
×
10
Hz.
[2]
5
diffraction
grating
having
8.00
×
10
lines
per
metre.
(iii)
The
spectrum
contains
a
double
yellow
line
of
to
show
that
the
maximum
kinetic
energy
of
the
wavelengths
18
hydrogen
589nm
and
590
c)
a)
Determine
the
angular
separation
of
the
two
lines
in
the
second
order
spectrum.
State
why
Sketch
a
graph
velocity
it
is
more
difcult
to
observe
the
line
when
viewed
in
the
rst
order
spectrum.
This
question
is
about
thin
lm
v
to
of
transparent
thin
lm
is
starting
the
Assuming
as
shown
in
the
the
[2]
variationwith
hydrogen
at
t
=
velocity
that
the
0.
atom
used
diagram
time
one
t
of
period
(There
is
no
need
to
of
add
[3]
of
the
hydrogen
atom
is
to
coat
its
frequency
of
oscillation
f
is
given
spectacle
expression
below.
k
_
1
_
coating, refractive
=
glass lens, refractive
√m
2π
index = 1.00
for
axis.)
motion
f
air, refractive
J.
interference.
sometimes
bythe
lenses
show
the
simpleharmonic,
A
10
[1]
d)
4.
×
double
valuesto
yellow
6.2
[4]
oscillation
b)
is
when
the
viewed
atom
nm.
index = 1.30
p
index = 1.53
where
k
is
hydrogen
the
force
atom
and
per
unit
the
carbon
displacement
atom
and
between
m
is
the
a
mass
p
ofa
proton.
(i)
Show
that
incoming
the
value
of
k
is
approximately
1
560
N
m
.
[1]
light
(ii)
Estimate,
using
your
answer
maximumacceleration
e)
boundary A
a)
State
the
phase
is
transmitted
(ii)
is
reected
at
at
is
classied
(i)
Describe
(ii)
Electromagnetic
what
which
occurs
boundary
boundary
A
to
into
light
is
transmitted
the
b)
Light
of
at
the
lm.
9.1
[1]
B.
boundary
A
from
the
lm
into
[1]
Determine
required
so
that
570
the
the
nm
in
air
smallest
reection
is
is
incident
thickness
of
minimized
on
the
for
the
coating
normal
incidence.
5.
Simple
a)
A
[2]
harmonic
body
is
conditions
harmonic
b)
In
a
and
necessary
model
the
for
the
greenhouse
equilibrium.
the
body
to
effect
State
the
execute
two
simple
I B
[2]
of
carbon
massesattached
104
and
from
motion.
simple
atom
motion
displaced
by
a
greenhouse
meant
by
a
the
atom.
[2]
gas.
a
methane
atom
can
spring.
Q u e s t I o n s
A
radiation
of
greenhouse
gas.
[2]
frequency
×
10
Hz
is
in
the
infrared
electromagnetic
spectrum.
theinformation
given
in
region
Suggest,
(b)(ii),
of
the
based
why
on
methane
is
[1]
air.
wavelength
coating.
a
(d)(i),
hydrogen
13
that
classied
(iii)
is
as
to
the
boundary B
change
(i)
Methane
of
molecule,
be
regarded
hydrogen
–
a
hydrogen
as
two
atom
w a v e
is
p h e n o m e n a
as
a
greenhouse
gas.
[2]
s
10
f i e l d s
Hl
P (  )
•
describing fields:   e
The
•
concept
the
of
eld
lines
gravitational
can
eld,
g,
be
used
around
a
to
visually
mass
(or
the
collection
This
of
test
masses)
•
the
The
represent:
electric
eld,
E,
around
a
charge
(or
collection
•
of
Magnetic
elds
can
page
In
cases
object
•
all
placed
at
a
gravitational
also
be
the
represented
eld
particular
eld
=
is
the
point
force
in
per
using
force
the
unit
per
eld
tes t
eld
lines
unit
test
•
(see
with:
p o i nt
An
is
N
kg
mas s
This
electric
eld
=
force
per
unit
test
point
positive
force
both
line,
the
angles
is
represented
gravitational
work
same
to
a
moved
method
the
by
the
direction
of
and
electric
elds,
as
a
the
eld
of
be
done
(force
and
distance
line,
no
work
will
be
done
(force
perpendicular).
mapping
needed
new
potential
will
direction)
are
energy
denes
gravitational
•
for
in
distance
consider
eld.
)
eld
alternative
to
the
moved:
are
right
and
1
(units:
at
point
that,
is
a
moved
of
lines.
means
object
along
charges).
61).
direction
eld
the
to
concepts
(see
elds
move
of
around
between
electric
an
object
points
potential
in
the
and
below).
charge
1
(units:
Forces
and
•
the
The
the
N
are
C
)
vectors
direction
and
of
magnitude
eld
more
page
lines
precise
eld
the
of
are
lines
force
the
to
represent
that
force
o ne
deni ti o n,
would
is
(fo r
eld
object
of
(gravitational
as
the
placed
the
eld.
the
In
potential
force
at
an
a
or
per
electric)
unit
particular
analogous
test
is
point
point
in
denition,
(gravitational,
V
,
or
test
a
test
how
object.
clo se
of
a
ux
)
is
dened
as
the
Potential
is
points,
and
A
energy
point
result
object
of
the
that
the
eld.
and
A
the
B,
in
an
at
A
there
per
If
will
be
mathematical
per
electric
the
eld)
test
is
at
the
on
a
two
test
(or
that
B
difference,
done
object.
eld
means
potential
potential
work
unit
gravitational
In
general,
moving
work
will
be
is
a
moving
charge
done.
different.
a
mass
between
When
work
Between
the
is
between
two
done,
points
two
points,
A
the
and
B,
∆V
object
points
must
as
it
moves
between
two
points
then
the
increase.
object
The
relationships
a
between
If
work
is
done
by
the
test
object
as
it
moves
between
the
two
points
then
the
full
are
between
the
two
points
must
decrease.
shown
Gravitational
page
in
and
a
positive
potential
energy
B,
potential
potential
on
on
Potential difference ∆V (electric and gravitational)
•
as
by
m a gn e t ic
e
has
magnitude
a
e xa m p le
•
g
electric,V
unit
the
by
112).
or electric)
dened
an
d eni t io n
Potential , V (gravitational
The
felt
rep r e s e nted
another
se e
both
be
potential
difference
between
two
points,
110.
work
potential,
V
∆V
=
g
done
moving
a
test
mass
___
energy
_
Gravitational
=
g
test
mass
mass
–1
Units
1
Units
of
V
=
J kg
of
∆V
=
J kg
g
g
Electric
potential
difference
between
two
points,
energy
_
Electric
potential,
V
=
e
charge
work
done
moving
a
test
charge
___
∆V =
e
test
1
Units
of
V
=
J C
(or
charge
volts)
e
1
Units
of
∆V
=
J C
or
V
(volts)
e
Thus
to
points
calculate
in
a
eld
W
=
q∆V
W
=
m∆V
the
we
work
done,
W,
in
moving
a
charge
q
or
a
mass
m
between
two
have:
e
g
f i e l d s
105
Hl
ep
e x amPles of equiPotentials
equiPotential surfaces
The
best
around
way
a
two
is
the
object
same.
dimensions
equipotential.
with
representing
charged
potential
In
of
the
A
to
These
they
good
contour
is
on
are
of
a
the
identify
would
way
lines
how
electric
the
called
be
potential
regions
where
equipotential
represented
visualizing
as
these
varies
The
diagrams
below
show
is
positively
+40 V
of
to
equipotential
lines
for
various
situations.
surfaces.
lines
lines
the
all
ve with units
1
charged
of J kg
sphere
start
map.
25
+
mass
contour lines
50
+30 V
3-d surface
+20 V
100
shallow
+10 V
200
steep
contour lines
contour lines
close
fur ther apar t
Equipotentials
point
The
contour
heights
that
are
high
lines
the
an
are
can
a
be
height.
low
that
right
the
represents
left.
Points
that
potential
gravitational
in
a
Each
are
and
the
line
points
potential.
gravitational
situation
up
up
sphere
that
are
a
Contour
a
low
lines
are
150 V
150 V
100 V
done
50 V
joining
have
the
potential.
30 V
20 V
charge
rightshows
10 V
1
200 J kg
the
and
points
represent
eld.
point
The
charge-conducting
changing
joins
high
a
eld.
drawn
electric
the
on
gravitational
electric
points
same
of
same
equipotential
same
Lines
up
at
on
landscape
represent
of
with
diagram
the
value
down
The
of
outside
mass.
equipotentials
for
40 V
an
1
150 J kg
isolatedpositive
point
1
100 J kg
charge.
Equipotentials
point
for
two
point
charges
(same
charge)
and
two
masses.
rel ationsHiP to field lines
There
lines
is
of
a
another.
line,
simple
relationship
equipotential
we
Imagine
stay
at
the
the
–
they
between
are
contour
same
electric
always
lines.
height
If
at
we
eld
lines
right
angles
move
along
in
the
gravitational
we
are
moving
and
to
a
one
+
contour
eld.
-
This
-80 V
80 V
does
to
not
the
require
work
gravitational
because
force.
Whenever
we
move
at
right
along
angles
an
60 V
-60 V
electric
40 V
equipotential
line,
we
are
moving
between
points
that
have
-40 V
the
20 V
same
electric
Moving
at
potential
right
angles
–
in
to
other
the
words,
electric
no
eld
is
work
the
is
being
only
way
done.
to
Equipotentials
avoid
doing
work
in
an
electric
eld.
Thus
equipotential
-20 V
0 V
for
two
point
charges
(equal
and
opposite
lines
charges).
must
be
at
right
angles
to
eld
lines
as
shown
below.
+V =
70 V
60 V
50 V
40 V
zero potential often
d
30 V
taken to be negative
20 V
terminal of battery
10 V
0 V
eld line
Equipotential
It
should
be
lines
noted
between
that
charged
although
the
parallel
correct
plates.
denition
of
equipotential
zero
potential
is
at
innity,
most
of
the
time
we
are
not
lines
really
interested
interested
that
in
in
some
in
the
the
actual
value
of
situations
the
(such
value
of
potential,
difference
as
the
in
we
are
potential.
parallel
only
This
conducting
means
plates)
it
always at 90°
is
Field
lines
and
equipotentials
are
at
right
angles
easier
setting
than
106
f i e l d s
to
sea
imagine
level
correctly
as
the
the
using
zero
zero
at
for
innity
a
different
point.
gravitational
for
the
zero.
This
contour
is
just
lines
like
rather
Hl
g p   p
gravitational Potential energy
It
is
easy
when
a
Earth’s
to
work
mass
out
moves
the
difference
between
two
in
gravitational
different
heights
energy
near
the
it
lost
The
difference
in
energies
=
m g(h
-
h
2
potential
must
value
point
)
in
•
are
this
two
important
derivation
strength
g
is
has
points
assumed
constant.
to
be
of
of
energy
the
negative
at
gravitational
space
is
mass,
moving
dened
a
in
m,
given
potential
as
the
was
towards
zero
point,
done
innity,
M,
the
and
potential
P
.
energy
work
at
mass
of
in
a
mass
at
moving
any
it
from
1
innity
There
energy
potential
energy
the
surface.
The
If
to
that
point.
The
mathematics
needed
to
work
this
out
note:
that
However,
the
gravitational
Newton’s
is
not
trivial
It
turns
since
the
force
changes
with
distance.
eld
theory
out
that
of
G M m
_
universal
with
gravitation
distance.
vertical
This
distance
states
that
equation
we
move
the
can
is
eld
only
not
MUST
be
very
Gravitational
CHANGE
used
if
the
equation
gives
zero
everyday
The
true
assumes
PE
at
the
situations
zero
of
that
the
surface
but
it
gravitational
of
is
the
Earth.
of
mass
m
=
-
(due
to
M)
large.
gravitational
not
energy
r
the
This
•
potential
potential
This
works
energy
of
is
the
a
scalar
path
quantity
taken
from
(measured
in
joules)
and
is
independent
innity.
for
fundamental.
potential
energy
is
taken
as
innity.
potential energy decreases as
zero of potential energy taken
gravitational force does work
to be at innity
M
m
F
4
F
m
3
F
m
2
F
m
1
as m moves towards M in the force on m increases
gravitational Potential
We
can
dene
the
energy
the
esc aPe sPeed
gravitational
potential
V
that
measures
The
escape
speed
of
a
rocket
is
the
speed
needed
to
be
able
to
g
per
unit
test
mass.
escape
the
getting
(work
W
_
V
to
gravitational
an
innite
attraction
distance
of
the
planet.
This
means
away.
done)
__
=
g
m
(test
We
mass)
know
that
gravitational
GM
_
potential
at
the
surface
of
a
planet
=
-
1
The
SI
units
of
gravitational
potential
are
J
kg
.
It
is
a
scalar
R
p
quantity.
(where
R
is
the
radius
of
the
planet)
p
Using
Newton’s
law
of
universal
gravitation,
we
can
work
out
This
the
gravitational
potential
at
a
distance
r
from
any
point
means
that
for
a
rocket
of
mass.
mass
m,
the
difference
between
GMm
_
its
V
energy
at
the
surface
and
at
innity
=
g
R
p
r
GMm
_
Therefore
the
minimum
kinetic
energy
needed
=
R
p
In
other
words,
GM
V
g
= r
1
GMm
_
2
m
(v
)
=
esc
R
2
p
so
______
This
formula
and
the
graph
also
works
for
spherical
masses
2GM
_
v
=
√(
esc
(planets
masses
etc.).
is
just
The
the
gravitational
addition
of
potential
the
as
individual
a
result
of
potentials.
lots
of
This
This
an
easy
sum
since
potential
potential due to
1
a
scalar
p
derivation
assumes
the
planet
is
isolated.
quantity.
potential due to
A
1
m
is
)
R
is
= -40 J kg
1
m
2
= -30 J kg
e x amPle
The
escape
speed
from
an
isolated
planet
like
Earth
(radius
of
6
Earth
R
=
6.37
×
10
m)
is
calculated
as
follows:
E
__________________________
11
24
2 × 6.67 × 10
× 5.98 × 10
___
v
=
esc
√(
)
6
6.37
×
10
m
overall potential
1
= (-40) + (-30) J kg
8
m
m
=
2
1
1

× 10
)
√(1.25
m
s
1
= -70 J kg
4
=
1.12
×
10
1
m
s
1
Once
you
another,
have
the
the
potential
difference
at
one
between
point
them
is
and
the
the
potential
energy
you
at
≈
The
to
move
a
unit
mass
between
the
two
points.
It
is
vast
the
path
km
s
majority
of
rockets
sent
into
space
are
destined
to
independent
orbit
of
11
need
the
Earth
so
they
leave
with
a
speed
that
is
less
than
the
taken.
escape
speed.
f i e l d s
107
Hl
o 
gravitational Potential gradient
energy of an orbiting s atellite
GMm
_
In
the
diagram
below,
a
point
test
mass
m
moves
in
a
We
already
know
that
the
gravitational
energy
=
r
gravitational
eld
from
point
A
to
point
B.
_____
1
The
difference
in
gravitational
potential,
The
ΔV
kinetic
energy
GM
_
2
=
m
g
v
but
v
=
√(
)
r
2
(Circular
average
force
×
distance
motion)
moved
______________________
=
=
-g×∆r
m
GM
_
1
∴
The
negative
doing
the
sign
work
direction
from
has
done
is
g.
is
because
directed
Since
g
is
directed
away
the
from
M
gravitational
towards
and
M,
thus
force
is
in
but
the
the
opposite
attractive,
kinetic
energy
m
=
=
r
2
force
So
total
energy
=
KE
+
GMm
_
1
r
2
PE
work
1
GMm
_
GMm
_
r
r
be
in
going
from
A
to
B,
so
the
potential
at
A
=
<
2
potential
at
B.
Note
GMm
_
1
=
to
r
2
that:
1
distance moved
•
In
the
orbit
the
magnitude
of
the
KE
=
magnitude
2
of
∆r
•
the
The
overall
must
have
have
mass M
PE.
energy
a
total
enough
of
the
energy
energy
to
satellite
less
is
than
escape
the
negative.
zero
(A
satellite
otherwise
Earth’s
it
would
gravitional
eld.)
distance r
•
A
In
order
orbit,
B
to
the
increase
move
total
in
from
energy
orbital
a
small
must
radius
radius
increase.
makes
the
orbit
To
to
be
total
a
large
precise,
energy
go
radius
an
from
a
F
average
large
average eld between A and B, g =
negative
number
to
a
smaller
negative
number
–
this
m
is
an
increase.
(towards M)
This
ΔV
=
-g
×
can
be
summarized
in
graphical
form.
Δr
g
ygrene
ΔV
_
g
=
∆r
ΔV
___
1
is
called
the
potential
gradient.
It
has
units
of
J
kg
kinetic energy
in orbit
1
m
∆r
1
(which
are
the
same
as
N
kg
2
or
m
s
).
orbital
total energy
The
gravitational
gradient.
elds
The
(see
eld
strength
equivalent
page
is
equal
relationship
to
also
minus
applies
the
for
potential
radius
electric
109).
gravitational potential
energy in orbit
The
WeigHtlessness
One
say
way
that
of
it
is
supporting
If
the
the
the
different
extreme
lift
value
weight
of
the
of
force
a
person
recorded
is
to
on
a
of
set
values
the
version
cable
up
breaks
in
a
lift,
they
depending
on
the
of
apparent
In
the
situations
lift
(and
occurs
if
passenger)
–2
accelerates
the
down
at
10
station
the
m
s
to
space
stay
in
resultant
.
The
in
same
an
and
to
be
the
orbiting
the
station,
force
the
of
of
space
astronaut
the
weightless
term
weightlessness
for
‘weight’,
objects
station
are
in
gravitational
in
it
the
is
free-fall
would
free-fall
pull
duration
better
on
also
to
of
call
the
this
fall.
Given
situation
together.
appear
weightless.
The
together.
the
astronaut
provides
the
needed
orbit.
force
centripetal
appear
ambiguity
astronaut
centripetal
these
would
possible
space
would
lift.
and
person
the
An
were
acceleration
An
the
scale.
scales
record
dening
This
causes
the
acceleration.
is
true
for
the
orbital path
2
accelerating down at 10 m s
gravitational
R = zero
satellite
and
pull
the
acceleration.
contact
2
resultant force
a = 10 m s
down = W
so,
There
force
satellite
once
and
on
the
is
the
velocity
no
between
again,
is a circle
satellite’s
the
astronaut
we
have
gravitational attraction on
W
apparent
weightlessness.
astronaut provides centripetal
no weight will
force needed to stay in orbit
be recorded
on scales
108
f i e l d s
e p   p
Hl
Potential and Potential difference
The
concept
of
electrical
potential
difference
Potential due to more tHan
between
two
points
was
introduced
one cHarge
on
page
105.
As
the
name
implies,
potential
difference
is
just
the
difference
If
between
the
potential
at
one
point
and
the
potential
at
another.
Potential
several
charges
potential
simply
a
measure
of
the
total
electrical
energy
per
unit
charge
at
a
given
space.
The
denition
is
very
similar
to
that
of
gravitational
at
up
point,
the
contribute
it
can
be
to
the
total
calculated
individual
potentials
by
due
to
the
potential.
individual
potential increases
a
point
adding
in
all
is
charges.
zero of potential
Q
3
r
3
taken to be at
as charge is moved
P
innity
in against repulsion
+Q
r
2
r
q
q
F
1
q
F
4
q
F
3
2
F
1
potential at point P
As q comes in the force on q increases.
= (potential due to Q
)
Q
1
If
the
total
work
done
in
bringing
a
positive
test
charge
q
from
innity
to
2
a
Q
1
point
in
dened
an
to
electric
eld
is
W,
then
the
electric
potential
at
that
point,
V,
)
+ (potential due to Q
)
2
is
3
be
W
_
V
+ (potential due to Q
The
=
electric
potential
at
any
point
outside
a
q
charged
The
units
for
potential
are
the
same
as
C
or
volts.
Q
_
V
=
4πε
r
o
This
equation
only
applies
the
units
for
potential
as
single
point
to
if
its
Q
a
charge.
sphere
is
exactly
the
same
difference:
V laitnetop
1
J
conducting
all
the
charge
had
been
concentrated
at
centre.
Potential and field strengtH
V =
4πε
r
o
potential dierence
= (V
V
B
A
)
potential = V
A
distance
potential = V
B
F
F
B
A
+Q
q
Potential inside a cHarged sPHere
Charge
•
distribute
Outside
as
•
will
if
all
Inside
the
the
sphere,
charge
the
itself
was
sphere,
uniformly
the
eld
on
lines
concentrated
there
is
no
net
the
outside
of
a
and
equipotential
at
point
a
q
B
at
the
contribution
conducting
surfaces
centre
from
the
of
are
the
A
sphere.
the
same
sphere.
charges
distance d
outside
Bringing a positive charge from A to B
the
sphere
and
the
electric
eld
is
zero.
The
potential
gradient
is
thus
also
means work needs to be done against
zero
meaning
that
every
point
inside
the
sphere
is
at
the
same
potential
–
the electrostatic force.
the
The
potential
graphs
at
below
the
sphere’s
show
how
surface.
eld
and
potential
vary
for
a
sphere
of
radius
direction of
a
at A and B
point charge
1
(a)
/ V m
Q (positive)
B
A
point
charge

direction of force
max
slope falls
2
applied by external
o as r
agent on test charge
q at A and B
a
1


The
Q
work
done
δW
=
- E q δx
[the
negative
sign
=
max
 4πε
2


r/m
is
a
because
the
direction
of
the
force
needed
opposite
to
the
direction
to
0
do
the
work
is
V/ V
Therefore
E
=
E]
δ W
_
1
(b)
of
q
δ
x
point
δ V
_
=
charge
-
In
δ W
_
[
δ
since
δ
V
=
q
x
]
words,
V
max
slope falls
electric
eld
=
-
potential
gradient
1
o as r
volt
_
Units
=
1
(V
m
1
),
N
C
metre
Q
=
 4πε

max


V
1
a
0
a
r/m
f i e l d s
109
e    p
Hl
comParison betWeen electric & gravitational field
Electrostatics
Force
can
be
Coulomb’s
Gravitational
attractive
law
q
–
for
q
2
=
attractive
Newton’s
law
m
2
F
=
for
point
masses
2
G
2
r
–
m
1
_
k
2
4πε
charges
always
_
=
E
Force
q
1
_
F
repulsive
point
q
1
or
2
r
r
o
Electric
eld
Gravitational
charge
electric
producing
mass
field
gravitational
=
F
k
2
q
g
2
4πε
r
o
2
1
=
=
r
m
2
2
test
Electric
charge
potential
due
test
to
a
point
field
Gm
1
1
=
=
producing
field
q
q
F
E
eld
field
Gravitational
charge
r
mass
potential
due
to
a
point
mass,
m
1
q
q
1
=
V
=
1
k
_
e
4πε
Gm
1
_
V
r
r
=
-
g
r
o
Electric
potential
Gravitational
gradient
=
_
g
-
=
-
∆r
Electric
∆r
potential
q
energy
q
1
=
qV
p
Gravitational
q
2
=
energy
GMm
_
2
_
=
k
E
e
4πε
potential
q
1
_
E
gradient
g
e
_
E
potential
∆V
∆V
=
mV
p
r
r
=
-
g
r
o
uniform fields
Field
A
strength
constant
is
equal
eld
thus
to
minus
the
potential
2.
gradient.
Constant
The
•
A
constant
will
•
In
potential
equate
3D
this
planes
to
a
that
are
gradient
xed
means
electrical
eld
means:
change
that
i.e.
in
a
given
spaced
in
distance
potential.
equipotential
equally
increase
surfaces
apart.
In
2D
will
be
at
electric
(e.g.
a
capacitor
in
the
middle
In
the
diagram
plates
equipotential
eld
is
V
in
–
between
see
page
charged
52)
is
parallel
effectively
plates
constant
section.
and
below,
the
the
potential
separation
of
difference
the
plates
is
across
d.
Thus
the
the
V
lines
will
be
equally
spaced.
electric
potential
gradient
is
and
the
constant
eld
in
the
d
V
centre
•
Field
lines
(perpendicular
to
equipotential
surfaces)
will
of
the
spaced
parallel
Constant
gravitational
gravitational
eld
near
the
surface
of
a
planet
is
At
the
The
units
V
m
1
and
N
C
are
and
can
both
be
used
for
electric
eld
between
E
the
surface
of
the
Earth,
the
eld
lines
cannot
remain
two
charged
uniform
throughout
effect.
is
the
parallel
plates
and
effectively
there
constant.
1
.
eld
plates
The
=
lines.
Strictly,
1.
E
d
equivalent
equally
plates,
be
will
will
be
an
edge
It
straightforward
to
show
be
that
at
the
edge,
the
eld
must
have
dropped
to
half
the
2
perpendicular
to
the
Earth’s
surface.
Since
g
=
9.81
m
s
,
the
value
1
potential
gradient
must
also
be
9.81
J
kg
in
the
m
.
that
are
1,000
m
apart
represent
but
modelling
the
eld
as
constant
Equipotential
everywhere
surfaces
centre,
1
changes
of
between
the
parallel
plates
with
the
edge
effects
potential
occurring
beyond
the
limits
of
the
plates
can
be
acceptable.
1
approximately
equal
to
10
kJ
kg
+V =
PE using PE = mgh
PE from 1st principles
zero at surface
zero at innity
70 V
60 V
50 V
40 V
zero potential often
d
height = 3 km
30 V

6.255 × 10
30 000 J
J
20 V
10 V
taken to be negative
terminal of battery
0 V
height = 2 km

6.25 × 10
20 000 J
J
Equipotentials
height = 1 km

6.253 × 10
10 000 J
J
PE ierence PE = 10 000 J

6.2603 × 10
0 J
surface of Ear th
110
f i e l d s
J
lines
between
charged
parallel
plates.
ib q – 
Hl
1.
Which
one
of
the
variation
of
the
potential
energy,
following
kinetic
GPE,
graphs
energy,
of
an
best
KE,
and
orbiting
represents
of
the
satellite
the
(ii)
Calculate
gravitational
at
a
distance
whilst
the
centre
of
the
Calculate
ygrene
ygrene
B.
KE
b)
(i)
the
of
the
Space
Shuttle
[2]
energy
Shuttle
What
forces,
inside
needed
to
put
the
the
into
if
orbit.
any,
Space
act
[2]
on
Shuttle
the
astronauts
whilst
in
orbit?
[1]
KE
(ii)
0
Explain
0
Shuttle
r
why
feel
astronauts
aboard
the
Space
weightless.
[2]
r
GPE
c)
GPE
ygrene
D.
ygrene
C.
speed
Earth?
Space
A.
the
orbit.
r
(iii)
from
in
Imagine
an
astronaut
Space
Shuttle,
and
Space
Shuttle.
By
approximations,
2
10
m
m
outside
from
making
calculate
the
the
exterior
centre
of
walls
mass
of
appropriate
assumptions
how
would
long
it
take
of
the
the
and
for
this
GPE
astronaut
force
of
to
be
pulled
gravity
alone.
back
to
the
Space
Shuttle
by
the
[7]
KE
0
0
5.
r
a)
The
diagram
below
shows
a
planet
of
mass
M
and
radius
R
p
r
KE
GPE
R
p
X
M
R
2.
The
diagram
between
two
below
illustrates
charged
parallel
+
some
equipotential
metal
+
lines
plates.
+
+
The
gravitational
distance
R
80 V
from
potential
the
centre
V
due
of
the
to
the
planet
planet
is
at
given
point
X
by
GM
_
V
=
R
60 V
where
0.1 m
G
is
the
universal
gravitational
constant.
40 V
Show
that
20 V
the
gravitational
potential
V
can
be
expressed
as
2
g
R
0
p
_
V
=
R
-
-
-
where
g
is
the
acceleration
of
free-fall
at
the
0
surface
The
electric
eld
strength
between
the
plates
6
NC
B.
8
NC
the
planet.
[3]
is
1
A.
of
b)
1
C.
600
NC
D.
800
NC
1
The
graph
below
due
to
planet
the
shows
varies
how
the
gravitational
with
distance
R
from
potential
the
V
centre
1
of
the
planet
for
values
of
R
greater
than
R
,
where
p
6
R
=
2.5
×
10
m.
p
3.
The
diagram
shows
equipotential
lines
due
to
two
objects
6
R/10
m
1
5
10
15
0
gk J
object 1
object 2
1
01/V
6
2
3
4
5
6
The
two
objects
could
be
7
A.
electric
B.
masses
charges
of
the
same
sign
only.
8
9
only.
10
C.
electric
D.
masses
charges
of
opposite
sign
only.
Use
or
electric
charges
of
any
sign.
(i)
4.
The
Space
the
Earth.
the
Space
6.0
×
Shuttle
The
orbits
shape
of
about
the
300
orbit
is
km
above
circular,
the
data
from
the
graph
to
[1]
the
and
surface
the
mass
determine
a
value
of
of
of
g
.
[2]
0
(ii)
show
that
the
minimum
energy
required
to
4
Shuttle
is
6.8
×
10
kg.
The
mass
of
the
24
10
Earth
raise
is
a
satellite
and
radius
of
the
Earth
is
6.4
×
10
mass
3000
kg
to
a
height
6
6
kg,
of
3.0
m.
×
10
m
above
the
surface
of
the
planet
10
is
a)
(i)
Calculate
the
gravitational
launch
and
change
in
potential
its
arrival
the
Space
energy
in
orbit.
about
1.7
×
10
J.
[3]
Shuttle’s
between
its
[3]
i B
Q u e s t i o n s
–
f i e l d s
111
11
E l E c t r o m a g n E t i c
i n d u c t i o n
ie eee e (e)
HL
inducEd Emf
When
a
production of inducEd Emf by rEl ativE
conductor
moves
through
a
magnetic
eld,
an
emf
is
motion
induced.
The
emf
induced
depends
on:
An
•
The
speed
of
the
wire.
emf
ux
is
are
induced
cut.
situation;
•
The
strength
•
The
length
of
the
magnetic
the
wire
in
the
magnetic
can
calculate
the
magnitude
the
of
the
induced
considering
an
electron
at
a
conductor
is
more
whenever
than
mathematical
just
a
lines
way
of
of
magnetic
picturing
the
denition.
magnetic
equilibrium
in
ux
eld
∆ϕ
is
perpendicular
passing
through
to
the
the
area
surface,
∆A
is
the
dened
in
emf
terms
by
a
eld.
magnetic
We
has
in
ux
eld.
If
of
it
But
the
of
the
magnetic
eld
strength
B
as
follows.
middle
∆ϕ
of
the
wire.
The
induced
electric
force
and
the
magnetic
_
force
∆ϕ
=
B
∆A,
so
B
=
∆A
are
balanced.
ϕ
In
negative end
a
uniform
eld,
B
=

A
B
B

An
F
alternative
name
for
‘magnetic
eld
strength’
is
‘ux
density’.
m
electric eld down wire
If
the
area
is
not
perpendicular,
but
at
an
angle
θ
to
the
eld
due to charge separation
lines,
potential
the
equation
becomes
2
ϕ
charge q
=
B
A
cos
θ
(units:
T
m
)
length l
dierence V
θ
F
is
the
angle
between
B
and
the
normal
to
the
surface.
e
B
B
v
Flux
1
due
to
emf,
F
=
E
×
q
=
e
(
)
×
force
due
to
movement,
F
be
measured
in
webers
(Wb),
dened
as
follows.
=
=
1
T
m
relationships
allow
us
to
calculate
the
induced
emf
ε
in
q
a
l
Magnetic
Wb
These
V
force
also
2
positive end
Electrical
can
moving
wave
is
terms
of
ux.
B q v
m
in a time ∆t:
V
So
B q v
=
(
l
q
)
l
V
=
B
l
v
N
As
no
current
is
owing,
the
emf
ε
=
ε
If
the
wire
was
part
of
a
complete
=
potential
S
difference
B l v
circuit
(outside
the
magnetic
area swept out ∆A = l∆x
∆x
eld),
the
emf
induced
would
cause
a
current
to
ow.
∆x
_
ε
=
B
l
v
since
v
B l ∆x
_
=
then
ε
=
∆t
× B
×
×
×
×
×
×
×
∆t
boundary
of B
but
l
∆x
=
∆A,
the
area
‘swept
out’
by
the
conductor
in
a
time
B ∆A
____
∆t
so
ε
=
∆t
b
c
∆ϕ
___
but
×
×
×
×
B
∆A
=
∆ϕ
so
ε
=
∆t
ex ternal agent
coil
emf
exer ts force F
In
words,
‘the
emf
induced
is
equal
to
the
rate
of
cutting
of
ε
×
×
×
×
×
×
a
×
×
×
ux’.
l
If
the
conductor
moved,
the
same
is
effect
kept
is
stationary
and
the
magnets
are
produced.
×
velocity v
×
d
×
E x amplE
induced
1
An
current I
×
×
×
aeroplane
that
can
be
Vertical
If
this
situation
was
repeated
with
a
rectangular
coil
with
each
total
section
emf
200
ab
would
generate
an
emf
equal
to
generated
m
generated
will
thus
across
magnetic
across
eld
=
10
×
30
×
B v l N
that
in
one
side
and
the
was
inside
emf.
112
the
the
other
The
would
of
side
the
two
situation
coil
(ab)
(cd)
is
magnetic
emfs
above,
is
a
moving
outside
eld,
would
current
through
the
each
oppose
eld.
side
one
only
the
If
ows
would
another
whole
generate
and
no
eld
coil
an
current
ow.
E l E c t r o m a g n E t i c
=
6
×
10
=
0.06
when
magnetic
the
=
wings
be
2
Note
Estimate
the
maximum
wings.
5
emf
=
.
its
10
T
(approximately)
Bvl.
5
ε
s
component
Earth’s
Length
The
at
N
of
turns,
ies
×
i n d u c t i o n
V
V
200
=
30
m
(estimated)
pd
le'   f' 
HL
transformEr-inducEd Emf
lEnz’s l aw
Lenz’s
law
states
that
An
emf
is
magnetic
‘The
direction
able
to
ow,
of
it
the
induced
would
oppose
emf
the
is
such
that
change
if
which
an
induced
caused
current
also
produced
eld
changes
in
a
with
wire
if
the
time.
were
it.’
If
the
turn
amount
of
a
coil
of
is
ux
ϕ,
passing
then
the
through
total
ux
one
linkage
(2)
(1)
with
I
all
N
turns
Flux
of
linkage
the
=
coil
N
is
given
by
ϕ
motion
The
universal
situations
S
N
N
rule
that
involving
applies
induced
to
all
emf
can
now
be
S
stated
as
motion
‘The
magnitude
proportional
to
of
an
the
induced
rate
of
emf
change
is
of
ux
linkage.’
Current induced in this direction,
∆ϕ
_
This
the force would be upwards
is
known
(left-hand rule)
∴ original motion would
Faraday’s
the magnet
combined
opposing motion.
law
can
be
explained
in
terms
of
the
conservation
of
energy.
The
and
Lenz’s
together
mathematical
generated
law
Faraday’s
law
ε
=
N
∆t
the induced eld would repel
be opposed.
Lenz’s
as
If current were induced this way,
in
in
the
statement
a
coil
law
of
N
can
be
following
for
the
turns
emf,
with
a
ε,
rate
of
∆ϕ
electrical
___
change
of
ux
through
the
coil
of
:
Δt
energy
generated
within
any
system
must
moved
through
result
from
work
being
done
on
the
∆ϕ
_
system.
When
current
ows,
a
conductor
is
a
magnetic
eld
and
an
ε
induced
=
-N
Δt
an
external
force
is
needed
to
keep
the
conductor
moving
(the
The
external
force
balances
the
opposing
force
that
Lenz’s
law
predicts).
The
does
work
and
this
provides
the
energy
for
the
current
to
another
change
the
that
case,
object
way,
the
caused
then
which
generated
if
a
it,
force
would
without
direction
then
it
would
an
would
be
generate
work
of
be
acting
generated
an
being
induced
even
to
that
greater
current
did
support
further
emf
–
not
the
oppose
change.
accelerated
electrical
and
the
the
number
of
rate
of
turns
change
is
of
Faraday’s
ow.
law
Put
on
external
ux
force
dependence
If
the
energy
and
the
change)
the
this
is
negative
Lenz’s
sign
(opposing
the
law.
was
moving
would
be
done.
applic ation of faraday ’s l aw to moving and rotating coils
There
are
moving
many
or
situations
rotating
coils.
involving
To
decide
magnetic
whether
elds
or
not
with
an
Example:
emf
A
is
generated
and,
if
it
is,
to
calculate
its
value,
the
physicist
Earth
procedure
•
coil
•
can
Choose
At
is
to
the
be
the
μT)
her
passes
hand
so
through
that
a
the
ring
magnetic
on
her
eld
of
the
hand.
of
time,
Δt,
over
which
the
motion
of
the
considered.
beginning
through
(50
used:
period
be
holds
following
one
of
turn
the
of
period,
the
coil,
work
ϕ
.
out
Note
the
that
ux
the
passing
shape
of
initial
the
coil
ϕ
•
At
=
the
one
is
not
relevant
just
the
cross-sectional
area.
BAcosθ
end
turn
of
of
the
the
period,
coil
ϕ
work
using
out
the
the
ux
passing
equation
above.
through
Note
that
nal
the
of
sense
the
the
of
eld
the
is
opposite
magnetic
the
same
eld
but
direction,
it
is
is
important.
passing
If
the
through
magnitude
the
coil
in
then
5
B = 5 ×
ϕ
=
•
=
the
ϕ
If
there
be
in
is
no
induced.
a
coil
of
change
nal
•
T
initial
Determine
Δϕ
10
-ϕ
nal
N
ux,
Δϕ:
initial
overall
If
in
ϕ
there
turns
∆ϕ
change
is
a
will
of
ux
change
be:
in
then,
ux
overall,
then
the
no
emf
emf
will
induced
In
0.1
s,
she
magnetic
Estimate
quickly
eld
the
of
emf
turns
the
her
Earth
no
generated
hand
through
longer
in
the
goes
90°
so
through
that
the
the
ring.
ring.
Answer:
_
ε
=
-N
Δt
2
Estimate
of
cross-sectional
5
ϕ
=
5
×
10
area
of
ring,
A
≈
4
×
10
1
cm
4
=
10
2
m
9
cos(0)
=
5
×
10
Wb
initial
ϕ
=
0
nal
9
∴
∆ϕ
=
5
×
10
Wb
9
∆ϕ
5 × 10
_
_
magnitude
of
ε
=
N
=
8
=
5
×
10
V
1
Δt
10
E l E c t r o m a g n E t i c
i n d u c t i o n
113
ae e (1)
HL
coil rotating in a magnEtic fiEld – ac
rms valuEs
If
the
output
of
an
ac
generator
is
connected
to
a
resistor
an
gEnErator
alternating
The
structure
of
a
typical
ac
generator
is
shown
current
will
ow.
A
sinusoidal
potential
difference
below.
means
a
sinusoidal
current.
coil (only one
eld lines
power,
turn shown)
2
cur ve (not a sin cur ve)
cur ve is a sin
P (= V × I )
P
B
o
C
average
A
power
N
S
D
time
P
o
=
2
carbon
brush
The
graph
shows
that
the
average
power
dissipation
is
half
the
slip rings (rotate with coil)
peak
power
dissipation
for
a
sinusoidal
current.
2
2
I
R
I
0
0
_
carbon
Average
power
P
_
=
=
2
(
√
brush
R
)
2
output
Thus
the
effective
current
through
the
resistor
is
2
√
ac
(mean
value
of
I
)
and
it
is
called
the
root
mean
square
generator
current
or
rms
current,
I
rms
The
coil
of
external
wire
force.
rotates
As
it
in
the
rotates
magnetic
the
ux
eld
linkage
due
of
to
the
an
coil
I
changes
0
_
I
with
time
current
and
to
induces
ow.
The
an
sides
emf
AB
(Faraday’s
and
CD
of
law)
the
causing
coil
=
√
force
opposing
the
motion
(Lenz’s
law).
The
work
the
coil
generates
electrical
ac
coil
rotating
at
constant
speed
will
values
square
for
value
voltage
current
that
is
or
being
used.
are
In
quoted,
Europe
it
is
this
the
root
value
is
energy.
230
A
currents)
done
mean
rotating
sinusoidal
2
experience
When
a
(for
rms
a
produce
a
V
,
whereas
in
the
USA
it
is
120
V
.
sinusoidal
V
0
_
induced
emf.
Increasing
the
speed
of
rotation
will
reduce
V
the
=
rms
√
time
period
induced
of
emf
the
(as
oscillation
the
rate
of
and
increase
change
of
the
ux
amplitude
linkage
is
of
2
the
1
increased).
P
=
V
I
rms
=
I
rms
V
0
0
fme decudni
2
P
=
I
max
V
0
0
V
V
V
R
rms
0
_
=
=
_
=
I
time
I
I
0
coil rotated
constant speed of
at double the
rotation means induced
speed
emf is sinusodial
rms
transformEr opEration
An
alternating
and
an
potential
alternating
iron core
difference
potential
is
put
difference
into
is
the
given
transformer,
out.
The
value
primary coil
of
or
the
output
potential
decreased)
by
difference
changing
transformer
increases
transformer
decreases
the
the
can
turns
voltage,
be
changed
ratio.
A
whereas
a
(increased
I
step-up
I
p
s
step-down
input ac
the
output ac
voltage.
voltage ε
voltage ε
p
The
following
method
•
The
for
sequence
calculating
output
turns
voltage
of
all
is
calculations
the
relevant
xed
by
the
provides
the
s
correct
values.
input
voltage
and
the
ratio.
number of turns N
number of turns N
p
•
The
value
current
of
the
(using
V
load
=
I
that
you
connect
xes
the
s
output
ε
R).
N
=
•
The
(P
value
=
V
of
the
output
of
the
input
power
is
xed
by
the
values
The
for
•
The
P
So
=
how
this
ideal
value
V
power
is
equal
to
the
output
power
•
transformer.
of
the
input
The
coil
current
can
now
be
calculated
(using
I).
•
p
does
the
transformer
manage
to
alter
the
voltages
in
on
and
hence
the
output
i n d u c t i o n
pd
an
alternating
induces
rate
increased
E l E c t r o m a g n E t i c
structure
alternating
This
and
way?
114
I
s
I).
value
an
N
s
Transformer
•
s
=
ε
above
I
p
p
an
of
across
magnetic
emf.
The
change
number
voltages
the
primary
alternating
of
are
of
eld
links
value
ux
turns
related
creates
magnetic
of
eld
with
the
linkage,
an
in
the
within
iron
emf
depends
increases
on
the
secondary.
by
the
turns
the
core.
secondary
induced
which
ac
the
The
ratio.
with
input
and
ae e (2)
HL
transmission of ElEctric al powEr
Transformers
efcient
play
a
very
transmission
of
important
electrical
role
in
power
the
over
lossEs in thE transmission of powEr
safe
and
large
In
distances.
of
addition
the
warm
•
If
large
amounts
of
power
are
being
distributed,
then
to
power
up,
power
supply
there
are
losses
lines,
also
associated
which
losses
with
cause
the
the
associated
resistance
power
with
lines
to
non-ideal
the
transformers:
currents
used
will
be
high.
(Power
=
V
I)
•
•
The
wires
must
cannot
dissipate
have
some
zero
resistance.
This
means
•
2
•
Power
dissipated
is
P
=
I
R.
If
the
current
is
large
then
of
transformer
result
Eddy
core.
will
be
very
Over
large
windings
in
the
(joule
heating)
transformer
warming
of
a
up.
distances,
currents
The
are
currents
unwanted
are
currents
reduced
by
induced
laminating
in
the
the
iron
core
large.
into
•
the
the
2
(current)
Resistance
they
power
the
power
wasted
would
be
individually
electrically
insulated
thin
strips.
very
•
Hysteresis
losses
cause
the
iron
core
to
warm
up
as
a
signicant.
result
•
The
solution
is
to
choose
to
transmit
the
power
at
a
potential
Only
a
small
Flux
losses
current
needs
to
only
A
very
high
are
100%
cycle
of
changes
to
its
magnetism.
caused
potential
efcient
by
if
magnetic
all
of
the
‘leakage’.
magnetic
A
ux
transformer
that
is
ow.
produced
•
continued
difference.
is
•
the
very
•
high
of
difference
is
much
more
efcient,
by
the
primary
links
with
the
secondary.
but
lamination
very
•
Use
dangerous
step-up
to
the
user.
transformers
to
increase
the
voltage
for
the
secondary
primary
transmission
for
the
stage
protection
and
of
then
the
end
use
step-down
transformers
user.
Current
diodE bridgEs
The
efcient
using
the
transmission
alternating
appropriate
current
V
is
of
electrical
(ac)
and
supplied.
power
is
transformers
Many
electrical
best
can
achieved
negative
ensure
devices
is
negative)
allowed
but
and
is
B
ow
from
A
prevented
from
following
is
to
to
B
(A
is
positive
from
B
to
and
A
(A
B
is
is
positive).
are,
V
rms
d
however,
designed
conversion
on
A
from
to
ac
operate
into
dc
is
using
called
direct
current
rectication
(dc).
The
which
relies
diodes.
diode
is
connected.
direction
allow
with
a
two-terminal
characteristics
An
ideal
(negligible
current
reverse
to
electrical
depending
diode
allows
resistance
ow
in
the
device
on
current
with
reverse
that
which
to
forward
has
way
ow
in
bias)
direction
different
around
the
but
(innite
it
is
forward
does
not
resistance
tnerruc edoid
electrical
O
bias).
Symbol:
diode voltage
B
A
allowed current direction
E l E c t r o m a g n E t i c
i n d u c t i o n
115
re   
HL
rEctific ation
1.
Half-wave
A
single
smoothing circuits
rectication
diode
will
Diode-bridge
convert
ac
into
a
pulsating
dc:
direction
a
circuits
(dc)
but
smoothing
(see
page
provide
still
device
117
for
a
pulsates.
is
current
In
required.
more
that
order
One
to
ows
in
achieve
possibility
is
a
a
one
steady
pd,
capacitor
details).
+
+
+
AC
load
+
output from
load
supply
rectifying circuit
smoothed half-wave rectication
voltage
across
voltage
smoothed output
load
across
load
time
time
In
half-wave
available
in
rectication,
the
negative
electrical
cycle
of
energy
the
ac
is
that
not
is
utilized.
smoothed full-wave rectication
2.
Full-wave
rectication
voltage
A
diode
bridge
(using
four
diodes)
can
utilize
all
the
across
electrical
energy
that
is
available
during
a
complete
unsmoothed output
cycle
load
as
shown
below.
time
A
+
Note
•
that:
The
output
output
load
is
still
uctuating
slightly;
this
is
known
as
the
ripple
AC
B
C+
•
The
capacitor
is
acting
as
a
short-term
store
of
electrical
supply
energy.
•
The
•
In
capacitor
order
to
is
constantly
ensure
a
slow
charging
discharge,
and
the
discharging.
value
of
the
D
capacitor
constant
C
needs
(see
to
page
be
chosen
118)
is
to
ensure
sufciently
that
the
time
large.
voltage
across
load
invEstigating a diodE-bridgE rEctific ation
time
circuit E xpErimEntally
In
the
diode
In
bridge
the
the
positive
half
from
negative
diode
of
cycle,
current
ows
through
the
A→C→B→D.
half
bridge
the
of
from
the
The
display
using
cycle,
current
ows
through
D→C→B→A.
The
y-input
Current
same
Diodes
•
The
•
always
ac
on
signal
positive
negative
•
The
During
side
each
through
control,
display
controls
pd
across
oscilloscope
allows
allows
a
the
changing
an
the
load
is
best
achieved
(CRO).
sensitivity
pd
appropriate
on
the
of
the
y-axis.
calibration
of
CRO
The
the
the
load
resistor
in
the
time
period
of
the
x-axis
oscillations.
the
is
fed
to
point
the
in
points
the
same
where
directions.
opposite
ends
join.
of
is
two
output
of
two
taken
from
the
junction
of
the
diodes.
is
taken
from
the
junction
of
the
diodes.
half-cycle
one
set
of
parallel-side
diodes
ch 1
che 1
ch 2
che 2
evy
p
evy
p
conducts.
1
time
1
base
set
oscillation
at
=
2.5
8
mS
cm
cm
on
screen
1
_
∴
frequency
=
=
0.02
116
to
time-
e be
sides
output
side
negative
positive
•
varying
ray
(C→B)
parallel
twodiodes
The
ows
direction.
•
of
the
that:
match
•
of
cathode
appropriately
base
Note
a
E l E c t r o m a g n E t i c
i n d u c t i o n
50
Hz
=
20
mS
to
ce
HL
The
c apacitancE
Capacitors
is
of
are
devices
proportional
to
the
proportionality
is
that
pd
can
store
across
called
the
the
charge.
The
capacitor
capacitance
V
charge
and
the
stored
q
constant
capacitance
different
•
C
The
of
a
parallel
plate
capacitor
depends
on
three
factors:
area
same
of
area
each
A
plate,
and
•
The
separation
•
The
material
the
of
A.
Each
plates
the
plate
overlap
plates,
is
one
assumed
another
to
have
the
completely.
d
Symbol:
dielectric
between
material.
the
plates
Different
which
is
materials
called
will
the
have
different
C
values
charge
in
coulombs
of
a
constant
permittivity
of
air
called
is
its
permittivity,
effectively
the
same
as
ε.
the
12
of
a
vacuum
(free
space),
ε
=
8.85
×
The
permittivity
2
10
C
1
N
2
m
.
0
q
C
Thepermittivity
=
of
all
substances
is
greater
than
ε
0
V
capacitance
in
The
farads
pd
in
relationship
is:
volts
εA
_
C
The
farad
(F)
measured
is
a
µF
,
in
very
nF
large
or
unit
and
practical
capacitances
A
F
=
1
C
to
be
dielectric
across
the
2.
parallel
material
dielectric
is
is
introduced,
induced.
This
change
increases
separation
the
capacitance.
of
the
pd
across
a
capacitance
allows
the
charge
calculated.
c apacitors in sEriEs and parallEl
The
a
V
measurement
stored
d
when
pF
.
1
1
=
are
effective
total
capacitance,
C
,
of
the
combination
In
of
total
C
1
capacitors
(C
,
C
1
,
C
2
,
etc.)
in
a
circuit
depends
on
whether
3
+q
the
capacitors
capacitor
are
joined
equation
can
together
be
used
in
on
series
or
individual
in
parallel.
capacitors
The
or
q
1
1
on
pd
the
V
combination.
1
q
q
total
1
_
C
q
and
C
C
2
_
=
total
_
=
,
C
1
=
,
2
etc.
2
V
V
total
V
1
+q
2
q
2
1.
In
2
series
C
C
1
+q
q
V
2
3
+q
q
+
pd
C
2
+q
q
+
C
+
3
+q
q
3
V
V
1
3
V
2
3
pd
V
3
V
total
The
charge
stored
in
each
capacitor
is
the
same,
q
and
the
pd
pds
across
the
total
the
individual
capacitors
add
together
to
total
pd
The
q
=
q
total
=
q
1
=
q
2
=
pd
V
=
V
total
+
V
1
+
in
give
3
the
V
q
q
total
q
q
1
_
C
1
total
=
V
total
q
3
q
_
q
_
=
+
=
∴
=
V
q
+
V
total
+
=
V
3
+
q
=
3
C
total
C
V
1
+
C
1
V
2
+
C
2
V
3
3
V
=
C
V
+
C
1
V
+
C
2
V
3
+
∴
⋯
C
=
C
+
1
C
+
⋯
2
2
capacitors
5
μF
,
10
μF
and
20
μF
are
added
if
three
combined
capacitance
capacitors
5
μF
,
10
μF
and
20
μF
are
added
in
in
parallel,
the
to
C
1
e.g.
series,
charges
1
_
C
three
V
2
parallel
if
=
q
parallel
=
the
together
3
∴
series
and
add
C
2
1
_
C
V
stored.
2
1
C
_
C
1
1
_
same,
capacitors
+
C
series
the
C
2
q
q
C
charge
1
total
_
is
individual
+
C
∴
capacitor
the
_
+
total
of
3
2
_
=
C
each
each
V
2
_
∴
across
q
3
stored
e.g.
V
give
the
combined
capacitance
is:
is:
C
=
5
+
10
+
20
=
35
μF
parallel
1
_
1
=
1
_
+
5
C
1
_
+
10
7
_
=
20
1
μF
20
series
20
_
∴
C
=
=
2.86
μF
series
7
E l E c t r o m a g n E t i c
i n d u c t i o n
117
c e
HL
The
product
and
is
initial
The
SI
pd
multipliers).
c apacitor (rc) dischargE circuits
If
the
two
resistor,
a
ends
of
current
a
charged
will
ow
capacitor
until
the
are
joined
capacitor
is
together
with
a
discharged.
τ
V
of
given
=
RC
the
is
called
symbol
τ
the
(the
time
constant
Greek
letter
for
the
circuit
tau).
RC
unit
for
τ
will
be
seconds
(NB:
care
needed
with
SI
0
t
-
+q
∴
q
0
q
=
q
τ
e
0
0
Since
the
charge,
current
the
I
and
following
the
pd
V
equations
are
both
also
proportional
to
the
apply:
t
-
I
when S is closed,
=
I
τ
e
0
switch S
t
-
V
=
V
the current I will ow
τ
e
0
Where
in direction shown
q
V
0
0
_
I
_
=
=
0
R
RC
Example
R
A
10
μF
capacitor
Calculate
During
the
discharge
the
value
of
maximum
the
I
the
is
discharged
time
constant
through
τ
for
the
a
20
kΩ
circuit
resistor.
and
(b)
the
process:
fraction
•
(a)
discharge
down
to
current,
I,
drops
from
an
of
charge
remaining
after
one
time
constant
initial
a)
τ
b)
After
=
RC
=
10
μF
×
20
kΩ
=
200
ms
zero
0
•
the
value
of
the
stored
charge,
q,
drops
from
an
one
time
constant,
initial
1
q
maximum
q
down
to
=
q
e
=
0.37q
0
zero
0
0
•
the
pd
value
of
across
the
the
pd
across
resistor),
V,
the
capacitor
drops
from
(which
an
initial
is
also
q/µC
the
maximum
V
0
q =
down
Applying
to
V
zero.
Kirchoff’s
law
around
the
loop
0
C
gives
q
0
=
IR
+
q
exponential
0
C
dq
_
Since
I
is
the
rate
of
ow
of
e
decay
charge,
dt
q
dq
___
0
=
R
+
O
dt
dq
q
_
_
=
-
RC
dt
This
has
stored.
given
t/ms
RC
C
the
The
rate
of
ow
solution
is
of
an
charge
proportional
exponential
decrease
to
of
the
charge
charge
time
charge
0
100%
1RC
37%
2RC
14%
3RC
5%
stored
by:
time
(s)
t
_
-
RC
q
=
q
e
capacitance
resistance
charge
remaining
118
2%
5RC
<1%
(Ω)
original
After
charge
E l E c t r o m a g n E t i c
4RC
(F)
0
i n d u c t i o n
5
time
constents,
the
capacitor
is
effectively
discharged
c e
HL
c apacitor charging circuits
If
the
is
charged.
two
ends
of
an
uncharged
capacitor
are
joined
together
with
a
resistor,
a
current
will
ow
until
the
capacitor
C
R
when S is closed, the current
switch S
I will ow in direction shown
until capacitor is charged
emf
During
the
charging
ε
process:,
•
the
value
of
the
charging
•
the
value
of
the
stored
current,
I,
drops
•
the
value
of
the
pd
across
the
capacitor,
•
the
value
of
the
pd
across
the
resistor
from
an
initial
maximum
I
down
to
zero
0
charge,
q,
increases
from
zero
up
to
a
nal
maximum
value,
q
0
V,
increases
drops
from
an
from
zero
initial
up
to
a
nal
maximum
q/C
ε
maximum
down
to
value,
ε
zero.
I/mA
ε
nal charge =
ε
initial current =
0c
q
R
0
I
0
1
exponential
q
1
(
e
)
0
growth
I
exponential
0
e
O
The
equation
for
the
RC
increase
of
charge
O
t/ms
on
the
capacitor
(which
decay
does
not
RC
need
to
be
t/ms
memorized)
is:
t
q
=
(
q
-
1
)
τ
e
0
EnErgy storEd in a chargEd c apacitor
A
charged
energy
capacitor
when
there
can
is
a
provide
a
potential
temporary
difference
V
store
of
across
electrical
V/V
the
V
0
V
0
capacitor.
one
plate
The
charge,
and
q
on
q,
that
the
is
other
stored
plate
is
as
distributed
shown
with
below.
+q
There
is
V
an
electric
eld
between
the
(nal potential)
on
1
0
plates.
V
2
+q
0
2
(average potential)
1
q
q
area =
0
V
0
2
O
pd
q
V
Q/C
0
2
In
the
charging
process,
as
more
charge
is
added
to
the
1
E
capacitor,
the
Thegraph
pd
across
(right)
it
shows
also
how
increases
with
charge
stored
in
the
the
pd
across
capacitor
the
energy
stored,
E,
is
the
1
2
=
2
C
CV
2
that
represented
by
both
charging
and
discharging
are
exponential
charging
the
If
a
circuit
is
arranged
in
which
a
capacitor
spends
area
equal
under
q
=
capacitor
during
processes.
process.Thetotal
1
qV
2
proportionally.
Note
varies
=
time
charging
and
discharging
through
the
same
value
thegraph.
resistor,
to
the
then
in
capacitor
the
discharging
the
capacitor
to
one
complete
during
time.
the
The
charge
up
cycle,
charging
result
to
more
time
over
the
than
several
same
E l E c t r o m a g n E t i c
charge
pd
as
it
will
loses
cycles
the
be
will
power
i n d u c t i o n
added
during
be
for
supply.
119
ib Qe – eee 
HL
1.
The
primary
of
an
ideal
transformer
has
1000
turns
and
the
5.
Two
loops
of
wire
are
next
to
1
secondary
the
input
100
turns.
power
is
12
The
current
in
the
primary
is
2
A
each
and
W
.
There
emf
Which
one
of
the
following
about
the
secondary
the
secondary
power
output
is
current
20
secondary
power
A
0.2
0.2
D.
20
in
variation
1.2
alternating
loop
loop
1
and
A
~
2.
A
12
A
in
with
loop
1
time
is
of
the
shown
as
W
1
in
each
of
the
graphs
below.
In
which
graph
120
W
12
W
represent
the
current
in
loop
line
2
2?
1
1
I
A
does
W
best
C.
of
to
output
line
B.
source
2
here.
true?
current
A.
a
shown
connected
ammeter
The
secondary
is
as
current
an
and
other
I
2
2.
This
question
is
about
electromagnetic
induction.
t
A
small
coil
is
placed
with
its
plane
parallel
to
a
long
2
current-carrying
wire,
as
shown
t
straight
(no
current)
A
below.
B
1
1
I
I
current-carrying
small coil
wire
t
t
2
2
C
6.
A
D
loop
eld.
A
of
A
4
cathode
potential
(i)
State
Faraday’s
law
of
electromagnetic
induction.
(ii)
Use
in
the
the
3.
The
the
through
to
explain
changes,
why,
an
when
emf
is
the
induced
current
in
coil.
diagram
between
[2]
law
wire
[1]
shows
magnetic
two
V / ecnereid laitnetop
a)
a
simple
poles.
brushes,
generator
Electrical
each
with
contact
touching
a
slip
the
is
coil
rotating
wire
Ω
of
negligible
resistor
ray
is
oscilloscope
difference
resistance
connected
measures
across
the
is
across
rotated
its
the
resistor
in
varying
as
a
magnetic
ends.
shown
induced
below.
2
1.5
1
0.5
0
0.05
0.15
0.25
time / s
0.35
–0.5
–1
maintained
ring.
–1.5
–2
a)
If
the
coil
above
N
is
how
rotated
at
potential
twice
the
speed,
difference
would
show
vary
on
the
with
axes
time.
[2]
S
b)
What
is
the
rms
difference,V
,
value
at
the
of
the
induced
original
potential
speed
of
rotation?
[1]
rms
c)
At
the
instant
when
the
rotating
coil
is
oriented
as
shown,
across
the
a
graph
showing
resistor
varies
with
how
time,
at
the
the
power
dissipated
original
speed
of
in
the
rotation.
a)
A
3μF
capacitor
is
charged
to
240
V
.
Calculate
the
brushes
chargestored.
A.
is
B.
has
its
maximum
C.
has
the
Estimate
same
constant
value
as
in
all
other
rms
direct
current
current
rating
would
of
an
electric
produce
the
heater
same
is
4A.
power
What
The
the
time
it
would
take
for
the
charge
have
calculated
connected
to
in
the
(a)
240
to
V
ow
through
mains
a
60
W
light
electricity.
60
electric
heater?
charged
W
240
V
capacitor
light
in
(a)
is
discharged
[2]
through
a
bulb.
dissipation
(i)
in
of
direction.
c)
The
amount
orientations.
bulb
reverses
the
value.
you
4.
[1]
zero.
b)
D.
[3]
the
7.
voltage
Draw
Explain
why
the
current
during
its
discharge
will
[2]
notbe
constant.
[2]
4
_
A.
√
C.
4
A
B.
4A
2A
D.
8A
(ii)
Estimate
time
taken
for
the
capacitor
to
2
√
(iii)
dischargethrough
the
Will
during
the
Explain
120
the
i B
Q u E s t i o n s
–
E l E c t r o m a g n E t i c
i n d u c t i o n
bulb
your
light
answer.
light
bulb.
[2]
discharge?
[2]
12
Q u a n t u m
HL
a n d
n u c l e a r
P h y S i c S
P 
Photoelectric effect
Under
(such
certain
as
conditions,
zinc),
electrons
einStein model
when
are
light
emitted
(ultra-violet)
from
the
is
shone
onto
a
metal
surface
Einstein
surface.
introduced
thinking
of
light
as
the
idea
being
of
made
up
of
particles.
More
detailed
experiments
(see
below)
showed
that:
His
•
Below
a
certain
threshold
frequency
f
,
no
photoelectrons
are
emitted,
explanation
was:
no
0
matter
how
long
one
•
waits.
Electrons
certain
•
Above
the
threshold
frequency,
the
maximum
kinetic
energy
of
depends
on
the
frequency
of
the
incident
The
number
of
electrons
emitted
depends
on
the
not
depend
escape
intensity
of
the
light
and
is
no
on
the
in
a
order
from
the
energy
surface.
is
called
This
the
work
does
of
the
metal
and
given
frequency.
the
There
need
energy
light.
function
•
surface
minimum
minimum
•
the
these
to
electrons
at
noticeable
delay
between
the
arrival
of
the
light
and
the
emission
symbol
ϕ
of
•
The
UV
light
energy
arrives
in
lots
electrons.
of
These
observations
cannot
be
reconciled
with
the
view
that
light
is
a
wave.
A
wave
little
packets
frequency
should
eventually
bring
enough
energy
to
the
metal
energy
–
the
of
packets
any
of
are
called
photons.
plate.
•
The
by
energy
the
being
in
each
frequency
used,
packet
of
UV
whereas
is
xed
light
the
that
number
is
of
StoPPing Potential e xPeriment
packets
window to
The
UV
vacuum
transmit UV
stopping
frequency
of
potential
UV
light
depends
in
the
on
linear
the
by
shown
in
the
graph
s
V laitnetop gnippots
micro-
ammeter
V
variable power supply
(accelerating pd)
per
of
second
the
is
xed
source.
The
energy
carried
by
a
photon
is
below.
given
G
arriving
intensity
way
•
(quar tz)
the
by
Planck’s
constant
34
6.63
E
×
10
J
=
frequency
energy
s
in
joules
light
in
of
Hz
frequency
threshold frequency, f
0
•
In
the
apparatus
above,
The
are
emitted
by
the
cathode.
They
accelerated
across
to
the
stopping
potential
is
a
measure
of
anode
potential
maximum
kinetic
energy
of
the
photon
is
electron
surface
potential
the
absorb
energy
of
different
the
large
enough,
enough
it
energy
gives
to
the
leave
the
difference.
Max
The
If
by
electrons.
the
electrons
photons.
are
the
then
Different
photoelectrons
between
cathode
KE
of
electrons
=
V
of
the
metal.
e
and
•
Any
‘extra’
energy
would
be
energy
_
anode
can
also
be
reversed.
[since
pd
=
retained
by
the
electron
as
kinetic
charge
In
this
situation,
the
electrons
energy.
are
and
decelerated.
At
a
certain
value
e
=
charge
on
an
electron]
of
•
potential,
the
stopping
potential,
V
1
∴
no
more
photocurrent
is
observed.
If
the
energy
_
2
mv
=
V
e
∴
v
=
√
the
have
before
at
been
brought
to
the
photon
will
still
is
too
gain
of
energy
but
it
will
this
soon
rest
share
arriving
the
electron
m
2
The
small,
amount
photoelectrons
of

2V e
,
it
with
other
electrons.
anode.
Above
high-intensity UV
the
incoming
threshold
energy
of
frequency,
photons
=
energy
photocurrent
needed
to
leave
the
surface
+
kinetic
energy.
low-intensity UV
In
symbols,
of same frequency
E
=
hf
ϕ
max
V
s
potential
hf
=
ϕ
+
E
or
hf
=
ϕ
+
V
e
max
This
means
against
that
stopping
a
graph
e x amPle
line
of
the
be
a
maximum
velocity
of
gradient
h
_____
is
frequency
should
e
straight
What
of
potential
electrons
2 KE
_
∴
emitted
from
a
zinc
surface
(ϕ
=
4.2
eV)
v
=
when
√
m
______________
illuminated
by
EM
radiation
of
wavelength
200
nm?
19
2 × 3.225 × 10
__
=
19
ϕ
=
4.2
eV
=
4.2
×
1.6
×
10
19
J
=
6.72
×
10
34
c
Energy
of
photon
=
h
J
√
31
9.1
×
10
8
6.63 × 10
× 3 × 10
___
=
5
=
8.4
×
10
1
m
s
7
2
λ
×
10
19
=
9.945
×
=
(9.945
10
J
19
∴
KE
of
electron
6.72)
×
10
J
19
=
3.225
×
10
J
Q u a n t u m
a n d
n u c l e a r
p h y s i c s
121
m ws
hl
of
Wave–Particle duality
The
photoelectric
light
can
behave
demonstrated
like
all
two
–
waves.
effect
like
it
of
light
particles,
reects,
So
what
waves
but
its
refracts,
exactly
is
clearly
wave
nature
diffracts
it?
It
demonstrates
and
seems
can
that
also
interferes
reasonable
be
just
to
it
but
as
nature
2.
If
ask
light
a
correct
wave
Physics
tries
imagining
or
answer
fundamental
and
to
is
it
to
a
particle?
matter
this
even
question
of
is
philosophical
understand
models
can
properties,
de
light
light
Broglie
its
and
’yes‘!
level,
explain
behaviour.
At
the
light
what
it
Sometimes
is
is.
it
and
most
just
can
it
complete.
helps
Light
to
is
particle
think
just
wave–particle
show
wave
We
do
helps
to
have
there
wave
should
hypothesis
properties
be
is
a
that
link
all
and
waves
between
moving
can
the
have
two
this
by
think
particles
matter
models.
associated
with
them.
This
matter
wave
have
can
be
a
correct
speed.
of
it
light.
as
a
This
particle,
dual
duality
properties,
can
particles
such
as
properties?
have
probability
function
associated
with
the
moving
This
wave
experiment
1.
At
very
these
model
‘yes’.
Most
having
does
small
not
gaps.
Once
a
people
denite
explain
In
again
order
they
why
to
imagine
size,
electrons
diffract
have
a
moving
shape,
they
dual
position
can
be
must
nature.
See
below.
high
energies:
situations,
negligible
is
particles
nature.
the
In
The
a
answer
little
through
the
compared
pc
rest
with
=
E
energy
their
of
the
energy
of
particles
can
be
motion.
‘matter
thought
example,
particle.
the
rest
energy
of
an
electron
(0.511
MeV)
is
of
negligible
a
sometimes
is
called
show
as
diffracted
light.
For
wave’
as
is
waves
the
electrons
de Broglie hyPotheSiS
If
of
and
model
questions.
Is
The
wave
electrons
Again
1.
a
neither
if
it
has
been
accelerated
through
an
effective
potential
The
difference
of
420
MV
to
have
kinetic
energy
of
420
MeV
.
In
2
(amplitude)
of
the
wave
at
any
given
point
is
a
measure
of
the
these
probability
of
this
wave
nding
the
particle
at
that
point.
The
wavelength
is
given
by
the
de
Broglie
the
total
energy
of
an
electron
MeV
.
The
de
Broglie
wavelength
of
420
MeV
420
photons
is
the
×
10
2.9
×
10
m
19
×
1.6
×10
E
2.
λ
=
6
for
is:
15
=
hc
_
=
pc
electrons
8
6.6 ×
10
×
3.0 ×
10
___
λ
hc
_
=
effectively
equation:
34
λ
is
of
420
matter
circumstances
wavelength
in
At
low
energies
m
In
these
situations
the
relationship
can
be
restated
in
terms
of
34
h
is
Plank’s
constant
=
6.63
×
10
J
s
1
the
8
c
p
is
the
is
speed
the
of
light
momentum
=
of
3.0
the
×
momentum
p
of
the
particle
measured
in
kg
m
s
(in
non-
1
10
m
s
relativistic
mechanics,
P
=
mass
×
velocity):
particle
h
λ
=
p
The
higher
equation
a
the
was
photon’s
energy,
the
introduced
wavelength
lower
on
from
the
page
its
69
de
as
Broglie
the
energy,
E.
wavelength.
method
In
order
of
This
calculating
for
the
wave
For
example,
electrons
accelerated
through
1
kV
would
gain
a
KE
16
of
1.6
×
10
J.
Since
KE
and
non-relativistic
momentum
are
2
p
nature
of
particles
to
be
observable
in
experiments,
the
particles
23
related
by
E
=
K
often
have
very
high
velocities.
In
these
situations
the
,
this
gives
p
=
1.7
×
10
1
kg
m
s
2m
proper
34
6.6 ×
10
__
calculations
are
relativistic
but
simplications
are
λ
possible.
=
11
=
3.9
×
10
m
23
1.7
electron diffraction e xPeriment
In
order
through
spacing
to
a
will
be
gap
in
electrons
show
diffraction,
of
the
crystal
atoms
impinges
diffracted
same
an
electron
order
as
provides
upon
its
according
to
the
gaps.
carbon
If
10
daviSSon and germer e xPeriment (1927)
‘wave’
must
wavelength.
such
powdered
×
a
then
travel
The
beam
the
atomic
The
diagram
and
Germer
below
shows
electron
the
principle
diffraction
behind
the
Davisson
experiment.
of
electrons
lament
wavelength.
movable
accelerating p.d.
screen
electron
~1000 V
detector
+
electron
~
beam
ϕ
scattered electrons
heater
powdered
vacuum
T
arget
graphite
The
circles
correspond
interference
carbon
takes
provides
to
place.
every
the
angles
They
are
possible
where
circles
constructive
because
orientation
of
the
gap.
A
powdered
higher
A
accelerating
potential
for
the
electrons
would
result
in
a
beam
are
momentum
for
each
electron.
According
to
the
de
of
scattered
the
wavelength
of
the
electrons
would
thus
would
mean
bigger
than
circles
would
the
constructive
122
that
the
size
wavelength
move
in
to
interference
of
so
there
smaller
are
Q u a n t u m
the
gaps
would
angles.
accurately
a n d
is
now
be
The
a
target
surface.
The
nickel
crystal.
intensity
of
The
these
electrons
scattered
their
depends
on
accelerating
the
speed
potential
of
the
electrons
difference)
and
(as
the
determined
angle.
proportionally
less
diffraction.
predicted
veried
strikes
the
decrease.
by
This
from
Broglie
electrons
relationship,
electrons
higher
angles
The
of
experimentally.
n u c l e a r
p h y s i c s
A
maximum
that
scattered
quantitatively
condition
from
intensity
agrees
adjacent
with
atoms
was
the
on
recorded
at
constructive
the
surface.
an
angle
interference
HL
a sp    ss
Different
introduction
As
we
have
already
absorption)
electron
seen,
provide
energy
atomic
evidence
levels.
See
spectra
for
page
the
69
(emission
quantization
for
the
levels.
and
of
for
the
laboratory
atomic
The
hydrogen:
1
=
H
wavelengths.
Balmer
spectrum
of
In
Swiss
found
1885
that
a
the
atomic
hydrogen
consists
schoolteacher
visible
wavelengths
called
tted
of
(
λ
Jakob
–
m
mathematical
(see
the
–
a
have
attempted
model
models
page
of
to
matter
describe
explain
was
the
the
these
Bohr
electrons
by
energy
model
using
125).
1
_
2
2
m
n
particular
Johann
a
modern
1
_
R
λ
emission
models
quantum
wavefunctions
set-up.
hydrogen SPectrum
The
rst
)
wavelength
whole
number
larger
than
2
i.e.
3,
4,
5
etc
formula.
These
wavelengths,
shown
to
be
just
wavelengths
known
one
that
all
of
as
the
several
had
Balmer
similar
similar
series,
series
formulae.
of
were
later
can
the
Lyman
For
the
Balmer
(n
possible
These
For
=
3),
each
be
series
series
Brackett
case
the
of
(n
n
=
constant
lines
=
4),
R
,
2.
(in
the
The
and
other
the
called
ultra-violet
series
Pfund
the
(n
range)
are
=
Rydberg
the
5)
n
=
1.
Paschen
series.
In
constant,
has
H
7
expressed
in
one
overall
formula
called
the
Rydberg
the
formula
one
unique
value,
1.097
×
10
1
m
.
e x amPle
The
diagram
emitted
below
when
an
represents
electron
some
falls
from
of
n
the
=
3
electron
to
n
=
energy
levels
in
the
hydrogen
atom.
Calculate
the
wavelength
of
the
photon
2.
energy level / eV
0
0.9
1.5
n = 3
‘allowed’ energy
levels
3.4
n = 2
ground state: n = 1
13.6
19
Energy
difference
in
levels
=
3.4
1.5
=
1.9
eV
=
1.9
×
1.6
×
10
19
J
=
3.04
×
10
J
19
3.04 × 10
__
E
Frequency
of
photon
f
=
=
14
=
4.59
×
10
Hz
34
h
6.63
×
10
8
c
Wavelength
of
photon
λ
3.00 × 10
__
=
=
7
=
6.54
×
10
m
=
654
nm
14
f
This
is
in
the
visible
part
of
4.59
the
×
10
electromagnetic
spectrum
and
one
wavelength
in
the
Balmer
series.
+
When
Pair Production and Pair annihil ation
Matter
and
absorption
place
page
be
a
but
•
in
radiation
or
emission
absorption
73,
for
every
The
other
e
an
is
not
by
restricted
matter
spectra,
matter
antimatter
of
are
radiation
emission
property
antiparticle
+
or
of
‘normal’
corresponding
every
interactions
particle
opposite.
electron,
that
which
For
e
(such
above).
particle
to
As
the
as
if
takes
introduced
exists,
has
they
the
there
same
mass
β
)
is
a
electron
two
combined
initially
sufcient
one
e
and
photons.
momentum
zero,
directions.
and
example:
(or
on
will
an
create
then
The
the
reverse
energy
can
antimatter).
a
positron
Each
of
two
the
Much
masses
of
the
particles
with
energy
of
the
particles
that
is
the
any
will
also
into
of
annihilate
has
a
a
be
of
energy
excess
pair
travelling
possible
pair
typically
momentum
electron–positron
photons
process
convert
e
photon
–
goes
going
in
opposite
photons
particles
into
into
of
(one
the
the
and
was
matter
rest
kinetic
positron,
have
been
created.
Typically
for
pair
+
(or
β
)
production
to
take
place,
the
photon
needs
to
interact
with
a
+
•
The
antiparticle
for
a
proton,
p
is
the
antiproton,
p
nucleus.
•
The
When
they
antiparticle
a
particle
annihilate
radiation.
certain
energy,
As
for
and
one
seen
on
conservations
momentum
a
its
neutrino,
ν
is
an
corresponding
another
page
and
and
and
78,
in
the
these
antineutrino,
antiparticle
mass
is
converted
annihilations
particular
the
ν
must
conservation
involved
that
meet
into
The
in
must
nucleus
the
take
momentum,
is
overall
place.
the
not
changed
conservation
Without
interaction
its
in
of
the
momentum
ability
could
not
interaction
to
‘absorb’
but
and
is
energy
some
of
the
occur.
obey
of
charge.
Q u a n t u m
a n d
n u c l e a r
p h y s i c s
123
HL
B    
The
Bohr model
Niels
Bohr
atom
and
orbits,
took
lled
there
the
in
are
standard
the
only
‘planetary’
mathematical
a
limited
model
details.
number
of
of
the
Unlike
hydrogen
makes
planetary
‘allowed’
second
predict
the
a
postulate
transition
=
E
electron.
Bohr
suggested
that
these
orbits
used
(with
radiation
between
stable
the
full
emitted
equation)
when
an
to
electron
orbits.
E
2
the
be
of
orbits
hf
for
can
wavelength
had
1
xed
4
m
e
1
_
e
multiples
of
angular
momentum.
The
orbits
were
_
quantized
by
terms
this
of
angular
momentum.
quantization
were
in
The
exact
energy
levels
agreement
h
2
)
2
n
0
predicted
with
(
2
2
8ε
in
1
_
=
n
1
2
c
the
but
f
=
λ
discrete
wavelengths
of
the
hydrogen
spectrum.
Although
this
4
agreement
with
experiment
is
impressive,
the
model
has
e
m
some
1
associated
with
λ
it.
8ε
(
3
ch
2
postulated
An
electron
The
only
where
does
stable
the
not
radiate
orbits
angular
energy
possible
for
when
the
momentum
of
in
a
electron
the
orbit
stable
are
)
2
is
should
be
noted
that:
orbit.
•
this
•
the
equation
is
of
the
same
form
as
the
Rydberg
formula.
ones
an
values
predicted
by
this
equation
are
in
very
good
integral
agreement
h
multiple
n
1
that:
It
•
2
n
0
Bohr
1
_
=
2
problems
1
_
e
_
∴
with
experimental
measurement.
34
of
where
h
is
a
xed
number
(6.6
×
10
J
s)
2π
•
called
Planck’s
constant.
the
Rydberg
constant
can
be
calculated
from
other
(known)
Mathematically
constants.
Again
the
agreement
with
experimental
data
is
good.
nh
_
vr
m
=
e
2π
[angular
The
momentum
is
equal
to
m
When
electrons
absorb)
move
between
•
stable
orbits
they
radiate
(or
the
same
other
atoms
•
=
if
of
energy.
F
centripetal
to
this
model
are:
vr]
e
•
limitations
the
or
rst
approach
elements,
ions
it
with
postulate
is
used
fails
more
to
to
than
(about
predict
predict
one
angular
the
the
emission
correct
spectra
values
for
electron.
momentum)
has
no
force
electrostatic
theoretical
2
e
_
justication.
v
m
2
e
_
=
∴
2
4πε
•
r
r
theory
predicts
that
electrons
should,
in
fact,
not
be
stable
0
in
nh
_
but
v
[from
=
2πm
1st
ε
n
orbits
around
a
nucleus.
Any
accelerated
electron
postulate]
r
should
e
2
circular
radiate
energy.
An
electron
in
a
circular
orbit
is
2
h
accelerating
so
it
should
radiate
energy
and
thus
spiral
in
to
0
_
∴
r
=
(by
substitution)
2
πm
the
nucleus.
•
it
is
unable
to
account
for
•
it
is
unable
to
account
for
e
e
Total
energy
of
electron
=
KE
+
PE
relative
intensity
of
the
different
lines.
2
1
where
KE
e
_
1
2
m
=
v
the
ne
structure
of
the
spectral
lines.
=
e
2
2
(4πε
r)
0
2
e
_
and
PE
=
-
[electrostatic
4πε
PE]
r
0
4
1
so
total
energy
E
=
e
_
-
e
_
=
-
n
2
4πε
2
r
8ε
0
This
•
nal
the
equation
electron
overall
it
is
has
e
m
2
shows
bound
n
2
h
0
that:
to
negative
2
(=
’trapped
by’)
the
proton
because
energy.
1
•
the
energy
of
an
orbit
is
proportional
to
–
.
In
electronvolts
2
n
13.6
_
=
E
–
n
2
n
The
nucle ar radii and nucle ar denSitieS
Not
s u r p r i s i n g l y,
more
massive
nuclei
have
larger
volume
of
4
V
Detailed
analysis
spherical
of
the
distribution
of
data
implies
positive
that
charge
the
with
nuclei
an
have
d e n s i t y.
The
results
are
consistent
nucleus,
4
3
=
πR
V,
of
radius,
R
is
given
by:
=
3
πAR
0
a
3
3
essentially
Where
constant
a
radii.
with
a
the
mass
number
A
is
equal
to
the
number
which
the
protons
and
neutrons
can
be
nucleons
3A
_
A
The
in
of
model
imagined
to
number
of
nucleons
per
unit
volume
=
=
3
be
V
4πAR
0
hard
spheres
that
are
bonded
tightly
together
in
a
3
_
sphere
=
3
4πR
of
constant
d e n s i t y.
A
nucleus
that
is
twice
the
size
of
a
0
27
3
smaller
nucleus
will
have
roughly
8
(=2
)
times
the
mass.
The
mass
density
The
can
nuclear
be
radius
modelled
R
by
of
the
element
with
atomic
mass
number
ρ
of
a
nucleon
=
Where
R
A
1.7
×
10
kg),
3 ×
1.7 ×
10
__
so
the
17
=
=
3
15
4πR
R
(≈
nuclear
27
3m
_
ρ
=
m
A
relationship:
1
R
is
is:
3
4π(1.2
×
10
2
×
10
3
kg
m
3
)
0
0
This
15
is
a
constant
roughly
equal
to
10
m
(or
1
is
a
vast
density
(a
teaspoon
of
matter
of
this
density
has
fm).
0
12
a
15
R
=
1.2
×
10
m
=
1.2
mass
≈
10
kg).
The
only
macroscopic
objects
with
fm.
0
density
e.g.
The
radius
of
a
uranium-238
nucleus
is
predicted
to
be
1
15
R
=
124
1.2
×
10
×
(238)
Q u a n t u m
3
m
=
7.4
a n d
fm
n u c l e a r
p h y s i c s
as
nuclei
are
neutron
stars
(see
page
200).
the
same
HL
t S    
•
Schrödinger model
Erwin
Schrödinger
waves
and
using
to
•
proposed
wave
give
a
The
(1887–1961)
an
mechanics.
physical
alternative
The
meaning
description
quantum
of
has
wavefunction
does.
ψ
is
•
At
a
complex
is
no
on
the
model
of
Copenhagen
to
the
particles
mechanics
wavefunction
•
built
in
concept
the
(matter
physical
of
hydrogen
of
and/or
a
wave
radiation)
but
the
than
atom
is
a
electron
varying
other
•
any
instant
different
The
and
The
at
square
any
of
of
in
of
a
of
The
the
how
with
wavefunction
of
nding
in
has
different
square
of
particle
•
When
in
an
a
wavefunction
wavefunctions
particle
within
of
develops
is
like
with
in
the
wave.
the
space
the
photon,
The
but
the
absolute
the
given
and
etc.)
standing
for
to
the
(electron
atom
is
or
at
by
that
waves
energy
of
ψ,|ψ
,
density
is
of
a
real
is
made
complete
be
on
reasons
the
observed
a
string
the
wavefunction
physical
to
be
have
same
is
particle
at
a
one
xed
not
true
is
for
point.
the
said
As
an
electron
moves
actual
likely
position
1s
orbital
in
lose
kinetic
in
as
a
be
‘cloud’
in
space
some
is
of
places
undened.
hydrogen
from
total
energies
orbitals.
it
is
the
In
more
for
the
general
likely
as
to
energy
because
they
be
that
energy
found
at
a
of
result
the
further
18
 = -2.18 × 10
J
1s
5
10
15
20
25
18
 = -0.545 × 10
J
2s
to
5
or
10
15
20
wavelength
the
from
have
electron
the
nucleus.
25
2p
electron
the
nucleus
10
15
20
25
it
18
 = -0.242 × 10
must
to
location.
for
away
xed
increased
5
wavefunctions.
the
possible
is
the
(electron
other
away
number
nding
are
different
distance
place.
the
will
value
cloud
electron
photon,
given
wavefunction
probability
observation
collapse,
visualized
values
There
this
other
amplitude
corresponding
be
more
the
2
•
its
is
This
square
space.
travelling
point
the
but
can
1s
time,
interacts
probability
etc.)
places,
It
in
ψ.
ytisned ytilibaborp
•
of
points
mathematics
orbital
density.
number
mathematics
time
this
way
Electron
at
in
electron
mechanics.
wavefunction
meaning
The
matter
interpretation
mathematics
terms
of
opposite
J
charges.
3s
Lower
a
kinetic
lower
energy
momentum
means
and
that
the
de
it
would
Broglie
be
travelling
relationship
with
predicts
a
5
longer
that
The
wavelength.
t
the
This
boundary
wavefunction
means
that
conditions
provides
a
the
have
way
of
possible
particular
nding
point
is
a
an
electron
measure
of
at
that
working
the
particular
out
probability
of
radius.
the
|ψ
nding
15
20
25
3p
shapes.
probability
5
2
of
10
wavefunctions
at
the
any
given
electron
10
15
20
25
at
3d
that
distance
away
from
the
nucleus
–
in
any
direction.
2
p(r)
=
| ψ
∆V
5
10
15
20
25
10
probability
space,
of
detecting
the
electrons
in
a
small
volume
radius (×10
of
Probability
The
it
wavefunction
hard
to
exists
visualize.
in
Often
all
the
three
dimensions,
electron
orbital
is
which
makes
pictured
as
atom.
The
exact
position
of
the
electron
is
not
known
but
The
where
it
is
more
likely
to
scale
on
wavefunction
functions
the
for
vertical
some
axis
is
orbitals
in
different
the
from
hydrogen
graph
is
central
to
quantum
mechanics
to
and,
graph.
in
we
principle,
know
density
a
The
cloud.
m)
∆V
should
be
applied
to
all
particles.
be.
Example:
In
Schrödinger’s
model
there
are
different
wavefunctions
depending
A
on
the
total
energy
of
the
electron.
Only
a
few
particular
particle
is
described
as
the
following
wavefunction:
energies
Ψ
in
in
the
at
any
an
have
ground
given
electron
nding
•
wavefunctions
only
The
it
state
time
of
at
these
a
of
distance
the
a
can
distance
orbital
The
the
energy
in
be
this
the
of
–13.6
model.
to
from
of
an
atom.
eV
,
The
but
the
–
electrons
An
its
electron
position
wavefunction
calculate
electron
nding
probability
conditions
within
of
used
away
for
boundary
energies
undened
probability
away.
t
total
energy
given
resulting
terms
has
is
this
that
particular
the
for
probability
of
nucleus.
can
the
be
described
electron
nding
the
at
a
in
certain
electron
at
orez si L edistuo
can
noitceted fo ytilibaborp
result
B
x
A
probability of
mid-point
detection outside
L
L is zero
L
a
2
given
distance
away
is
shown
in
the
graph
below.
ǀΨǀ
% / ytilibaborp
100
80
60
x
A
40
B
20
0
The
0
particle
will
not
be
detected
at
the
mid
point
and
the
5
distance from nucleus / 10
10
m
probability
of
detection
Q u a n t u m
a n d
at
A
=
probability
n u c l e a r
of
detection
p h y s i c s
at
B.
125
t hsb   pp   ss
hl
 s
The
heiSenBerg uncertainty PrinciPle
The
Heisenberg
fundamental
limit
measurement.
quantum
of
impossible
a
a
arises
and
given
less
vice
not
because
as
a
precisely
They
the
of
any
the
of
and
the
momentum
is
it
position
known
variables
was
and
in
are
is
this
called
quantities
mathematical
by
exactly
precise
relationship
linking
these
uncertainties.
is
to
It
has
of
can
never
out
been
a
the
(in
–
are
profound.
physical
e.g.
world
Newton’s
principle)
deterministic.
single
even
and
us
not
a
various
this
precision
to
Before
was
laws.
make
best
A
absolute
future.
is
of
the
of
theories
allows
the
results
takes
work
lack
introduced,
mechanics
position
future
this
theory
about
the
principle
of
was
deterministic
probabilities
its
h
_
∆x∆p
described
Quantum
momentum
the
theory
predictions
(or
that
implications
quantum
deterministic
of
ability
precisely
linked
physical
showed
position
a
nature
the
He
more
are
of
result
the
The
identies
accuracy
experimenter.
exactly
versa.
principle
possible
limit
measure
the
and
conjugate
is
the
simultaneously.
determined,
There
This
any
to
particle
instant,
to
mechanics
otherwise)
of
uncertainty
possible
further.
range
suggested
of
a
determined
of
science
only
we
for
its
would
The
at
ever
gives
cannot
particle
precisely.
possibilities
that
cannot
It
outcomes.
Since
momentum
be
It
experiment.
the
uncertainty
know
any
The
predict
us
the
given
best
we
time,
can
do
future.
allow
us
to
≥
4π
calculate
As
∆x
The
uncertainty
in
the
measurement
of
the
Heisenberg
The
uncertainty
Measurements
of
energy
in
the
and
measurement
time
are
also
so
long
as
himself
said,
it
we
is
know
not
the
the
present
conclusion
exactly.
of
this
position
suggestion
∆p
future
of
that
is
wrong
but
the
premise.
momentum
linked
variables.
h
_
∆E∆t
≥
4π
∆E
The
uncertainty
in
the
measurement
of
energy
∆t
The
uncertainty
in
the
measurement
of
time
This
eStimateS from the uncertainty PrinciPle
Example
calculation:
The
position
of
a
proton
is
the
measured
11
with
an
accuracy
uncertainty
in
of
the
±
1.0
×
proton’s
10
m.
position
What
1.0
s
is
the
calculation
right
(ground
minimum
later?
2.
order
state
Impossibility
is
of
of
of
a
very
rough
magnitude
electron
an
in
H
electron
estimate
for
the
atom
but
correctly
electron’s
is
existing
13.6
kinetic
predicts
energy
eV).
within
a
nucleus
of
anatom.
h
_
∆x∆p
h
_
≥
∴
∆x
×
m∆v
≥
4π
4π
The
above
calculation
can
be
repeated
imagining
an
electron
14
being
trapped
inside
the
nucleus
of
size
10
m.
If
conned
34
h
_
∆v
6.63× 10
___
≥
=
27
4π×1.67×10
4πm∆x
to
11
a
space
this
small,
the
electron’s
kinetic
energy
would
be
×1.0×10
8
estimated
to
be
a
factor
of
10
times
bigger.
An
electron
with
1
=
3200
m
s
an
Thus
The
uncertainty
uncertainty
general
in
position
principle
principles
but,
to
after
can
1.0
also
quote
be
s
=
3200
applied
Richard
m
to
=
3.2
km
illuminate
Feynman
energy
nucleus
some
3.
Estimate
on
Physics,
volume
III,
1963),
‘[the
of
be
taken
too
seriously;
the
idea
is
right
but
application]
the
order
it
of
100
would
lifetime
spectral
of
MeV
have
an
cannot
enough
electron
in
be
bound
energy
an
to
excited
to
a
escape.
analysis
is
linewidth
associated
with
an
energy
atom’s
state.
emission
must
spectrum
not
the
thus
(Feynman
The
lectures
of
and
not
is
usually
taken
to
be
very
small
–
only
discrete
very
wavelengths
are
observed.
As
a
result
of
the
uncertainty
accurate’.
principle,
1.
Estimate
When
then
the
of
an
the
size
the
energy
electron
is
the
uncertainty
in
atom,
for
its
an
known
uncertainty
of
of
a.
its
If
electron
to
be
Δx
equate
momentum
an
conned
position
we
in
can
there
within
must
the
be
atom.
be
two,
an
less
this
estimated
atom,
than
means
the
with
any
with
the
is
as:
as
h
_
∆p
≈
we
take
this
be
very
given
small.
is,
estimate
can
uncertainty
be
in
range
and
difference
An
state
however,
limited
transition
energy
excited
the
linewidth
a
of
thus
the
between
of
the
made
energy,
not
proportional
ΔE,
the
to
the
average
of
a
lifetime,
the
in
electron
the
momentum
(∆p≈p),
the
as
a
value
equations
of
for
h
_
the
∆E∆t
≈
classical
7
mechanics
can
estimate
the
kinetic
energy
of
the
If
electron:
∆E
=
∆t
≈
5
×
10
eV
2
34
2
p
6.6 ×
10
___
h
_
_
E
=
≈
7
K
2
2m
32π
2
4π
ma
×
5
×
10
19
×
1.6
10
10
The
so
diameter
the
of
a
estimation
hydrogen
of
the
atom
kinetic
34
(6.6
×
10
is
approximately
energy
is:
≈
126
19
=
2
32π
1
31
×
9.3×10
m,
2
≈
K
10
)
___
E
associated
of
levels
an
associated
involved
electron
uncertainty
transition
Δt,
4πa
uncertainty
of
two
lifetime
using
Practically,
uncertainty
the
4π
momentum
zero.
wavelengths
is
in
principle
inversely
h
_
≈
4π∆x
If
very
the
the
will
10
×
(10
1.5
×
10
J
2
)
eV
Q u a n t u m
a n d
n u c l e a r
p h y s i c s
=
6.6
×
10
≈
1
ns
×10
in
the
excited
state:
t , p b  s 
hl
 pbb
Heisenberg’s
the
uncertainty
quantum
considered
is
is
the
less
than
relationship
phenomenon
a
particle
energy
of
that
it
is
can
be
tunnelling.
trapped
needs
to
used
The
because
escape
(U
).
to
explain
situation
its
In
energy
The
energy
being
is
E
strong
classical
less
being
than
of
the
the
force
emitted
total
within
formed
the
inside
alpha
potential
nucleus.
the
particle
energy
If
uranium
is
4.25
needed
we
imagine
nucleus,
it
MeV
to
which
escape
an
can
the
alpha
only
particle
escape
0
physics,
from
if
the
a
particle
does
potential
not
barrier
have
then
enough
it
will
energy
always
to
escape
remain
by
trapped
tunnelling
the
very
through
long
half-life
the
potential
must
mean
barrier.
that
the
In
this
example,
probability
of
the
2
inside
with
two
the
a
system.
total
walls
An
energy
that
are
example
of
4
1.2
J
m
would
bouncing
high.
In
be
up
a
500
and
order
to
g
tennis
down
get
ball
tunnelling
process
taking
place
(given
walls,
mgh
=
the
0.5
trapped
×
by
tennis
10
the
×
ball
1.2
=
needs
6
J.
to
have
Since
it
a
over
one
potential
only
has
4
)
must
be
very
low.
between
of
wavefunction of
the
ψ
by
J
energy
it
must
of
repulsive Coulomb
alpha par ticle
potential ∝
1/r
remain
walls.
U
In
an
equivalent
U
an
atom
needed
to
with
energy
escape),
the
situation
E
which
rules
of
is
(e.g.
less
an
electron
than
quantum
the
physics
trapped
)r(V ygrene
inside
microscopic
energy
mean
0
that
it
is
now
possible
wavefunction
to
zero
when
is
it
for
the
continuous
meets
the
particle
and
sides
to
does
of
the
escape!
not
drop
The
particle’s
immediately
potential
well
but

X
r
nuclear
the
surface
amplitude
has
a
decreases
nite
width
exponentially.
then
the
This
means
wavefunction
does
that
if
the
continue
barrier
on
the
attractive
other
side
there
is
of
the
barrier
(with
reduced
amplitude).
Therefore
nuclear
a
probability
that
particle
will
be
able
to
escape
despite
potential
not
having
does
not
enough
use
up
energy
any
of
to
the
do
so.
Escaping
particle’s
total
the
potential
well
energy.
Example 2 – tunnelling electron microscope
In
a
scanning
tunnelling
microscope,
a
very
ne
metal
tip
classically forbidden region
is
a
scanned
sample
close
metal
to,
but
not
surface.
touching
There
is
a
(separated
potential
by
a
few
difference
nm),
between
U
0
the
probe
and
the
surface
but
the
electrons
in
the
surface
do
not
E
have
enough
energy
to
escape
the
potential
energy
barrier
as
par ticle energy
represented
however,
by
take
wavefunction
the
work
place
of
an
and
function
a
ϕ.
Quantum
tunnelling
electron
at
the
current
surface
tunnelling
will
will
ow
extend
as
can,
the
beyond
incoming par ticle
the
wavefunction
metal
electrical
par ticle wavefunction
depends
past the barrier
used
Ψ
to
surface.
current
on
the
Some
will
be
electrons
measurable.
separation
visualize
atomic
will
of
the
tip
tunnel
The
and
the
value
the
gap
of
the
surface
and
current
and
can
be
structure.
Ψ
incident
exit
Reduced probability, but not
reduced energy!
An
explanation
principle.
greater
In
total
can
order
be
for
energy
offered
the
(E
+
in
terms
particle
ΔE
=
U
to
).
of
the
escape
The
it
uncertainty
would
particle
can
need
sample
a
‘disobey’
0
tip
the
law
energy
of
conservation
ΔE
provided
uncertainty
it
principle
of
energy
‘pays
it
by
back’
‘borrowing’
in
a
time
Δt
an
amount
such
that
of
the
applies:
tunnelling
h
_
∆E∆t
current
≈
4π
path of
The
longer
the
barrier,
the
more
time
it
takes
the
particle
to
the probe
tunnel.
possible
Increased
tunnelling
uncertainty
in
the
time
will
reduce
the
maximum
energy.
scanning tunnelling
surface
Example 1 – alpha decay
microscope (STM)
The
protons
within
nuclei
energy.
4.5
and
For
billion
and
It
form
emitted
by
alpha
overall
has
emitting
a
particles
there
is
a
half-life
an
alpha
already
release
of
of sample
exist
of
about
particle:
4
Th
90
that
uranium-238
decays
234
U→
92
when
example
years.
238
neutrons
+
α
2
Q u a n t u m
a n d
n u c l e a r
p h y s i c s
127
t s
hl
the nucleuS – Size
In
the
gold
example
below,
e x amPle
alpha
particles
are
allowed
to
bombard
atoms.
If
the
α
particles
approach
to
the
have
gold
an
energy
nucleus
(Z
of
=
4.2
79)
19
As
they
approach
the
(2
×
1.6
×
10
MeV
,
is
the
given
closest
by
19
)
(79
×
1.6
×
10
)
____
12
gold
nucleus,
force
of
they
repulsion.
feel
If
a
4
×
π
×
8.85
×
10
×
r
an
alpha particles
6
alpha
particle
directly
for
is
the
=
heading
nucleus,
4.2
×
19
×
10
1.6
×
10
it
19
2 × 1.6 × 10
× 79
___
∴
nucleus
will
be
back
reected
along
the
r
=
12
straight
4
×
π
×
8.85
×
6
10
×
4.2
×
10
same
14
=
5.4
×
10
m
closest approach, r
path.
It
will
close
as
it
collides
Alpha
have
can.
with
Note
the
particles
energy.
As
got
that
none
nucleus
are
they
as
–
emitted
come
in
of
they
the
do
from
they
alpha
not
their
gain
particles
have
actually
enough
source
with
electrostatic
energy.
a
nucle ar Sc attering e xPeriment involving
known
potential
electronS
energy
and
lose
kinetic
energy
(they
slow
down).
At
the
Electrons,
closest
approach,
the
alpha
particle
is
temporarily
electrons
and
all
its
energy
is
as
leptons,
do
not
feel
the
strong
force.
High-energy
stationary
have
a
very
small
de
Broglie
wavelength
which
can
be
potential.
of
q
the
right
order
to
diffract
around
small
objects
such
as
nuclei.
q
2
____
Since
electrostatic
energy
=
,
4πε
and
we
know
q
,
the
charge
The
diffraction
pattern
around
a
circular
object
of
diameter
D
1
r
0
on
an
alpha
particle
and
q
,
the
charge
on
the
gold
nucleus
we
has
its
rst
minimum
at
an
angle
θ
given
by:
2
can
calculate
r.
λ
sin
θ
≈
D
[Note
that
this
small
angle
approximation
is
usually
not
de viationS from rutherford Sc attering in
appropriate
to
use
to
determine
the
location
of
the
minimum
high energy e xPerimentS
intensity
Rutherford
scattering
is
modelled
in
terms
of
the
around
repulsion
between
the
alpha
particle
and
the
but
this
is
being
used
to
give
an
approximate
answer
coulomb
target
a
spherical
object.
A
more
exact
expression
nucleus.
that
is
λ
sometimes
used
for
circular
objects
is
sin
θ
=
1.22
]
D
At
relatively
low
energies,
detailed
analysis
of
this
model
High
accurately
predicts
the
relative
intensity
of
scattered
energy
at
however,
given
the
angles
scattered
of
scattering.
intensity
At
departs
high
from
(400
MeV)
electrons
are
directed
at
a
target
alpha
containing
particles
carbon-12
nuclei:
energies,
predictions.
Fixed scattering angle,
electron beam
range of alpha
xed
θ
par ticle energies
detector
detector
thin sample
α
60°
+
scattering
The
results
are
shown
below:
angle
208
Pb
lead target
intensity of
82
diracted
The scattered intensity
electrons
derettacs fo ytisnetni evitaler
depar ts from the Rutherford
(logarithmic
°06 ta selcit rap ahpla
scattering formula at about
scale)
27
.5 MeV
2
10
35
diraction angle (θ)
10
alpha
The
rst
The
de
minimum
is
θ
=
35°
energy
Broglie
wavelength
for
the
electrons
is
effectively:
in MeV
34
hc
_
λ
8
6.6 ×
10
×
3.0 ×
10
___
=
=
Eisberg,
R.
M.
and
20
Porter,
25
C.
E.,
30
Rev.
35
Mod.
400
40
Phys.
33,
190
high
energies
the
alpha
particles
are
beginning
to
10
1.6
×
begin
enough
to
nucleus
box
on
128
have
in
the
to
the
an
more
target
effect.
detail,
In
nucleus
order
high
to
for
the
strong
investigate
energy
electrons
nuclear
=
θ
15
=
sin
5.4
×
35
get
the
can
size
be
force
of
to
the
used
(see
right).
Q u a n t u m
a n d
n u c l e a r
p h y s i c s
×
10
15
close
3.1
19
×
3.1 × 10
__
λ
_
≈
sin
these
×
15
(1961)
D
At
15
=
6
E
15
So
radius
of
nucleus
≈
2.7
×
10
m
10
m
10
m
HL
n  s   
energy le velS
226
The
The
energy
levels
in
a
nucleus
are
higher
than
the
energy
levels
decay
of
222
Ra
into
226
Rn
of
Ra
88
the
a
electrons
gamma
are
but
photon
observed.
two
the
nuclear
is
principle
emitted
These
the
from
energies
energy
is
levels
same.
the
nucleus
correspond
in
the
When
to
same
only
the
way
an
alpha
discrete
difference
that
particle
the
or
energies
between
photon
energies
α
correspond
to
the
difference
between
two
atomic
energy
levels
(4.59 MeV)
Beta
particles
energies.
the
In
are
this
observed
case
case
of
beta
amount
of
energy
to
there
minus
is
have
continuous
another
decay)
released
a
that
in
the
particle
shares
decay
the
is
spectrum
(the
antineutrino
energy.
xed
α
of
by
Once
the
(4.78 MeV)
in
again
the
222
difference
Rn*
excited state
between
the
nuclear
energy
levels
involved.
The
beta
particle
and
γ
the
photon
86
ground
antineutrino
can
take
varying
proportions
of
the
energy
available.
222
(0.19 MeV)
The
Rn
state
antineutrino,
however,
is
very
difcult
to
observe
the
undetectable
needed
and
the
to
existence
particle,
account
(angular)
decay
decay
of
the
for
properly
a
the
difference
It
‘missing’
momentum
mass
virtually
neutrino.
mathematically.
involving
requires
when
is
energy
analysing
Calculations
mean
that
for
the
how
much
energy
is
→
ν
has
form
place,
decay.
For
example,
isotope
+
β
+
ν
an
antineutrino
mentioned
radioactive
namely
before,
decay
positron
can
decay.
another
also
In
take
this
we
available
an
equation
1
been
of
full
is:
0
He
is
The
tritium
2
where
As
existence.
of
3
H
1
in
0
0.5
a
proton
within
the
nucleus
1
decays
beta
its
decay
3
decay,
know
86
below).
snortcele fo rebmun
accepting
beta
box
conrmed
neutrinoS and antineutrinoS
Understanding
(see
into
a
neutron
and
the
antimatter
of
energy / MeV
version
hydrogen,
tritium,
decays
as
of
an
electron,
a
positron,
which
follows:
3
The
3
3
H
→
distribution
of
the
electrons
is
emitted.
0
He
1
energy
+
β
2
emitted
in
the
beta
decay
of
bismuth-210.
1
1
1
p
→
mass
difference
for
the
decay
kinetic
energy
of
these
electrons
0
n
1
The
The
+
+
β
0
+
ν
+1
is
is
+
In
between
–2
19.5
keV
c
.
This
means
that
the
zero
and
1.17
this
case,
the
should
have
19.5
keV
of
kinetic
The
neutrino
(and
antineutrino)
must
In
fact,
a
few
beta
particles
are
electrically
neutral.
Its
mass
would
with
this
energy,
but
all
the
others
to
be
very
small,
or
even
zero.
It
is
less
about
than
half
this.
this
The
value
average
and
energy
there
antineutrino
the
decays
seem
gamma
to
follow
photon.
a
is
All
similar
is
away
no
to
the
detect.
excess
One
of
energy
the
but
it
is
triumphs
very
of
beta
particle
pattern.
to
be
physics
able
to
of
the
design
last
e.g.
hard
the
19
Ne
→
antimatter
+
9
14
C
0
F
10
14
the
century
experiments
is
form
neutrino.
→
+
β
+
ν
+1
0
N
6
accompanying
,
neutrino.
carries
19
have
a
have
of
emitted
by
be
The
energy.
β
MeV
.
beta
accompanied
particles
positron,
+
7
β
+
ν
1
was
that
mathematicS of e xPonential dec ay
The
basic
relationship
that
In (N)
denes
e x amPle
exponential
decay
as
a
random
process
is
intercept =
The
expressed
as
follows:
ln (N
0
gradient = -λ
)
is
half-life
10
days.
of
a
radioactive
Calculate
the
isotope
fraction
of
a
dN
_
∝
– N
sample
dt
The
that
remains
after
25
days.
t
constant
of
proportionality
T
between
=
10
days
2
the
rate
of
decay
and
the
number
of
nuclei
ln 2
_
λ
available
to
decay
is
called
the
=
decay
T
2
constant
and
given
the
symbol
λ.
Its
units
λt
N
=
N
e
2
=
0
1
are
1
time
i.e.
6.93
×
10
1
day
1
s
or
yr
etc.
-λt
If
t
=
T
dN
_
N
1
=
N
e
0
=
-λN
2
N
_
dt
Fraction
remaining
=
N
The
solution
of
this
equation
N
=
N
2
(6.93
e
of
a
source,
A,
A
=
A
=
N
=
λN
=
e
useful
to
take
natural
=
0.187
=
18.7%
logarithms:
∴
(N)
=
λT
1
∴
In
½
=
-λT
=
-
-λt
ln
25)
0
2
e
0
2
is
×
e
λt
e
0
It
10
λT
0
So
dt
λt
=
×
e
_
dN
_
activity
A
=
N
0
The
0
=
λt
N
N
0
_
is:
ln
(N
e
1
)
2
0
-λt
=
ln
(N
)
+
ln
(e
)
∴
0
-λT
1
In
½
2
=
ln
(N
=
ln
(N
)
λt
)
λt
ln
(e)
0
∴
ln
(N)
=
(since
ln
(e)
=
of
ln
is
N
of
the
vs
t
form
will
2
ln 2
_
∴
This
ln
1)
0
y
give
=
a
c
+
mx
so
a
straight-line
graph
T
1
2
=
λ
graph.
Q u a n t u m
a n d
n u c l e a r
p h y s i c s
129
iB Qss – q   pss
hl
1.
The
diagrams
show
wavefunction
Ψ
horizontal
axis
electron
the
is
of
in
the
variation
four
all
with
different
four
diagrams
uncertainty
in
distance
electrons.
the
is
the
x
The
same.
momentum
of
the
scale
For
the
R
on
the
those
which
this
constant
From
c)
of
the
constant
Ψ
particles
will
not
the
that
enter
matter
in
the
Geiger
determining
tube.
the
Explain
decay
sample.
[1]
largest?
e)
a)
of
why
graph,
determine
a
value
for
the
decay
λ.
[2]
Ψ
The
x
0
d)
0
f)
Dene
the
g)
Derive
a
h)
Hence
Ψ
Ψ
and
0
x
now
wishes
to
calculate
the
half-life.
x
0
b)
physicist
x
the
half-life
of
relationship
half-life
a
radioactive
between
the
substance.
decay
[1]
constant
λ
τ.
calculate
[2]
the
half-life
of
this
radioactive
isotope.
4.
2.
The
diagram
represents
the
available
energy
levels
of
This
How
many
emission
lines
could
result
from
question
biography
between
these
energy
of
about
the
quantum
Schrödinger
concept.
contains
the
following
sentence:
electron
‘Shortly
transitions
is
an
A
atom.
[1]
after
de
Broglie
introduced
the
concept
of
matter
waves
levels?
in
1924,
Schrödinger
began
to
develop
a
new
atomic
theory.‘
energy
a)
Explain
the
determines
b)
Electron
the
term
the
‘matter
diffraction
existence
waves’.
wavelength
of
of
provides
matter
State
such
what
evidence
waves.
quantity
waves.
What
to
is
[2]
support
electron
diffraction?
5.
ground
Light
is
maximum
state
[2]
incident
on
kinetic
a
clean
energy
metal
KE
surface
of
the
in
a
vacuum.
electrons
ejected
The
from
max
the
A.
3
B.
6
C.
8
D.
A
medical
physicist
wishes
to
investigate
the
decay
of
the
A
Geiger–Müller
sample
of
the
determine
counter
isotope,
as
for
different
values
of
the
frequency
is
its
used
decay
to
constant
detect
incident
f
light.
and
radiation
measurements
are
shown
plotted
below.
half-
from
2.0
91
a
and
)J
life.
isotope
measured
a
The
radioactive
is
12
of
3.
surface
shown.
01(
voltage supply
1.5
xam
and counter
radioactive
1.0
K
Geiger–Müller
source
tube
0.5
a)
Dene
the
activity
of
a
radioactive
sample.
[1]
0.0
4.0
Theory
predicts
sample
should
that
the
activity
A
of
the
isotope
in
4.5
5.0
5.5
6.0
6.5
7
.0
7
.5
8.0
the
14
decrease
exponentially
with
time
t
f
according
(10
H z)
-λt
to
the
equation
A
=
A
e
,
where
A
0
and
b)
λ
is
the
decay
Manipulate
straight
this
line
variables
if
on
for
equation
a
semi-log
the
is
the
activity
at
t
=
0
0
constant
axes.
the
into
a
graph
State
a)
Draw
a
line
of
best
b)
Use
the
graph
(i)
the
Planck
(ii)
the
minimum
t
for
the
plotted
data
points.
[1]
isotope.
form
is
what
which
plotted
will
with
variables
give
to
determine
a
constant
[2]
appropriate
should
be
plotted.
electron
[2]
from
energy
the
required
surface
of
to
the
eject
metal
an
(the
work
function).
The
Geiger
emitted
by
the
of
particles
as
a
graph
counter
detects
source.
detected
of
ln
R
as
a
The
a
versus
proportion
physicist
function
t,
as
of
the
records
of
time
shown
t
the
and
count-rate
plots
the
c)
R
data
below.
Explain
Does
the
plot
consistent
show
with
an
that
the
experimental
exponential
law?
data
briey
how
accounts
for
from
surface
the
c)
[3]
particles
the
incident
the
Einstein’s
fact
of
light
is
that
this
less
no
electrons
metal
than
photoelectric
if
a
the
Explain.
[1]
6.
certain
of
value.
Thorium-227
days
to
(Th-227)
form
undergoes
radium-223
a-decay
(Ra-223).
A
with
[3]
a
sample
half-life
of
of
Th-227
1
)
5
2
has
an
initial
s / R( nI
Determine
50
1
7.
activity
the
of
activity
3.2
of
×
the
10
Bq.
remaining
1
2
3
4
The
role
of
angular
momentum
in
Geiger
activity
A
counter
of
i B
the
does
sample,
not
but
measure
rather
Q u e s t i o n s
b)
Pair
c)
Quantum
the
Bohr
model
for
[3]
production
and
ahnihilation
[3]
5
t / hr
The
after
[4]
hydrogen
0
thorium-227
days.
Explain:
a)
130
emitted
frequency
are
18
d)
are
theory
–
the
the
tunnelling
total
count-rate
Q u a n t u m
a n d
n u c l e a r
p h y s i c s
[3]
13
O p T i O n
a
–
R e l a T i v i T y
Rr r
The
ObseRveRs and fRames Of RefeRence
The
proper
treatment
understanding
thinking
The
reasons
pages,
from
this
but
two
is
A
not
case
we
sitting
the
sitting
the
the
a
of
so
chair
in
are
to
relativity
developed
simple.
will
In
this
we
apply
way.
that
of
the
Newton’s
relative
like
all
velocity
the
velocities
laws
to
are
small
different
to
object’s
motion
(or
the
view
the
think
this
mean
point
motion.
that
must
situation.
Sun’s
in
lack
by
is
possible
frames
frame
be
of
of
to
formalize
reference.
reference
recorded
in
The
to
of
the
it)
of
This
depends
be
The
view
they
true,
Earth
the
example
on
are
but
this
another
without
taking
is
to
out
frame
the
called
The
Galilean
simplest
(S
and
as
shown
S')
able
be
to
reference.
is
in
orbit
person
the
shows
that
use
the
of
between
the
two
The
that
an
different
measurement
measurements
reference.
theory
of
relativity
…or moving at great velocity?
observer.
in
of
reference
Since
measurements
one
would
equations
into
frame
event.
y'
=
y;
z'
=
the
will
z;
t'
can
relative
be
=
the
record
the
motion
is
position
along
the
and
time
x-axis,
of
most
same:
t
that
an
event
is
stationary
according
to
one
frame,
it
will
be
frames
will
record
consideration
according
to
the
other
frame
–
the
transformations.
situation
with
to
9.
far,
at
moving
are
of
page
so
this
If
do
enough
frames
on
book
why
Each
relationship
idea
work
this
an
Galile an TRansfORmaTiOns
It
considered
in
follow
see
Is this person at rest…
an
was
mechanics
following
logically
order
treatment,
assumes
means
different
the
They
what
probably
of
viewing
be
in
and
calculation
This
an
place.
point
must
involves
completely
consider
rst
from
of
a
assumptions.
chair
their
way
Sun,
theory
time
change
need
in
velocities
surprisingly
in
from
only
the
in
and
straightforward
Indeed
around
this
are
motion
person
rest.
is
in
for
large
Einstein’s
space
they
the
object
of
about
of
one
to
consider
frame
(S')
is
two
moving
frames
past
the
of
reference
other
one
different
values
between
the
for
two
the
is
x
measurement.
given
The
transformation
by
(S)
x'
=
x
vt
below.
We
t = zero (two frames on top of one another)
can
use
velocities.
these
The
equations
frames
will
to
formalize
agree
on
any
the
calculation
velocity
of
measured
in
y
the
y
or
z
direction,
but
they
will
disagree
on
a
velocity
in
the
frame S (stationary)
y′
x-direction.
Mathematically,
frame S′
u'
=
u
v
1
For
example,
if
the
moving
frame
is
going
at
4
m
s
,
then
velocity v
an
object
15
m
moving
in
the
same
direction
at
a
velocity
of
1
s
as
recorded
in
the
stationary
frame
will
be
measured
as
1
x
travelling
at
Newton’s
3
11
m
s
in
the
moving
frame.
x′
laws
of
motion
describe
how
an
object’s
motion
is
t = later
effected.
y
frame S
y′
laws
all
is
An
that
assumption
the
observers.
time
Time
(Newton's
interval
is
the
Postulates)
between
same
for
all
two
underlying
events
frames
and
is
the
the
these
same
for
separation
frame S′
between
the
same
events
will
physical
also
laws
be
will
the
same
apply
in
in
all
all
frames.
As
a
result,
frames.
velocity v
x
x′
piOn dec ay e XpeRimenTs
In
1964
Centre
the
an
for
speed
experiment
Nuclear
of
at
the
Reseach
gamma-ray
failuRe Of Galile an TRansfORmaTiOn equaTiOns
European
(CERN)
photons
If
measured
that
the
the
speed
of
Galilean
light
has
the
transformation
same
value
equations
for
all
cannot
observers
work
for
(see
box
on
left)
then
light.
had
velocity of bicycle, v
been
the
speed
also
is
produced
to
be
of
light
moving
consistent
with
independent
high
The
by
degree
of
of
called
gamma-ray
and
at
the
found
the
the
moving
these
speed
speed
speed
of
of
of
to
photons
light.
light
its
close
This
being
source,
to
a
accuracy.
experiment
particle
particles
analysed
the
the
neutral
photons.
decay
pion
Energy
into
of
a
two
considerations
Light leaves the torch
meant
that
the
pions
were
known
to
Light arrives at the observer
be
at velocity c with respect
moving
faster
than
99.9%
of
the
speed
of
at velocity c (not v + c).
light
to the person on the bicycle.
and
the
speed
of
the
photons
was
8
be
2.9977
±
0.0040
×
10
1
m
s
measured
to
The
theory
of
relativity
attempts
to
work
out
o p t i o n
A
what
–
has
gone
wrong.
r e l A t i v i t y
131
mx’ to
maXwell and The cOnsTancy Of The speed Of liGhT
In
1864
Royal
James
Society
mathematical
was
known
electric
the
two
–
the
E,
Maxwell
London.
form
at
eld
Clerk
in
that
time
but
it
His
elegantly
about
also
Maxwell’s
are
summarized
equations.
These
in
new
magnetic
a
‘rules’
four
not
of
only
eld
unifying
B
at
in
the
The
a
–
what
and
link
predict
to
the
as
nature
be
electric
technical
allows
between
known
the
changing
the
through
the
electromagnetic
equations
equations
theory
encapsulated
expressed
the
The
a
were
proposed
electromagnetism.
interactions
presented
ideas
way
space.
the
The
speed
predicted.
electric
which
and
they
magnetic
saying
physics
of
It
and
of
all
this
of
elds
is
how
that
these
electromagnetic
turns
out
magnetic
that
this
constants
elds
be
the
through
elds
waves
can
of
move
the
space
propagate
propagate
(including
done
in
medium
light)
terms
of
through
travel.
____
of
1
_
electromagnetic
c
waves.
=
√
ε
µ
0
Most
people
know
that
light
is
an
electromagnetic
wave,
This
it
is
quite
hard
to
understand
what
this
actually
means.
equation
wave
involves
the
oscillation
of
matter,
whereas
wave
involves
the
oscillation
of
electric
important
elds.
The
diagram
below
attempts
to
show
is
need
that
to
the
be
understood
speed
of
light
in
is
detail.
The
independent
the
velocity
of
the
source
of
the
light.
In
other
words,
a
and
prediction
magnetic
not
idea
an
of
electromagnetic
does
A
only
physical
0
but
from
Maxwell’s
equations
is
that
the
speed
of
light
in
this.
a
z
vacuum
This
an
has
the
prediction
of
inconsistency
same
the
that
value
for
constancy
cannot
be
all
of
observers.
the
speed
reconciled
of
light
with
highlights
Newtonian
oscillating electric eld
mechanics
the
speed
forced
and
(where
addition
of
of
light
as
to
be
resultant
relative
speed
speed
measured
long-held
time
the
the
by
assumptions
the
of
of
the
light
source).
about
would
source
the
and
be
the
Einstein’s
equal
to
relative
analysis
independence
of
space
rejected.
y
x
b)
cOmpaRinG elecTRic and maGneTic fields
Electrostatic
one
forces
another.
aspects
nature
of
of
and
magnetic
Fundamentally,
one
force
which
–
eld
the
is
forces
however,
appear
they
electromagnetic
observed
depends
very
are
different
just
different
interaction.
on
the
to
The
observer.
For
example:
a)
A
Two
identically
velocities
An
observer
charge
moving
at
right
angles
to
a
magnetic
in
the
charged
this
observer
charges
charged
according
as
a
to
frame
particles
sees
solely
the
particles
a
moving
laboratory
of
reference
will
force
see
of
electrostatic
the
that
of
is
parallel
reference.
moving
particles
repulsion
in
with
frame
at
with
rest.
between
Thus
the
two
nature.
eld.
F
E
An
observer
respect
to
in
a
the
frame
of
reference
magnetic
eld
that
will
is
at
explain
rest
the
with
force
acting
+q
on
the
charge
(and
its
acceleration)
in
terms
of
a
magnetic
force between 2 stationary charges is electrostatic
force
(F
=
Bqv)
that
acts
on
the
moving
charge.
M
+q
stationary
magnetic eld into paper
F
E
initial force on moving
An
observer
in
a
frame
of
reference
where
the
laboratory
moving
charge is magnetic
X
X
X
X
is
at
rest
will
see
the
total
force
between
the
two
charges
charge
as
An
observer
respect
to
on
the
an
induced
in
charge
magnetic
a
the
frame
charge
(and
electric
its
of
reference
will
initial
force
that
explain
the
is
acceleration)
that
results
at
rest
initial
from
in
with
force
terms
the
acting
combination
charges
its
of
are
own
frame.
of
cutting
a
electrostatic
currents
magnetic
Each
magnetic
of
charge
eld
and
eld
and
is
thus
which
moving
will
and
each
is
moving
stationary
in
the
experience
a
charge
in
other’s
Moving
the
creates
laboratory
stationary
magnetic
force.
ux.
F
E
X
X
X
X
X
X
X
X
initial force on stationary
& F
M
+q
force between 2 moving objects is a combination
stationary
charge is electric
X
X
X
of electric and magnetic
X
charge
+q
moving magnetic eld
F
E
132
magnetic.
o p t i o n
A
–
r e l A t i v i t y
& F
M
s rtt
pOsTul aTes Of special Rel aTiviTy
The
special
theory
of
relativity
assumptions
or
postulates.
be
be
wrong,
shown
to
If
then
is
based
either
the
on
of
two
these
theory
of
This
fundamental
postulates
relativity
could
would
the
When
discussing
relativity
we
need
to
be
even
usually
precise
with
our
use
of
technical
two
the
postulates
speed
important
technical
phrase
is
an
inertial
frame
This
means
a
frame
of
reference
in
which
the
inertia
(Newton’s
laws)
apply.
Newton’s
the
laws
do
not
apply
is
frames
either
of
reference
stationary
or
so
an
moving
inertial
with
frame
is
a
laws
rst
be
constant
important
idea
to
between
grasp
being
is
that
there
stationary
is
and
no
Newton’s
laws
link
forces
and
resultant
force
on
an
object
then
its
it
could
mean
that
special
in
a
relativity
vacuum
is
are:
the
same
constant
for
all
of
physics
are
leads
the
on
same
from
for
all
inertial
Maxwell’s
observers.
equations
veried.
The
second
postulate
and
seems
frame
reasonable
differentiate
–
particularly
between
being
at
since
rest
Newton’s
and
moving
laws
at
do
constant
fundamental
moving
at
If
both
are
accepted
as
being
true
then
we
need
to
start
constant
accelerations.
If
there
acceleration
will
be
about
space
and
time
in
a
completely
different
way.
If
is
in
no
of
light
experimentally
thinking
velocity.
of
postulate
velocity.
difference
or
velocity.
not
An
rest
velocity
observers
completely
that
at
in
can
accelerating
is
laws
The
of
object
constant
of
•
reference.
the
at
terms.
inertial
One
that
moving
more
•
than
mean
is
be
The
wrong.
could
object
doubt,
we
need
to
return
to
these
two
postulates.
zero.
simulTaneiTy
One
example
everyday
of
of
how
simultaneity.
If
two
are
simultaneous.
are
simultaneous
to
all
observers
demonstrate
The
two
the
pulses
ends
carriage
is
of
are
As
is
know
to
but
this
that
observer.
will
–
We
is
experimenter
carriage
the
understanding
one
far
as
rest.
happen
not
the
an
positioned
that
the
they
of
the
she
is
in
in
the
the
pulses
the
is
say
if
be
simple
velocity.
ends
the
we
should
A
exactly
reect
is
that
experimenter
constant
the
expect
case!
disrupt
us
together
experimenter
Since
relativity
around
normally
consider
at
of
world
observer,
is
towards
mirrors
at
events
moving
light
the
would
this
to
is
postulates
of
a
train.
to
train.
sends
back
of
1st pulse hits back wall
a
out
Mounted
towards
concerned,
the
they
events
simultaneous
middle
She
middle,
that
two
way
in
pulses leave together
our
concept
the
at
the
whole
2nd pulse hits front wall
experimenter
that:
•
the
pulses
were
•
the
pulses
hit
sent
•
the
pulses
returned
the
out
simultaneously
mirrors
simultaneously
simultaneously.
pulses arrive together
pulses leave
together
Interestingly,
the
arriving
at
observer
on
the
platform
does
see
the
beams
pulses arrive
back
the
same
time.
The
observer
on
the
platform
at mirrors
will
know
that:
together
•
the
pulses
•
the
left-hand
were
sent
pulse
out
hit
simultaneously
the
mirror
before
the
right-hand
pulse
pulses return
•
the
pulses
returned
simultaneously.
together
In
general,
point
in
events
simultaneous
space
that
take
simultaneous
The
situation
observer
(on
will
the
seem
platform).
travel
at
constant
speed
speed
as
far
is
at
different
towards
means
as
that
he
times.
the
beam
the
very
–
different
This
both
The
and
so
left-hand
the
reection
right
will
watched
observer
beams
concerned,
if
they
end
of
hand
happen
knows
are
a
that
travelling
must
the
end
on
by
hit
carriage
is
the
light
the
must
same
mirrors
is
moving
to
be
place
one
events
that
simultaneous
at
different
observer
take
to
all
points
but
not
place
at
the
observers
in
space
same
whereas
can
simultaneous
be
to
another!
stationary
at
the
will
moving
away.
left-hand
end
Do
not
dismiss
fanciful
to
rely
event
to
on
is
these
be
tried
the
rst
ideas
out.
because
The
use
postulate.
of
This
the
a
experiment
pulse
of
conclusion
light
is
seems
allowed
valid
too
us
whatever
considered.
This
rst.
o p t i o n
A
–
r e l A t i v i t y
133
lort trorto
lORenT z TRansfORmaTiOn e X ample
lORenT z facTOR
The
formulae
that
depends
observers,
for
on
special
the
relativity
relative
all
involve
velocity
a
between
We
factor
different
v
can
a pply
dene
the
Lorentz
factor,
γ
as
L o r e ntz
shown
the
measure s
train
observer
We
the
situation
on
on
the
page
the
tr a ns for ma t io n
1 33.
S uppos e
car r ia g e
p l a tf o r m
to
be
me a s ur es
e qua t io ns
the
5 0. 0
the
to
t he
e x pe ri me n t
m
lon g
sp ee d
of
an d
the
on
the
tr a in
follows:
8
to
be
2.7
×
1
10
m s
(0.90
c)
to
the
right.
In
thi s
s i tua ti on,
1
_
γ
=
4
γ ,rotcaf ztneroL
2

v
1
√
At
2
c
low
the
velocities,
Lorentz
factor
approximately
to
one
–
effects
is
equal
relativistic
are
negligible.
we
know
the
according
to
time s
the
(t)
and
locations
e x p e r i me nte r
on
2
the
experimenter
1.
According
Time
1
to
taken
carriage
is
on
the
for
the
is
experimenter
each
given
p la tfor m
pulse
to
the
speed
of
fr a m e
on
the
reach
mirror
=
time
taken
to
the
8.33
×10
for
each
denes
a
∆t
frame
by
of
reference
different
to
complete
and
different
of
space
coordinates
and
time.
In
frame
S,
=
1.67
to
an
According
to
the
experimenter
on
and
take
place
at
a
given
given
time
position
(t).
(x,
y
Observers
and
in
z
disagree
on
the
numerical
values
for
γ
=
relative
Galilean
transformations
equations
(page
2
_
1
c
to
calculate
what
an
observer
in
a
second
frame
1
_
=
=
we
know
the
values
in
one
frame
but
assume
2.29
√
0.19
allowed
will
that
taken
for
LH
pulse
to
reach
mirror
at
end
of
carriage
record
is
if
2
c
=
Time
us
-
√
2
uniform
coordinates.
131)
S'),

(0.9c)
1
√
√
1 0.81
The
(frame
=
coordinates)
these
platform
1
__
1
__
motion
the
the
event
2
a
s
×10
)

v
with
round
events
according
a
the
is:
8
×10
1
_
measurements
associated
of
7
=
total
2.
characterized
be
end
s
pulse
experimenter
(3.0
will
at
S),
×10
50.0
_
be
(frame
speed of light, c
lORenT z TRansfORmaTiOns
observer’s
train
8
3.0
journey
can
and
8
=
v
light.
observer
S)
S'
It
near
Total
An
measured
( f r am e
by:
25.0
_
innity
are
t ra i n
3
∆t
approaches
( x)
the
given
by:
the
v∆x
measurement
of
time
is
the
same
in
both
frames.
Einstein
has
∆t'
=
(LH
γ
(
pulse )
∆t
)
2
c
shown
that
this
is
not
correct.
8
where
∆t
=
8.33
8
×10
s,
v
=
-2.7
1
×10
m
s
(relative
clock in frame S and clock in frame S′ are synchronized
velocity
of
platform
is
moving
to
the
left)
and
∆x
=
-25.0
m
to t = t′ = zero w hen frames coincide.
(pulse
moving
to
left)
(two frames on top of one another)
8
(
∴
∆t'
=
(LH
2.29
(
pulse)
8.33
2.7
×10
)
×
(
25.0)
___
8
y
×10
-
8
(3.0
×
10
)
2
)
frame S (stationary)
y′
7
=
1.91
=
1.9
7
×10
1.72
×
10
frame S′
8
velocity v
Time
is
taken
given
for
×10
RH
s
pulse
to
reach
mirror
at
end
of
carriage
by:
v∆x
∆t'
x
=
(RH
γ
(
pulse)
∆t
)
2
c
x′
8
(
=
time′ = t′
2.29
(
8.33
2.7
×
10
)
×
25.0
__
8
time = t
×10
-
8
(3.0
×
10
)
2
)
y
7
=
frame S
1.91
×10
7
+
1.72
×
10
7
=
3.63
×
10
s
y′
Note that the time taken by each pulse is different – they do not arrive
(x, y, z, t)
frame S′
simultaneously
according
to
the
experimenter
on
the
platform.
(x′, y′, z′, t′)
The
return
time
for
the
LH
pulse
is
the
same
as
the
time
velocity v
taken
for
the
RH
to
initially
reach
the
mirror
(in
each
8
case,
x
So
∆x
total
=
25.0
time
m
and
taken
for
∆t
LH
=
8.33
pulse
×
to
10
s)
return
to
centre
of
x′
carriage
is
8
Because
the
frames
were
synchronized,
the
observers
total
agree
time'
=
(LH
on
the
measurements
measurements
made
of
by
y
z.
To
different
Lorentz
transformations.
dened
above.
The
and
These
derivation
between
observers
all
of
switch
involve
these
we
the
the
need
to
Lorentz
equations
is
other
use
the
factor,
not
This
γ,
is
both
as
the
=
γ(x
vt);
Δx'
=
γ
(Δx
=
γ
(
t
2
;
Δt'
=
γ
(
Δt
2
reverse
transformations
consequence
of
×10
7
=
3.82
× 10
same
as
the
experimenters
total
time
observe
the
taken
return
for
of
the
the
RH
pulses
to
The
taken
above
for
calculates
the
transformation,
)
round
that
can
also
trip
be
is
for
3.82
frame
applied
×10
S',
to
the
s.
the
The
frame
S ')
the
being
relative
also
apply.
velocity
of
These
frame
are
S
just
(with
In
this
to
its
situation,
in
the
opposite
direction.
∆x
(in
frame
S)
=
0
as
pulse’s
the
pulse
journey.
returns
a
starting
position.
total
∆t'
=
pulse )
γ
(
∆t
2
)
=
γ∆t
c
vx'
_
=
γ(x'
+
vt');
t
=
γ
(
t'
+
2
7
)
c
134
be
total
respect
(either
x
so
Lorentz
v∆x
to
s
pulse
c
c
The
3.63
7
time
vΔx
____
)
7
+
required.
vΔt);
vx
_
t'
×10
simultaneous.
Check:
x'
1.9
pulse )
o p t i o n
A
–
r e l A t i v i t y
=
2.29
×
1.67
×
10
7
=
3.82
×
10
s
vot to
velOciTy addiTiOn
When
two
observers
value.
The
calculation
cOmpaRisOn wiTh Galile an
measure
each
other’s
velocity,
they
will
always
agree
on
the
equaTiOn
of
relative
velocity
is
not,
however,
normally
straightforward.
The
For
example,
an
observer
might
see
two
objects
approaching
one
another,
as
top
line
of
the
relativistic
addition
shown
of
velocities
equation
can
be
compared
below.
with
velocity = 0.7c
the
Galilean
calculation
velocity = 0.7c
u'
At
=
low
give
of
u
same
equation
for
the
velocities.
v
values
the
equation
relative
only
of
v
these
value.
starts
two
The
to
equations
Galilean
fail
at
high
velocities.
At
high
velocities,
equation
than
c,
gives
person A
stationary obser ver, C
(rst frame S)
each
object
predict
be
the
than
The
that
case
the
has
the
as
the
speed
situation
a
relative
relative
of
velocity
velocity
Lorentz
factor
of
0.7
between
can
c,
the
the
only
be
the
relative
the
Galilean
answers
of
relativistic
velocity
that
greater
one
is
always
less
speed
of
than
light.
A
r e l A t i v i t y
(second frame S′)
Galilean
two
transformations
objects
worked
out
would
for
be
1.4
objects
c.
would
This
travelling
cannot
at
less
light.
considered
is
one
frame
moving
frame S (stationary)
y
while
give
person B
the
If
a
can
relative
to
another
frame
at
velocity
v
frame S′ (moving)
y′
velocity v
x
Application
of
the
Lorentz
x′
transformation
gives
the
equation
used
to
move
between
frames:
u
v
_
u'
=
uv
_
1
2
c
u'
–
the
velocity
under
secondframe,
u
–
the
velocity
rstframe,
v
In
–
the
each
of
velocity
these
x-direction.
velocity
If
under
of
the
cases,
be
in
the
x-direction
as
measured
in
the
consideration
in
the
x-direction
as
measured
in
the
S
a
second
positive
something
should
consideration
S'
is
frame,
moving
substituted
S',
velocity
into
in
the
as
measured
means
the
motion
negative
in
the
along
rst
the
x-direction
frame,
S
positive
then
a
negative
equation.
Example
In
the
light.
u'
example
So
=
u’
is
relative
u
=
0.7
v
=
-0.7
above,
person
two
A
’s
velocity
objects
velocity
of
as
approached
measured
approach
–
to
be
each
in
other
person
with
B’s
70%
frame
of
of
the
speed
of
reference.
calculated
c
c
1.4 c
_
u' =
note
(1
+
the
sign
in
the
brackets
0.49)
1.4 c
_
=
1.49
=
0.94
c
o p t i o n
–
135
irt tt
spaceTime inTeRval
Relativity
has
shown
that
pROpeR Time, pROpeR lenGTh & ResT ma ss
our
Newtonian
ideas
of
space
a)
Proper
time
interval
Δt
0
and
time
are
incorrect.
Two
inertial
observers
will
generally
When
disagree
on
their
measurements
of
space
and
time
but
expressing
agree
on
a
measurement
of
the
speed
of
light.
Is
else
upon
which
they
will
the
relativity,
consider
a
good
way
everything
spacetime.
From
as
of
imagining
different
one
what
‘events’
observer’s
length
of
taken
time
that
between
a
events
rework
is
(for
giving
proper
time
is
the
time
as
measured
in
out
a
agree?
frame
In
the
there
light),
anything
time
they
example
will
the
point
in
of
is
going
on
something
view,
is
to
where
space.
called
It
the
turns
observer
events
out
could
to
take
be
the
correctly
place
at
shortest
record
for
the
same
possible
the
point
time
in
that
any
event.
three
measuring how long a rework lasts
co-ordinates
further
time
(x,
y
and
‘coordinate’
(t).
An
event
z)
is
is
can
dene
required
a
given
to
a
position
dene
point
its
in
space.
position
specied
by
these
One
in
Moving frame measures a
longer time for the rework
four
since in this frame the
coordinates
(x,
y,
z,
t).
rework is moving.
As
a
result
would
for
thing
is
be
all
of
is
best
of
Lorentz
expected
these
that
2
come
up
with
measurements
two
2
x
observers
2
y
normal
transformation,
another
totally
–
(x',
will
observer
different
y',
z',
agree
t').
on
numbers
The
amazing
something.
This
mathematically:
2
-
to
four
these
stated
(ct)
On
the
2
z
axes,
=
(ct ')
2
-
Pythagoras’s
2
x'
2
y'
z
theorem
shows
us
that
Clock that is stationary with the
the
__________
2
quantity
2
√ (x
+
rework measures the proper
2
y
+
z
)
is
equal
to
the
length
of
the
line
from
time for which it lasted.
2
the
In
origin,
other
so
2
(x
words,
+
it
2
y
is
+
z
the
2
)
is
equal
separation
2
(Separation
in
to
space)
2
=
(the
in
+
of
the
line)
.
space.
2
(x
length
2
y
+
z
)
If
A
is
moving
running
past
2
l
2
2
= x
+ y
2
A.
slowly
+ z
b)
This
for
Proper
past
slowly
B
for
means
B.
Both
length
then
A.
B
From
that
A
views
will
A
’s
will
are
think
point
think
that
of
time
view,
that
B
time
is
is
is
moving
running
correct!
L
0
l
z
As
before,
different
different
observers
measurements
depending
on
their
for
relative
will
the
come
length
motions.
up
of
The
with
the
same
proper
object
length
x
of
an
object
object
y
is
at
is
the
length
recorded
in
a
frame
where
the
rest.
Moving frame measures
The
two
observers
agree
about
something
very
similar
a shor ter length for the
to
this,
but
it
includes
a
coordinate
of
time.
This
can
be
rework’s diameter since
thought
of
as
the
separation
in
imaginary
four-dimensional
the rework is moving
in this frame.
spacetime.
2
(Separation
in
spacetime)
in
spacetime)
2
=
2
(ct)
2
x
2
y
z
or
2
(Separation
2
=
In
(time
1
dimension,
2
this
2
(ct')
2
separation)
(x')
(space
is
simplied
2
=
separation)
to
2
(ct)
(x)
OTheR invaRianT quanTiTies
In
addition
box
to
above),
the
all
spacetime
observers
interval
agree
on
between
the
values
two
of
events
three
(see
other
Ruler that is stationary with the
quantities
or
with
associated
reference
•
Proper
time
•
Proper
length
to
with
a
the
given
interval
separation
object.
between
These
two
events
rework measures the proper length
for its diameter.
are:
Δt
0
c)
Rest
mass
m
0
L
0
The
•
Rest
mass
are
four
always
observer.
velocity.
136
quantities
constant
There
mechanics,
of
mass
depends
on
relative
m
0
These
measurement
that
are
are
are
and
said
do
not
additional
also
o p t i o n
to
be
vary
–
with
quantities,
invariant
A
invariant
e.g.
a
as
they
change
not
of
associated
electric
charge.
r e l A t i v i t y
with
Once
again
it
measurement
taken
other
frames.
mass
A
possible
as
measured
particle’s
rest
is
in
in
mass
a
important
the
The
frame
rest
frame
does
to
of
not
the
mass
where
distinguish
of
the
change.
object
an
object
object
the
from
is
is
at
all
its
rest.
T to
liGhT clOck
A
light
clock
deRivaTiOn Of The effecT fROm fiRsT
is
an
pRinciples
imaginary
device.
A
beam
If
of
light
bounces
‘tick’
two
mirrors
–
we
imagine
the
t
is
the
time
by
the
light
light
As
one
on
with
their
one
light
stationary
clock
clock.
In
then
frame,
a
moving
clock
runs
slowly
and
t'
this
is
‘tick’
of
time
between
‘ticks’
on
the
moving
clock:
t'
is
greater
the
than
t
the
clock.
shown
the
a
is
observer
‘ticks’
between
l
bounces
stationary
between
time
stationary
taken
a
between
path
light
in
the
taken
clock
derivation
by
that
light
is
pulse leaves bottom mirror
in
moving
l
l
at
constant
longer.
the
velocity
We
speed
know
of
l′
is
that
light
is
xed
‘tick’
so
the
time
‘ticks’
must
effect
on
a
also
–
between
moving
be
that
the
vt ′
clock
longer.
In
This
moving
clocks
the
the
time
clock
t
,
has
moved
on
a
distance
=
v t'
_________
run
slow
–
is
called
time
2
Distance
dilation
travelled
by
the
light,
l'
=
√ ((vt')
2
+
l
)
pulse bounces o top mirror
l'
The
time
between
t'
bounces
=
c
Δt
is
the
proper
time
________
for
2
0
√ (vt')
2
+
l
__
this
clock
in
the
frame
=
c
where
the
clock
is
at
rest.
2
‘tick’
t'
2
2
v
t'
+ l
_
2
∴
=
2
c
2
2
t'
l
v
2
∴
1
(
-
)
2
=
2
c
c
2
l
2
but
=
t
=
t
2
c
pulse returns to bottom mirror
2
v
2
∴
t'
(
1
2
2
)
c
1
_
or
deRivaTiOn Of effecT fROm lORenT z
t' =
______
×
t
or
t'
=
γ
t
2
v
√
TRansfORmaTiOn
If
frame
S
is
a
frame
point
in
space,
two
events
take
place
at
the
then
the
time
interval
between
events
must
be
the
proper
time
interval,
Δt
equation
is
true
for
all
measurements
of
time,
whether
these
they
two
2
c
where
This
same
1
have
been
made
using
a
light
clock
or
not.
.
0
Time
dilation
is
then
a
direct
consequence
of
the
Lorentz
transformation:
νΔx
_
Δt'
=
γ
Δt
(
-
)
2
c
Where
Δt
=
Δt
,
(the
proper
time
interval)
and
Δx
=
zero
0
(same
∴
point
time
in
space)
interval
in
frame
S',
Δt'
=
γ∆t
0
o p t i o n
A
–
r e l A t i v i t y
137
lgt otrto   to or t 
rtt
effecT Of lenGTh cOnTRacTiOn
Time
is
not
the
only
measurement
that
is
e X ample
affected
by
relative
motion.
There
is
another
An
unstable
particle
has
a
lifetime
of
8
relativistic
effect
separation
direction.
called
between
The
length
two
contraction.
points
contraction
is
in
in
space
the
According
contracts
same
if
direction
to
there
as
the
a
is
(stationary)
relative
relative
observer,
motion
in
the
4.0×10
that
moving
motion.
a)
Its
b)
The
s
at
in
its
98%
lifetime
of
in
length
own
the
the
rest
frame.
speed
of
in
it
light
laboratory
travelled
If
is
calculate:
frame.
both
frames.
_________
moving frame
1
__
γ
a)
=
√
∆t
2
1
(0.98)
=
5.025
=
γ∆t
=
5.025
0
8
×
4.0
×
10
7
=
2.01
×
10
s
Length contracts along direction
b)
In
the
laboratory
frame,
the
particle
moves
of motion wh en compared
Length
with stationary frame.
=
speed
×
=
0.98
×
=
59.1
m
time
8
3
×
–7
10
×
2.01
×
10
stationary frame
In
the
particle’s
frame,
the
laboratory
moves
59.1
_
∆l
=
γ
Length
contracts
by
the
same
proportion
as
time
dilates
–
the
Lorentz
factor
is
=
once
again
used
in
the
equation,
but
this
time
there
is
a
division
rather
than
11.8
m
a
(alternatively:
multiplication.
length
=
speed
×
8
L
=
0.98
×
3
=
11.8
m)
×
time
8
10
×
4.0
×
10
0
_
L
=
γ
deRivaTiOn Of lenGTh cOnTRacTiOn fROm lORenT z TRansfORmaTiOn
When
we
measure
recording
the
instant
time
the
position
length
of
of
each
a
moving
end
of
the
object,
object
then
at
one
we
are
given
where
the
length
L
object
is
at
rest,
we
will
be
measuring
the
proper
:
0
of
according
to
that
frame
of
reference.
In
other
Δx'
words
the
time
interval
measured
in
frame
S
between
these
=
γ(Δx
-
vΔt)
two
Where
Δx'
=
L
(the
proper
length)
and
0
events
will
be
zero,
Δt
=
0.
In
this
case,
the
length
measured
Δx
Δt =
is
the
length
of
the
moving
object
L
zero (simultaneous measurements of position of end of object)
.
0
∴
Length
contraction
is
then
a
direct
consequence
of
Length
in
frame
S',
L
the
=
γ(L)
0
L
Lorentz
transformation,
as,
if
we
move
into
the
frame,
S',
0
_
L
=
γ
Without
The muOn e XpeRimenT
relativity,
no
muons
would
be
expected
to
reach
the
6
Muons
are
leptons
a
massive
(see
page
78)
–
they
can
be
thought
of
surface
as
at
all.
A
particle
with
a
lifetime
of
2.2
×
10
8
more
the
laboratory
2.2
×
version
but
they
of
an
electron.
quickly
decay.
They
Their
can
be
created
average
in
lifetime
is
is
travelling
expected
near
to
the
travel
speed
less
6
6
at
10
s
as
measured
in
the
frame
in
which
the
muons
are
rest.
Muons
(2.2
The
are
also
created
high
up
(10
km
above
the
surface)
×
a
light
(3
×
kilometre
which
1
10
m
before
s
)
would
be
decaying
8
10
×
moving
speed
in
of
than
s
3
×
10
muons
means
that
=
are
the
660
m).
effectively
Lorentz
moving
factor
is
of
×
‘clocks’.
Their
high
high.
________
the
atmosphere.
Cosmic
rays
from
the
Sun
can
cause
them
1
_
γ
to
be
created
travel
with
towards
the
huge
Earth
velocities
some
of
–
perhaps
them
0.99
decay
but
c.
As
=
they
there
is
still
=
√
7.1
2
1
0.99
a
6
Therefore
detectable
number
of
muons
arriving
at
the
surface
of
the
an
average
lifetime
2.2
10
s
in
the
muons’
Earth.
frame
as
a
of
reference
stationary
will
be
observer
time
on
the
dilated
Earth
to
is
a
longer
time
concerned.
as
From
far
this
‘shower of
cosmic rays from Sun
frame
of
Many
muons
the
reference
surface
–
will
this
they
still
is
will
last,
decay
exactly
but
what
on
average,
some
is
will
7.1
make
times
it
longer.
through
to
observed.
6
In
the
They
a
tm
o
s
p
h
e
re
the
muons’
make
Earth)
it
is
frame
down
they
to
moving
the
with
exist
for
surface
respect
2.2
×
10
because
to
the
s
the
on
average.
atmosphere
muons.
This
(and
means
10 km
that
the
atmosphere
will
be
length-contracted.
The
10
km
some muons decay
before reaching surface
distance
as
measured
by
an
observer
on
the
Earth
will
only
some muons
10
___
reach surface
Ear th
be
=
o p t i o n
A
–
r e l A t i v i t y
km.
A
signicant
number
of
muons
7.1
enough
138
1.4
for
the
Earth
to
travel
this
distance.
will
exist
long
st gr (mo gr) 1
•
spaceTime diaGRams
Spacetime
separation
diagram
a
is
visual
Measurements
actual
We
so
way
can
be
of
introduced
on
representing
taken
from
the
page
the
136.
A
beam
geometry.
diagram
to
Whatever
spacetime
•
The
calculate
cannot
represent
limit
represent.
of
The
all
the
four
simplest
space
dimensions
number
and
of
on
representation
one
of
the
dimensions
time
as
has
shown
one
of
diagram,
space
only
the
advance
are
is
of
from
being
used,
by
represented
by
proper
for
the
frame
separation
shown
that
axes
light
traveller’s
usually
dimension
of
calculated
values.
we
we
was
of
time
overall
convention,
line
any
at
two
they
45°
to
traveller
separation
reference,
between
a
in
can
can
path
be
of
a
axes.
be
spacetime.
remained
events
the
the
In
the
stationary
calculated
so
as
below.
one
below.
e X ample 1 Of spaceTime diaGRams
emit
par ticle at rest
The
advance
events
par ticle with constant speed
of
proper
A→B→C→D
spacetime
time
can
be
for
the
journey
calculated
from
between
the
the
values
on
the
diagram.
time/yr
par ticle which star ts fast
D
6
and then slows down
5
space
An
object
line
in
(moving
or
stationary)
is
always
represented
as
4
a
spacetime.
C
Note
•
light
that:
The
values
on
the
spacetime
diagram
are
as
would
be
2
measured
by
the
by
an
vertical
observer
whose
worldline
is
represented
axis.
B
•
The
t.
vertical
An
axis
alternative
means
that
both
in
is
the
to
above
plot
axes
can
spacetime
(speed
have
of
the
diagram
light
same
×
is
time),
units
time
ct.
(m,
1
This
light-
A
years
or
equivalent).
–0.5
–1
1
2
3
4
space/ly
A
journey
through
spacetime
2
Journey
Space
separation
Time
separation
(Spacetime
2
(x)/ly
(t)/yr
separation)
2
(ct)
(x)
Advance
of
proper
time
according
to
2
/ly
traveller
/
yr
_________
2
2
(ct)
(x)
_
t'
√
=
c
____
2
A→B
0.0
1.0
B→C
1.5
2.0
C→D
2.5
3.0
2
1
0
=
√
1
1.00
=
1.00
=
1.32
=
1.66
____
4
2.25
=
√
1.75
1.75
____
The
total
advance
according
dilation
The
to
an
(see
of
page
alternative
proper
observer
time
whose
for
the
9
traveller
worldline
is
a
is
1.00
vertical
+
line
6.25
1.32
on
+
this
=
√
2.75
1.66
=
3.98
spacetime
yr.
This
diagram.
compares
This
2.75
with
difference
the
is
an
advance
of
example
6.0
of
years
time
137).
journey
direct
from
A
→
D
shows
a
greater
elapsed
proper
time.
2
Journey
Space
separation
Time
separation
(Spacetime
2
(x)/ly
(t)/yr
(ct)
separation)
2
(x)
Advance
of
proper
time
according
to
traveller
/
yr
_________
2
/ly
2
2
(ct)
(x)
_
t'
=
√
c
1.0
A→D
This
is
always
true.
A
direct
6.0
worldline
always
36
has
a
greater
1
=
amount
√
35
of
elapsed
proper
time
than
o p t i o n
an
A
35
=
indirect
–
5.92
worldline.
r e l A t i v i t y
139
st gr 2
c alcul aTiOn Of Time dil aTiOn and
e X ample 2 – cuRved wORldline
lenGTh cOnTRacTiOn
Time
dilation
and
length
contraction
B
are
quantitatively
represented
diagrams.
Refer
a)
dilation:
to
diagram
on
on
spacetime
page
139.
15
Time
In
the
journey
12
direct
14
from
B
→
C,
the
relative
velocity
11
between
13
the
traveller
1.5
and
the
stationary
is
=
2.0
10
observer
ly
_____
0.75
c.
The
Lorentz
12
gamma
yrs
9
factoris:
11
1
_
γ
=
1
__
=
______
√
=
________
2
1.51
10
8
2
√
v
1
0.75
1
time
2
9
c
7
8
The
journey
takes
2
yrs
according
to
the
6
observer
at
measured
rest.
by
This
the
means
traveller
will
∆t
_
∆t
=
γ∆t
⇒
∆t
0
the
proper
7
as
5
be:
6
5
2.0
_
=
=
0
time
=1.32
γ
yr
1.51
4
4
as
shown
in
the
table
on
page
139.
3
3
1
2
rest
2
the
journey
length
from
B→C
 proper time
1
be
1.5
ly.
contracted
The
to
journey
will
be
advance
2

=
1
to

increase
2
in time


measures
increase in


2
 in space

at

observer

The

contraction:

Length

b)
length
be
O
1
2
3
4
5
6
7
8
9
10
L
1.5
_
0
_
L
=
=
=
γ
The
relative
the
traveller’s
taken
makes
shown
space
velocity
time
traveller
ly
1.51
the
This
0.99
to
frame
the
to
be
of
go
travel
from
of
c
×
0.75
c,
B
C,
→
reference,
distance
0.75
is
from
is
according
1.32
yr
=
and
1.32
to
Proper
in
than
time
the
along
proper
a
curved
time
worldline
along
the
from
straight
event
O
to
event
from
O
to
B.
line
B
is
smaller
yr.
the
0.99
ly
as
above.
T t  rox 1
As
mentioned
preference
effect
to
on
page
136,
different
(moving
clocks
the
inertial
run
theory
of
observers
slowly)
is
relativity
–
the
always
time
the
gives
no
This
dilation
same.
to
to
the
‘twin
paradox’.
In
this
imaginary
a
time
situation,
on
Earth
twins
compare
while
the
their
other
views
twin
of
time.
undergoes
a
One
This
very
twin
star
and
back
fast
is
a
relativistic
of
far
as
the
twin
on
trip
out
to
because
observer.
will
think
the
This
that
The
difference
Earth
is
concerned
the
other
twin
is
means
time
that
has
the
been
twin
that
running
remains
slowly
for
on
the
When
aged
they
correct
according
effect
relative
–
time
is
velocity
running
between
at
different
the
two
rates
twins
of
the
distance
between
and
them.
meet
up
again,
the
returning
twin
as
in
far
ageing
as
both
is
of
relative.
them
Neither
are
twin
concerned,
is
getting
time
at
the
normal
rate.
It’s
just
that
the
moving
has
been
twin
the
that
she
has
been
away
for
a
shorter
time
than
the
other
as
recorded
by
the
twin
on
the
Earth.
should
The
have
is
a
time
twin.
it
that:
a
thinks
Earth
but
Remember
again.
passing
moving
the
not
younger;
As
prediction,
formula.
remains
•
distant
strange
two
because
identical
very
dilation
This
•
leads
seems
the
paradox
is
that,
according
to
the
twin
who
made
the
less.
journey,
the
before
the
twin
twin
left
on
on
the
the
Earth
Earth
was
should
moving
have
all
aged
the
less.
time
and
Whose
so
version
after
of
time
The
is
correct?
solution
equations
two
back
would
case
then
twin
situation
The
on
140
o p t i o n
A
–
r e l A t i v i t y
up
This
are
the
141.
constant
one
Earth
be
of
of
is
no
has
are
from
only
relative
them
external
situation
must
comes
relativity
in
involve
the
on
paradox
again,
resolution
page
the
special
observers
meet
The
to
of
forces
not
realization
motion.
would
longer
the
symmetrical
and
have
For
to
the
turn
acceleration.
symmetrical
accelerated
so
for
her
that
when
the
the
twins
to
around.
If
this
the
view
is
the
twins.
of
the
correct.
the
twin
paradox
using
a
spacetime
diagram
is
T rox 2
In
ResOlvinG The Twin paRadOX usinG spaceTime
order
to
to
send
check
light
whose
signals
version
every
of
year.
time
The
‘is
correct’,
spacetime
they
agree
diagram
for
diaGRams
this
The
diagram
below
is
a
spacetime
diagram
for
a
journey
to
situation
in
the
Earth’s
frame
of
reference
is
shown
below
a
(left).
distant
planet
followed
by
an
immediate
return.
Note
According
to
the
twin
remaining
on
that
signals
•
the
distance
to
the
planet
=
3.0
relative
velocity
of
traveller
is
sent
is
and
no
paradox;
received;
the
they
agree
travelling
on
the
twin
has
number
aged
of
less
than
ly
the
•
there
Earth:
0.6
twin
that
stayed
on
Earth.
c
A
more
complicated
spacetime
diagram
can
be
drawn
for
the
3.0
___
•
each
leg
•
Total
of
the
journey
takes
=
5.0
yr
reference
frame
of
the
outbound
traveller
(below
right).
Note
that:
0.6
journey
time
=
10.0
yr
•
The
rst
vertical
The
gamma
factor
•
=
has
the
travelling
=
_______
worldline
stationary.
When
the
travelling
twin
turns
round,
she
leaves
her
1.25
original
2
frame
of
reference
and
changes
to
a
frame
where
2
√
v
1
-
0.6
3
1
the
2
Earth
is
moving
towards
her
at
c
according
to
c
(=
0.6
c).
5
•
So
twin’s
1
_
_
=
______
√
i.e.
years
is
1
_
γ
four
the
twin
undertaking
the
Her
relative
velocity
towards
the
Earth
with
respect
to
journey:
her
original
frame
of
reference
can
be
calculated
from
the
5.0
____
•
each
leg
of
the
journey
takes
=
4
.0
yr
15
1.25
velocity
transformation
equations
as
c
(=
0.88
c)
back.
17
•
Total
journey
time
=
8.0
yr
•
In
this
frame
of
reference,
the
total
time
for
the
round
trip
3.0
____
•
the
distance
to
the
planet
=
=
2.4
ly
would
1.25
be
measured
as
12.5
yr
4.8
___
•
relative
velocity
of
Earth
=
=
0.6
c
8.0
reference frame
13
of outbound traveller
8
reference frame
10
of Ear th
12
9.2
10
8
9
11
8.4
7
9
t (yr)
8
7
10
inbound traveler
7
.8
8.4
v = –3/5 c
8
9
7
.6
7
inbound traveller
6
6.8
light signals
6
v = -15/1
7 c
7
8
from Ear th
6.8
6
5
7
6
)ry( t
Ear th
5
5
6
v = -3/5 c
5
4
5
4
annual
Ear th
4
signals
v = 0
3
4
from Ear th
3.2
3.2
annual
3
signals
3
2.4
from
2.4
outbound
2
traveller
2
traveller
2
lines of
2
1.6
v = 0
1.6
outbound
1
simultaneity
for traveller
traveler
1
1
0.8
1
0.3
v = 3/5 c
x (ly)
x (ly)
8
7
6
5
4
o p t i o n
3
A
2
–
1
r e l A t i v i t y
141
st gr 3
RepResenTinG mORe Than One ineRTial fRame
Mathematically
transformation
for
the
above
calculations,
process
the
to
agree
following
with
must
the
Lorentz
apply:
On The s ame spaceTime diaGRam
•
The
Lorentz
transformations
describe
how
The
space
and
time
in
one
frame
can
be
angle
converted
into
S')
situation
in
observed
each
frame
in
of
another
frame
reference
can
of
be
and
the
reference.
visualized
and
the
x
by
(see
page
141
for
diagrams
for
each
frame
of
is
also
possible
same
spacetime
ct')
moving
to
represent
diagram.
at
A
two
frame
inertial
S’
frames
on
(coordinates
x'
the
relative
constant
velocity
+v
The
frame
The
S
(coordinates
same
(that
The
x
is,
worldline
to
x
and
Lorentz
and
ct).
ct,
applies
The
as
well
to
principles
as
both
to
x'
sets
and
of
are
transformation
is
made
by
by
A
given
for
as
The
worldline
the
angle
for
the
origin
between
axis.
It
the
x'
is:
c
)
frame
S'
rather
than
the
spacetime
axes
for
frame
S
x
by
axes
the
in
axes
in
S'
are
different
to
the
scales
S.
value
is
represented
compared
with
by
the
ct
a
greater
length
on
the
ct'
length
on
the
x'
axis.
A
given
value
is
represented
by
a
greater
axes
axis
when
The
ratio
compared
with
the
x
axis.
ct').
changing
position
has
used
follows:
coordinate
the
of
the
and
ct
at
of
the
measurements
on
the
axes
depends
on
the
coordinate
velocity
between
the
frames.
The
equation
(which
worldline.
does
•
as
v
(
the
when
relative
system
(the
same
to
•
•
axis
and
according
•
•
tan
scales
used
axis
a
ct'
the
examples).
•
is
=
reference
•
It
is
using
1
spacetime
the
axis
The
θ
separate
ct
the
axis
measurements
between
measurements
of
of
right
angles
not
need
to
be
recalled)
is:
______
to
2
v
1
one
another
as
normal.
of
units
c
_
=
√
ct
•
The
spacetime
axes
for
+
2
ct'
_
ratio
frame
S'
has
its
x'
and
ct'
axes
both
2
v
1
2
c
angled
a
in
beam
towards
of
the
x
=
ct
line
(which
represents
a
path
of
light.
ct'
•
The
coordinates
of
a
spacetime
event
in
S
are
read
from
the
ct
xand
•
The
ct
axes
drawing
x'
directly.
coordinates
and
lines
ct'
of
a
spacetime
parallel
to
the
ct'
event
and
x'
in
S'
are
axes
measured
until
they
hit
by
the
axes.
x'
Frame S
Frame S'
ct
ct'
(x', ct') = (0, 1)
light
(x, ct) = (γv/c, γ)
(x', ct') = (1, 0)
θ
(x, ct) = (γ
γ
v/c)
θ
x
D
Summary
•
At
◊
C
greater
the
S'
speed:
axes
swing
towards
the
x
=
ct
line
as
the
angle
θ
increases.
◊
x'
the
ct'
with
and
the
ct
x'
axes
and
x
are
more
stretched
when
compared
axes.
B
•
Events
that
are
simultaneous
in
S
are
that
are
simultaneous
in
S'
on
the
same
horizontal
A
line.
•
θ
Events
the
x
1.
Events
A
&
B
simultaneous
are
in
simultaneous
frame
S'
(A
in
frame
occurs
S
before
but
are
not
B)
2
tan
θ
=
=
0.25
8
∴
2.
3.
relative
velocity
of
frames
S’
Events
C
&
D
occur
at
same
Events
C
&
D
occur
at
different
A
pulse
of
according
events
142
B
light
to
or
emitted
both
by
frames
and
location
=
in
A
0.25
frame
locations
event
of
S
in
arrives
reference.
It
A
–
S'.
frame
at
r e l A t i v i t y
S
event
cannot
C.
o p t i o n
c
D
arrive
at
x'
axis.
are
on
a
line
parallel
to
HL
m  rg
2
ma ss and eneRGy
E = mc
The
most
famous
equation
in
all
of
physics
is
surely
Einstein’s
2
mass–energy
from?
By
length
relationship
now
need
to
it
should
be
E
=
not
viewed
mc
be
in
a
,
a
but
where
surprise
different
does
that
way,
if
it
time
then
so
Mass
be
come
and
equation
and
how
does
energy
converted
the
can
KE
=
equivalent.
mass
always
numbers
1
as
energy.
are
into
be
are
and
vice
used,
This
means
versa.
but
substituted.
one
that
Einstein’s
needs
to
Newtonian
energy
can
mass–energy
be
careful
equations
about
(such
2
mv
or
momentum
=
mv)
will
take
different
forms
2
when
According
constant
at
all
have
to
should
to
Newton’s
acceleration.
do
be
is
laws,
If
this
achievable
apply
a
–
a
constant
was
even
constant
force
always
faster
force
and
produces
true
then
than
theory
is
applied.
a
any
light.
relativity
velocity
All
The
energy
energy
we
E
needed
and
can
to
be
create
a
particle
calculated
from
at
the
rest
is
rest
called
the
rest
mass:
0
2
wait.
E
=
0
yticolev
If
constant force – velocity as
this
m
c
0
particle
is
given
a
velocity,
it
will
have
a
greater
total
energy.
predicted by Newton
2
E
=
γm
c
0
speed of light, c
constant acceleration
time
In
practice,
object
gets
this
starts
less
to
and
does
not
approach
less
even
if
happen.
the
the
As
speed
force
soon
of
is
as
light,
the
the
speed
of
an
acceleration
constant.
yticolev
constant force – velocity as
predicted by Einstein
speed of light, c
acceleration decreases
as speed gets close to c
time
The
the
force
is
object
still
must
relativistic
doing
still
equation
work
be
is
(=
gaining
needed
force
×
kinetic
for
distance),
energy
therefore
and
a
new
energy:
2
E
=
γ m
c
0
Note
that
some
textbooks
compare
this
equation
with
the
2
denition
of
rest
energy
(E
=
m
0
concept
of
relativistic
mass
c
)
in
order
to
dene
a
0
that
varies
with
speed
(m
=
γm
).
0
The
current
The
preferred
adopt
a
new
IB
syllabus
approach
relativistic
does
is
to
not
see
formula
encourage
rest
for
mass
this
as
kinetic
approach.
invariant
and
to
energy:
2
Total
energy
=
rest
energy
+
kinetic
energy
=
γm
c
0
2
rest
energy
=
m
c
0
2
so,
kinetic
energy
E
=
K
(γ
-
1)m
c
0
o p t i o n
A
–
r e l A t i v i t y
143
Rtt ot  rg
HL
equaTiOns
The
of
laws
of
energy
concepts
uniTs
conservation
still
apply
often
in
have
of
momentum
relativistic
to
be
rened
and
conservation
situations.
to
take
However
into
SI
the
account
the
units
of
viewing
space
and
be
it
is
example,
dened
p
=
as
in
the
Newtonian
product
of
in
to
these
use
equations.
other
units
Sometimes,
instead.
the
atomic
scale,
the
joule
is
a
huge
unit.
Often
the
time.
electronvolt
For
applied
useful
new
At
ways
can
however,
mechanics,
mass
and
momentum
p
is
gained
velocity.
by
(eV)
one
difference
of
is
used.
electron
1
volt.
if
One
it
electronvolt
moves
is
through
a
the
energy
potential
Since
energy
mv
difference
__
Potential
difference
=
charge
In
relativity
to
be
it
has
a
similar
form,
but
the
Lorentz
factor
needs
19
1
taken
into
eV
=
1
V
×
1.6
×
10
C
consideration.
19
p
=
γ
m
=
1.6
is
too
×
10
J
v
0
In
The
momentum
of
an
object
is
related
to
its
total
energy.
fact
the
2
E
2
=
p
mechanics,
2
c
2
+
m
the
relationship
can
be
stated
small
a
unit,
so
the
standard
SI
In
multiples
relativistic
electronvolt
are
used
as
1
4
keV
=
1000
eV
c
0
6
1
In
Newtonian
and
mechanics,
momentum
the
relationship
between
MeV
=
10
eV
etc.
energy
Since
is
mass
and
comparable
2
energy
units
for
are
equivalent,
mass.
The
it
makes
equation
that
sense
links
to
the
have
two
p
_
E
=
2
(E
2m
=
mc
speed
Do
not
be
tempted
to
use
the
standard
Newtonian
2
)
of
the
situation
is
relativistic,
then
you
need
to
is
a
new
unit
included
in
for
the
mass
unit
–
so
the
MeV
that
no
c
.
The
change
of
equations–
number
if
denes
light
use
is
needed
when
switching
between
mass
and
energy
–
the
2
If
relativistic
a
particle
of
mass
of
5
MeV
c
is
converted
completely
be
It
into
equations.
energy,
the
energy
released
would
2
possible
In
a
to
use
similar
keV
way,
the
5
MeV
.
would
also
be
2
c
or
GeV
easiest
c
as
unit
a
for
unit
for
mass.
momentum
is
the
1
MeV
c
.
which
b)
e X ample
The
Large
Centre
Electron
for
Nuclear
total
energies
with
positrons
of
/
Positron
Research
about
90
(LEP)
collider
(CERN)
GeV
.
These
at
the
accelerates
electrons
electrons
links
is
positive
are
moving
in
identical
charge.
The
the
in
opposite
rest
mass
positrons
to
have
direction
electrons
the
same
as
these
two
carry
energy
unit
to
energy
particles,
use
and
if
using
the
equation
momentum.
estimate
their
relative
velocity
of
to
since
γ
so
large
collide
shown
but
best
relativistic
relative
Positrons
the
approach.
European
then
For
This
as
velocity
≃
c
below.
a
the
electrons.
c)
Electron
What
is
the
total
momentum
of
the
system
(the
two
Electron
particles)
●
before
the
collision?
●
zero
Total
a)
energy
=
90
GeV
Use
the
equations
(i)
the
velocity
of
of
Total
special
an
relativity
electron
(with
energy
to
=
90
GeV
calculate,
respect
to
the
d)
The
collision
causes
new
particles
to
be
created.
laboratory);
(i)
Total
energy
=
90
GeV
=
90000
Estimate
2
mass
=
0.5
∴
≃
c
MeVc
∴
γ
maximum
total
rest
mass
possible
for
the
MeV
new
Rest
the
=
18000
particles.
Total
(huge)
energy
available
=
180
GeV
2
(ii)
the
v
momentum
of
∴
an
electron
(with
respect
to
max
total
rest
mass
possible
=
180
GeVc
the
laboratory).
(ii)
2
p
2
c
2
=
E
2
m
Give
one
reason
why
your
answer
is
a
maximum
4
c
0
Above
2
≃
E
≃
90
1
p
144
GeVc
o p t i o n
A
–
r e l A t i v i t y
assumes
that
particles
were
created
at
rest
Rtt  x
HL
paRTicle acceleRaTiOn and elecTRic chaRGe
e X ample
In
An
6
a
particle
charged
basic
accelerator
particles
principle
are
is
to
(e.g.
a
linear
accelerated
pass
the
up
accelerator
to
charged
very
or
high
particles
cyclotron),
energies.
through
a
The
electron
Calculate
is
its
potential
differences
and
each
time,
the
particle’s
through
a
pd
of
1.0
×
10
V
.
velocity.
series
6
Energy
of
accelerated
gained
=
1.0
×
10
=
1.6
×
10
=
m
19
×
1.6
×
10
J
total
13
energy
increases
as
a
result.
of
a
charge
The
increase
in
kinetic
J
energy
2
E
(ΔE
)
as
a
result
q
passing
through
a
potential
0
31
c
=
9.11
×
8
10
×
(3
×
10
2
)
0
K
14
difference
V
is
given
by:
=
8.2
×
10
J
13
qV
=
ΔE
∴
Total
energy
=
1.6
×
=
2.42
10
14
+
8.2
×
10
K
13
×
10
J
13
2.42 × 10
__
∴
γ
=
=
2.95
14
8.2
×
10
______
1
_
velocity
=
1
-
c
√
=
The
phOTOns
Photons
the
are
speed
particles
of
light,
c.
that
have
Their
a
total
zero
rest
energy
mass
and
and
their
travel
frequency
f
relativistic
momentum,
at
is
2
E
2
=
p
must
2
+
m
c
that
also
links
apply
total
to
energy,
E
and
photons:
4
2
c
0.94
equation
p,
2
γ
c
0
linked
by
Planck’s
constant
h:
The
E
=
hf
rest
photon
mass
a
hf
E
p
of
photon
is
zero
so
the
momentum
of
a
is:
=
=
h
=
c
c
λ
2
Note
e X ample: dec ay Of a piOn
neutral
A
typical
pion
(π
2
)
is
a
meson
of
in
this
example,
total
energy
2m
c
,
so
γ
=
2
so
v
0
0
A
that
rest
mass
m
=
135.0
MeV
c
.
=
0.866
c
0
mode
of
decay
is
to
convert
into
two
photons:
Each
photon
will
have
a
total
energy
of
11
0
π
→
2γ
135
MeV
and
a
=
2.16
×
10
J
1
The
wavelength
of
these
photons
can
be
calculated:
momentum
photons
a)
Decay
at
will
of
135
MeV c
.
The
wavelengths
of
the
be:
rest
8
c
λ
If
the
have
pion
half
was
the
at
rest
total
when
energy
it
of
decayed,
the
each
=
67.5
MeV
=
1.08
×
=
67.5
×
10
=
3.0 ×
10
__
34
h
=
6.63
×
10
×
11
would
E
2.16
×
10
15
pion:
=
6
E
photon
9.21
×
10
m
19
×
1.6
×
10
J
Initial
total
momentum
for
the
pion
in
the
forward
direction
11
Planck’s
10
constant
J
can
can
be
used
to
calculate
the
be
calculated
from
2
wavelength
E
2
=
p
2
c
4
2
+
m
c
0
of
one
of
the
photons:
2
2
p
c
2
=
2
E
m
4
c
2
=
(4
1)m
0
4
c
0
c
E
=
h
1
p
λ
=
√
3
m
c
=
1.73
×
135.0
=
233.8
MeV
c
0
8
c
λ
=
3.0 ×
10
__
34
h
=
6.63
×
10
So
×
conservation
of
momentum
in
forward
direction
is:
11
E
1.08
×
10
233.8
=
2
×
135
×
cos
θ
14
=
1.84
×
10
m
233.8
_
∴
cos
θ
=
∴
θ
=
=
0.866
270
The
momentum
Conservation
emitted
photon
b)
Decay
in
add
the
was
to
give
initially
means
directions.
together
the
pion
momentum
opposite
while
Suppose
of
of
a
The
that
total
total,
zero
the
as
it
was
photons
momentum
once
again,
of
at
will
of
rest.
30°
be
each
zero.
moving
pion
was
moving
forward
when
it
decayed
2
with
a
total
emitted
as
energy
shown
270.0
MeV c
;
the
photons
will
be
below:
photon 1
after
before
θ
θ
pion
photon 2
o p t i o n
A
–
r e l A t i v i t y
145
Gr rtt – t   r
HL
pRinciple Of equivalence
One
of
Einstein’s
example
There
below
are
two
‘thought
considers
possible
experiments’
an
observer
situations
•
The
rocket
could
be
far
•
The
rocket
could
be
at
away
rest
to
a
how
closed
an
observer’s
view
of
the
world
would
change
if
they
were
accelerating.
The
spaceship.
compare.
from
on
considers
inside
the
any
planet
surface
of
a
but
accelerating
forwards.
planet.
dropped
rocket at rest
dropped object
accelerating
will ‘fall’
for ward
object will fall
on planet
towards oor
towards oor
planet
astronaut feels a force when rocket is
astronaut feels a force when rocket is
accelerating for ward
at rest on the surface of a planet
Although
these
situations
seem
completely
different,
the
observer
inside
the
rocket
would
interpret
these
situations
as
being
identical.
This
is
Einstein’s
and
From
principle
down
the
near
a
a
‘principle
reference
of
gravitational
of
massive
equivalence’
equivalence,
body
–
a
postulate
that
states
(see
it
can
page
eld
the
In
principle
should
rocket
both
of
inertial
bend
that
the
be
deduced
that
light
rays
cases
point
that
the
rocket
is
was
in
and
rays!
a
are
bent
beam
equivalence
light
allows
observer
the
of
there
of
diagrams
would
exactly
suggests
There
beam
is
a
light
1
a
that
gravitational
window
high
up
in
enter.
and2,
the
see
the
light
the
small
upwards
that
small
to
opposite
accelerating
is
no
difference
between
an
accelerating
frame
of
in
a
gravitational
eld
(see
below)
and
that
time
slows
147).
bendinG Of liGhT
Einstein’s
that
eld.
(see
But
no
observer
shining
on
window.
is
diagram3)
wall
at
however,
then
light
would
opposite
Einstein’s
difference
observer
an
the
If,
of
is
follow
a
diagram
in
a
the
hit
a
small
principle
between
point
of
an
path
4.
effect
This
in
the
equivalence
accelerating
gravitational
curved
on
wall
below
the
point
window.
a
eld.
If
this
gravitational
does
states
that
observer
is
true
eld
there
and
then
as
is
inertial
light
shown
should
in
happen!
the
light hits
nal position of
wall
light hits
window when
opposite
below window
window
light hits below
light hits
window
light hits
nal position
as rocket has
of window
speeded up
window in an
wall
rocket
original
window
accelerating
when
opposite
rocket
moves
position
light hits
accelerating
view
rocket and in a
upwards at
of window
inside
stationary rocket
constant
rocket
in a gravitational
velocity
eld
1
rocket at rest
2
in space
146
o p t i o n
rocket moving with
constant velocity
A
–
r e l A t i v i t y
3
rocket accelerating
upwards
4
rocket at rest in a
gravitational eld
Grtto r t
HL
cOncepT
The
can
red
maThemaTics
general
be
of
relativity
experimentally
shift
words
theory
a
–
clocks
clock
on
tested.
slow
the
down
ground
makes
One
in
a
oor
other
such
predictions
effect
is
gravitational
of
a
This
gravitational
eld.
building
that
will
In
other
run
gravitational
worked
out
frequency
for
∆f
a
is
time
dilation
uniform
given
effect
can
gravitational
be
mathematically
eld
g.
The
change
in
by
slowly
g∆h
∆f
___
_
=
when
compared
with
a
clock
in
the
attic
–
the
attic
is
further
2
f
away
from
the
centre
of
the
c
Earth.
where
f
is
the
frequency
g
is
the
gravitational
emitted
eld
at
the
source
strength
(assumed
to
be
constant)
∆h
is
the
height
c
is
the
speed
difference
of
and
light.
A clock on the ground
oor runs slow when
e X ample
compared with a
A
clock in the attic
the
UFO
travels
Earth
radio
at
signal
(i)
What
(ii)
If
the
the
is
a
at
such
height
of
a
of
speed
200
frequency
the
of
frequency
signal
was
observer
to
km
110
remain
above
MHz
received
reected
frequency
of
back
the
is
by
to
above
the
sent
the
to
point
on
surface.
the
A
UFO.
UFO?
Earth,
return
one
Earth’s
what
signal?
would
Explain
be
your
answer.
8
(i)
f
=
1.1
g
=
10
∆h
=
2.0
∆f
=
×
10
Hz
–2
The
same
seen
that
away
effect
a
from
can
be
imagined
gravitational
a
mass
(for
eld
in
a
affects
example
the
different
light.
Sun),
If
way.
light
the
is
We
have
of
s
5
shone
photons
m
×
10
m
light
5
10 × 2.0 × 10
__
must
be
increasing
their
gravitational
potential
energy
as
∴
they
move
away.
This
means
that
they
must
be
decreasing
8
×
8
(3
×
10
1.1
×
10
Hz
2
)
their
–3
total
energy.
photon,
be
less
the
than
Since
frequency
observed
the
is
a
frequency
emitted
measure
away
of
from
the
the
energy
source
of
=
a
2.4
×
10
received
=
1.1
Hz
=
109999999.998
≈
1.1
8
must
∴
f
×
10
–3
–
2.4
×
10
frequency.
Hz
8
(ii)
The
return
Therefore
frequency
signal
it
will
as
×
10
will
be
arrive
Hz
gravitationally
back
at
exactly
blue
the
shifted.
same
emitted.
At the top of the
building, the photon
has less energy, and
so a lower frequency,
than when it was at
the bottom.
The
oscillations
clock.
the
An
clock
of
observer
on
the
the
at
light
the
ground
can
top
oor
be
of
to
imagined
the
be
building
running
as
the
pulses
would
of
a
perceive
slowly.
o p t i o n
A
–
r e l A t i v i t y
147
sor tg 
HL
e vidence TO suppORT GeneRal Rel aTiviTy
Bending of star light
The
predictions
One
main
physicist
The
called
idea
was
Arthur
the
visible
the
Moon
not
blocks
same
all
stars
relativity,
the
experiment
of
because
the
recorded
light
at
just
bending
Eddington
behind
stars
the
general
are
the
of
of
prediction
a
in
those
light
by
a
of
special
relativity,
gravitational
eld.
seem
One
so
of
strange
the
rst
that
we
need
experiments
to
strong
experimental
check
this
effect
evidence.
was
done
by
a
1919.
was
the
like
of
to
measure
Sun
from
is
the
different
so
the
bright.
Sun.
time,
If
the
the
deection
During
a
positions
stars
that
of
light
solar
of
(from
eclipse,
the
stars
appeared
a
star)
during
near
as
however,
the
the
edge
a
result
stars
total
of
the
are
of
the
visible
eclipse
Sun
Sun’s
mass.
during
were
would
the
During
few
compared
appear
to
with
have
the
minutes
the
day,
when
positions
moved.
actual position of star
apparent position
Sun
of star
Moon
not to scale!
Ear th
The
angle
of
the
shift
of
these
stars
turned
out
to
be
exactly
apparent position
usual position of star in sky
during eclipse
(compared with others)
the
angle
as
predicted
by
Einstein’s
general
theory
of
relativity.
Gravitational lensing
The
bending
very
bend
of
extreme
around
the
path
effects.
the
of
light
Massive
galaxy
as
or
the
galaxies
shown
warping
can
of
deect
spacetime
the
light
(depending
from
quasars
on
which
(or
other
description
very
you
distance
prefer)
sources
can
of
also
light)
so
produce
that
the
some
rays
below.
image of quasar
massive galaxy
quasar
obser ver
image of quasar
not to scale!
In
this
strange
situation,
the
galaxy
is
acting
like
a
lens
and
we
e vidence TO suppORT GRaviTaTiOnal Red shif T
can
observe
multiple
images
of
the
distant
quasar.
Atomic clock frequency shift
Because
they
are
so
sensitive,
comparing
the
difference
in
Pound–Rebka–Snider experiment
time
The
decrease
in
the
frequency
of
a
photon
as
it
climbs
recorded
a
gravitational
eld
can
be
measured
in
the
measurements
been
successfully
experiments
to
photons
were
to
achieved
do
Pound–Rebka
need
this
was
be
on
The
after
sensitive,
many
done
experiment.
measured
very
in
and
frequencies
they
but
occasions.
1960
ascended
is
of
or
they
One
of
Physical
Laboratory
Tower
at
measurement
original
greater
148
Pound–Rebka
Harvard
accuracy
by
Pound
o p t i o n
A
experiment
and
–
was
clocks
can
provide
of
gravitational
red
shift.
One
of
the
a
clocks
called
taken
to
high
remains
on
the
altitude
will
faster.
ground.
by
a
The
rocket,
clock
whereas
that
is
at
a
the
second
higher
one
altitude
the
run
the
gamma-ray
Global positioning system
descended
the
global
positioning
system
to
be
so
accurate,
general
University.
relativity
The
atomic
have
For
Jefferson
identical
laboratory.
is
The
two
out
direct
of
by
repeated
Snider.
r e l A t i v i t y
with
the
must
satellite's
be
taken
orbit.
into
account
in
calculating
the
details
of
cr tr o t
HL
spacetime.
effecT Of GRaviTy On spaceTime
The
Newtonian
forces
between
thinking
changes
spacetime
both
set
travel
two
about
in
the
is
off
way
of
masses.
gravity
shape
caused
from
describing
is
In
not
general
to
think
(warping)
by
mass.
different
gravity
of
in
it
as
about
on
terms
relativity
of
a
spacetime.
Think
points
is
the
the
two
but
four
going
of
As
has
been
dimensions
on
by
of
explained,
it
spacetime.
representing
It
spacetime
is
is
very
hard
easier
as
a
at
to
to
imagine
picture
what
is
two-dimensional
sheet.
as
warping
travellers
Earth’s
the
the
way
force,
The
of
of
who
equator
and
north.
On the surface of the Ear th, two travellers
spacetime represented by at sheet
who set o parallel to one another…
Any
you
of
mass
have
spacetime
around
P
Q
would
itself.
…may eventually meet:
the
P
they
They
of
travel
could
attraction
Earth
so
they
this
between
consequence
travellers
north
explain
of
have
their
the
to
will
them
surface
move
paths
get
coming
in
come
or
of
closer
they
the
Earth
straight
in
could
closer
terms
explain
being
lines
can
Sun.
explain
be
the
(or
bends)
warping
used
The
the
Earth
shortest
warps
greater
to
describe
diagram
situation.
orbits
possible
the
path
below
The
Sun
in
spacetime.
that
takes
the
The
place.
orbit
of
represents
Sun
warps
because
it
spacetime.
is
more
This
the
Earth
how
Einstein
spacetime
travelling
This
turns
mass
warping
out
around
along
to
be
a
Q
and
together
the
The
curved
As
present
the
together.
of
it
a
as
curved.
across
path.
the
force
a
The
surface
of
the
together.
Sun
Einstein
curved
showed
by
mass.
how
The
spacetime
becomes.
spacetime
or
in
spacetime
more
Moving
other
could
matter
you
objects
words,
they
be
thought
have,
follow
take
the
the
the
of
as
more
being
curved
curvature
shortest
path
•
Mass
‘tells’
•
Spacetime
spacetime
how
to
curve.
of
‘tells’
matter
how
to
move.
with
spacetime
in
applic aTiOns Of GeneRal Rel aTiviTy TO The univeRse a s a whOle
General
affect
relativity
each
cosmology
The
Very
sections
and
large
the
mass
holes.
now
of
of
fundamental
allows
the
the
are
relativity
astrophysics
of
holes
may
the
evidence
can
cosmic
searching
predicts
Experimental
to
understanding
far-reaching
Universe
presence
black
astronomers
General
is
This
development
elements
and
other.
option
be
at
the
the
of
in
to
the
objects
created
in
about
the
the
Universe
future
interact
development
and
fate
of
and
the
thus
how
Universe
–
they
see
D).
detail.
of
Many
are
many
current
aspects
(e.g.
its
large-scale
structure,
the
creation
of
the
predicted.
galaxies.
General
relativity
predicts
how
these
may
interact
with
matter
evidence.
gravitational
existence
how
be
radiation)
centres
appropriate
existence
for
(option
modelled
background
exist
for
predictions
of
these
waves
waves
associated
is
being
with
high
energy
events
such
as
the
collision
of
two
black
sought.
o p t i o n
A
–
r e l A t i v i t y
149
b o
HL
At
descRipTiOn
When
a
gravity
for
density
star
of
the
would
be
of
the
Sun,
all
of
its
it
collapse
down
details).
The
matter
and
in
more
becomes
the
thus
star.
described
the
stopping
this
dwarf.
the
star
on
itself
of
more
and
it
of
the
fuel,
(see
the
the
force
greater
relativity,
spacetime
The
severe
near
a
masses
can
of
stop
more
astrophysics
the
gravitational
general
curved.
more
the
contracts
greater
terms
terms
more
nuclear
more
the
In
and
collapsing
collapsing
then
up
more
becoming
mass
used
collapsing
spacetime
If
has
makes
option
near
star
it
the
mass
eld
and
folds
in
is
greater
the
than
this
contraction.
more
over
warped
itself.
until
What
concentrated
we
do
not
Spacetime
into
is
a
know
around
eventually
left
is
point
it
called
–
the
of
any
the
process
mass
becomes
a
black
that
becomes
so
hole.
great
All
that
the
singularity
this
collapsing
curvature
of
depending
on
the
star.
is
less
electrons
than
play
contraction.
an
The
about
1.4
times
important
star
that
is
part
left
the
in
is
mass
of
the
eventually
called
a
white
spacetime with
If
collapsing
star
is
greater
than
this,
the
electrons
extreme curvature
cannot
times
halt
the
neutrons
a
mass
play
neutron
more
the
contraction.
of
an
star.
extreme
the
Sun
can
important
The
A
the
also
role
curvature
than
contracting
be
and
of
mass
stopped
the
star
spacetime
curvature
near
a
–
of
this
that
near
white
up
is
a
to
three
time
left
the
is
called
neutron
star
is
dwarf.
schwaR zchild Radius
The
curvature
extreme
can
be
since
that
of
attracted
nothing
forces
are
deected
so
spacetime
nothing,
into
can
the
travel
extreme
near
a
not
hole,
faster
that
black
near
even
a
black
light,
but
light.
would
be
is
can
The
singularity
so
escape.
nothing
than
light
hole
can
Matter
get
out
gravitational
severely
hole.
event horizon
photon sphere
At
a
particular
distance
Schwarzchild
velocity
is
predicts
that
given
by
equal
the
from
radius,
to
the
the
we
the
get
speed
escape
centre,
to
of
velocity
a
light.
v
called
point
the
where
the
Newtonian
from
a
mass
M
escape
mechanics
of
radius
r
is
formula
_____
2GM
_
v
photon sphere
=

√
r
black hole
If
the
escape
velocity
Schwarzchild
If
you
were
to
approach
a
black
hole,
the
gravitational
radius
is
the
speed
would
be
of
light,
given
c,
then
the
by
forces
2GM
_
R
on
you
the
would
photon
increase.
sphere.
The
This
rst
thing
consists
of
a
of
interest
very
thin
would
shell
of
captured
in
orbit
around
the
black
hole.
As
we
turns
in,
the
gravitational
forces
increase
and
so
the
out
that
proper
at
that
distance
also
longer
this
called
e X ample
of
a
black
radius
hole
that
has
the
same
mass
as
for
be
able
reason
crossing
object
size
is
general
also
correct
relativity.
If
if
we
we
and
get
closer
to
the
singularity,
to
crossing
the
the
event
approaching
the
communicate
a
with
Schwarzchild
horizon.
black
the
hole
An
would
Universe
radius
observer
see
time
object.
30
our
Sun
(1.99
×
10
kg).
The
11
observed
time
dilation
is
worked
out
from
30
2 × 6.67 × 10
× 1.99 × 10
___
R
∆t
=
0
Sch
8
(3
×
10
_
2
∆t
)
=
______
R
S
_
=
2949.6
=
2.9
√1
m
r
km
where
150
o p t i o n
A
use
cross
the
we
would
increases.
For
the
of
escape
no
Calculate
equation
equations
Schwarzchild
velocity
this
fall
the
further
2
c
light
It
photons
=
S
be
–
r e l A t i v i t y
r
is
the
distance
from
the
black
hole.
is
outside.
sometimes
watching
slowing
an
down
ib qto – oto a – rtt
1.
In
the
laboratory
particle
an
of
electric
the
the
frame
parallel
current.
alpha
wire
alpha
2.
travels
are
in
the
The
alpha
b)
The
laboratory
line
the
is
as
moving
rocket
of
drift
Explain
frame
of
slow
reference,
velocity
the
moving
wire
of
origin
reference
that
the
the
of
the
alpha
and
carries
station
velocity
electrons
force
on
away
station
observer
each
in
platform.
platform
T
is
strike
Observer
midway
sitting
travels
in
to
between
the
middle
both
S
is
the
of
standing
two
the
on
strikes,
train.
the
while
Light
from
observers.
the
of
0.5 c
[2]
from
speed
the
[2]
rockets
viewed
at
frame
the
a
metal
particle
identical
Earth
reference,
stationary
this
and
identical.
particle
a)
Two
In
particle
of
to
are
moving
Earth.
0.80
from
c
along
Rocket
1
relative
rocket
1
at
is
to
the
same
moving
the
speed
away
Earth
0.60
c
straight
from
and
rocket
relative
2
b)
to
If
observer
S
on
observations
1
the
that
simultaneously,
will
conclude
station
the
two
explain
that
they
concludes
lightning
why
did
from
strikes
observer
not
occur
T
his
occurred
on
the
train
simultaneously.
[4]
rocket 1
0.80 c relative
0.60 c relative
to Ear th
to rocket 1
c)
Which
d)
What
on
e)
a)
Calculate
using
the
velocity
of
rocket
2
relative
to
the
What
be
train,
will
the
will
the
T
conclude
distance
according
be
the
platform,
occurred
between
to
T
distance
and
the
to
T
the
and
[1]
scorch
according
between
according
rst?
to
marks
S?
scorch
according
[3]
marks
to
S?
[2]
the
(i)
Galilean
transformation
(ii)
relativistic
equation.
transformation
HL
[1]
equation.
[2]
5.
b)
will
the
on
Earth,
strike
Comment
on
your
answers
in
(a).
In
a
laboratory
experiment
two
identical
particles
(P
and
Q),
each
[2]
of
rest
mass
m
,
collide.
In
the
laboratory
frame
of
reference,
0
3
c)
The
rest
mass
of
rocket
1
is
1.0
×
10
kg.
Determine
the
they
relativistic
kinetic
energy
of
rocket
1,
as
measured
the
an
observer
on
are
both
moving
at
a
velocity
of
2/3
c.
The
situation
before
by
Earth.
collision
is
shown
in
the
diagram
below
.
[3]
Before:
3.
The
spacetime
diagram
below
shows
two
events,
A
and
B,
2/3 c
as
observed
in
a
reference
frame
S.
Each
event
emits
a
P
signal.
Use
the
diagram
to
calculate,
according
to
frame
The
time
between
event
A
and
event
Q
S,
a)
a)
B
In
the
The
time
taken
for
the
light
signal
leaving
event
A
what
at
the
position
of
event
B.
The
location
of
a
stationary
observer
who
receives
signal
from
events
A
reference,
total
momentum
is
the
total
energy
of
P
and
Q?
[1]
what
simultaneously
same
collision
can
be
of
P
and
viewed
Q?
[3]
according
to
P’s
frame
the
of
light
of
the
[2]
The
c)
frame
is
to
(ii)
arrive
laboratory
[2]
(i)
b)
2/3 c
light
reference
as
shown
in
the
diagrams
below.
with
velocity = v
receiving
the
light
signal
from
event
B.
[2]
P (rest)
d)
The
velocity
event
A
and
of
a
moving
event
B
frame
occurred
of
reference
in
Q
which
simultaneously.
[4]
b)
ct / ly
6
In
P’s
frame
of
reference,
(i)
what
is
Q’s
(ii)
what
is
the
total
what
is
the
total
(iii)
velocity,
v?
[3]
momentum
energy
of
of
P
P
and
and
Q?
[3]
Q?
[3]
5
c)
B
As
a
result
formed,
of
but
the
the
collision,
total
many
energy
of
particles
the
and
particles
photons
depends
are
on
4
the
frame
reference
of
reference.
agree
or
Do
the
disagree
on
observers
the
in
number
each
of
frame
particles
of
and
3
photons
formed
in
the
collision?
Explain
your
answer.
[2]
A
2
6.
The
concept
slower
as
of
they
gravitational
approach
a
red-shift
black
indicates
that
clocks
run
hole.
1
a)
Describe
(i)
0
1
2
3
4
5
6
Relativity
and
is
meant
by
red-shift.
[2]
7
x / ly
4.
what
gravitational
(ii)
spacetime.
(iii)
a
black
[1]
hole
with
reference
to
the
concept
of
simultaneity
spacetime.
a)
State
two
postulates
of
the
special
theory
of
relativity.
b)
Einstein
proposed
a
‘thought
experiment’
along
A
particular
lines.
Imagine
a
train
of
proper
length
100
through
a
station
at
half
the
speed
of
light.
at
two
of
lightning
the
train,
strikes,
leaving
one
at
scorch
the
front
marks
on
and
one
both
has
a
Schwarzschild
radius
R.
A
the
distance
of
measures
2R
the
from
time
the
event
between
horizon
two
of
events
the
to
be
10
s.
There
at
that
for
a
person
a
very
long
way
from
the
black
hole
the
the
rear
a
hole
Deduce
are
hole
m
black
passing
black
the
person
following
[2]
[2]
time
between
the
events
will
be
measured
as
12
s.
train
i B
Q u e s t i o n s
–
o p t i o n
A
–
r e l A t i v i t y
151
[1]
14
o p t i o n
B
–
e n g i n e e r i n g
p h y s i C s
t   
ConCepts
The
Translational
complex
motion
of
a
rigid
body
can
be
analysed
as
Every
combination
of
two
types
of
motion:
translation
and
these
types
of
motion
are
studied
separately
in
Rotational
particle
this
the
same
in
the
object
Every
instantaneous
(pages
9
and
particle
moves
in
a
in
the
circle
object
around
the
study
velocity
guide
motion
rotation.
has
Both
motion
a
same
axis
of
rotation
65).
Displacement,
in
s,
measured
Angular
m
displacement,
measured
Velocity,
change
v,
of
is
the
rate
of
Angular
displacement
rate
of
in
radians
velocity,
change
ω,
of
in
m
is
the
angle
1
measured
θ,
[rad]
1
s
measured
in
rad
s
ds
v
=
dθ
_
dt
mg
ω
=
dt
Acceleration,
change
of
a,
is
the
velocity
rate
of
Angular
measured
the
acceleration,
rate
of
α,
is
change
of
angular
measured
in
rad
2
in
m
2
s
velocity
dv
_
a
A
bottle
bottle
thrown
follows
addition
the
of
linear
mass
angles
of
the
The
of
the
is
object.
air
predicted
about
described
all
these
the
by
centre
one
(or
motion
is
apply
apply
to
to
circular
of
angular
mechanics
of
velocity,
circular
ω,
has
motion
already
(see
centre
to
the
frequency
of
rotation
by
the
When
angular
been
about
and
a
of
a
The
2π
motion
of
the
translational
rotational
motion
a)
Translational
the
wheel
and
forward
wheel
the
can
at
each
be
constant
have
analysed
rotational
velocity
different
as
the
v,
the
velocities.
addition
of
motion.
motion
is
bicycle
is
moving
forward
at
velocity
v
so
the
formula:
centre
velocity
v.
of
All
mass
points
has
on
forward
the
translational
wheel’s
rim
have
motion
a
f
translational
angular
and
moving
on
the
of
=
is
points
wheel’s
ω
linear
bicycle
different
introduced
66)
following
dt
e x ample: BiCyCle wheel
using
The
linked
=
In
Comparison
motion
page
α
dt
velocities
the
and
=
the
axes.
described
velocities
of
motion.
displacements,
angular
quantities
mass
more)
quantities
Rotational
of
projectile
using
these
displacement),
all
–
rotation.
concept
with
the
rotates
motion
(angular
axis
as
accelerations;
accelerations;
given
path
bottle
Translational
and
a
through
s
dω
_
component
forward
at
velocity
v
velocity
translational component of
velocity ν
equations of uniform angul ar aCCeleration
The
denitions
acceleration
equations
equations
of
can
(page
of
average
be
11).
An
constant
Translational
linear
rearranged
velocity
to
derive
equivalent
angular
average
constant
rearrangement
linear
acceleration
derives
b)
Rotational
motion
The
wheel
is
at
constant
the
rotating
around
the
motion
Rotational
a
rim
motion
have
a
angular
tangential
velocity
ω.
Angular
Initial
u
Initial
v
Final
component
displacement
axis
points
of
of
on
rotation
velocity
v
the
(=
wheel’s
rω)
angular
velocity
θ
ω
tangential
i
velocity
All
ν
s
Final
central
acceleration.
Displacement
velocity
and
the
angular
velocity
component of
ω
f
velocity ν
Time
taken
Acceleration
t
Time
a
Angular
taken
t
acceleration
α
ν
[constant]
v
=
u
+
at
[constant]
ω
=
ω
f
+
αt
i
ν
1
s
=
ut
+
c)
1
2
at
θ
=
ω t
+
Combined
motion
2
αt
i
2
2
The
2
v
2
=
u
2
+
2as
ω
2
=
f
ω
+
the
2αθ
motion
vector
of
the
different
addition
of
the
points
above
on
two
the
wheel’s
rim
is
components:
i
Point at top of wheel is
(v
+
(ω
u)t
s
+
f
_
ω
)t
i
_
=
θ
2
moving with instantaneous
=
2
velocity of 2ν, for ward
Point in contact with
Point at side of
ground is at rest.
with instantaneous velocity of
Instantaneous
velocity is zero
152
o p t i o n
B
–
E n g i n E E r i n g
p h y s i c s
2ν, at 45°
wheel is moving
to the horizontal
t   
rel ationship Between line ar and rotational
quantities
When
an
object
additional
is
just
rotating
translational
about
motion
of
a
the
xed
object,
axis,
all
and
the
there
is
c)
Accelerations
The
total
two
components:
linear
acceleration
of
any
particle
is
made
up
of
no
individual
a)
The
centripetal
acceleration,
a ,
(towards
the
axis
r
particles
values
that
of
make
linear
acceleration.
up
that
object
displacement,
They
do,
have
linear
however,
all
different
velocity
share
the
instantaneous
and
of
linear
same
rotation
–
see
page
65),
also
known
as
the
acceleration.
instantaneous
Tangential
velocity
Angular
values
of
angular
acceleration.
The
displacement,
link
between
angular
these
velocity
values
radial
and
involves
velocity
angular
the
distance
2
v
from
the
axis
of
rotation
to
the
particle.
2
_
a
=
r
=
rω
r
instantaneous velocity
V
1
par ticle 1
Rotation about
Centripetal
acceleration
Distance
from
axis
of
axis. All par ticles
have same
rotation
(along
the
to
particle
radius)
m
1
instantaneous
r
1
angular velocity
V
2
b)
An
additional
tangential
acceleration,
a ,
which
results
t
, instantaneous
ω
from
velocity
r
2
an
angular
acceleration
taking
place.
If
α
=
0,
then
m
2
a
=
0.
t
axis
Instantaneous
acceleration
Angular
acceleration
of rotation
(along
(into the page) par ticle 2
V
1
the
tangent)
≠ V
2
a
a)
Distance
on
travelled
circular
=
rα
t
Displacements
Angular
displacement
Distance
path
rotation
=
from
to
axis
of
particle
rθ
The
total
acceleration
of
the
particle
can
be
found
by
vector
______
4
addition
Distance
rotation
b)
Instantaneous
Linear
from
to
axis
of
these
two
components:
a
=
r√ ω
2
+
α
of
particle
velocities
instantaneous
Angular
velocity
(along
velocity
the
tangent)
v
=
Distance
rotation
ω
from
to
axis
of
particle
o p t i o n
B
–
E n g i n E E r i n g
p h y s i c s
153
t   b
the moment of a forCe: the torque Γ
A
particle
when
the
particle
is
is
in
equilibrium
vector
zero
sum
(see
of
if
all
page
its
the
16).
acceleration
external
In
this
is
forces
zero.
This
acting
situation,
all
on
the
occurs
force F
the
forces
r
pass
real
through
objects
create
a
called
The
do
single
not
turning
the
torque
a
is
always
effect
moment
the
moment
Greek
or
point
and
pass
about
of
the
a
Γ
of
a
to
zero.
through
given
force
uppercase
torque
sum
axis.
or
letter
force,
the
the
The
same
The
about
point
turning
torque.
gamma,
F
forces
The
and
θ
O
on
θ
can
effect
is
symbol
axis of
for
rotation
r
Γ.
an
axis
is
dened
⊥
as
perpendicular
the
product
of
the
force
and
the
perpendicular
distance
from
distance from O
the
axis
of
rotation
to
the
line
of
action
of
the
force.
to line of action of F
moment
or
torque
force
line of action of F
=
Fr⊥
Note:
•
perpendicular
The
torque
energy
•
Γ
=
Fr
sin
The
can
the
quantity
to
be
a
pair
of
points
is
the
of
a
system
produce
equal
but
of
a
about
forces
turning
that
all
In
this
axes
has
effect.
anti-parallel
application.
by
torque
rotation
calculations,
the
the
measured
as
is
in
N
m,
but
only
joules.
clockwise
of
this
directed
the
is
being
can
of
into
be
the
rotation.
opposite
of
that
direction
axis
is
the
out
forces
no
A
common
acting
situation,
drawn
resultant
the
force
example
with
to
is
different
resultant
perpendicular
but
torque
the
If
a
resultant
(page
17).
object
(page
16)
the
force
When
then
or
anticlockwise
treated
torque
In
the
considered.
the
the
a
vector
example
paper.
direction,
as
If
the
torque
For
the
vector
considered
above,
force
vector
F
the
was
would
be
paper.
we
as
acts
know
this
on
there
it
an
is
no
to
be
means
its
object
then
resultant
in
it
must
force
translational
acceleration
accelerate
acting
on
an
equilibrium
must
be
zero.
plane
Similarly,
dened
both
rotational and transl ational equiliBrium
is
same
are
expressed
any
of
with
in
directed
Couples
of
vector
applied
does
of
axis
along
torque
which
be
direction
purposes
couple
energy
θ
about
A
and
also
distance
if
there
is
a
resultant
torque
acting
on
an
object
then
forces.
it
O
F
must
be
in
have
an
angular
rotational
external
torques
acceleration,
equilibrium
acting
on
the
only
α.
if
object
Thus
the
is
an
vector
object
sum
will
of
all
the
zero.
arbitrary
x
axis
If
an
in
object
static
is
not
moving
equilibrium.
and
This
not
must
rotating
mean
then
that
it
the
is
said
object
to
is
be
in
d
both
For
rotational
rotational
α
In
that
to
F
=
2D
0
∴
and
∑
Γ
problems
there
the
is
plane
problems,
translational
equilibrium.
equilibrium:
=
(in
no
0
the
torque
being
three
x-y
plane),
about
any
considered
axis
it
is
one
sufcient
axis
(parallel
directions
(x,
y
to
and
the
z)
to
show
perpendicular
z-axis).
would
In
need
3D
to
be
considered.
For
translational
a
In
=
2D
0
three
=
F(x +
=
F
d
d) -
∑
F
force
axis
clockwise
–
is
sufcient
two
directions
to
different
(x,
y
and
z)
show
that
directions.
would
there
In
need
3D
to
5 N
3 N
E n g i n E E r i n g
f N
axis into
the
p h y s i c s
=
example
2.25
N
above,
for
rotational
is
equilibrium:
no
problems
be
x
f
B
0
F
In
o p t i o n
it
in
This result is independent of position of axis, O
154
equilibrium:
=
problems,
resultant
Torque of forces
∴
considered.
eb 
(a)
Centre of gravity
plank
balances
if
pivot
is
in
middle
centre of gravity
The
effect
object
of
can
object’s
gravity
be
on
treated
centre
of
all
as
a
the
different
single
force
parts
acting
of
at
the
the
There is no moment about
gravity.
W
the centre of gravity.
If
an
of
object
gravity
object
is
trivial
be
an
–
is
not
it
is
outside
object
of
will
uniform
be
in
the
uniform,
possible
the
then
for
object.
from
a
shape
middle
an
and
of
density,
the
nding
its
object’s
centre
Experimentally,
point
and
it
is
free
to
the
object.
If
position
if
of
centre
the
is
(b)
gravity
you
of
gravity
will
always
end
up
move,
below
rotates
clockwise
if
pivot
is
to
the
then
the
left
to
suspend
W
the
(c)
centre
plank
not
point
plank
rotates
anticlockwise
if
pivot
is
to
the
right
of
suspension.
W
e x ample 1
10 m
•
All
forces
at
•
You
do
you
calculate
(for
the
•
You
need
•
When
not
an
axis
have
have
to
zero
choose
moment
the
pivot
about
as
the
that
axis
axis.
about
which
4 m
6 m
R
2
R
torques,
but
it
is
often
the
simplest
thing
to
do
1
W
c
W
b
that
, weight of car
•
, weight of bridge
reason
to
object
Newton’s
AND
remember
solving
an
in
above).
the
sense
two-dimensional
is
laws
in
rotational
still
apply.
translational
(clockwise
problems
equilibrium
Often
an
equilibrium.
it
anticlockwise).
sufcient
about
object
This
or
is
is
can
any
in
to
ONE
show
axis.
rotational
provide
a
simple
When a car goes across a br idge, the forces (on the bridge) are
way
of
nding
an
unknown
force.
as shown.
•
The
weight
of
an
object
can
be
considered
to
be
concentrated
T
aking moments about right-hand suppor t:
at
its
centre
of
gravity.
clockwise moment = anticlockwise moment
•
(R
1
× 20 m) = (W
b
× 10 m) + (W
c
2
× 20 m) = (W
b
× 10 m) + (W
c
the
lines
problem
of
order
T
aking moments about left-hand suppor t:
(R
If
only
involves
three
non-parallel
forces,
the
× 4 m)
action
to
be
in
of
all
the
forces
rotational
must
meet
at
a
single
point
in
equilibrium.
× 16 m)
Also, since bridge is not accelerating:
R
1
When
+ R
solving
2
= W
b
problems
P
+ W
c
to
do
R
with
rotational
equilibrium
W
remember:
3
b)
e x ample 2
A
ladder
friction)
a)
of
length
at
an
Explain
friction
forces
5.0
angle
why
of
the
between
m
leans
30°
to
ladder
the
against
the
can
smooth
wall
and
stay
the
What
stay
in
place
if
there
is
in
meet
the
between
(no
vertical.
only
ground
a
must
at
a
point
minimum
the
ladder
if
in
equilibrium
coefcient
and
the
of
ground
static
for
fraction
the
ladder
to
place?
is
ladder.
R
H
(a) The reaction from the wall,
Rw
Rw and the ladder ’s weight
(b)
R
P
Equilibrium conditions:-
R
g
v
(
)
W
=
R
1
R
=
R
2
R
h =
Wx
3
v
meet at point P. For
(
)
H
w
equilibrium the force from
wall
moments
30°
5 m
w
the ground, Rg must also
about Q
pass through this point
h
(for zero torque about P).
F
f
∴
∴
R
g
≤
µ
s
R
Rg is as shown and has
R
H
≤
µ
s
Rv
a horizontal component
W
R
w
(i.e. friction must be acting)
using
µ
&
s
≥
W
60°
Q
x
3
µ
s
≥
x
2.5 cos 60
=
h
5.0 sin 60
ground
∴
o p t i o n
B
–
E n g i n E E r i n g
µ
s
≥
0.29
p h y s i c s
155
n’ c  –    
ne wton’s seCond l aw – definition of moment of inertia
F
•
Every
particle
in
the
object
has
the
same
angular
xed axis of
acceleration,
α
rotation
angular
tangential
The
acceleration
dened
moment
of
inertial,
I,
of
an
object
about
a
particular
axis
is
acceleration
α
by
the
summation
below:
a
t
O
the
moment
of
inertia
distance
from
the
of
axis
the
or
particle
rotation
par ticle
rigid body
2
I
Newton’s
second
law
as
applied
to
one
particle
in
a
rigid
=
∑
mr
body
mass
Newton’s
second
law
applies
to
every
particle
that
makes
up
a
of
and
motion.
small
the
must
In
the
particles
resultant
also
apply
diagram
each
force
if
the
above,
with
that
a
the
mass
acts
on
object
object
m
one
is
F
is
undergoing
is
the
particle.
made
up
tangential
The
other
radial
not
included.
F
=
component,
m
a
For
=
this
cannot
produce
particle
we
can
angular
apply
in
individual
the
object
rotational
of
lots
of
component
Note
that
moment
•
A
scalar
•
Measured
•
Dependent
of
inertia,
I,
is
of
quantity
component,
2
the
an
large
particle
object
acceleration
Newton’s
so
second
it
in
kg
2
m
(not
kg
m
)
is
on:
law:
◊
The
mass
◊
The
way
◊
The
axis
of
the
object
mrα
t
2
so
torque
Similar
up
the
Γ
=
(mrα)r
equations
object
can
and
be
=
mr
created
summed
this
mass
is
distributed
α
for
all
the
particles
that
of
rotation
being
considered.
make
together:
Using
this
denition,
equation
1
becomes:
2
∑
Γ
=
∑
2
mr
resultant
α
torque
2
or
∑
Γ
=
α∑
mr
external
in
N
angular
acceleration
in
rad
s
m
(1)
ext
Note
that:
Γ
•
Newton’s
third
law
applies
and,
when
summing
up
all
=
I α
the
2
moment
torques,
forces
the
internal
between
external
torques
particles)
torques
are
(which
must
sum
result
to
from
zero.
the
Only
of
inertia
in
kg
m
internal
This
the
is
Newton’s
compared
left.
to
F
second
=
law
for
rotational
motion
and
can
be
ma
moments of inertia for different oBjeCts
Equations
for
moments
of
inertia
in
different
Object
situations
Axis
of
do
not
need
moment
rotation
to
be
memorized.
of
Object
Axis
inertia
of
moment
rotation
of
inertia
thin ring (simple wheel)
Sphere
through
centre,
2
perpendicular
to
mr
r
plane
m
2
through
centre
2
mr
5
thin ring
through
a
1
2
mr
r
2
diameter
m
disc and cylinder (solid ywheel)
through
centre,
1
perpendicular
to
Rectangular lamina
2
mr
2
Through
the
r
plane
centre
m
of
mass,
2
2
l
+
h
_
perpendicular
to
m
(
)
12
l
thin rod, length d
through
the
plane
of
the
centre,
perpendicular
lamina
h
1
_
m
2
md
12
to
rod
d
i.
e x ample
A
torque
of
30
N
m
acts
on
a
wheel
with
moment
of
what
is
the
angular
velocity
of
the
wheel?
inertia
ii.
how
fast
is
a
point
on
the
rim
moving?
2
600
kg
m
.
The
wheel
starts
off
at
rest.
30
_
Γ
a)
a)
What
b)
The
angular
acceleration
is
Γ
=
I
α
⇒
α
=
produced?
=
2
=
I
5.0
×
10
2
b)
wheel
has
a
radius
of
40
cm.
After
1.5
i.
ω
=
αt
=
5.0
×
10
1
×
90
=
4.5
minutes:
1
ii.
156
o p t i o n
B
–
E n g i n E E r i n g
p h y s i c s
v
=
r
ω
=
0.4
×
2
rad
600
4.5
=
1.8
m
s
rad
s
s
r c
energy of rotational motion
Energy
considerations
complicated
done.
the
In
problems.
the
object
often
absence
will
be
provide
When
of
any
stored
as
a
Conservation of angul ar momentum
simple
torque
resistive
acts
solutions
on
torque,
rotational
kinetic
an
the
to
object,
work
In
work
done
is
exactly
linear
the
same
motion
to
way
that
Newton’s
laws
can
be
applied
to
derive:
on
•
the
concept
of
the
•
the
relationship
impulse
of
a
force
energy.
F
between
impulse
and
change
in
momentum
•
the
law
of
conservation
of
linear
momentum,
F
then
Newton’s
laws
can
be
applied
to
angular
situations
θ
to
derive:
r
P
•
The
axis of rotation
concept
Angular
which
Calculation
of
work
done
by
a
the
the
situation
above,
a
As
a
result,
an
force
F
is
applied
and
the
done,
W,
is
angular
calculated
as
displacement
of
θ
occurs.
=
F
the
×
Γ
(distance
=
I
α
we
along
know
of
torque
and
the
time
for
acts:
shown
arc)
that
=
W
F
=
ΓΔt
varies
given
to
with
an
time
object
then
can
be
the
total
angular
estimated
from
the
area
below:
×
=
impulse
torque
rθ
=
Γθ
time.
the
This
given
Using
impulse:
product
The
under
W
the
object
impulse
work
angular
is
torque
If
rotates.
the
torque
angular
In
of
impulse
to
graph
is
an
showing
analogous
object
as
a
the
to
variation
estimating
result
of
a
of
the
torque
total
varying
with
impulse
force
(see
Iαθ
page23).
We
can
apply
the
constant
angular
acceleration
equation
to
•
substitute
for
The
relationship
angular
2
between
angular
impulse
and
change
in
αθ:
momentum:
2
ω
=
ω
f
+
2αθ
i
angular
impulse
angular
momentum
applied
to
an
object
=
change
of
2
2
ω
ω
f
∴
W
(
=
I
1
i
___
_
)
-
=
means
that
we
have
=
the
object
i
2
an
equation
for
rotational
The
law
of
conservation
of
angular
momentum.
KE:
The
1
E
by
Iω
f
2
•
This
experienced
2
Iω
2
2
1
2
total
angular
momentum
of
a
system
remains
constant
2
I
ω
K
Work
provided
2
rot
done
by
the
torque
acting
on
object
=
change
no
resultant
external
torque
acts.
in
Examples:
rotational
KE
of
object
a)
The
total
KE
is
equal
to
the
sum
of
translational
KE
and
A
skater
body
rotational
KE
=
translational
1
Total
KE
=
KE
1
2
Mv
+
rotational
KE
that
For
a
single
The
linear
tangential
spinning
their
on
a
vertical
moment
of
axis
inertia
down
by
their
drawing
the
This
mass
of
allows
the
their
arms
is
mass
no
to
be
in
redistributed
so
from
the
axis
of
longer
rotation
thus
at
a
signicant
reducing
Σmr
2
Ex tended arms mean
Bringing in her arms
larger
decreases her moment
radius and smaller
velocity of ro tation.
of iner tia and therefore
particle
increases her rotational
momentum,
speed
angular
arms.
distance
Iω
angul ar momentum
linear
is
reduce
2
2
+
2
The
can
KE:
their
Total
who
the
v
is
m
p,
a
particle
of
mass
dened
as
the
m
which
has
velocity.
a
v
momentum,
momentum
of
L,
about
is
the
axis
of
moment
of
the
rotation
2
Angular
momentum,
For
a
larger
The
angular
rotation
is
L
=
L
of
L
=
(mv)r
=
(mrω)r
=
(mr
)ω
object
momentum
dened
an
object
about
an
axis
of
as
2
Angular
L
Note
•
=
momentum,
∑(mr
)ω
Iω
that
total
angular
momentum,
a
vector
(in
the
same
a
vector
for
calculations)
2
•
measured
in
•
dependent
kg
m
way
that
a
L,
is:
torque
is
considered
to
be
1
s
or
N
m
s
b)
total
on
angular
all
rotations
momentum
taking
of
a
place.
planet
For
example,
orbiting
a
star
the
would
involve:
The
a
Earth–Moon
result
between
acts
to
the
spinning
of
the
planet
about
an
axis
through
centre
of
mass
the
the
oceans
the
orbital
the
star.
angular
the
produces
movement
and
Earth.
Earth’s
Earth’s
spin
angular
tides
of
This
on
in
the
water,
provides
its
own
momentum.
oceans.
friction
a
axis
The
torque
and
As
exists
that
thus
angular
momentum
means
that
there
conservation
must
be
a
and
corresponding
◊
system
relative
the
of
planet’s
the
reduce
reduces
◊
of
momentum
about
an
axis
increase
in
the
orbital
angular
momentum
through
of
the
Earth–Moon
separation
o p t i o n
B
–
is
slowly
system.
As
a
result,
the
Earth–Moon
increasing.
E n g i n E E r i n g
p h y s i c s
157
s  b
summary Comparison of equations of line ar and rotational motion
Every
equation
for
linear
motion
has
a
corresponding
Linear
Physics
principles
A
Newton’s
Work
second
law
F
external
done
W
m
=
energy
E
force
The
on
a
value
determined
by
point
of
object
A
the
the
F
resultant
causes
mass
and
of
s
=
W
=
m
Conservation
of
momentum
v
E
P
=
F
p
=
m
The
v
v
total
linear
momentum
constant
force
Resultant
provided
of
no
a
analysing
approach
is
to
appropriate
a)
Graph
This
not
at
In
is
the
10).
any
Thus
is
the
value
determined
resultant
by
of
the
object
the
moment
torque.
I
ω
2
P
=
Γ
L
=
I
The
ω
ω
total
constant
acts.
angular
force
m
Moment
Acceleration
a
Angular
Displacement
s
Velocity
v
Linear
p
momentum
the
linear
no
of
resultant
a
system
external
remains
torque
acts.
Mass
situation,
momentum
provided
torque
of
Γ
inertia
I
acceleration
α
Angular
displacement
θ
Angular
velocity
ω
Angular
momentum
L
e x ample
simplest
situation
A
and
use
solid
of
cylinder,
angle
30°
as
initially
shown
at
in
rest,
the
rolls
down
diagram
a
2.0
m
long
slope
below:
relationships.
to
a
situation,
useful
equal
is
extended
2
=
Resultant
equivalent
equivalent
instant
the
an
The
θ
F
displacement
linear
represent
any
the
equivalent
angular
graph
time.
rotational
imagine
the
of
any
system
resultant
proBlem solving and graphiC al work
When
Γ
rot
external
used
on
K
remains
Symbols
and
1
2
Momentum
=
torque
acceleration.
α
2
K
Power
I
external
acceleration
inertia
Γ
motion
rotational
angular
force.
a
1
Kinetic
is
resultant
=
Rotational
acceleration.
acceleration
the
equivalent:
motion
resultant
causes
angular
vs
graph
the
the
gradient
of
of
linear
area
quantity
to
time
and
the
angular
the
graph
gradient
instantaneous
an
displacement
under
of
velocity
2.0 m
vs
does
the
(see
displacement
line
page
vs
30°
time
graph
gives
the
instantaneous
angular
velocity.
The
b)
Graph
of
angular
velocity
vs
mass
of
Calculate
This
graph
is
equivalent
to
the
cylinder
is
m
and
the
radius
of
the
cylinder
a
graph
of
linear
velocity
the
velocity
of
the
cylinder
at
the
bottom
of
the
vs
Answer:
time.
In
the
linear
situation,
the
area
under
the
graph
Vertical
represents
the
distance
gone
and
the
gradient
of
the
line
height
instant
page
10).
is
equal
Thus
the
to
the
area
instantaneous
under
an
acceleration
angular
fallen
by
cylinder
lost
=
1
vs
gained
1
2
=
mv
+
gives
the
total
angular
displacement
gradient
of
an
angular
velocity
vs
time
the
instantaneous
angular
graph
but
I
2
=
mR
(cylinder)
v
Graph
of
torque
vs
ω
=
time
R
2
This
graph
is
equivalent
to
a
graph
of
force
vs
time.
1
In
⇒
KE
gained
2
=
mv
1
mR
_
2
2
2
v
_
+
2
2
the
linear
situation,
the
area
under
the
graph
total
impulse
given
to
the
object
which
is
equal
to
the
1
2
=
mv
+
2
change
of
momentum
of
the
object
(see
page
23).
area
under
the
total
the
torque
vs
time
graph
2
mv
4
Thus
3
the
2
=
represents
mv
4
angular
impulse
given
to
the
object
which
Conservation
is
equal
to
the
change
of
angular
of
energy
momentum.
3
⇒
mgh
2
=
mv
4
____
gh
_
∴
v
=
4
√
3
___________
4 × 9.8 × 1.0
__
=
√
3
1
=
158
o p t i o n
B
–
R
represents
1
the
E n g i n E E r i n g
m
2
acceleration.
and
c)
1.0
2
2
gives
=
and
1
the
sin30
Iω
2
graph
2.0
mgh
(see
velocity
KE
time
=
at
PE
any
is
R.
time
p h y s i c s
3.61
m
s
see
page
156
slope.
tc   cc
definitions
Historically,
situations.
powerful
The
the
study
These
intellectual
terms
used
of
laws,
need
the
behaviour
otherwise
of
known
to
be
laws
led
of
to
some
very
fundamental
thermodynamics,
provide
concepts
the
that
modern
are
applicable
physicist
with
to
a
many
set
of
other
very
explained.
Most
system
macroscopic
of
the
thermal
time
when
The
If
surroundings
its
we
energy
are
and
this
focusing
the
can
our
(see
context
it
of
do
the
gas
as
work
behaviour
a
or
whole.
work
of
In
can
an
ideal
terms
be
of
done
gas
in
work
on
it.
particular
and
In
situations,
energy,
this
the
context,
gas
the
we
can
gas
focus
gain
can
or
be
on
the
lose
seen
as
a
system
surroundings.
surroundings
In
studying
behaviour
thermodynamic
Q
gases
the
tools.
Thermodynamic
Heat
ideal
as
study
For
on
the
example
behaviour
the
of
expansion
an
of
a
ideal
gas
gas,
then
means
everything
that
work
is
else
done
can
by
be
the
called
gas
on
the
below).
heat
refers
to
the
transfer
HOT
of
a
the
quantity
system
This
of
and
transfer
thermal
its
energy
between
thermal energy ow
surroundings.
must
be
as
a
result
of
thermal
a
HOT
temperature
COLD
energy
difference.
ow
thermal energy ow
HOT
Work
W
In
1.
this
context,
work
done
work
=
refers
force
×
to
the
macroscopic
transfer
2.
distance
of
energy.
work
done
F
For
=
example
potential
difference
×
current
×
time
heater
F
compression
This
is
just
example
When
a
gas
is
compressed,
work
is
done
on
the
work
the
Internal
U
in
(∆U
=
energy
change
done
internal
gas
on
it.
is
compressed,
When
a
gas
the
surroundings
expands
it
does
on
the
being
gas.
do
work
on
surroundings.
The
the
a
work
gas
power supply
When
another
of
internal
energy
intermolecular
page
can
be
forces
thought
and
the
of
as
kinetic
the
energy
energy
held
due
to
within
the
a
system.
random
It
motion
is
of
the
the
sum
of
the
molecules.
PE
due
to
See
26.
energy)
This
is
different
the
system,
the
overall
to
the
which
total
would
energy
also
of
system
include
with
motion
of
the
system
and
internal
any
PE
due
to
external
forces.
energy U
In
thermodynamics,
in
internal
it
is
the
changes
velocity (system also has kinetic energy)
v
energy
that
are
being
h
considered.
a
gas
must
(e.g.
is
If
the
increase.
liquid
change
internal
increased,
of
→
A
then
change
gas)
internal
energy
its
also
of
height (system also has
temperature
of
gravitational potential energy)
phase
involves
a
energy.
The
its
Internal
of
an
energy
ideal
monatomic
The
gas
gas
internal
changes
produces
energy
from
the
T
same
of
to
an
(T
ΔT.
3
page
30),
E
=
K
T
B
its
the
depends
internal
only
on
energy
temperature
energy
is
of
a
system
is
not
the
same
as
energy
temperature.
changes
related
to
from
the
When
U
to
(U
average
the
+
temperature
ΔU).
kinetic
The
energy
of
same
per
an
ΔU
ideal
always
molecule
(see
R
=
T,
2
gas
ΔT)
Since
3
k
2
ideal
+
total
internal
the
internal
energy
U,
is
the
sum
of
the
total
random
kinetic
energies
of
the
N
A
molecules:
3
U
=
nN
E
A
=
nRT
K
[n
=
number
of
moles;
N
=
Avogadro’s
constant]
A
2
o p t i o n
B
–
E n g i n E E r i n g
p h y s i c s
159
w  b   
Work
work done during e xpansion
at Constant pressure
Whenever
a
gas
expands,
it
is
done
W
=
force
=
F∆x
F
×
distance
doing
force
_
Since
work
of
on
the
its
gas
surroundings.
is
changing
all
If
the
the
pressure
time,
area
then
F
calculating
is
the
complex.
amount
This
is
of
work
because
we
force
the
=
pressure
=
pA
constant
done
therefore
pressure p
cannot
∆x
assume
of
×
a
work
constant
done
distance).
(work
If
the
in
done
=
the
force
pressure
must
also
is
constant
then
If
and
we
can
∆V
work
done
=
p∆V
the
the
force
calculate
if
a
gas
positive)
the
positive)
work
done.
p V diagrams and work done
It
is
often
useful
thermodynamic
is
that
the
shown
area
to
represent
process
under
on
the
the
a
pV
graph
changes
that
diagram.
An
represents
happen
to
important
the
work
a
gas
during
reason
done.
The
for
a
choosing
reasons
for
to
this
below.
p erusserp
A
B
area of strip
p
= p∆V
= work done
in expansion
area under graph
= work done in expanding
fr om state A to state B
∆V
This
turns
out
to
be
generally
volume V
true
for
A
expanding from
any
thermodynamic
p erusserp
p erusserp
work done by gas
process.
work done by
A
state A to state B
to state C
atmosphere as
gas contracts
from state C to
state D to state A
B
D
C
C
volume V
160
o p t i o n
B
–
increases
its
volume
(∆V
F
is
is
constant
pA∆x
=
changes
change.
So
pressure
=
A∆x
force
so
then
W
but
equation
E n g i n E E r i n g
volume V
p h y s i c s
do
this
are
then
the
gas
does
work
(W
is
t    c
Q
first l aw of thermodynamiCs
There
rst
are
law
three
is
simply
conservation
energy
Q
happen
As
is
=
a
a
is
laws
statement
applied
to
a
to
∆U
system,
or
it
of
the
combination
energy
energy
Q
as
given
(or
internal
fundamental
of
can
of
the
thermodynamics.
principle
system.
then
one
both).
do
If
an
of
The
work
of
two
of
things
can
If
+
must
∆U
increase
it
its
is
W
W
to
are
remember
all
negative,
this
is
increasing.
If
it
If
is
is
taken
what
from
the
the
signs
system’s
of
these
‘point
then
then
positive,
thermal
thermal
is
then
(The
negative,
this
energy
is
going
into
the
energy
is
going
out
of
the
the
internal
temperature
the
internal
positive,
then
the
gas
energy
the
energy
temperature
surroundings.(The
of
of
system
is
of
the
is
the
gas
is
of
gas
the
system
increasing.)
system
is
is
decreasing.)
doing
work
on
expanding.)
symbols
If
They
is
decreasing.(The
W
important
mean.
positive,
If
the
It
is
system.
thermal
conserved
∆U
this
system.
energy
amount
system
If
The
of
it
is
negative,
the
surroundings
are
doing
work
on
view’.
the
system.
(The
gas
is
contracting.)
ide al ga s proCesses
A
gas
the
can
diagrams
1.
undergo
changes
can
any
be
number
represented
represent
a
type
of
of
on
different
a
process
2.
Isochoric
types
of
change
pressure–volume
called
a
or
process.
diagram
reversible
and
Four
important
the
rst
law
of
3.
Isothermal
processes
are
thermodynamics
considered
must
below.
apply.
To
be
In
each
precise,
case
these
process.
Isobaric
4.
Adiabatic
(isovolumetric)
In
In
an
isochoric
process,
also
the
gas
has
isochoric
below
a
constant
The
shows
diagram
an
isobaric
expansion
constant
an
decrease
has
pressure.
an
process,
diagram
shows
gas
process
in
pressure.
In
an
the
isothermal
gas
has
a
diagram
A
B
shows
In
an
adiabatic
there
The
below
isothermal
process
constant
temperature.
p erusserp
below
a
The
isobaric
p erusserp
volume.
the
called
isovolumetric
an
is
energy
an
expansion
the
gas
no
transfer
and
if
the
between
the
surroundings.
that
process
thermal
This
gas
means
does
work
it
A
must
in
result
internal
in
decrease
energy.
compression
p erusserp
is
a
or
A
rapid
expansion
approximately
B
A
adiabatic.
volume V
volume V
done
sufcient
Isobaric
change
Isothermal
=
constant,
or
T
=
is
because
there
time
for
is
not
thermal
change
energy
p
B
This
quickly
constant,
to
be
exchanged
or
with
the
surroundings.
V
=
pV
=
constant
positive
Q
positive
an
positive
(T ↑)
∆U
p erusserp
∆U
W
=
zero
constant,
positive
W
positive
or
p
=
constant
T
Q
below
adiabatic
shows
expansion
(volumetric)
change
V
diagram
T
Q
Isochoric
constant
The
volume V
A
negative
B
∆U
negative
(T↓)
volume V
W
zero
Adiabatic
Q
change
zero
e x ample
∆U
A
monatomic
adiabatic
gas
doubles
expansion.
its
What
5
5
3
3
is
volume
the
as
a
change
result
in
of
W
pressure?
For
p
V
1
=
p
1
negative
(T↓)
an
a
positive
monatomic
gas,
the
V
2
2
equation
for
an
adiabatic
5
p
V
3
2
1
_
=
p
1
(
process
)
V
is
5
2
3
pV
=
constant
5
3
=
∴
nal
pressure
0.5
=
0.31
=
31%
of
initial
pressure
o p t i o n
B
–
E n g i n E E r i n g
p h y s i c s
161
sc   c  
seCond l aw of thermodynamiCs
Historically
stated
in
shown
the
many
to
be
second
law
different
equivalent
of
ways.
to
All
one
entropy and energy degradation
thermodynamics
of
these
has
versions
been
can
Entropy
be
of
principle
thermal
there
is
energy
nothing
into
to
useful
a
property
that
expresses
the
disorder
in
the
another.
The
In
is
system.
stop
the
work.
In
complete
practice,
conversion
a
gas
can
is
not
details
linked
system.
are
to
[S
not
the
=
important
number
k
of
but
the
possible
entropy
S
of
arrangements
a
system
W
of
the
ln(W)]
B
continue
to
expand
forever
–
the
apparatus
sets
a
physical
Because
limit.
Thus
the
continuous
conversion
of
thermal
molecules
roughly
into
work
requires
a
cyclical
process
–
a
heat
are
in
random
motion,
one
would
expect
energy
equal
numbers
of
gas
molecules
in
each
side
of
a
engine.
container.
W
An arrangement
like this
is much
more likely
T
than one like
T
hot
cold
Q
Q
hot
cold
The
Carnot showed
In other words there
that Q
must be thermal energy
number
set-up
hot
> W
‘wasted’ to the cold reser voir.
realization
leads
to
possibly
the
simplest
the
second
law
of
thermodynamics
(the
of
arranging
the
molecules
that
the
is
arranging
right
greater
entropy
to
of
the
than
get
the
molecules
the
the
number
set-up
system
on
on
the
the
to
of
get
the
ways
left.
right
is
of
This
greater
formulation
than
of
ways
the
means
This
of
on
the
entropy
of
the
system
on
the
left.
Kelvin–Planck
In
any
to
increase.
random
process
the
amount
of
disorder
will
tend
formulation).
No
heat
engine,
operating
in
a
cycle,
can
take
in
its
surroundings
and
totally
convert
it
into
other
words,
the
total
entropy
will
always
heat
increase.
from
In
The
entropy
change
∆S
is
linked
to
the
thermal
∆Q
work.
___
energy
change
∆Q
and
the
temperature
T.
(∆S
=
)
T
Other
possible
formulations
include
the
following:
thermal energy ow
No
heat
pump
can
low-temperature
reservoir
transfer
reservoir
without
work
thermal
to
a
being
energy
from
a
high-temperature
done
on
it
∆Q
(Clausius).
T
hot
Heat
The
ows
concept
of
from
hot
entropy
objects
leads
to
to
one
cold
nal
T
cold
objects.
version
of
the
∆Q
∆Q
second
law.
decrease of entropy =
increase of entropy =
T
T
hot
The
entropy
of
the
Universe
can
never
cold
decrease.
When
thermal
object,
overall
energy
the
ows
total
from
entropy
a
has
hot
object
to
a
colder
increased.
e x amples
In
The
rst
and
second
laws
of
thermodynamics
both
must
many
concept.
to
all
situations.
Local
decreases
of
entropy
are
possible
as
elsewhere
there
is
a
corresponding
A
refrigerator
is
an
example
of
a
idea
of
is
energy
shared
degradation
out,
the
more
is
a
useful
degraded
becomes
–
it
is
harder
to
put
it
to
use.
For
example,
can
be
the
increase.
internal
1.
the
energy
more
so
it
long
situations
The
apply
heat
energy
that
is
‘locked’
up
in
oil
released
when
pump.
the
oil
the
form
It
thermal energy taken from
is
is
not
burned.
of
In
the
thermal
feasible
to
end,
energy
get
it
all
–
the
energy
shared
released
among
many
will
be
in
molecules.
back.
ice box and ejected to
surroundings
3.
Water
freezes
which
the
receiving
at
0
°C
entropy
the
because
increase
latent
heat)
this
of
the
equals
is
the
temperature
surroundings
the
entropy
at
(when
decrease
of
source of work
the
water
molecules
becoming
more
ordered.
It
would
not
is the electric
freeze
at
a
higher
temperature
because
this
would
mean
energy supply
that
A
2.
It
the
overall
entropy
of
the
system
would
decrease.
refrigerator
should
be
possible
theoretical
system
boat
around
to
for
design
increasing temperature of surroundings
a
propelling
a
-2 °C
based
atmosphere
reservoir
could
The
the
be
could
and
cold
used
as
movement
water
heat
be
the
0 °C
2 °C
The
hot
the
sea
reservoir.
boat
the
as
from
cold
the
be
engine.
used
water
the
of
would
a
ICE
through
work
ICE/WATER
since
done.
WATER
MIX
since
since
This
to
is
possible
work
warmed
for
BUT
ever.
and
the
it
The
cannot
sea
continue
would
atmosphere
be
entropy
entropy
entropy
entropy
entropy
entropy
decrease
increase
decrease
increase
decrease
increase
would
of ice
be
cooled
and
eventually
there
formation
be
162
no
temperature
<
of
surroundings
difference.
o p t i o n
B
–
of ice
=
of
of ice
>
of
would
E n g i n E E r i n g
p h y s i c s
formation
surroundings
formation
surroundings
h    
In
he at engines
A
central
engine.
thermal
The
are
concept
A
heat
used
in
is
order
examples
of
heat
any
to
combustion
generate
generalized
study
engine
is
that
work.
in
It
a
A
and
in
block
shown
uses
a
a
is
the
source
converts
car
energy
engines.
engine
thermodynamics
device
do
electrical
heat
of
heat
the
power
diagram
of
into
work.
turbines
station
representing
in
pressure
of
thermal
a
this,
hot
and
the
energy
the
some
thermal
reservoir
isobaric
must
the
been
must
A
different
ejected
decrease
in
have
been
isovolumetric
expansion).
have
isovolumetric
energy
(during
to
a
pressure
cold
and
increase
amount
reservoir
the
isobaric
a
isobaric expansion
A
B
total work
done by
the gas
COLD
done W
reser voir
T
ENGINE
hot
do
compression).
are
below.
reser voir
T
to
from
(during
that
work
HOT
order
taken
heat
p erusserp
both
to
the
engine
energy
internal
in
thermal
thermal
energy
energy
Q
Q
isovolumetric
isovolumetric
increase in
decrease in
pressure
pressure
cold
C
hot
D
isobaric compression
cold
volume V
The
Heat
thermal
efciency
of
a
heat
engine
is
dened
as
engine
work done
____
η
In
this
context,
temperature
the
word
source
(or
reservoir
sink)
of
is
used
thermal
to
imply
energy
.
a
=
constant
Thermal
(thermal
energy
This
be
taken
from
the
hot
reservoir
without
causing
the
energy
taken
from
hot
reservoir)
can
temperature
is
equivalent
to
of
rate
of
doing
work
____
the
hot
reservoir
to
change.
Similarly
thermal
energy
can
be
given
to
η
=
(thermal
the
cold
reservoir
An
ideal
gas
can
without
be
used
increasing
as
a
heat
its
power
to
its
a
simple
starting
enclosed
by
example.
conditions,
the
cycle
but
The
the
represents
engine.
The
four-stage
gas
the
pV
diagram
right
has
cycle
done
amount
returns
work.
of
The
work
the
heat
heat
pump
pump
reservoir
to
a
reservoir)
useful work done
__
η
=
input
area
done.
The
cycle
of
maximum
changes
possible
that
results
efciency
is
in
a
heat
called
the
engine
with
Carnot
the
cycle
Carnot CyCles and Carnot theorem
is
a
causes
hot
work
heat
engine
thermal
energy
reservoir.
must
be
being
In
to
order
run
be
in
reverse.
moved
for
this
to
from
be
A
a
The
cold
achieved,
Such
done.
input
HOT
COLD
work ∆W
reser voir
Carnot
theoretical
an
cycle
heat
reser voir
represents
engine
idealized
p erusserp
mechanical
hot
gas
he at pumps
A
from
temperature.
energy
represents
taken
the
with
engine
is
cycle
the
of
processes
maximum
called
a
for
possible
Carnot
a
efciency.
engine
A
Q
hot
thermal energy taken in
B
area = work done
HEAT
T
T
hot
cold
by gas during
PUMP
D
Carnot cycle
thermal energy
thermal
thermal
C
energy
energy
Q
Q
given out
Q
cold
hot
Heat
cold
V
pump
Once
again
Carnot
an
ideal
thermodynamic
used
in
time
an
the
heat
gas
can
processes
engine,
anticlockwise
be
can
but
circuit
used
be
the
as
a
heat
exactly
the
processes
will
are
represent
pump.
same
all
the
The
ones
as
It
cycle
consists
of
an
p erusserp
•
Isothermal
•
Adiabatic
•
Isothermal
total work
•
Adiabatic
done on
The
opposite.
cycle
of
ideal
gas
undergoing
the
following
processes.
were
expansion
(A
→
B)
This
expansion
(B
→
C)
processes.
compression
(C
→
D)
isobaric compression
A
D
the gas
compression
temperatures
maximum
of
possible
the
(D
hot
→
A)
and
efciency
cold
that
reservoirs
can
be
x
the
achieved.
isovolumetric
isovolumetric
The
efciency
decrease in
of
a
Carnot
engine
can
be
shown
to
be
T
increase in
cold
_
η
=
1
(where
T
is
in
kelvin)
Carnot
pressure
T
pressure
hot
An
B
at
C
engine
20
°C.
operates
The
at
300
maximum
°C
and
possible
ejects
heat
theoretical
to
the
surroundings
efciency
is
isobaric expansion
293
_
volume V
η
=
1
=
0.49
=
49%
Carnot
573
o p t i o n
B
–
E n g i n E E r i n g
p h y s i c s
163
f  
HL
The
symbol
representing
density
is
the
Greek
letter
normal
pressure
definitions of density and pressure
rho,
ρ.
The
force
ΔF
_
p
average
density
of
a
substance
is
dened
by
the
following
=
equation:
ΔA
average
density
m
•
_
ρ
area
mass
Pressure
is
a
scalar
quantity
–
the
force
has
a
direction
but
=
the
V
pressure
does
not.
Pressure
acts
equally
in
all
directions.
volume
2
•
•
Density
is
a
scalar
The
SI
unit
of
pressure
is
N m
2
or
pascals
(Pa).
1 Pa
=
1 N m
quantity.
5
3
•
The
SI
units
of
density
are
3
•
Densities
can
also
be
•
Atmospheric
•
Absolute
pressure
≈
10
Pa
kg m
quoted
in
g cm
(see
conversion
pressure
is
the
actual
pressure
at
a
point
in
a
factor
uid.
Pressure
gauges
often
record
the
difference
between
below)
absolute
3
•
The
density
of
water
is
pressure
and
atmospheric
pressure.
Thus
if
a
3
1 g cm
=
1,000 kg m
5
difference
Pressure
at
any
point
in
a
uid
(a
gas
or
a
liquid)
is
pressure
gauge
gives
a
reading
of
2
×
10
Pa
for
a
dened
5
gas,
interms
of
smallarea,
the
force,
ΔA,
that
ΔF
,
that
contains
acts
the
normally
(at
90°)
to
pressure
separated
ρ
,
then
by
the
in
a
a
uid
increases
vertical
pressure
with
distance,
d,
difference,
absolute
pressure
of
the
gas
is
3
×
10
Pa.
a
point.
variation of fluid pressure
The
the
in
Δp,
BuoyanCy and arChimedes’ prinCiple
depth.
a
uid
If
two
of
between
points
constant
these
two
are
Archimedes’
density,
in
points
magnitude
is:
a
uid,
it
principle
states
experiences
to
the
a
weight
that
when
buoyancy
of
the
uid
a
body
upthrust
is
immersed
equal
displaced.
B
in
=
ρ V
f
f
density
of
uid
∆p
gravitational
=
eld
g
f
strength
ρ
f
22N
pressure
The
of
total
the
difference
pressure
pressure
at
due
a
acting
to
given
at
the
depth
depth
in
surface
1
7N
12N
depth
a
liquid
is
the
(atmospheric
addition
pressure)
density
and
the
additional
pressure
due
to
the
depth:
B
Atmospheric
pressure
density
of
B
of uid
1
2
uid
depth
P
=
P
+
ρ
0
volume of
f
W
W
W
uid displaced
uid displaced
(w = 5N)
(w = 10N)
(a)
Note
•
that:
Pressure
depth
is
be
expressed
head)
in
approximately
column
•
can
(or
As
of
a
the
mercury
pressure
is
in
known
same
(Hg)
as
or
dependent
terms
a
on
of
liquid.
exerted
10
the
equivalent
Atmospheric
m
by
a
760
column
depth,
the
of
A
pressure
mm
high
water.
pressures
at
consequence
principle
object
is
of
that
displaces
weight
of
a
this
oating
its
own
uid.
two
weight of uid displaced
points
that
are
at
the
same
horizontal
level
in
the
same
=
liquid
that
must
liquid
be
the
and
same
the
provided
liquid
is
they
are
connected
total weight of duck
by
static.
atmospheric pressure
pa sC al’s prinCiple
the water column exer ts
Pascal’s
principle
states
that
the
pressure
applied
to
an
a pressure at B equal to
h
excess gas
enclosed
liquid
is
transmitted
to
every
part
of
the
liquid,
the excess pressure of
pressure P
whatever
A
B
design
solids
•
The
pressure
this
means
is
independent
that
liquids
will
of
the
always
cross-sectional
nd
their
own
area
–
level.
of
respond
When
a
uid
is
in
hydrostatic
when
all
the
equilibrium
forces
on
a
solid
one
end
will
be
it
takes.
and
when
given
it
volume
is
of
at
rest.
uid
hydraulic
to
This
principle
(e.g.
other
on
the
solids
transmit
systems
and
is
is
central
different
to
to
the
how
forces.
object
its
exerted
liquids
happens
shape
many
Incompressible
hydrostatiC equiliBrium
A
the
the gas supply: P = hρg
an
end
is
incompressible
held
restraining
transmit
in
place,
stick)
then
is
the
pushed
same
at
force
object.
forces
whereas
incompressible
pressures.
This
are
applied force F
A
2
balanced.
Typically
external
forces
(e.g.
gravity)
are
balanced
load =
F ×
(eor t)
A
1
by
a
pressure
gradient
across
the
volume
of
uid
(pressure
load platform
increases
with
depth
–
see
above).
downward force due to
piston of
piston of area A
2
pressure from uid above
area A
1
volume of uid
weight of uid
W
upward force due to
contained in volume
pressure from uid below
164
o p t i o n
B
–
E n g i n E E r i n g
hydraulic liquid
p h y s i c s
f   – B c
HL
•
the ide al fluid
In
most
real
following
create
be
uid
dene
model.
ow
an
This
is
extremely
ideal
simple
uid
that
model
complicated.
can
can
be
be
used
later
Is
non-viscous
converted
The
of
to
rened
to
•
the
ow)
box
uid:
incompressible
–
thus
its
density
will
be
•
constant.
into
of
a
not
as
of
a
steady
uid.
below).
Does
–
a
result
thermal
viscosity
Involves
realistic.
ideal
Is
properties
simple
more
An
•
a
situations,
real
See
have
(as
these
page
uid
See
ow,
page
no
167
energy
for
gets
the
denition
uid.
ow
Under
of
energy.
167
angular
opposed
to
conditions
for
an
a
turbulent,
the
analysis
momentum
–
ow
of
it
is
or
turbulent
does
chaotic,
laminar
not
(see
ow.
rotate.
l aminar flow, stre amlines and the
Continuity equation
speed ν
2
When
of
the
ow
the
ow
uid
is
of
can
said
to
a
liquid
have
be
is
steady
different
laminar
if
or
laminar,
instantaneous
every
particle
different
velocities.
that
passes
parts
The
through
speed ν
1
a
given
is
point
made.
place
has
The
when
the
same
opposite
the
of
particles
velocity
laminar
that
pass
whenever
ow,
the
observation
turbulent
through
a
given
ow,
point
takes
have
a
area A
area A
1
2
wide
variation
of
velocities
depending
on
the
instant
when
the
density ρ
1
observation
is
made
(see
page
density ρ
boundary
167).
(streamlines)
A
streamline
laminar
given
ow
point
is
the
means
in
the
path
that
uid
taken
all
by
a
particles
must
follow
particle
that
the
in
pass
same
the
uid
through
and
In
a
a
time
Δt,
of
the
tangent
to
a
streamline
gives
streamline.
the
mass,
m
,
entering
the
cross-section
The
direction
m
=
ρ
A
1
v
1
∆t
1
the
mass,
m
,
leaving
the
cross-section
A
2
instantaneous
velocity
that
the
particles
of
the
uid
have
=
ρ
2
point.
No
uid
ever
crosses
a
streamline.
Thus
a
A
2
v
2
∆t
2
collection
Conservation
of
streamlines
tubular
tube
can
region
through
of
its
together
uid
ends
dene
where
and
a
uid
never
tube
only
of
ow.
enters
through
its
and
is
2
at
m
that
is
1
of
Similarly
the
A
1
1
direction
the
This
of
mass
applies
to
this
tube
of
ow,
so
is
leaves
the
ρ
A
1
This
sides.
v
1
is
=
ρ
1
A
2
an
v
2
ideal
2
uid
and
thus
incompressible
meaning
ρ
=
ρ
1
A
v
1
This
a
uid
ows
into
a
1
is
The
uid
must
end
up
v
2
the
narrow
section
of
a
The
pipe:
or
Av
=
,
so
2
constant
2
continuity
Bernoulli
work
•
A
equation
the Bernoulli equation
the Bernoulli effeCt
When
=
moving
at
a
higher
speed
done
equation
and
the
results
from
conservation
of
a
consideration
energy
when
an
of
the
ideal
uid
(continuity
changes:
equation).
•
This
means
the
uid
must
have
been
•
its
speed
(as
a
result
of
a
change
in
cross-sectional
area)
•
its vertical height as a result of work done by the uid pressure.
accelerated
forwards.
The
equation
along
any
a
quantity
that
is
always
of
vertical
uid
height
particles
uid
uid
higher pressure
lower pressure
higher pressure
lower speed
higher speed
lower speed
constant
streamline:
speed
density
of
identies
given
1
pressure
2
_
ρv
+
ρgz
+
p
=
constant
2
gravitational
density
•
This
means
there
must
be
a
pressure
difference
forwards
with
eld
of
a
lower
in
the
pressure
wider
in
the
narrow
section
and
a
higher
pressure
section.
Note
that:
1
Thus
an
increase
in
uid
speed
must
be
associated
with
strength
uid
a
•
The rst term (
2
ρv
), can be thought of as the dynamic pressure.
2
decrease
in
uid
pressure.
This
is
the
Bernoulli
effect
–
the
•
greater
the
speed,
the
lower
the
pressure
and
vice
The
last
two
terms
(ρgz
+
p),
can
be
thought
of
as
the
static
versa.
pressure.
•
Each
term
in
the
2
N
•
The
the
m
last
Pa,
of
J
the
Bernoulli
B
has
several
possible
units:
.
above
units
leads
gravitational
unit
volume
o p t i o n
m
–
to
a
new
interpretation
for
equation:
KE
per
equation
3
,
+
per
PE
unit
+
pressure
=
constant
volume
E n g i n E E r i n g
p h y s i c s
165
B – 
HL
appliC ations of the Bernoulli equation
a)
Flow
out
of
a
container
d)
Pitot
tube
to
determine
the
speed
of
a
plane
A
A
pitot
two
liquid
tube
is
separate
attached
facing
forward
on
a
plane.
It
has
tubes:
streamline
direction
small static
of air ow
pressure openings
density ρ
h
B
arbitrary zero
impact
opening
static
To
calculate
can
apply
the
speed
Bernoulli’s
of
uid
owing
equation
to
the
out
of
a
container,
streamline
shown
pressure
we
tube
above.
total
At
A,
p
=
At
B,
p
=
atmospheric
and
v
=
zero
pressure
tube
atmospheric
and
v
=
?
•
1
+
ρgz
+
p
=
front
hole
(impact
opening)
is
placed
in
the
constant
airstream
1
∴
The
2
ρv
2
0
+
hρg
+
p
called
2
=
ρv
+
0
+
and
the
measures
stagnation
the
total
pressure),
pressure
P
p
(sometimes
.
T
2
___
v
=
•
The
side
•
The
difference
hole(s)
measures
the
static
pressure,
P
.
√ 2gh
between
P
and
P
,
is
the
dynamic
T
b)
Venturi
tubes
pressure.
A
Venturi
meter
calculated
from
allows
a
the
rate
of
ow
of
a
measurement
of
pressure
uid
to
The
Bernoulli
equation
can
be
used
to
calculate
airspeed:
be
1
difference
P
P
T
2
=
ρv
s
2
between
two
different
cross-sectional
areas
of
a
pipe.
________
2(P
P
)
T
_
to metal end
area A
v
=
√
ρ
constriction
e)
of area a
Aerofoil
(aka
airfoil)
ν
A
dynamic lift F
B
pressure P
1
ow of (e.g.) water,
h
density ρ
1
ν
1
manometer liquid
aerofoil
(e.g. mercury),
density ρ
2
ν
2
•
The
pressure
calculated
difference
by
taking
between
readings
of
A
Δh
and
B
and
can
ρ
be
from
air ow
the
2
attached
manometer:
pressure P
2
P
P
A
•
This
=
∆hρ
B
value
g
2
and
measurements
of
A,
a
and
ρ
allows
the
Note
that:
1
uid
speed
at
A
to
be
calculated
by
using
Bernoulli’s
•
equation
and
the
equation
of
Streamlines
closer
together
above
the
aerofoil
imply
a
continuity
decrease
above
in
the
cross-sectional
area
of
equivalent
tubes
of
ow
aerofoil.
2∆hρ
g

2
________
v
=
2
√[
A
ρ
(
a
1
)
•
]
Decrease
in
increased
•
The
rate
of
ow
of
uid
through
the
pipe
is
equal
to
A
×
v
cross-sectional
velocity
continuity).
v
>
Fragrance
spray
•
Since
v
>
v
1
b.
Constriction in tube causes low pressure
•
region as air travels faster in this section
Bernoulli
different
the
below-pressure zone
•
squeeze-
When
c.
(height
can
difference
and forms little droplets
Squeezing
as it enters the air jet
bulb
forces air
through
tube
166
o p t i o n
B
–
tube
the
of
E n g i n E E r i n g
p h y s i c s
implies
(equation
of
P
can
be
used
difference
not
to
calculate
relevant)
the
pressure
which
can
support
aeroplane.
attack
become
and
ow
aerofoil
2
the
of
of
is
too
great,
turbulent.
the
This
ow
over
reduces
Liquid is drawn up tube
by pressure dierence
a.
of
angle
surface
bulb
<
1
equation
weight
area
above
2
P
2
ow
v
1
c)
of
leads
to
the
plane
‘stalling’.
the
the
upper
pressure
vc
HL
A)
definition of visCosity
An
ideal
uid
different
into
are
does
layers
thermal
needed
of
not
uid.
energy
to
resist
As
a
during
maintain
a
the
relative
result
there
laminar
steady
motion
is
ow
rate
of
no
and
ow.
Tangential
stress
relative
between
conversion
no
external
Ideal
uids
of
velocity ∆v
area of contact A
work
forces
are
retarding force
-F
non-
accelerating force
viscous
whereas
external
force
is
acceleration).
layers
of
a
real
uids
needed
Viscosity
uid
which
to
is
are
maintain
an
are
viscous.
a
internal
moving
In
a
viscous
steady
friction
with
rate
uid,
of
ow
between
different
a
F
steady
(no
different
The
tangential
stress
is
dened
as:
velocities.
F
τ
The
denition
of
the
viscosity
of
a
uid,
η,
(Greek
letter
Nu)
=
A
is
2
in
terms
of
two
new
quantities,
the
tangential
RH
side).
stress,
τ,
•
and
Units
of
tangential
stress
are
N
m
or
Pa
Δv
the
velocity
gradient,
(see
B)
Velocity
gradient
Δy
The
coefcient
of
viscosity
tangential
η
is
as:
y
velocity
stress
F⁄A
_
__
η
dened
=
=
velocity
(v
gradient
+
v)
Δv⁄Δy
v
2
•
The
units
•
Typical
of
η
values
are
at
N
s
1
m
room
or
kg
y
1
m
s
or
Pa
s
v
temperature:
3
◊
Water:
◊
Thick
1.0
×
10
Pa
s
2
•
syrup:
Viscosity
is
1.0
very
×
10
Pa
sensitive
to
s
changes
of
temperature.
The
For
a
class
of
uid,
called
Newtonian
uids,
velocity
gradient
is
dened
as:
experimental
Δv
_
velocity
measurements
gradient
(e.g.
show
many
that
pure
tangential
liquids).
stress
For
is
these
proportional
uids
the
to
gradient
=
velocity
coefcient
Δy
of
1
•
viscosity
is
constant
provided
external
conditions
remain
law
•
predicts
the
viscous
drag
force
F
that
of
velocity
gradient
are
s
constant.
stokes’ l aw
Stokes’
Units
acts
on
The
uid
is
columns
a
innite
of
uid
in
can
volume.
be
Real
affected
by
spheres
the
falling
proximity
through
of
the
D
perfect
sphere
when
it
moves
through
a
walls
uid:
•
uid at this point moves
The
of
the
size
than
of
the
container.
the
size
particles
of
the
of
the
uid
is
very
much
smaller
sphere.
with body (boundary layer)
The
forces
on
a
sphere
falling
through
a
uid
at
terminal
sphere has
velocity
are
as
shown
below:
uniform
velocity
D
v
uid upthrust
F
viscous drag
F
r
driving
equal opposing
force
viscous drag
sphere
sphere
velocity
innite expanse
v
r
density ρ
of uid η
uid
pull of
Drag
force
acting
on
sphere
in
viscosity
N
of
uid
in
Pa
s
density σ
Ear th
F
=
6
W
D
1
radius
of
sphere
in
m
velocity
of
sphere
in
m
s
At
terminal
velocity
v ,
t
Note
Stokes’
law
assumes
that:
W
=
U
+
F
D
•
The
speed
of
the
sphere
is
small
so
F
that:
=
U
W
D
◊
the
ow
of
uid
past
the
sphere
is
4
streamlined
6πηrv
=
3
πr
(ρ
σ)g
t
3
◊
there
is
no
slipping
between
the
uid
and
the
sphere
2
2r
(ρ
-
σ)g
__
∴
v
=
t
9η
speed
turBulent flow – the re ynolds numBer
Streamline
ow
rates
ow
the
only
ow
occurs
becomes
at
low
uid
ow
rates.
At
high
bulk
Reynolds
of
ow
radius
vrρ
R
Note
is
extremely
uid
ow
down
a
becomes
pipe,
number,
difcult
R,
a
predict
turbulent.
useful
which
to
is
number
dened
the
When
to
as:
exact
conditions
considering
consider
is
the
of
uid
=
η
It
pipe
density
_
turbulent
laminar
of
number
turbulent:
viscosity
of
uid
that:
when
uid
•
The
Reynolds
number
•
Experimentally,
uid
turbulent
R
does
not
have
any
units
–
it
is
just
a
ratio.
ow
Reynolds
o p t i o n
B
when
–
>
ow
2000
is
often
but
laminar
precise
E n g i n E E r i n g
when
predictions
R
<
are
p h y s i c s
1000
and
difcult.
167
fc c  c (1)
HL
Heavy
damping
Damping
opposite
involves
direction
particle.
As
resistive
(or
energy.
As
the
a
to
frictional
the
particle
oscillates,
dissipative)
the
total
force
direction
force
energy
it
and
of
the
that
of
does
so
is
always
motion
of
work
the
an
is
the
oscillating
against
particle
particle
in
forces
completely
taken
this
be
loses
proportional
to
the
damping
(e.g.
for
the
or
prevent
the
overdamping
SHM
taking
the
particle
place
in
oscillations
to
return
to
involves
a
large
viscous
from
zero
resistive
liquid)
taking
and
place.
displacement
can
The
can
time
again
long.
Critical
damping
involves
an
intermediate
value
for
resistive
2
(amplitude)
with
of
the
SHM,
the
amplitude
decreases
exponentially
time.
force
such
that
displacement
x ,tnemecalpsid
Examples
with
π
2π
4π
ω
ω
ω
time, t
is
a
time
taken
minimum.
critically
moving
tnemecalpsid
exponential envelope
of
the
and
the
particle
Effectively
damped
pointers
for
systems
door
closing
to
there
is
include
return
no
to
zero
‘overshoot’.
electric
meters
mechanisms.
overdamped
critical
damping
time
The
above
system
is
is
small
cycle.
example
said
so
a
The
to
time
oscillations
be
small
shows
the
effect
of
underdamped)
fraction
period
continue
of
for
of
the
the
a
light
where
total
the
energy
oscillations
signicant
damping
is
is
not
number
resistive
force
removed
affected
of
(the
cycles.
each
and
the
The
time
overshoot
underdamped
taken
for
the
oscillations
to
‘die
out’
can
be
long.
•
natural frequenCy and resonanCe
If
a
system
is
temporarily
position,
the
be
natural
at
the
system
will
displaced
oscillate
frequency
of
from
as
a
its
The
equilibrium
result.
vibration
This
of
◊
oscillation
the
system.
the
will
and
systems
with
It
is
its
own
rst
the
can
have
natural
choose
the
from
to
by
chosen
provided
is
to
possible
we
with
tap
you
tend
also
that
you
hear
a
wine
note
for
possible
glass
a
with
short
modes
a
knife,
while.
of
will
Complex
vibration
force
a
system
subjecting
the
it
to
This
to
a
changing
periodic
system.
combination
oscillate
of
When
natural
at
any
force
driving
each
this
and
frequency
that
force
driving
forced
forced
values
oscillations
of
the
depends
natural
on:
frequency
and
the
frequency
For
it
frequency.
frequency.
a
of
a
many
outside
applied,
rim
the
varies
must
the
amount
of
damping
noitallic so fo edutilpma
oscillate
if
of
comparative
driving
◊
example,
amplitude
be
frequency
present
in
the
system.
light damping
oscillations
increased damping
take
the
place
which
amplitude
produces
of
the
complex
transient
transient
oscillations
‘die
oscillations.
down’,
a
Once
steady
heavy damping
condition
is
achieved
•
The
system
•
The
amplitude
in
which:
oscillates
of
at
the
the
driving
forced
frequency.
oscillations
is
xed.
Each
cycle
driving frequency, f
driving
energy
is
dissipated
as
a
result
of
damping
and
the
driving
natural frequency, f
natural
force
does
energy
of
work
the
on
the
system
system.
remains
The
overall
result
is
that
the
Resonance
constant.
force
at
oscillation
Typical
q faCtor and damping
The
degree
quality
of
factor
denition
damping
or
Q
is
measured
factor.
It
is
a
by
ratio
a
quantity
(no
units)
called
and
the
occurs
exactly
of
the
orders
Car
the
of
when
same
a
system
is
frequency
subject
as
the
to
an
natural
oscillating
frequency
of
system.
magnitude
for
suspension:
different
Q-factors:
1
3
the
Simple
pendulum:
10
is:
3
Guitar
string:
10
7
energy
stored
Excited
atom:
10
__
Q
=
2π
energy
lost
per
cycle
When
Since
the
energy
is
amplitude
of
amplitude
with
Q
approximately
factor
is
the
stored
proportional
oscillation,
time
can
be
to
the
measurements
used
equal
to
to
of
calculate
the
square
of
the
decreasing
the
number
of
Q
factor.
is
The
In
a
system
energy
all
used
this
is
in
provided
to
resonance
by
overcome
situation,
the
Q
the
the
and
driving
its
resistive
factor
can
amplitude
frequency
be
forces
is
that
calculated
cause
as:
oscillations
energy
stored
__
that
are
completed
before
damping
stops
the
oscillation.
Q
=
2π
×
resonant
frequency
×
power
168
o p t i o n
B
–
E n g i n E E r i n g
p h y s i c s
constant,
during
loss
one
cycle
damping.
rc (2)
HL
pha se of forCed osCill ations
After
transient
relationship
oscillations
between
have
these
two
died
down,
oscillations
the
is
frequency
complex
of
and
the
forced
depends
on
oscillations
how
close
equals
the
the
driven
driving
system
frequency.
is
to
The
phase
resonance:
phase lag
φ/rad
driven vibration
1
period behind
2
π
driven vibration
π
1
2
period behind
4
heavy damping
light damping
0
f/Hz
in phase
natural
forcing
e x amples of resonanCe
Comment
Vibrations
in
machinery
When
other
in
operation,
sections
amplitude
a
Quartz
oscillators
A
truck’s
the
generator
Radio
receivers
Electrical
circuits
natural
a
the
components
the
driving
Many
be
the
chosen
The
the
circuit’s
radio
at
natural
region.
of
be
as
a
the
natural
frequency
Radiation
atmosphere.
an
of
causes
(using
the
for
that
is
the
of
for
e.g.
the
at
at
to
a
drive
known
to
a
forces
natural
on
the
frequency,
particular
eld
an
the
is
engine
the
speed
crystal
in
at
the
voltage
its
own
from
natural
systems.
The
molecules
removed,
oscillating
frequency.
oscillate.
resistors
The
its
the
The
driving
changing
frequency
particular
by
be
When
are
of
the
provided
will
it
to
the
that
in
have
an
receives.
be
increase
in
will
Adjusting
adjusted
driving
their
aerial
to
equal
frequency
amplitude
and
stations.
arranging
amplitude
of
that
frequency
oscillations
readily
inductors)
(electrons)
waves
station.
molecules
can
radio
other
the
and
charges
natural
radio
sounds
free
the
electrical
the
the
generate
microprocessor
water
of
causes
Earth
more
driving
to
increased.
allows
their
oscillation
regular
equal
When
to
used
charges
dominate
which
is
waves
all
particular
will
the
this
frequency
a
is
high.
eld.
added
capacitors,
circuit
by
are
oscillations.
produce
92
electric
clocks
means
frequency,
from
dangerously
and
temperature
signal
page
in
provide
frequency
vibrate.
accurate
which
frequency
of
to
crystal
that
connected
emitted
See
the
provided
natural
get
electronics
electrical
result
station’s
may
placed
machinery
driving
seen
designed
of
of
the
electromagnetic
the
instruments
its
of
force
i.e.
frequency
musical
driven
–
can
force
if
energy,
frequency
driving
be
provide
produce
driving
energy
the
effect
a
with
kinetic
can
force
devices
provides
equals
Greenhouse
is
a
If
vibration
movements
ovens
parts
Appropriate
microwaves
feel
instruments
feels
These
eld
moving
machinery.
mirror
oscillate.
Microwave
own
Musical
view
mechanical
electric
the
the
particular
crystal
will
frequency.
Microwave
a
rear
quartz
crystal
of
of
for
of
a
column
the
greenhouse
absorbed
by
of
air
oscillations
gases
the
is
in
or
to
the
a
string
to
increase.
infra-red
greenhouse
gases
in
details.
o p t i o n
B
–
E n g i n E E r i n g
p h y s i c s
169
iB q –  B –  c
5
1.
A
sphere
rest
down
plane,
of
of
it
the
given
mass
an
has
m
and
inclined
fallen
sphere,
v,
a
radius
plane.
vertical
when
it
r
rolls,
When
it
distance
arrives
at
without
slipping,
reaches
h.
the
the
Show
base
base
that
of
the
the
from
of
4.
the
a
and
speed
incline
In
diesel
a
temperature
changes
is
its
engine,
listed
starting
air
of
is
initially
27
below.
At
°C.
The
the
at
a
air
end
of
pressure
of
undergoes
the
cycle,
1
×
the
the
10
cycle
air
is
Pa
of
back
at
conditions.
by:
1
An
2
A
adiabatic
3
An
4
A
a)
Sketch,
compression
to
1/20th
of
its
original
volume.
original
volume.
_____
10gh
_
v
=
[4]
√
brief
isobaric
expansion
to
1/10th
of
its
7
adiabatic
expansion
back
to
its
original
volume.
2
2.
A
ywheel
of
moment
of
inertia
0.75
kg
m
is
accelerated
cooling
down
at
constant
volume.
1
uniformly
a)
from
Calculate
rest
the
to
an
angular
resultant
torque
speed
of
acting
8.2
on
rad
the
s
in
6.5
s.
with
undergoes.
during
this
time.
Calculate
the
the
Accurate
cycle
values
of
are
changes
not
that
the
gas
required.
[3]
[2]
b)
b)
labels,
ywheel
rotational
kinetic
energy
of
the
If
the
pressure
ywheel
after
the
adiabatic
compression
has
risen
6
to
6.6
×
10
In
which
Pa,
calculate
the
temperature
of
the
gas.
[2]
1
when
it
rotates
at
8.2
rad
s
[2]
c)
c)
The
radius
applied
of
on
the
the
ywheel
is
15
circumference
cm.
and
A
breaking
brings
it
to
angular
Calculate
3.
A
xed
speed
the
mass
of
of
value
a
gas
pressure
the
shown
V
cycle
of
rad
the
s
in
and
in
[2]
that
it
of
is
four
processes:
(i)
is
work
done
on
(ii)
is
work
done
by
(iii)
does
the
gas?
[1]
from
revolutions.
changes
such
diagram
2
force.
various
volume
the
exactly
breaking
undergoes
temperature,
p
8.2
the
force
rest
1
an
of
d)
taken
round
Explain
to
this
ignition
how
cycle
the
of
of
the
the
2nd
gas?
[1]
air-fuel
law
of
mixture
take
thermodynamics
place?
[1]
applies
changes.
[2]
below.
aP
HL
01/erusserp
5
5.
X
With
the
aid
of
diagrams,
explain
2.0
a)
What
is
b)
The
c)
Pascal’s
d)
An
meant
Bernoulli
by
laminar
ow
effect
principle
1.0
Z
Y
ideal
uid
[8]
3
6.
Oil,
of
viscosity
0.35
Pa
s
and
density
0.95
g
cm
,
ows
1
3
1.0
2.0
3.0
4.0
volume/10
5.0
through
3
Deduce
The
following
sequence
of
processes
takes
place
A
of
radius
20
cm
at
a
velocity
of
2.2
m
s
whether
the
ow
is
laminar
or
.
turbulent.
pendulum
clock
maintains
a
constant
amplitude
[4]
by
means
cycle.
of
X
pipe
during
7.
the
a
m
→
Y
the
gas
expands
absorbs
energy
at
constant
from
a
temperature
reservoir
and
and
does
the
450
J
gas
an
electric
available
for
power
the
supply.
The
following
information
is
pendulum:
of
2
Maximum
kinetic
Frequency
of
energy:
5
×
10
2
Hz
J
work.
Y
→
Z
the
gas
is
compressed
and
800
J
of
thermal
energy
Q
transferred
Z
→
X
the
gas
from
a)
Is
there
the
b)
Is
the
X
→
Use
d)
What
e)
How
What
The
in
→
the
to
Y
change
process
quantity
overall
is
to
→
in
Y
30
reservoir.
stage
energy
the
or
by
absorbing
energy
Calculate:
→
→
gas
the
its
gas
the
than
work
the
450
The
driving
b)
The
power
frequency
needed
done
process
J?
on
Explain.
the
[2]
gas
[3]
energy
of
the
gas
Z?
[2]
X?
is
absorbed
Explain
by
by
your
the
the
gas
answer.
area
[2]
enclosed
by
value.
a
a)
[2]
heat
done
by
engine
the
gas
is
dened
during
a
as
cycle
____
Efciency
=
total
If
this
p
engine
170
V
cycle
represents
determine
i B
energy
the
absorbed
the
cycle
efciency
Q u E s t i o n s
–
of
during
for
the
a
a
cycle
particular
heat
engine.
o p t i o n
of
the
power
supply
[3]
during
Z.
of
work
during
more
represented
efciency
of
[2]
internal
energy
Z
Estimate
net
by
determine
thermal
graph?
a
Explain.
equal
process
the
to
initial
internal
Y?
process
the
gas
its
absorbed
graph
much
during
X
than,
the
is
during
g)
less
the
during
the
change
energy
Y
the
to
factor:
reservoir.
processes
c)
f)
a
a
from
returns
oscillation:
is
B
–
heat
[2]
E n g i n E E r i n g
p h y s i c s
to
drive
the
clock.
[3]
15
o p t I o n
C
–
I m a g I n g
I ri
In
Ray dIagRams
If
an
be
object
is
placed
in
front
of
a
plane
mirror,
an
image
order
diagram
will
to
is
nd
the
location
and
nature
of
this
image
a
ray
needed.
formed.
image
object
upright
same size
as object
laterally
inverted
The
process
is
as
follows:
The
•
Light
sets
off
in
all
directions
from
every
part
of
the
•
(This
is
a
result
of
diffuse
reections
from
a
source
image
of
the
Each
ray
of
according
•
These
light
to
rays
the
can
that
law
be
arrives
of
at
the
mirror
is
The
the
location
rays
are
received
of
the
image
assumed
to
by
reection
distance
in
behind
a
plane
the
mirror
mirror
as
is
the
always:
object
is
front
reected
•
upright
•
the
(as
opposed
to
being
inverted)
reection.
by
an
seen
have
by
same
size
as
the
object
(as
opposed
to
being
magnied
observer.
or
•
same
light.)
in
•
formed
object.
the
observer
travelled
in
arises
straight
diminished)
because
•
laterally
•
virtual
inverted
(i.e.
left
and
right
are
interchanged)
lines.
(see
below).
Re al and vIRtual Images
The
image
mirror
term
is
is
formed
by
described
used
to
reection
as
a
virtual
describe
in
a
images
concave
plane
image.
created
mirror
This
when
object
rays
but
of
in
In
the
to
be
not,
at
light
fact
seem
they
do
example
coming
of
come
not
from
pass
above,
from
course,
to
the
behind
actually
a
single
through
rays
the
pass
of
that
light
mirror.
behind
point
real image
point.
seem
They
the
do
mirror
all.
The
opposite
of
a
virtual
image
is
a
real
(a) real image
image.
In
actually
this
pass
case,
the
through
a
rays
of
single
light
(b) vir tual image
do
point.
I
Real
images
mirrors,
but
they
example,
if
you
can
or
be
by
look
formed
by
formed
lenses.
into
the
plane
converging
by
For
concave
O
surface
of
a
spoon,
you
will
see
an
image
point
of
yourself.
This
particular
image
is
object
•
Upside
down
•
Diminished
•
Real.
diverging
vir tual point
rays
image
metsys lacitpo
mirrors
be
metsys lacitpo
concave
cannot
I
real
point
image
rays
O
point
object
stICk In wateR
The
image
leaving
formed
water
is
so
as
a
result
of
commonly
the
seen
refraction
that
most
of
light
people
forget
air
that
will
that
the
objects
appear
the
bent
rays
travelling
are
in
if
that
a
made
it
is
seem
placed
arrive
straight
to
at
in
one’s
strange.
water.
eyes
A
The
must
straight
brain
have
stick
water
assumes
been
line.
A
straight
when
stick
placed
in
appears
bent
The
image
of
the
end
of
the
pen
is:
water
•
Nearer
pen
•
o p t i o n
to
the
actually
surface
than
the
is.
Virtual.
C
–
i m a g i n g
171
Cri 
ConveRgIng lenses
A
converging
lens
brings
air
parallel
rays
into
one
focus
air
point.
normal
converging lens
normal
parallel rays
refraction at
glass
1st surface
refraction at
2nd surface
The
rays
of
light
are
all
brought
together
in
one
point
because
of
focal point
the
as
particular
a
collection
that
The
reason
both
that
surfaces
this
of
the
happens
is
the
refraction
that
takes
place
any
spheres
at
lens.
A
thin
will
The
power
of
a
lens
of
of
that
converge
lens
with
the
lens.
Any
different-shaped
lens
converging
compared
poweR of a lens
shape
glass
surfaces
light
will
the
has
into
always
lens
can
blocks.
formed
one
be
one
focus
thicker
It
from
be
can
thought
be
of
shown
sections
of
point.
at
the
centre
when
edges.
wave model of Image foRmatIon
measures
the
extent
to
region in which waves are
which
lens
light
bends
is
the
focal
length.
lens,
P,
P
=
is
bent
light
The
the
by
the
more
lens.
and
denition
reciprocal
of
of
the
A
higher
thus
the
has
power
a
power
focal
smaller
of
a
length,
O
f :
I
1
f
real image (point to which
object (source
f
is
the
focal
length
measured
in
m
wave energy is concentrated)
of wave energy)
1
P
is
the
power
dioptres
A
lens
and
of
has
lenses
of
the
lens
measured
in
m
or
(dpt)
Formation
power
a
focal
are
=
placed
approximately
+5
length
dioptre
of
close
20
is
real
together
When
their
The
terms
two
centre
•
The
The
of
lenses
need
and
to
this
of
curvature
of
be
each
centre
the
images
that
they
form,
The
the
some
focal
the
of
of
surface
of
a
curvature
lens
for
makes
the
lens
it
part
of
surface
axis
(ignoring
diffraction)
lens.
the
the
line
Technically
two
going
it
directly
joins
the
through
centres
A
axis
lens
(principal
axis
are
will
to
focus)
which
brought
thus
rays
to
have
a
of
that
focus
focal
a
lens
were
after
point
is
the
the
the
•
The
•
The
focal
and
of
has
length
the
linear
(height)
surfaces.
no
of
passing
on
each
focal
is
the
distance
magnication
the
between
the
image
and
m,
the
is
the
size
ratio
image
magnication,
m
of
the
i
_
=
size
h
o
lens
centre of
172
o p t i o n
C
–
focal point
i m a g i n g
the
side.
centre
h
size
=
object
f
on
the
through
between
(height)
_
c
to
f
of
the
point.
units.
linear
cur vature
point
parallel
a
is
lens
is
point
principal
principal
dened.
sphere.
principal
middle
refraction
thin
•
curvature
sphere.
by
powers
lens.
•
image
add.
analysing
technical
a
converging
cm.
defInItIons
When
of
c
principal axis (PA)
the
size
object.
It
I ri i c 
ImpoRtant Rays
lens
distant object
In
order
position
to
determine
of
the
image
the
nature
created
of
a
real image
and
inver ted
given
O
object,
we
need
to
construct
a
f
scaled
diminished
PA
ray
do
diagram
this,
taken
as
the
we
by
of
the
set-up.
concentrate
three
paths
on
particular
taken
by
In
the
rays.
two
of
order
to
I
f
paths
As
soon
these
rays
object at 2f
have
the
been
other
constructed,
rays
can
be
the
paths
inferred.
of
all
These
f
O
important
rays
are
described
below.
PA
f
Converging
I
real image
lens
inver ted
1.
Any
ray
that
was
travelling
same size
parallel
to
refracted
on
the
the
principal
towards
other
side
the
of
axis
focal
the
will
be
object bet ween 2f and f
point
lens.
f
O
PA
real inver ted
f
magnied
I
f
object at f
PA
f
f
O
PA
2.
Any
focal
ray
that
point
travelled
will
be
through
refracted
the
f
vir tual image
parallel
upright
to
the
principal
axis.
image at innity
object closer than f
I
f
f
O
PA
PA
f
3.
Any
the
ray
f
that
centre
of
goes
the
vir tual image
through
lens
will
upright
be
magnied
undeviated.
Converging
lens
images
possIble sItuatIons
f
PA
A
ray
diagram
can
be
constructed
as
follows:
f
•
An
upright
•
The
•
This
•
The
the
paths
of
locates
bottom
top
of
arrow
two
the
of
the
if
it
is
real
•
if
it
is
upright
•
if
it
is
magnied
its
exact
should
image
still
An
be
of
image
rays
the
must
axis
from
top
be
of
on
represents
the
the
the
top
of
the
the
object.
object
are
constructed.
image.
principal
axis
directly
above
(or
below)
of
the
image
created
would
include
the
following
information:
virtual
or
inverted
or
diminished
position.
be
also
important
the
•
It
principal
image.
full
or
the
position
A
•
description
on
noted
consists
formed
observer
that
of
even
if
receiving
the
all
important
the
some
other
of
parallel
the
rays
rays
rays
rays
sees
are
from
are
an
just
the
used
object.
blocked
image
to
In
locate
the
image.
particular,
the
The
image
real
will
off.
located
in
the
far
distance
(at
innity).
o p t i o n
C
–
i m a g i n g
173
ti  i
lens equatIon
There
is
a
lIne aR
mathematical
method
of
locating
the
image
formed
by
a
lens.
An
analysis
of
the
angles
magnIfIC atIon
involved
shows
that
the
following
equation
can
be
applied
to
thin
spherical
lenses:
In
1
1
=
cases,
linear
1
+
v
f
all
magnication,
u
h
v
i
_
m
=
=
-
u
h
o
height
of
image
__
m
=
height
of
object
h
object
i
_
=
h
o
f
f
v
=
-
u
image
For
real
images,
negative
and
m
is
image
is
inverted
For
virtual
positive
images
and
upright
object distance u
Suppose
u
f
=
25
=
10
would
mean
v
cm
cm
1
This
image distance
1
that
1
1
_
=
u
f
5
_
1
_
=
v
-
=
10
3
_
2
_
-
25
50
=
50
50
50
_
In
other
word,
v
=
=
16.7
cm
i.e.
image
is
real
3
16.7
_
In
this
case
m
=
=
-1.67
and
inverted.
10
Re al Is posItIve
Care
needs
convention
to
be
has
taken
to
be
with
virtual
images.
•
Distances
are
taken
to
be
positive
•
Distances
are
taken
to
be
negative
•
Thus
the
a
virtual
lens
as
image
the
The
equation
does
work
but
for
this
to
be
the
is
if
represented
actually
if
traversed
apparently
by
a
by
traversed
negative
value
for
the
by
v
–
light
the
in
ray
light
other
(i.e.
ray
distances
to
(distances
to
words,
it
will
be
object.
image
object
f
f
object
distance u
negative image
distance v
Suppose
u
=
10
cm
f
=
25
cm
1
This
would
mean
1
that
1
=
v
1
_
1
_
25
10
=
f
2
_
5
_
50
50
=
u
other
word,
v
=
=
-
-16.7
cm
i.e.
image
3
16.7
_
In
this
case
m
=+
=
+1.67
and
upright
10
174
o p t i o n
C
–
i m a g i n g
is
3
_
=
50
_
In
case,
the
following
followed:
virtual
-
50
real
object
virtual
on
the
and
objects
same
image).
and
side
images).
of
image
m
is
is
diri 
When
dIveRgIng lenses
A
to
diverging
all
come
lens
spreads
from
one
parallel
focus
rays
point
on
apart.
the
These
other
rays
side
of
the
constructing
important
appear
other
lens.
1.
ray
Any
rays
paths
ray
ray
whose
that
can
be
was
diagrams
paths
are
inferred)
travelling
for
diverging
known
(and
lenses,
from
the
which
all
are:
parallel
to
the
principal
axis
will
concave lens
be
refracted
the
away
from
a
focal
point
on
the
incident
side
of
lens.
focal point
PA
f
2.
Any
ray
other
that
side
of
is
f
heading
the
lens,
towards
will
be
the
focal
refracted
so
point
as
to
on
be
the
parallel
to
focal length
the
The
at
reason
both
centre
that
this
surfaces.
when
A
happens
diverging
compared
with
is
the
lens
the
refraction
will
always
that
be
takes
thinner
principal
axis.
place
at
the
edges.
PA
f
f
defInItIons and ImpoRtant Rays
Diverging
lenses
converging
Centre
of
have
lenses
for
curvature,
the
all
same
of
principal
the
analogous
following
axis,
focal
denitions
as
3.
terms:
point,
focal
Any
ray
that
goes
through
the
centre
of
the
lens
will
be
undeviated.
length,
linear
magnication.
f
Note
that:
PA
•
The
focal
which
to
•
As
rays
come
the
point
that
after
focal
length
of
is
the
were
is
on
parallel
passing
point
a
point
the
to
through
behind
diverging
is
the
the
the
lens
principal
axis
principal
f
from
axis
appear
lens.
diverging
lens,
the
focal
negative
Images CRe ated by a dIveRgIng lens
Whatever
the
focal
the
position
point
and
the
of
the
lens
object,
on
the
a
diverging
same
side
of
lens
the
will
lens
always
as
the
create
an
upright,
diminished
and
virtual
image
located
between
object.
If you look at an object through a concave lens,
If you move the object fur ther out, the image will not
it will look smaller and closer.
move as much.
object
f
f
v
u
f
object
image
image
object inside focal length
object outside focal length
The
For
thin
lens
example,
equation
if
an
will
object
is
still
work
placed
at
providing
a
distance
one
2l
remembers
away
from
a
the
negative
diverging
focal
lens
of
length
focal
of
a
length
diverging
l,
the
lens.
image
can
be
calculated
as
follows:
Given:
u
1
=
2l,
1
=
-l,
v
=
?
1
+
u
f
=
v
1
f
1
=
1
-
v
1
_
=
u
f
3
_
1
-
l
=
2l
2l
2l
∴
v
=
-
3
1
This
is
a
virtual
diminished
and
upright
image
with
m
=
+
3
o p t i o n
C
–
i m a g i n g
175
Cri  iri irrr
geometRy of mIRRoRs and lenses
The
geometry
of
the
paths
of
rays
after
Image foRmatIon In mIRRoRs
reection
by
a
(1) Concave
spherical
concave
or
convex
mirror
is
exactly
analogous
to
the
object at innity
paths
of
rays
difference
is
through
that
converging
mirrors
reect
or
all
diverging
rays
lenses.
backwards
The
only
whereas
real
rays
pass
through
lenses
and
continue
forwards.
PA
inver ted
F
2f
diminished
(a) Convex lens
I
object between innity and 2f
PA
O
real
PA
F
2f
inver ted
diminished
f
I
(b) Concave mirror
object at 2f
O
real
PA
2f
F
inver ted
same size
PA
I
object between 2f and f
f
O
real
inver ted
(c) Concave lens
PA
2f
F
magnied
I
object at f
PA
vir tual
O
upright
2f
PA
f
image at
F
innity
object between f and mirror
(d) Convex mirror
vir tual
2f
PA
upright
F
magnied
PA
(2) Convex
object at innity
f
This
analogous
equations
detail
for
with
behaviour
lenses
the
sign
can
means
be
used
that
all
(with
conventions)
with
the
denitions
suitable
and
attention
to
I
vir tual
mirrors.
upright
PA
An
additional
through
(or
(located
at
back
along
important
towards)
twice
the
the
same
ray
the
for
centre
focal
mirrors
of
length).
is
the
curvature
This
ray
ray
of
will
that
the
be
travels
diminished
F
2f
mirror
reected
path.
object near lens
vir tual
PA
F
176
o p t i o n
C
–
i m a g i n g
2f
upright
diminished
t i ii 
angul aR sIze
ne aR and faR poInt
The
human
distances
describe
point
•
from
the
and
The
eye
can
the
focus
eye.
possible
the
far
distance
Two
range
point
to
objects
the
at
terms
of
If
different
are
useful
distances
–
the
bring
then
near
occupies
we
point
is
the
an
see
a
subtends
distance.
near
we
to
it
object
in
bigger
a
closer
more
visual
larger
to
detail.
us
(and
This
angle.
is
The
our
eyes
because,
technical
are
as
still
the
term
able
object
for
this
to
focus
on
approaches,
is
that
the
it)
it
object
angle.
distance
objects are the
between
the
eye
and
the
nearest
object
that
same size
can
or
be
brought
help
from,
into
for
clear
focus
example,
(without
lenses).
It
is
strain
also
angle subtended
known
By
as
‘least
convention
normal
•
the
The
distance
be
is
taken
to
of
distinct
be
25
cm
vision’.
for
vision.
between
can
it
distance
the
to
the
eye
brought
far
and
into
point
the
is
the
furthest
focus.
This
is
distance
object
taken
that
to
be
close
distant
angle subtended
innity
for
normal
vision.
object
object
by distant object
1.
angul aR magnIfIC atIon
The
angular
dened
as
normally
the
the
and
optical
context.
the
magnication,
It
same
ratio
the
between
angle
instrument.
should
as
the
M,
be
that
The
noted
linear
of
the
its
an
optical
angle
image
‘normal’
that
the
that
instrument
an
object
subtends
situation
angular
as
a
is
In
subtends
result
depends
on
magnication
Image
The
of
formed
this
resulting
seen
at
innity
arrangement,
by
the
image
relaxed
the
will
object
be
is
placed
formed
at
at
the
innity
focal
and
point.
can
be
eye.
the
is
not
magnication.
h
θ
=
i
f
top
rays from object at
θ
o
specied distance
h
θ
i
f
bottom
θ
i
top
eye focused
on innity
f
rays from nal image
formed by optical
θ
In
i
this
case
the
angular
magnication
would
be
instrument
h
θ
f
i
M
=
D
=
=
innity
h
θ
f
o
D
bottom
This
can
is
the
smallest
value
that
the
angular
magnication
be.
θ
i
Angular
magnication,
M
2.
=
Image
formed
at
near
point
θ
o
In
The
largest
visual
angle
that
an
object
can
occupy
is
when
this
placed
at
the
near
point.
This
is
often
taken
as
the
the
object
is
placed
nearer
to
the
lens.
it
The
is
arrangement,
resulting
virtual
image
is
located
at
the
near
point.
This
‘normal’
arrangement
has
the
largest
possible
angular
magnication.
situation.
θ
i
o
h /D
h
i
i
D
1
a
u
v
f
1
1
1
M
h
θ
1
+
=
θ
o
D
h/D
h
-
⇒
1
=
=
h
θ
o
a
D
D
D
+
=
∴
h
i
a
f
1
f
D
h
θ
A
simple
lens
can
increase
the
angle
subtended.
It
is
usual
to
i
f
consider
two
possible
θ
f
situations.
i
a
D
D
So
the
magnitude
of
M
=
near
+
1
point
f
o p t i o n
C
–
i m a g i n g
177
arri
spheRIC al
A
lens
some
a
is
said
perfect
are
to
reason,
Spherical
have
point
point
spherical
describe
a
image.
do
not
fact
of
slightly
different
inner
a
be
In
general,
are
•
of
a
The
a
of
way
lens
objects
•
the
A
at
a
effect
this
is
The
effect
point
(as
46)
the
effect
smaller
is
and
for
the
to
can
could
be
the
a
for
The
be
ray striking inner regions
not
of
small
effect:
ray striking outer regions
in
range of focal points
effect.
longer
be
works
outer sections
for
of lens not used
away.
given
lens
technical
the
total
made
can
a
There
this
only
a
a
altered
distance
term
aperture.
amount
by
of
diffraction
The
light
aper ture
(see
worse.
be
by
spherical)
again
no
down
axis
into
the
shape
the
is
point.
be
for
effects
mirrors
on
focus
reduced
that
to
striking
This
reducing
aperture.
would
objects
opposed
the
be
those
lens.
course,
stopping
reduced
page
of
to
outer
distortion
correct
particular
disadvantage
is
lens
used
brought
perfect
of
particular
can
decreasing
for
ways
a
be
that
images.
the
from
will
than
would,
spherical.
will
same
object
to
term
for
produce
lenses
striking
point
if,
not
perfect
the
barrel
the
as
is
the
rather
shape
such
of
with
possible
does
reality,
lens
focus
point
light,
several
the
In
rays
spherical
confused
circle
that
regions
to
aberration
produce
aberration
the
regions
the
an
object
eliminated
using
a
mirror.
reduced
For
by
for
all
parabolic
mirrors,
using
a
aperture.
Spherical
aberration
ChRomatIC
Chromatic
to
describe
aberration
the
fact
that
is
the
rays
of
term
used
different
white light
colours
will
be
brought
to
a
slightly
V
different
The
focus
refractive
point
index
by
of
the
the
same
red
lens.
material
used
violet
to
make
the
frequencies
lens
of
is
different
for
R
different
light.
R
violet
A
point
image
object
of
will
different
produce
a
blurred
red
colours.
V
The
effect
can
given
colours
using
two
a
types
an
of
eliminated
(and
different
compound
called
be
lens.
reduced
for
materials
This
achromatic
glass
for
produce
two
all)
to
compound
but
do
not
suffer
from
red
focus
focus
up
lens
The
is
two
opposite
dispersion.
Mirrors
violet
by
make
doublet.
equal
white light
converging lens
of crown glass
Canada balsam
(low dispersion)
cement
diverging lens
chromatic
of int glass
aberration.
(high dispersion)
Achromatic
178
o p t i o n
C
–
i m a g i n g
doublet
t c icrc  ric c
Compound mICRosCope
A
compound
forms
(the
a
real
microscope
magnied
eyepiece
lens)
consists
image
which
virtual
magnied
angular
magnication
acts
image.
is
In
of
of
as
two
the
a
lenses
object
magnifying
normal
–
the
being
lens.
adjustment,
objective
viewed.
The
this
This
rays
lens
real
from
virtual
and
this
image
the
image
real
is
eyepiece
can
then
image
arranged
be
lens.
travel
to
be
The
rst
considered
into
the
located
at
as
lens
the
eyepiece
the
near
(the
object
lens
point
objective
for
and
so
the
they
that
lens)
second
form
lens
a
maximum
obtained.
objective
construction
lens f
eyepiece lens f
o
e
top half
construction
line
line
of object
B
real image formed
f
h
by objective lens
o
θ
i
θ
O
f
i
h
e
1
eye focused on near point to
vir tual image
see vir tual image (in practice
h
of top half of
2
it would be much nearer to the
object
eyepiece lens than implied here)
M
D
h
2
h
θ
D
i
M
=
=
h
2
h
2
_
1
_
=
_
=
h
θ
h
=
h
linear
magnication
produced
by
eyepiece
×
linear
magnication
produced
by
objective
h
1
o
D
a stRonomIC al telesCope
An
astronomical
distant
object
magnifying
telescope
being
lens.
adjustment,
this
also
viewed.
The
rays
virtual
consists
Once
from
image
this
is
of
again,
real
two
this
image
arranged
to
lenses.
real
travel
be
In
image
this
can
into
located
at
case,
then
the
the
be
objective
considered
eyepiece
lens
and
lens
as
forms
the
they
a
real
object
form
a
for
but
the
diminished
eyepiece
virtual
lens
magnied
image
acting
image.
of
as
In
the
a
normal
innity.
objective
eyepiece lens
lens
f
f
o
e
construction
line
parallel rays all from
real image formed in mutual
top of distant object
focal plane of lenses
f
e
θ
eye focused on innity
i
f
θ
o
o
θ
o
h
1
θ
i
vir tual image
at innity
h
θ
M
f
f
i
=
o
_
=
_
=
h
θ
f
e
o
f
The
length
of
the
telescope
≈
f
o
+
f
e
o p t i o n
C
–
i m a g i n g
179
aric rci c
CompaRIson of RefleCtIng and RefRaCtIng
ne wtonIan mountIng
A
small
at
mirror
is
placed
on
the
principal
axis
of
the
mirror
telesCopes
to
A
refracting
form
a
real
telescope
uses
diminished
an
objective
image
of
a
(converging)
distant
object.
lens
This
reect
the
image
formed
to
the
side:
to
image
is
concave
then
viewed
by
the
eyepiece
lens
(converging)
which,
acting
small at mirror
mirror
as
a
simple
nal
In
magnifying
glass,
produces
a
virtual
but
magnied
image.
an
analogous
mirror
distant
set
up
so
object.
view
as
it
Thus
mirrors
way,
a
as
form
to
This
would
are
reecting
image,
be
a
to
in
uses
diminished
however,
produced
used
telescope
real,
would
front
produce
a
of
be
the
a
of
difcult
concave
viewable
F
concave
image
image
o
a
to
mirror.
that
F'
o
can,
like
the
refracting
telescope,
be
viewed
by
the
eyepiece
eyepiece lens
lens
(converging).
Once
magnifying
glass
image.
common
the
All
Two
Newtonian
telescopes
and
again
made
to
eyepiece
the
mountings
mounting
are
the
produces
and
have
virtual,
for
acts
but
reecting
the
as
simple
telescopes
Cassegrain
large
a
magnied,
apertures
nal
are
mounting.
in
order
to:
C a ssegRaIn mountIng
a)
reduce
b)
collect
diffraction
effects,
and
A
enough
light
to
make
bright
images
of
low
convex
The
mirror
mirror
has
is
a
mounted
central
on
hole
the
to
principal
allow
the
axis
of
image
telescopes
are
reecting
The
because:
convex
mirror
will
add
to
the
angular
magnication
achieved.
•
Mirrors
•
It
do
not
suffer
from
chromatic
aberration
concave
is
difcult
to
get
a
uniform
refractive
index
throughout
small convex mirror
a
mirror
large
•
volume
Mounting
a
of
glass
large
lens
is
harder
to
achieve
than
mounting
F
a
•
large
Only
mirror.
one
Reecting
surface
telescopes
needs
can
to
be
easily
the
right
suffer
shape.
damage
the
mirror
eyepiece
lens
surface.
180
to
o p t i o n
C
–
i m a g i n g
ex t
F
o
the
to
viewed.
sources.
Large
small
mirror.
power
be
Ri c
sIngle dIsh RadIo telesCopes
A
single
to
a
to
form
dish
radio
reecting
an
telescope
telescope.
image,
the
waves
are
reected
that
the
receiver
up
is
specic
study
of
the
the
wavelengths
naturally
Rather
much
by
operates
longer
curved
radio
under
occurring
than
in
RadIo InteRfeRometRy telesCopes
a
very
reecting
visible
wavelengths
receiving
waves
can
emission
of
dish.
be
observation
radio
similar
The
from
are
The
light
stars,
some
pick
used
to
than
and
other
astronomical
objects
between
about
10
m
and
received
distance
any
at
two
apart
creates
of
the
of
called
a
a
radio
telescope
interferometry.
(or
but
more)
radio
pointing
virtual
individual
in
radio
can
This
be
telescopes
the
same
telescope
improved
process
analyses
that
are
direction.
that
is
This
much
larger
telescopes.
galaxies,
technique
is
complex
as
it
involves
collecting
signals
wavelengths
from
of
resolution
principle
effectively
The
quasars
a
signals
antenna
to
angular
using
radio
tuned
and
way
two
or
more
radio
telescopes
(an
array
telescope)
1mm.
in
one
central
individual
location.
antenna
The
needs
arrival
to
be
of
each
carefully
signal
at
calibrated
an
against
a
Radio telescope
single
shared
reference
signal
so
that
different
signals
can
be
incoming radio
combined
as
though
they
arrived
at
one
single
antenna.
When
waves
the
signalsfrom
they
that
interfere.
is
the
The
equivalent
single
radio
to
maximum
different
result
in
is
telescopes
to
create
resolution
telescope
whose
a
are
combined
(though
diameter
added
not
is
in
together,
telescope
sensitivity)
approximately
to
a
equal
Radio waves reect
the
separations
of
the
antennae.
o the dish and
The
principle
can
be
extended,
in
a
process
called
Very
focus at the tip.
Long
Baseline
Interferometry,
to
allow
recordings
of
Receivers detect and
radio
signals
(originally
made
hundreds
of
km
apart)
to
be
amplify radio signals.
synchronized
scientists
Diffraction
effects
can
signicantly
limit
the
accuracy
a
signals.
the
radio
telescope
Increasing
telescope’s
that
more
the
ability
power
can
can
locate
diameter
to
resolve
be
individual
of
a
radio
different
received
(see
sources
telescope
sources
resolution
from
within
different
a
millionth
countries
of
to
a
second
thus
collaborate
to
allowing
create
a
with
virtual
which
to
of
radio
telescope
of
huge
size
and
high
resolving
power.
radio
improves
and
on
ensure
page
101).
CompaRatIve peRfoRmanCe of e aRth-bound and s atellIte-boRne telesCopes
The
•
following
SB
observations
better
•
•
to
are
•
SB
observations
•
SB
facilities
•
The
•
There
are
is
a
great
signicantly
is
an
not
There
•
SB
telescopes
•
EB
optical
not
can
suffer
subject
deal
added
of
telescopes
and/or
added
EM
in
(SB)
absorptions
correct
radiation
their
light
continual
space
cost
to
of
only
in
build
due
telescopes
to
the
can
Earth’s
be
made:
atmosphere
that
hinder
EB
observations,
giving
for
many
atmospheric
effects
making
new
ground-based
telescopes
similar
pollution
wear
getting
and
wider
operate
(UV
,
and
exists
the
this
/
and
radio
tear
for
long
wavelength
repairs
night
as
SB
interference
radio)
are
absorbed
by
the
Earth’s
atmosphere
so
SB
a
/
a
result
into
limit
on
alterations
to
variations
whereas
of
as
the
a
result
Earth’s
of
nearby
atmosphere
human
activity.
(storms
etc.).
telescopes.
telescope
places
temperature
at
IR
wavelengths.
debris
effecting
withstand
can
effectively
from
to
difcultly
to
of
from
expensive
need
satellite-borne
telescopes.
possibility
damage
more
•
SB
only
and
interference
techniques
do
of
from
(EB)
telescopes.
wavelengths
the
possibility
SB
some
signicant
telescopes
Earth-based
free
for
computer
resolution
Many
about
are
resolution
Modern
in
points
SB
orbit
and
their
a
SB
than
size
controlling
and
telescope
EB
telescopes
it
remotely,
meaning
that
SB
telescopes
are
weight.
once
operational.
telescopes.
can
operate
at
all
times.
o p t i o n
C
–
i m a g i n g
181
fir ic
•
optIC fIbRe
Optic
page
bres
45)
make
a
to
ray
between
the
ray
angle,
bre
As
the
shown
walls
ray
of
will
a
bre.
a
is
path.
reection
The
transparent
So
bre
always
internal
certain
along
the
total
long
as
always
remain
the
the
is
by
than
bre
angle
the
even
if
the
of
critical
the
•
This
medical
images
called
to
bouncing
incident
greater
within
idea
bre
In
carry
(see
an
type
bre.
Bundles
from
inside
of
optic
the
bres
body.
can
This
be
used
instrument
to
is
endoscope.
of
optic
Cladding
surrounds
the
world.
back
the
of
bre
a
is
known
material
bre.
This
as
with
cladding
a
a
step-index
lower
optic
refractive
protects
and
index
strengthens
bre.
right).
page
index
the
of
of
along
travel
wall
(see
on
principle
light
light
the
bent
refractive
the
guide
of
the
on
is
use
n
45,
is
the
relation
given
between
critical
angle,
c,
and
by
1
_
n
=
sin
Two
•
c
important
In
the
into
is
uses
of
optic
communication
pulses
used
for
of
light
are:
industry.
that
telephone
bres
can
Digital
then
travel
communication,
data
can
along
cable
be
the
TV
encoded
bres.
This
etc.
types of optIC fIbRes
The
simplest
Technically
bre
this
optic
is
is
known
a
step-index
as
a
bre.
multimode
index
index
output
prole
pulse
pulse
step-
n2
index
bre.
Multimode
refers
to
the
fact
that
light
can
n1
take
different
some
page
paths
distortion
183).
The
of
down
signals
the
bre
(see
(multimode)
which
waveguide
can
result
in
dispersion,
graded-index
bre
is
multimode step-index
an
improvement.
prole
in
speeds
depending
the
This
bre
uses
a
meaning
on
their
graded
that
refractive
rays
distance
travel
from
at
the
index
different
n2
centre.
n1
This
has
pulse.
the
Most
graded
narrow
effect
bres
index.
core
–
The
a
of
reducing
used
in
data
optimum
the
spreading
communications
solution
singlemode
out
is
to
step-index
have
of
have
a
bre
the
a
very
multimode graded-index
n2
n1
singlemode step-index
182
o p t i o n
C
–
i m a g i n g
diri, i  i i ic r
mateRIal dIspeRsIon
The
refractive
frequency
reason
it
of
that
passes
index
of
any
waveguIde dIspeRsIon
substance
electromagnetic
white
through
light
a
is
depends
radiation
dispersed
triangular
on
the
considered.
into
different
If
This
is
colours
the
when
prism.
the
optical
called
a
pulse
along
is
light
travels
along
an
optical
bre,
different
frequencies
at
slightly
different
speeds.
This
means
that
if
the
light
involves
out
as
the
bre.
a
square
a
range
wave
of
will
frequencies,
tend
to
then
spread
a
out
pulse
as
it
of
a
reections.
signicant
or
modal
bre
This
is
diameter,
that
can
another
cause
dispersion.
shorter
means
than
that
a
rays
the
The
path
from
path
that
a
process
stretching
of
length
involves
particular
will
not
all
arrive
at
the
same
time
because
of
the
different
source
distances
of
centre
a
dispersion
will
pulse
travel
has
multipath
the
multiple
As
bre
waveguide
that
travels
they
have
travelled.
starts
along
B
This
process
is
known
as
material
dispersion
C
A
cladding
before transmission
The
after transmission
problems
of
bres.
optical
same
used
These
order
of
As
light
travels
The
scattered
that
entered
The
in
or
arrives
the
of
optic
by
end
bre.
amount
decibels
an
absorbed
at
the
along
the
of
The
attenuation
(dB).
The
glass.
the
signal
bre,
bre
is
The
is
said
is
some
than
be
is
energy
of
the
the
can
be
light
intensity
energy
that
attenuated.
measured
attenuation
the
intensity
less
to
of
on
given
a
from
scale
path
of
therefore
series
of
–
a
5
10
be
µm)
km
40
the
so
very
the
narrow
wavelength
that
along
The
have
led
singlemode)
length
dB.
is
dispersion
(or
have
as
directly
factors
attenuation
processes:those
scattering
by
bres
core
there
the
of
cores
of
the
the
only
(of
the
light
one
being
effective
bre.
this
overall
algebraic
is
to
step-index
bre
optic
cable
attenuation
sum
of
the
resulting
individual
attenuations.
The
logarithmic
a
modal
magnitude
attenuation
would
by
monomode
(approximately
transmission
attenuatIon
caused
development
paths
that
in
an
takes
the
glass
the
wavelength
optic
caused
absorbs
by
place
the
bre
is
a
result
impurities
in
light.
the
These
in
glass
last
of
the
and
two
several
glass,
the
the
extent
factors
are
general
to
which
affected
by
I
attenuation
(dB)
=
10log
of
light
used.
A
typical
the
overall
attenuation
is
I
o
shownbelow:
I
is
I
the
is
intensity
the
of
intensity
the
of
output
the
power
original
measured
input
power
in
W
measured
in
W
1
o
negative
in
power.
been
See
page
It
is
A
positive
means
that
attenuation
the
signal
would
has
imply
been
that
the
reduced
signal
amplied.
188
common
for
to
another
quote
example
of
the
the
attenuation
For
example,
use
per
of
unit
the
decibel
length
scale.
as
1
measured
causes
The
in
an
dB
input
km
.
power
attenuation
per
of
100
unit
mW
length
is
5
to
km
of
bre
decrease
calculated
to
as
optic
1
cable
rep noitaunetta
has
attenuation
6
mk Bd / htgnel tinu
A
5
4
3
2
1
mW
.
0.6
follows:
0.8
1.0
1.2
1.4
1.6
1.8
wavelength / µm
3
attenuation
attenuation
=
10
per
log
unit
(10
1
/10
length
2
)
=
10
=
-20
=
-20
=
-4
log
(10
)
dB
dB/5
km
1
dB
km
C apaCIty
Attenuation
noIse, amplIfIeRs and ReshapeRs
causes
information
that
bre.
often
This
is
capacity
of
an
can
upper
be
sent
stated
an
in
optical
limit
to
along
a
terms
bre
of
=
the
amount
particular
its
bit
of
type
digital
Noise
of
scatterings
optical
capacity.
rate
×
to
distance
to
1
A
bre
with
a
capacity
of
80
Mbit
s
km
can
Mbit
s
1
along
a
1
km
length
of
bre
but
a
4
km
that
in
take
any
electronic
place
within
circuit.
an
optical
Any
dispersions
bre
will
also
or
add
noise.
amplier
correct
only
20
Mbit
increases
the
effect
of
the
signal
strength
attenuation
–
and
these
are
thus
also
will
tend
sometimes
called
regenerators.
An
amplier
will
also
increase
any
noise
s
that
along
inevitable
transmit
1
80
the
An
is
has
been
added
to
the
electrical
signal.
reduce
the
effects
of
noise
of
1s
length.
A
reshaper
by
can
returning
transitions
the
signal
between
to
the
a
series
allowed
o p t i o n
C
on
and
a
0s
digital
with
signal
sharp
levels.
–
i m a g i n g
183
C  cici
The
table
below
shows
some
common
Options
Wire
pairs
(twisted
Two
pair)
the
communication
for
wires
communication
can
sender
links.
connect
and
Uses
Very
receiver
of
Advantages
simple
systems
communication
e.g.
Very
simple
information.
For
simple
link
between
to
a
Coaxial
cables
an
noise
transfer
and
interference.
information
at
the
a
highest
microphone,
to
example
Unable
a
disadvantages
cheap.
intercom
Susceptible
copper wire
and
and
amplier
rates.
and
loudspeaker.
This
arrangement
reduces
electrical
of
two
wires
interference.
Coaxial
to
cables
transfer
are
signals
used
Simple
Less
A
central
wire
is
surrounded
TV
aerials
to
TV
by
wire
of
the
second
wire
in
the
form
Historically
they
straightforward.
susceptible
to
noise
compared
receivers.
to
copper
and
from
simple
wire
pair
but
noise
still
a
are
problem.
an
tube
outer
or
cylindrical
mesh.
An
copper
standard
for
telephone
insulator
underground
links.
insulation
separates
the
two
wires.
copper mesh
outside insulation
Wire
links
can
frequencies
but
be
the
MHz
of
The
about
more
A
sent
would
intervals
0.5
to
for
a
typical
down
need
of
1
GHz
frequencies
wire.
signal
cable
at
higher
attenuated
length
carry
up
will
given
100
low-loss
repeaters
approximately
km.
upper
coaxial
limit
cable
is
for
a
single
approximately
1
140
Optical
bres
Mbit
Laser
light
signals
with
s
can
down
be
used
optical
approximately
frequency
limit
as
to
send
bres
the
telecommunication
same
cables
is
volume
data
Compared
and
transfer
including
to
equivalent
coaxial
capacity,
cables
optical
with
bres:
of
•
have
a
•
are
•
cost
•
allow
higher
transmission
capacity
video
much
smaller
in
size
and
weight
data.
attenuation
bre
high
digital
1GHz.
The
Long-distance
less
than
in
an
in
a
less
optical
for
a
wider
possible
spacing
of
coaxial
regenerators
cable.
The
repeaters
even
distance
can
between
easily
hundreds)
of
be
tens
(or
•
kilometres.
offer
immunity
to
electromagnetic
interference
•
suffer
from
(signals
in
another
•
are
very
negligible
one
cross
channel
talk
affecting
channel)
suitable
for
digital
data
transmission
•
provide
•
are
good
quiet
when
–
security
they
carrying
do
not
large
hum
even
volumes
of
data.
There
•
are
the
some
repair
of
disadvantages:
bres
is
not
a
simple
task
•
regenerators
thus
184
o p t i o n
C
–
i m a g i n g
are
potentially
more
less
complex
reliable.
and
x-r
HL
IntensIty, qualIty and attenuatIon
The
effects
intensity
of
X-rays
and
the
on
matter
quality
of
depend
the
on
two
ba sIC x-Ray dIspl ay teChnIques
things,
the
X-rays.
The
basic
(for
example
more
•
The
intensity,
I,
is
the
amount
of
energy
per
unit
area
principle
than
other
carried
by
the
The
quality
spread
of
of
potentially
image.
If
the
It
energy
still
be
X-ray
photons
cause
is
the
If
is
attenuated
attenuation
by
will
be
the
absorbed
in
simple
given
the
by
all
from
then
the
beam
in
it
way
spreads
scattering
is
of
to
the
said
an
out.
and
to
are
the
parts
(for
is
that
the
some
X-ray
example
body
beam
skin
and
parts
much
darkens
when
a
beam
white
areas
of
tissue).
X-rays
dominant
ones
for
low-energy
show
up
on
an
is
X-ray
shone
on
picture.
X-ray beam
forming
to
T
wo
as
and
the
be
X-ray
beam,
it
processes
photograph
of
photoelectric
sharpness
of
an
X-ray
image
is
a
measure
of
how
easy
it
is
X-rays.
to
scattering
bones
beam.
X-ray
the
so
the
The
effect
lm
beam.
tissues
contributing
these
absorbed,
the
name
present
remove
nothing
as
matter,
is
is
are
without
to
beam
there
beam
that
harm
desirable
of
attenuated.
will
the
wavelengths
Low-energy
body
imaging
attenuate
X-rays.
them
•
X-ray
will
that
Photographic
is
of
bones)
see
X-ray
photoelectric
eect
and
the
edges
of
beams
the
will
result
contrast
different
be
will
and
organs
scattered
be
to
blur
sharpness.
or
in
the
the
To
different
patient
nal
help
types
being
image
reduce
of
and
this
tissue.
scanned
to
reduce
effect,
a
metal
X-ray
electron
lter
grid
is
added
below
the
patient:
photon
low  X-ray
X-ray beam
light
electron
low  X-ray
photon
photon
photon
Simple
scattering
between
zero
affects
and
30
X-ray
photons
that
have
energies
keV
.
patient
•
In
the
photoelectric
energy
the
to
atom.
cause
It
effect,
one
will
of
result
the
the
in
incoming
inner
one
of
X-ray
electrons
the
to
outer
has
be
enough
ejected
electrons
from
‘falling
metal grid
down’
into
this
energy
level.
As
it
does
so,
it
releases
some
X-ray lm
light
energy.
energies
This
process
between
zero
affects
and
100
X-ray
photons
that
have
keV
.
Alternatively
Both
attenuation
transmission
of
processes
radiation
result
as
in
shown
a
in
near
the
exponential
diagram
enhance
below.
given
energy
of
X-rays
and
given
material
there
will
be
thickness
that
reduces
the
intensity
of
the
X-ray
X-rays
that
This
is
known
as
the
half-value
be
used
to
detect
and
cause
the
ionizations,
intensity
used
they
needs
are
to
dangerous.
be
kept
to
This
an
absolute
by
minimum.
50%.
can
a
means
certain
software
For
Since
a
computer
edges.
This
can
be
done
by
introducing
something
to
thickness
intensify
(to
noissimsnart X
techniques
enhance)
of
the
image.
There
are
two
simple
the
energy
enhancement:
100%
•
When
X-rays
radiated
as
strike
visible
an
intensifying
light.
The
screen
photographic
lm
can
is
re-
absorb
I
x
= e
this
extra
light.
The
overall
effect
is
to
darken
the
image
in
I
0
50%
the
•
In
areas
an
X-rays
image-intensier
screen
and
emitted
half-value
where
produce
from
a
are
still
tube,
light.
present
the
This
X-rays
light
photocathode.
(see
strike
causes
These
page
a
187).
uorescent
electrons
electrons
are
to
be
then
thickness of absorber, x
accelerated
towards
uorescent
screen
an
anode
where
they
strike
another
thickness
The
attenuation
allows
coefcient
us
to
calculate
the
thickness
of
material.
The
µ
is
intensity
a
constant
of
equation
the
is
as
that
X-rays
=
I
give
off
light
to
produce
an
image.
mathematically
given
any
follows:
ma ss attenuatIon CoeffICIent
µx
I
and
An
e
alternative
way
of
writing
the
equation
for
the
attenuation
0
coefcient
The
relationship
between
the
attenuation
coefcient
and
is
shown
below:
the
μ
half-value
thickness
(ρ)
is
I
=
I
ρx
e
0
μ
µ x1
=
ln
2
Where
ρ
is
the
density
of
the
substance.
In
this
format,
is
ρ
2
μ
x1
The
half-value
thickness
The
natural
The
attenuation
of
the
material
(in
m)
known
as
the
mass
known
as
the
area
attenuation
coefcient
,
and
ρx
is
ρ
2
ln
2
log
of
2.
This
is
the
number
density
or
mass
thickness
0.6931
μ
2
Units
of
mass
attenuation
coefcient,
1
µ
coefcient
(in
m
=
m
1
kg
ρ
)
2
Units
µ
depends
on
the
wavelength
of
the
X-rays
–
short
of
area
density,
ρx
=
kg
m
wavelengths
μ
are
highly
are
easily
penetrating
and
these
X-rays
are
hard.
Soft
ρx
x-rays
(ρ)
I
=
I
e
0
attenuated
and
have
long
wavelengths.
o p t i o n
C
–
i m a g i n g
185
x-r ii ci
HL
1)
Intensifying
The
screens
arrangement
page
185
are
3)
of
the
shown
intensifying
screens
described
on
below.
X-rays emerging from patient
Tomography
Tomography
is
a
photograph
focus
the
All
is
patient.
achieved
by
technique
on
a
other
regions
moving
that
certain
the
makes
region
are
source
or
the
blurred
of
X-ray
‘slice’
out
X-rays
through
of
and
focus.
the
This
lm
together.
cassette front (plastic)
motion
X-ray tube
front intensifying screen: phosphor
double-sided lm
about
12 mm
rear intensifying screen: phosphor
pivot point
plane of cut
felt padding
cassette
A
B
With
a
simple
X-ray
photograph
it
is
hard
to
identify
X-ray table
lm
problems
within
soft
tissue,
for
example
in
the
gut.
There
A′
are
two
general
techniques
aimed
at
improving
B′ B′′
A′
B′
this
motion
situation.
2)
Barium
meals
An
In
a
barium
its
progress
between
Typically
meal,
along
the
gut
the
of
barium
of
the
a
the
and
patient
sulfate.
dense
gut
substance
can
be
The
asked
result
to
is
swallowed
monitored.
surrounding
is
is
tissue
swallow
an
is
a
increase
image.
The
and
contrast
increased.
harmless
in
the
solution
sharpness
extension
out
a
pulse
detector.
around
can
in
o p t i o n
C
–
i m a g i n g
of
information
a
of
CT
and
the
X-ray
the
a
tomography
or
X-rays
patient
analyse
terms
basic
scan
about
The
reconstruct
186
of
tomography
a
set
levels
source
and
scan.
the
information
the
this
are
repeated.
recorded
of
and
the
tube
reaching
detectors
is
a
detectors
X-radiation
‘map’
attenuation.
computed
set-up
sensitive
the
process
3-dimensional
X-ray
of
of
and
is
In
is
A
each
then
rotated
computer
able
inside
sends
collects
to
of
the
body
uric ii
HL
ultRa sound
The
is
limit
of
of
a- and b-sC ans
human
higher
hearing
frequency
Typically
ultrasound
the
range.
MHz
The
than
used
is
about
this
in
is
20
known
medical
velocity
of
kHz.
as
Any
that
ultrasound.
imaging
sound
sound
is
through
just
soft
from
within
tissue
There
are
an
A-scan
is
as
a
two
ways
of
ultrasound
presenting
probe,
the
(amplitude-modulated
graph
of
signal
strength
the
information
A-scan
scan)
versus
or
the
presents
time.
the
The
gathered
B-scan.
The
information
B-scan
1
approximately
used
are
Unlike
of
1500
the
X-rays,
m
order
s
of
meaning
a
few
ultrasound
is
that
typical
wavelengths
millimetres.
not
ionizing
(brightness-modulated
the
so
it
can
be
used
brightness
of
a
dot
scan)
of
uses
light
on
the
a
signal
for
imaging
inside
the
body
–
with
pregnant
to
affect
screen.
very
organ
to scan display
safely
strength
women
for
pulse
example.
of
The
emitting
basic
and
principle
receiving
is
to
pulses
use
of
a
probe
that
ultrasound.
is
The
capable
ultrasound
ver tebra
is
reected
The
time
where
at
any
taken
the
boundary
for
these
boundaries
between
reections
must
be
different
allows
us
types
to
of
work
echo
tissue.
out
A-scan display
located.
boundaries
abdominal wall
langis fo
htgnerts
reections from
ver tebra
organ
time
B-scan display
probe
A-scans
organs
path of ultrasound
is
The
acoustic
density,
ρ,
impedance
and
the
of
speed
the
an
a
of
substance
sound,
is
the
product
of
the
several
at
one
which
can
where
known
If
body
image
process
useful
well
required.
of
aCoustIC ImpedanCe
are
is
be
=
a
arrangement
precise
B-scans
time,
all
represent
achieved
are
the
a
taken
lines
section
using
a
of
the
internal
measurement
of
can
the
be
distance
section
assembled
through
large
of
same
the
number
into
body.
of
This
transducers.
c
(i)
Z
the
and
(ii)
ρc
(iii)
placenta
probe
2
unit
Very
of
Z
strong
between
=
kg
m
1
s
reections
two
impedances.
take
substances
This
can
place
that
cause
when
have
some
the
very
boundary
different
is
acoustic
difculties.
foetal
•
In
order
for
the
ultrasound
to
enter
the
body
in
the
limbs
rst
skull
place,
the
there
needs
patient’s
skin.
to
be
An
no
air
air
gap
gap
between
would
cause
the
probe
almost
all
(ii)
and
of
(i)
the
(iii)
ultrasound
to
ultrasound
is
to
•
the
Very
be
achieved
density
dense
reection
reected
of
objects
and
by
tissue)
straight
putting
between
(such
multiple
as
back.
a
gel
the
bones)
images
can
The
or
oil
probe
can
be
transmission
(of
and
similar
the
cause
a
created.
of
density
skin.
strong
These
need
Building
to
be
recognized
and
a
picture
from
a
series
of
B-scan
lines
eliminated.
original
2nd reection reected
reection
by bone back to probe
ChoICe of fRequenCy
The
the
•
choice
choice
Here,
that
of
frequency
between
the
can
resolution
be
of
ultrasound
resolution
imaged.
and
means
Since
the
to
use
can
be
seen
as
attenuation.
size
of
ultrasound
the
is
a
smallest
wave
object
motion,
path of
diffraction
ultrasound
small
effects
object,
we
will
be
must
taking
use
a
place.
small
In
order
wavelength.
to
If
image
this
a
was
organ
the
only
factor
to
be
considered,
the
frequency
chosen
beam reected
would
be
as
large
as
possible.
from bone
•
Unfortunately
attenuation
increases
as
the
frequency
of
2nd reection back
ultrasound
war
is
used,
back.
If
it
increases.
will
this
frequency
all
was
be
the
chosen
If
very
high
absorbed
only
would
and
factor
be
frequency
as
none
to
be
small
will
ultrasound
be
reected
considered,
as
the
possible.
pIezoeleCtRIC CRystals
These
and
They
one
quartz
can
be
also
crystals
used
generate
crystal
is
change
with
used
an
pds
for
shape
when
alternating
when
pd
receiving
generation
and
to
an
electric
generate
sound
current
pressure
detection.
ows
ultrasound.
waves
so
On
balance
between
frequency
is
about
the
the
is
200
frequency
two
chosen
extremes.
often
such
It
that
wavelengths
of
has
turns
the
to
out
part
of
ultrasound
o p t i o n
C
–
be
somewhere
that
the
the
best
body
away
choice
being
from
i m a g i n g
the
of
imaged
probe.
187
Ii ci
HL
Mathematically,
Rel atIve IntensIty le vels of ultRa sound
The
relative
intensity
are
compared
the
decibel
using
levels
the
of
ultrasound
decibel
scale
between
(dB).
As
its
two
points
name
Relative
intensity
level
in
bels,
suggests,
intensity
level
of
ultrasound
at
measurement
point
_____
unit
is
simply
one
tenth
of
a
base
unit
that
is
L
called
=
log
I
intensity
the
bel
(B).
The
decibel
scale
is
level
of
ultrasound
at
reference
point
logarithmic.
I
1
or
Relative
intensity
level
in
bels
=
log
I
0
Since
1
bel
=
10
dB,
I
1
Relative
intensity
level
in
decibels,
L
=
10
log
I
I
0
•
nmR
Nuclear
process
images
or
Magnetic
but
of
one
sections
dangerous
tumours
Resonance
that
in
is
through
techniques.
the
brain.
(NMR)
extremely
It
the
It
is
body
of
is
a
useful.
very
It
without
particular
involves
the
use
provide
any
use
of
a
in
eld
in
conjuction
with
a
large
pulse
of
nuclei
radio
can
resonance.
detailed
invasive
•
detecting
After
by
the
The
pulse,
emitting
outline,
the
uniform
The
nuclei
process
•
The
spin
of
is
of
atoms
as
these
a
property
means
that
called
they
spin.
The
time
over
can
These
The
radio
act
like
tiny
•
The
will
tend
to
line
up
in
a
strong
magnetic
eld.
•
The
to
signal
(protons)
nuclei
a
the
a
spin
return
to
Larmor
process
frequency,
called
transition.
their
lower
energy
state
waves.
which
radio
waves
are
emitted
is
called
the
time
waves
processed
magnets.
•
make
nuclei
at
in
follows:
have
nuclei
applied
energy
eld.
•
•
is
this
protons
the
radio
relaxation
In
waves
absorb
non-uniform
•
magnetic
a
the
complicated
can
If
emitted
produce
analysis
is
and
the
their
NMR
targeted
relaxation
scan
at
times
can
be
image.
the
hydrogen
nuclei
present.
number
of
H
composition
so
different
energy
the
from
nuclei
applied
varies
with
tissues
the
extract
chemical
different
amounts
of
signal.
RF generator
•
Thus
RF
signal
forces
protons
to
make
a
spin
transition
and
eld
S
gradient
◊
coils
The
gradient
eld
allows
determination
of
the
point
from
RF coil
which
receiver
◊
The
body
the
photons
proton
tissue
at
spin
the
are
emitted.
relaxation
point
where
time
the
depends
radiation
is
on
the
type
of
emitted.
N
permanent magnet
relaxation time
oscilloscope
•
They
do
not,
particular
high
•
The
however,
way
that
frequency
particular
magnetic
called
perfectly
called
the
same
frequency
eld
the
–
is
and
the
Larmor
line
up
–
precession.
as
of
the
they
This
frequency
precession
particular
oscillate
happens
of
radio
depends
nucleus
in
at
a
a
very
waves.
on
involved.
the
It
is
frequency
•
CompaRIson between ultRa sound and nmR
The
•
following
NMR
points
imaging
ultrasound
brought
•
as
the
brought
required
NMR
NMR
expensive
a
and
at
quality
a
are
the
of
when
very
machine
patient
but
is
and
image
2-dimensional
–
process
to
of
can
3-dimensional
compared
bulky
easy
point
Ultrasound
reections
noted:
measurements
to
produces
produces
very
be
equipment
Ultrasound
be
•
to
is
can
is
time
perform
care)
rely
scan,
and
on
•
with
patient
needs
to
consuming.
(equipment
can
skill
be
be
of
ultrasound
can
•
wave
•
energies
ultrasound
with
the
radio
At
the
radio
diagnostics
•
Detail
•
NMR
produced
by
NMR
is
greater
than
by
useful
for
very
delicate
areas
of
body
e.g.
NMR
can
188
patients
be
more
have
to
remain
very
still,
ultrasound
dynamic.
o p t i o n
C
–
i m a g i n g
cavitation
will
tissue.
in
ultrasound
cause
avoid
used
absorb
The
this
but
that
used
the
in
and
multiple
energy
energy
associated
associated
NMR.
NMR
the
energy
frequencies
possibility
the
the
there
energy
–
easily
image.
as
can
is
no
cause
production
and
and
can
danger
of
as
small
damage
intensities
much
of
heating.
used
for
possible.
The
strong
magnetic
elds
used
in
NMR
present
and
or
problems
brain.
for
•
energy
greater
body
of
ultrasound.
•
particularly
some
which
surrounding
scan.
is
the
clarity
frequencies
can
bubbles
enter
the
frequencies
but
Ultrasound
not
carry
the
gas
typically
do
reduce
with
resonance
repeated
operator.
Both
waves
can
images
patients
with
surgical
implants
/
pacemakers.
Ib qi – i C – ii
1.
For
each
the
nal
of
diagrams
a)
An
A
and
object
focal
b)
the
following
image
formed.
situations,
Solutions
locate
should
and
describe
found
using
The
scale
of
mathematically.
is
placed
length
diverging
14
7
cm
in
front
of
a
concave
mirror
of
b)
(i)
focal
length
12.0
cm
is
placed
at
point
of
a
converging
lens
of
focal
length
8.0
object
cm.
is
placed
object
focal
16.0
cm
in
front
of
the
converging
the
graph
lens.
plotted
A
b)
6.0
is
an
student
is
given
tube
a)
length
cm
in
order
cm.
to
two
make
a
the
length
She
nds
A
simple
the
cm
18
of
behind
converging
a
a
convex
lens
the
lenses,
A
of
lens
focal
rst
and
by
which
she
can
a
determine
lengths
[2]
to
be
of
lens
A
10
cm
as
Focal
length
of
lens
B
50
cm
a
diagram
arranged
in
the
to
show
tube
in
how
order
should
depth
to
a
layer
of
gel
is
applied
transmitter/receiver
below
the
the
and
the
and
between
the
the
skin.
[2]
pulse
time
strength
lapsed
of
the
between
reected
the
pulse
(ii)
the
for
focal
time
that
the
pulse
is
received,
t
follows:
25
50
75
100 125
150 1
75
200 225 250 275 300
t / µs
the
to
lenses
make
a
should
Indicate
be
telescope.
pulses
on
A,
B
the
diagram
and
C
and
the
origin
of
the
reected
D.
[2]
Your
The
mean
speed
in
tissue
in
this
and
muscle
of
the
include:
each
lens;
points
for
each
lens;
the
position
of
the
eye
when
the
telescope
is
in
use.
On
your
diagram,
mark
the
location
of
the
The
formed
in
the
Is
the
image
seen
through
above
depth
d
of
the
organ
beneath
l
of
the
organ
O.
1.5
×
graph,
the
1
10
ms
estimate
skin
and
.
the
the
scan
is
known
as
[4]
an
A-scan.
State
one
intermediate
tube.
in
which
a
B-scan
differs
from
an
A-scan.
[1]
[1]
d)
d)
the
is
data
above
way
image
from
scan
[4]
c)
c)
used
Using
length
(iii)
pulses
being
3
labels
its
C
ultrasound
(i)
d
nd
D
(iii)
diagram
the
also
B
(ii)
Draw
nd
and
A
0
length
to
[4]
[4]
and
lens.
Focal
is
skin
length
lens.
B,
the
of
telescope.
each
focal
front
convex
cm
method
of
in
second
additional
Describe
focal
18.0
why
against
stinu evitaler
2.
placed
/ htgnerts eslup
3.0
is
scan
below
An
transmitted
An
O
the
is
c)
particular
I
ultrasound
On
focal
this
labelled
Suggest
[4]
of
of
organ
length,
cm.
lens
purpose
the
the
telescope
upright
State
one
advantage
and
one
disadvantage
of
using
or
ultrasound
upside-down?
as
opposed
to
using
X-rays
in
medical
[1]
diagnosis.
e)
Approximately
how
long
must
the
telescope
tube
be?
6.
3.
Explain
what
is
meant
[2]
[1]
a)
State
and
explain
which
imaging
technique
is
normally
by
used
a)
Material
dispersion
b)
Waveguide
c)
Spherical
d)
Chromatic
e)
dispersion
f)
aberrations
g)
A
Cassegrain
Total
mounting
Internal
Step-index
(i)
to
detect
(ii)
to
examine
A
km
km
length
of
[2
optical
bre
has
an
attenuation
each]
a
graph
parallel
through
of
below
beam
a
.
ampliers
A
as
5
mW
signal
represented
is
by
sent
the
along
the
diagram
the
bone
growth
wire
using
two
below.
optical bre
input power
shows
of
the
X-rays
thickness
ytisnetni
15
dB
aberrations
1
4
broken
[2]
of
a
fetus.
[2]
bres
The
4.
a
reection
x
of
variation
after
it
has
of
the
been
intensity
I
of
transmitted
lead.
20
15
output
10
= 5 mW
gain = 20 dB
gain = 30 dB
5
Calculate
0
a)
the
overall
gain
of
b)
the
output
power.
the
system
0
2
4
6
8
10
12
x / mm
[2]
b)
(i)
Dene
half-value
thickness,
x
estimate
x
.
[2]
2
HL
(ii)
Use
the
graph
to
for
this
beam
in
2
lead.
5.
This
question
is
about
ultrasound
(iii)
a)
State
used
a
in
typical
value
medical
for
the
[2]
scanning.
frequency
of
Determine
the
reduce
intensity
scanning.
diagram
below
shows
an
ultrasound
transmitter
placed
in
contact
of
lead
transmitted
required
to
20%
to
of
its
with
the
value.
[2]
and
(iv)
receiver
the
[1]
initial
The
thickness
ultrasound
A
second
metal
has
a
half-value
8
mm.
thickness
x
skin.
2
for
this
radiation
of
Calculate
what
d
thickness
intensity
of
of
this
the
metal
is
required
transmitted
beam
to
by
reduce
80%.
the
[3]
O
ultrasound transmitter
and receiver
layer of skin and fat
l
i B
Q u e s t i o n s
–
o p t i o n
C
–
i m a g i n g
189
16
o p t i o n
D
–
a S t r o p h y S i c S
ojs   vs (1)
Sol ar SyStem
We
live
Each
such
on
the
planet
as
is
dwarf
diameter
/
Earth.
kept
in
This
its
planets
is
one
of
elliptical
like
eight
orbit
Pluto
or
planets
by
the
that
orbit
gravitational
planetoids
also
the
Sun
–
attraction
collectively
between
this
the
system
Sun
and
is
known
the
as
planet.
the
Solar
Other
System.
smaller
masses
exist.
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
4,880
12,104
12,756
6,787
142,800
120,000
51,800
49,500
58
107.5
149.6
228
778
1,427
2,870
4,497
km
8
distance
to
Sun
/
×
10
m
Relative positions of the planets
Jupiter
Venus
Uranus
Mars
Saturn
Ear th
Neptune
Sun
Mercury
Some
Mercury
of
objects
these
planets
orbiting
(including
around
them
the
called
Earth)
have
moons.
Our
other
Moon
small
is
8
3.8
×
10
An
asteroid
m
away
is
a
and
its
small
diameter
rocky
is
body
about
that
1/4
drifts
of
the
around
Earth‘s.
the
Solar
Venus
System.
Jupiter
with
There
–
the
are
many
asteroid
another
planet
orbiting
belt.
is
An
the
Sun
asteroid
known
as
a
between
on
a
Mars
collision
and
course
meteoroid.
Ear th
Small
meteors
atmosphere
Mars
on
Earth.
can
be
vaporized
(‘shooting
The
bits
stars’)
that
due
to
whereas
arrive
are
the
friction
larger
called
ones
with
can
the
land
meteorites
Sun
Comets
are
mixtures
elliptical
orbits
of
around
rock
the
and
Sun.
ice
(a
Their
‘dirty
‘tails’
snowball’)
always
point
in
very
away
Jupiter
from
the
Sun.
Saturn
Vie w from one pl ace on e arth
If
we
look
up
at
the
night
sky
we
see
the
stars
–
many
of
these
Uranus
‘stars’
The
the
are,
stars
in
in
Milky
fact,
our
other
own
galaxies
galaxy
but
appear
they
as
a
are
very
band
far
across
away.
the
sky
–
Way.
Neptune
Patterns
of
the
of
sky
stars
have
have
been
been
identied
labelled
as
the
and
88
different
different
regions
constellations.
Relative sizes of the planets
Stars
in
a
constellation
are
not
necessarily
close
to
one
another.
Over
the
period
of
a
night,
the
constellations
seem
to
rotate
nebul ae
around
In
many
constellations
there
are
diffuse
but
relatively
one
which
are
called
nebulae.
These
are
of
dust,
hydrogen,
helium
and
other
ionized
gases.
is
M42
otherwise
known
as
the
rotation
is
a
result
of
the
of
the
Earth
about
its
own
axis.
top
of
Orion
this
nightly
rotation,
there
is
a
slow
change
in
An
thestars
example
apparent
interstellar
On
clouds
This
large
rotation
structures
star.
and
constellations
that
are
visible
from
one
night
Nebula.
tothe
next.
to
rotation
the
Planetary
This
variation
of
the
systems
over
Earth
have
the
about
been
period
the
of
one
year
is
due
Sun.
discovered
around
many
stars.
Vie w from pl ace to pl ace on e arth
If
of
you
the
move
night
latitude.
same,
190
O p t i O n
D
–
A s t r O p h y s i c s
from
sky
The
but
total
you
place
that
is
to
pattern
will
see
place
visible
of
around
over
the
different
a
the
year
Earth,
changes
constellations
sections
of
the
is
the
section
with
always
pattern.
the
ojs   vs (2)
maximum
During one Day
height
Hemisphere
The
most
stars
of
remains
stars
sky
important
have
have
the
same
been
been
observation
is
that
from
one
night
identied
and
88
labelled
as
the
the
to
pattern
the
next.
different
different
of
the
at
midday.
Sun
is
in
a
At
this
time
southerly
in
the
Northern
direction.
the
Patterns
regions
of
constellations.
the
A
Sun
particular
The
pattern
constellations
They
appear
Hemisphere
It
is
is
to
not
appear
rotate
to
refer
to
in
the
move
around
everything
common
always
over
one
seems
to
same
place,
the
period
direction.
rotate
measurements
to
In
about
the
however.
of
the
the
‘xed
one
night.
Northern
pole
stars’
star.
the
looking south
east
patterns
always
some
of
the
constellations.
appears
stars
rise
to
rotate
above
The
around
the
xed
the
horizon
background
pole
and
star.
some
of
During
stars
set
west
stars
the
night,
beneath
it.
During the ye ar
Every
pole star
to
night,
each
portion
looking nor th
of
night
same
Sun
rises
movement
slow
The
Sun
the
east
is
continued
and
sets
in
the
location
sky
from
change
that
night
returns
continues
to
is
to
in
the
the
to
same
pole
visible
night.
back
rise
have
of
above
Over
exactly
the
star
east
(and
the
the
the
and
relative
the
horizon)
period
same
set
positions
thus
in
of
a
year
position.
the
west,
but
east
during
the
day.
the
the
west,
year
goes
from
winter
into
summer,
the
arc
gets
bigger
The
and
in
the
slightly
this
as
The
constellations
but
changes
west
the
other,
reaching
the
Sun
climbs
higher
in
the
sky.
its
unitS
When
the
comparing
distances
astronomical
unit
on
the
(AU),
astronomical
the
parsec
(pc)
scale,
or
the
it
can
light
be
quite
year
unhelpful
(ly).
See
to
page
remain
193
for
in
the
SI
units.
Possible
denition
of
the
other
rst
units
two
of
include
these.
15
The
light
away.
year
Our
Universe
is
the
galaxy
is
13.7
distance
is
about
billion
travelled
100,000
light
years
by
light
in
light
years
any
in
given
band
observing
of
light
can
constellations.
across
the
the
be
This
night
night
seen
a
(or
became
The
the
are
‘way’)
known
as
(9.5
×
nearest
centre
direction
faint
crossing
‘path’
sky
sky
year
The
10
m).
galaxy
The
is
next
about
a
nearest
million
star
light
to
our
years
Sun
away
is
about
and
4
the
light
years
observable
direction.
The
the milk y way gal ax y
When
one
across.
of
of
galaxy
is
orbiting
our
the
galaxy
rotating
the
lies
constellation
–
centre
all
of
in
the
the
the
a
Sagittarius.
stars
galaxy
result
of
their
attraction.
250
The
million
mutual
period
gravitational
of
orbit
is
about
years.
as
plan view
side view
Sun
the
Milky
seeing
that
are
a
some
make
too
stars.
in
is
Way.
up
far
The
band
What
of
our
away
that
be
that
our
are
of
galaxy
seen
they
of
they
galactic nucleus
rotation
individual
appear
has
to
be
a
Sun
spiral
direction
stars
but
as
galaxy
100 000 light years
actually
millions
own
to
reason
is
the
you
disc
shape.
globular clusters
The
Milky
Way
galaxy
the uniVerSe
26
Stars
are
grouped
together
in
stellar
clusters.
1.5
These
3
can
be
open
containing
10
stars
e.g.
located
in
the
disc
(=
×
15
10
the
m
billion
light
years)
visible
Universe
5
of
is
our
just
galaxy
one
of
or
globular
the
billions
containing
of
stars
in
10
our
stars.
galaxy
Our
Sun
(the
22
5
Milky
Way
galaxy).
The
galaxy
rotates
with
a
period
×
10
local
m
group
of
(=
5
million
light
years)
of
galaxies
8
about
2.5
Beyond
×
our
10
years.
galaxy,
there
are
billions
of
other
galaxies.
21
10
Some
of
them
are
grouped
together
into
clusters
m
or
(=
super
clusters
space
(like
stars)
appears
the
of
galaxies,
gaps
but
between
the
the
vast
majority
planets
or
100,000
light
our
galaxy
our
Solar
years)
of
between
13
Everything
to
be
empty
together
is
–
essentially
known
as
the
a
vacuum.
Universe
10
m
(=
O p t i O n
0.001
D
–
light
years)
System
A s t r O p h y s i c s
191
t   ss
equilibrium
energy flow for StarS
The
stars
energy
this
is
term.
are
is
emitting
the
fusion
referred
The
combustion).
1
4
to
→
0
1
+
2
energy.
into
but
reaction,
reaction
The
helium.
burning’
nuclear
the
of
See
it
not
source
page
this
a
is
for
all
196.
not
chemical
a
The
this
(such
outer
as
stars,
is
mass
in
layers
been
the
It
of
Sun
radiating
might
the
equilibrium
+
core
the
is
be
energy
imagined
should
Sun
stable
a
have
long
this
the
ago.
there
outward
is
past
the
forced
time
because
between
for
that
away
Like
a
4½
powerful
the
other
hydrostatic
pressure
and
the
2ν
1
inward
The
has
years.
reactions
precise
one
Sun
billion
Sometimes
+
e
2
deal
hydrogen
a
Overall
He
great
‘hydrogen
is
4
p
of
as
reaction
a
of
the
products
is
less
than
the
mass
of
the
reactants.
gravitational
force
(see
page
164).
Using
2
ΔE
=
Δm
c
we
9
4
×
10
can
work
out
that
This
takes
place
the
Sun
is
losing
mass
at
a
rate
of
outward radiation
1
kg
s
.
in
the
core
of
a
star.
Eventually
all
this
‘pressure’
26
energy
The
is
radiated
structure
from
inside
a
the
star
surface
does
–
not
approximately
need
to
be
10
known
J
in
every
second.
detail.
convective zone
inward pull
surface
of gravity
radiative
zone
core
A
stable
star
is
in
equilibrium
(nuclear reactions)
binary StarS
A
Both stars are
night 0
Our
Sun
is
a
single
star.
Many
‘stars’
actually
turn
out
to
be
two
moving
(or
more)
stars
in
orbit
around
each
other.
(To
be
precise
to obser ver
B
orbit
around
their
common
centre
of
mass.)
These
are
at 90°
they
called
obser ver
binary
stars.
night 12
Star A is moving
A
centre of mass
towards
obser ver
B
whereas star B is
light from A
moving away
will be blue shifted
obser ver
light from B
B
will be red shifted
binary stars – two stars in orbit around
night 24
their common centre of mass
There
are
different
spectroscopic
1.
A
visual
separate
categories
and
binary
stars
of
binary
star
–
A
visual,
obser ver
eclipsing.
is
one
using
a
that
can
be
distinguished
as
two
3.
An
eclipsing
brightness
telescope.
shows
2.
A
spectroscopic
the
spectrum
wavelengths
An
example
of
light
show
of
star
a
this
is
from
periodic
is
shown
identied
from
the
the
Over
time
‘star’.
shift
or
splitting
in
a
the
periodic
star
light
is
identied
from
variation.
the
An
from
‘star’.
the
Over
example
of
analysis
time
this
the
is
of
the
brightness
shown
below.
analysis
ssenthgirb
of
binary
binary
of
the
frequency.
(below).
wavelength
night 0
time (nights)
The
Each wavelength
night 12
of
explanation
its
are
orbit,
of
one
equal
for
star
the
gets
brightness,
‘dip’
in
in
brightness
front
then
of
this
the
is
that
other.
would
If
cause
as
the
the
a
total
splits into two
brightness
A
B
A
B
A
B
A
to
drop
to
50%.
separate
B
star B
wavelengths.
When one star blocks the light
night 24
coming
from the other star, the
overall brightness is reduced
The
explanation
effect.
As
towards
a
star
is
shifted.
192
a
for
result
the
of
Earth
moving
When
it
the
its
and
shift
orbit,
the
moving
O p t i O n
frequencies
stars
sometimes
towards
is
in
the
D
–
they
Earth,
away
,
are
it
its
involves
sometimes
are
moving
spectrum
will
be
red
the
star A
Doppler
moving
away
.
will
be
shifted.
A s t r O p h y s i c s
When
blue
result
stars
obser ver
S 
principleS of me a Surement
As
you
their
move
relative
objects
from
positions.
appear
Objects
that
one
to
are
position
As
move
very
far
as
when
far
to
mathematicS – unitS
another
you
are
do
not
change
concerned,
compared
away
objects
with
appear
far
to
The
near
for
situation
close
that
stars
is
gives
shown
rise
to
a
change
in
apparent
position
below.
objects.
move
at
distant stars
all.
You
can
demonstrate
this
effect
by
closing
one
eye
and
close star
moving
you
when
This
head
example
compared
distant
can
your
(for
side
tip
with
of
to
side.
your
objects
An
object
nger)
that
are
will
far
that
is
appear
away
(for
near
to
to
move
example
a
building).
apparent
used
galaxy.
from
the
to
All
θ
movement
measure
stars
the
appear
is
known
distance
to
move
as
to
parallax
some
over
the
of
and
the
period
the
stars
of
a
θ
effect
in
our
night,
but
stellar
some
stars
appear
to
move
in
relation
to
other
stars
over
the
distance
period
of
a
year.
d
parallax angle
θ
θ
Sun
Ear th (January)
If carefully obser ved, over the period
Ear th
of a year some stars can appear to move
orbit of Ear th
1AU
(July)
between two ex tremes.
The
reason
moved
position
has
movement
The
for
over
closer
this
the
meant
when
a
apparent
period
star
of
that
a
a
close
compared
is
to
the
movement
year.
This
star
with
Earth,
a
is
will
have
more
the
that
change
the
in
an
distant
greater
will
Earth
has
observing
be
of
parallax
in
star’s
if
apparent
set
The
stars.
a
we
out
know
the
angle,
position
the
can
from
(distance
be
the
distance
distance
the
θ,
over
measured
period
from
the
the
Earth
from
of
a
by
Earth
to
Earth
the
to
observing
year.
to
From
the
star,
the
changes
trigonometry,
Sun,
we
can
work
since
Sun)
__________________________
tan
parallax
Since
all
the
=
(distance
stars
are
very
distant,
this
effect
is
a
very
small
parallax
angle
will
be
very
small.
It
is
usual
to
from
Sun
to
Star)
one
Since
and
θ
shift.
θ
is
a
very
small
angle,
tan
θ
≈
sin
θ
≈
θ
(in
radians)
quote
1
__________________________
This
parallax
angles
not
in
degrees,
but
in
seconds.
An
angle
of
means
that
θ
∝
1
(distance
second
of
arc
('')
is
equal
to
one
sixtieth
of
1
minute
of
arc
In
and
1
minute
of
arc
is
equal
to
one
sixtieth
of
a
other
words,
terms
of
angles,
3600''
=
=
angle
and
Earth
distance
to
star)
away
If
we
use
the
right
units
we
can
are
end
up
inversely
with
a
1°
very
360°
parallax
degree.
proportional.
In
from
(')
1
full
simple
relationship.
The
units
are
dened
as
follows.
circle.
The
distance
from
the
Sun
to
the
Earth
is
dened
to
be
one
11
astronomical
show
that
a
unit
star
(AU).
with
a
It
is
1.5
parallax
e x ample
×
10
angle
of
m.
Calculations
exactly
one
second
16
of
arc
must
be
3.08
×
10
m
away
(3.26
light
years).
This
18
The
star
alpha
Eridani
(Achemar)
is
1.32
×
10
m
away.
distance
Calculate
its
parallax
is
dened
to
be
one
parsec
(pc).
The
name
‘parsec’
angle.
represents
‘parallax
angle
of
one
second’.
18
d
=
1.32
×
10
m
If
distance
=
1
pc,
θ
=
1
second
If
distance
=
2
pc,
θ
=
0.5
18
1.32 × 10
___________
=
pc
second
etc.
16
3.08
×
10
1
_________________________
=
42.9
Or,
pc
distance
in
pc
=
(parallax
angle
in
seconds)
1
_____
parallax
angle
=
1
42.9
d
=
p
=
0.02''
The
that
for
parallax
are
less
stars
measure
use
the
though
method
than
that
at
it
is
The
not
It
is
=
1
used
to
distances
an
SI
SI
measure
parsecs.
common,
standard
strictly
parsecs
be
100
greater
accurately.
unit.
1000
are
can
about
The
becomes
however,
prexes
can
stellar
parallax
too
to
also
distances
angle
small
to
continue
be
used
to
even
unit.
kpc
6
10
parsecs
=
1
Mpc
O p t i O n
D
etc.
–
A s t r O p h y s i c s
193
ls
luminoSity anD apparent brightneSS
The
total
The
SI
power
units
received
radiated
are
by
an
watts.
by
This
observer
is
on
a
star
very
the
is
called
different
Earth.
The
its
to
luminosity
the
(L).
power
It
is
thus
possible
apparent
power
received
for
brightness.
two
It
very
all
different
depends
on
stars
how
to
far
have
away
the
the
same
stars
are.
per
close star
unit
area
is
called
the
apparent
brightness
of
the
star.
The
SI
(small luminosity)
2
units
If
are
two
then
Stars
W
stars
the
m
were
one
are,
.
at
with
the
the
however,
at
same
distance
greater
away
luminosity
different
distances
from
would
from
be
the
the
Earth
distant star
brighter.
Earth.
(high luminosity)
The
2
brightness
is
inversely
proportional
to
the
(distance)
.
Two
stars
have
can
have
different
the
same
apparent
brightness
even
if
they
luminosities
area 4A
area A
alternatiVe unitS
The
x
SI
units
for
luminosity
and
brightness
have
already
x
been
introduced.
brightness
of
In
stars
practice
using
astronomers
the
apparent
often
compare
magnitude
the
scale.
As distance increases, the brightness decreases since
A
magnitude
1
star
is
brighter
than
a
magnitude
3
star.
This
the light is spread over a bigger area.
measure
distance
brightness
x
b
The
of
of
brightness
magnitude
different
account.
scale
stars,
is
sometimes
can
also
provided
Astronomers
be
the
quote
shown
used
to
distance
values
of
on
star
compare
to
the
star
absolute
maps.
the
is
luminosity
taken
into
magnitude
in
b
2x
order
to
compare
luminosities
on
afamiliar
scale.
4
inverse square
b
3x
9
b
4x
16
b
5x
25
and so on
L
_____
apparent
brightness
b
=
2
4πr
e x ample on luminoSity
bl ack-boDy raDiation
3
The
star
Betelgeuse
has
a
parallax
angle
of
7.7
×
10
arc
7
seconds
and
Calculate
its
an
apparent
brightness
of
2.0
×
10
Stars
can
m
luminosity.
.
luminosity
and
to
Betelgeuse
analysed
of
a
star
temperature
Wien’s
Distance
be
as
perfect
emitters,
or
black
bodies.
The
2
W
law
can
is
related
according
be
used
to
to
to
its
the
relate
brightness,
surface
Stefan–Boltzmann
the
wavelength
at
area
law.
which
the
d
intensity
1
is
a
maximum
to
its
temperature.
See
page
90
for
=
p
more
details.
1
__________
=
pc
Example:
3
7.7
×
10
e.g.
=
129.9
pc
=
129.9
×
So
our
the
sun’s
temperature
wavelength
at
is
which
5,800k
the
intensity
3
2.9 × 10
__________
16
3.08
×
10
m
maximum
is
λ
=
=
max
5800
18
=
4.0
×
10
m
2
L
=
194
b
×
4πd
31
=
4.0
×
O p t i O n
10
D
W
–
A s t r O p h y s i c s
500
nm
of
its
radiation
is
at
a
S s
abSorption lineS
The
radiation
from
stars
is
not
a
perfect
continuous
spectrum
–
there
are
particular
wavelengths
that
are
‘missing’.
bands of wavelengths
emitted by the Sun
‘missing’ wavelength
wavelength
violet
red
The
missing
of
number
a
that
the
The
elements.
from
is
the
means
star
star
–
is
that
at
was
Although
we
The
place
have
in
its
are
a
to
it
in
the
seems
the
way
outer
in
in
sensible
spectra
to
atmosphere,
would
still
A
assume
be
this
absent
if
outer
telling
layers
what
of
the
us
to
the
stars
classify
same
class.
give
stars
type
of
out
by
spectrum
Historically
these
its
receding
stars
will
now
know
exist
that
surface
be
be
red
blue
to
the
spectrum.
shifted
Earth
Light
will
show
from
stars
light
from
whereas
a
that
Doppler
are
approaching
shifted
Stefan–bolt zmann l aw
spectra
spectral
are
were
these
relative
in
class.
allocated
just
of
given
to
a
light.
Stars
the
This
that
same
different
allows
emit
a
spectral
letter,
The
Stefan–Boltzmann
black
body.
body
The
different
letters
also
(per
unit
important
law
links
area)
to
correspond
the
the
relationship
but
total
power
temperature
is
radiated
of
the
by
black
that
4
power
radiated
∝
T
=
σ A T
to
In
different
moving
will
Total
we
is
absorption
layers.
different
their
that
in
star.
elements
cl a SSific ation of StarS
Different
star
shift
space.
the
of
absorption
Earth’s
wavelengths
analysed
taking
least
correspond
concerned
incorrect.
absorption
This
the
of
elements
assumption
light
wavelengths
symbols
we
have,
temperatures.
4
Total
The
seven
main
spectral
classes
(in
order
of
power
radiated
decreasing
Where
surface
temperature)
are
O,
B,
A,
F
,
G,
K
and
M.
The
main
σ
spectral
classes
can
be
is
a
constant
called
the
8
σ
Class
Effective
surface
temperature/K
30,000–50,000
=
5.67
×
10
2
W
constant.
4
m
K
Colour
2
A
O
Stefan–Boltzmann
subdivided.
is
the
surface
area
of
the
emitter
(in
m
)
blue
T
is
the
absolute
temperature
of
the
emitter
(in
kelvin)
8
B
10,000–30,000
A
7,500–10,000
blue-white
e.g.
The
radius
of
the
Sun
=
6.96
×
10
m.
2
white
Surface
F
6,000–7,500
yellow-white
G
5,200–6,000
yellow
K
3,700–5,200
orange
If
area
temperature
=
4π r
=
5800
10
=
6.09
×
10
2
m
K
4
then
total
power
radiated
=
σ A T
8
=
5.67
×
10
18
×
6.09
×
10
4
M
2,400–3,700
×
red
(5800
)
26
Spectral
many
classes
text
do
not
need
to
be
mentioned
but
are
used
=
in
books.
The
radius
of
the
star
r
is
linked
to
3.9
its
×
10
surface
W
area,
A,
using
2
the
•
Summary
If
we
the
know
star
•
the
•
the
and
the
distance
work
chemical
surface
to
a
star
we
can
analyse
the
light
from
out:
•
composition
temperature
(by
analysing
(using
a
the
absorption
measurement
of
spectrum)
λ
equation
A
=
the
luminosity
the
distance
the
surface
.
(using
measurements
of
the
brightness
and
away)
area
temperature
4π r
of
and
the
the
star
(using
the
luminosity,
Stefan–Boltzmann
the
surface
law).
and
max
Wien’s
law
–
see
page
90)
O p t i O n
D
–
A s t r O p h y s i c s
195
nsss
Stell ar typeS anD bl ack holeS
The
source
There
are
Type
Red
of
of
energy
however,
for
object
giant
our
other
Sun
types
is
of
the
As
the
the
dwarf
name
stars
As
fusion
the
of
to
hydrogen
known
variables
These
are
hence
are
into
to
means
forced
a
as
that
stars
total
neutrons.
and
when
that
useful
Neutron
of
stars
these
They
place,
light
stars
turn
elements
hot.
luminosity.
very
This
stars
out
these
They
suggests,
taking
give
cool.
some
name
longer
Neutron
of
are
suggests,
comparatively
Cepheid
that
helium.
exist
in
the
This
is
also
true
for
many
other
stars.
Universe.
Description
stars
comparatively
White
fusion
object
it
are
This
stars
white
a
little
a
the
collapse
The
density
the
of
a
in
one
to
is
be
just
mass
the
them
are
neutron
a
star
is
It
is
then
help
of
calculate
is
Since
stages
that
have
the
not
enormous.
the
larger
as
a
a
they
down.
variation
distance
to
stars.
of
of
energy
and
is
in
They
is
will
no
cease
their
brightness
are
–
quite
average
and
rare
but
luminosity.
galaxies.
gravitational
stars
it
are
dwarf
some
The
they
Fusion
Eventually
star.
atoms
neutron
are
white,
stars.
variation
the
they
source
are
mass
brown
of
red,
The
larger.
smaller
regular
composed
are
star.
even
Since
size
mass
Rotating
a
cooling
known
in
are
some
is
they
for
colour.
for
brightness
some
star
in
to
oscillation
of
colour.
stages
observed
an
neutron
white
remnant
period
to
in
possible
supergiants
nal
hot
remnants
of
and
cold.
to
red
later
Red
the
a
They
due
between
use
and
the
size
of
sufciently
post-supernova
and
size
of
hydrogen.
unstable.
can
in
one
small
be
dwarf
link
astronomers
are
be
than
to
thought
is
large
to
are
out
becomes
is
there
other
turn
a
are
out
it
is
have
pressure
essentially
been
has
composed
identied
as
pulsars
Black
holes
Black
the
holes
are
the
gravitational
See
page
general
nuclei
of
of
takes
be
of
how
place
candidates
in
the
SL
do
process
proton
this
by
is
cycle
In
creation
is
as
a
Sun
do
this
known
p–p
need
but
the
come
take
is
an
of
larger
object
mass
whose
stars.
escape
There
velocity
is
is
no
for
two
any
of
these
positively
or
close
place.
helium
enough
reactions
charged
nuclei)
for
Obviously
to
need
will
The
the
to
interactions
they
take
particles
star
1
1
0
1
to
of
stop
light.
stable
the
of
inward
But
how
did
at
high
temperature
size
the
because
radiation
gravitational
is
pull.
the
cloud
of
gas
get
to
be
one
a
in
the
rst
place?
means
that
they
must
be
at
a
high
of
the
mean
an
hence
If
a
large
cloud
of
hydrogen
is
enough,
then
these
can
place
nuclear
cloud
take
comes
gravitational
together,
potential
increase
in
temperature.
kinetic
In
the
energy
loss
must
energy
simple
and
terms
the
hot
molecules
speed
up
as
they
fall
in
reactions
spontaneously.
The
the
centre
to
form
a
proto-star.
power
0
+
e
a
pressure
by
gas
H +
speed
to
repel
temperature.
proton–
2
H →
the
balanced
As
This
cycle
1
H +
mechanism
than
remains
outward
towards
1
step 1
known
greater
another.
to
HL
information.
as
remnant
result
(hydrogen
result
reaction
not
order
place,
of
nucleosynthesis.
overall
candidates,
need
or
the
elements
reactions
recalled
One
for
different
ssion
Details
name
The
150.
main Sequence StarS
The
post-supernova
collapse.
+
1
ν
0
radiated
power
+
by
the
temperature
e
star
released
is
in
is
balanced
these
by
reactions
effectively
Once
the
–
constant.
the
ignition
can
remain
See
page
has
stable
205
for
taken
for
place,
billions
more
the
of
star
years.
details.
p
n
p
p
ν
2
step 2
1
H +
1
3
H →
1
0
He +
2
F
γ
g
0
n
F
g
p
n
γ
p
p
F
g
p
F
g
3
step 3
3
He +
2
4
He →
2
2
p
1
p
p
p
p
p
n
n
p
p
p
the proton
196
proton cycle (p
O p t i O n
D
With sucient KE,
under gravity gives
nuclear reactions
molecular KE
can take place.
1
He + 2
n
n
collapse of cloud
–
p cycle)
A s t r O p h y s i c s
cloud of gas
t h s–rss d
h–r Diagram
The
point
of
classifying
the
Hertzsprung–Russell
the
•
various
diagram.
types
Each
of
dot
stars
on
is
the
to
see
if
diagram
any
patterns
represents
a
exist.
A
useful
different
star.
way
The
of
making
following
this
axes
comparison
are
used
to
is
the
position
dot.
The
vertical
axis
is
the
luminosity
of
the
star
as
compared
with
the
luminosity
of
the
Sun.
It
should
be
noted
that
the
scale
is
logarithmic.
•
The
the
The
horizontal
star
result
axis
a
scale
of
decreasing
temperature.
Once
again,
the
scale
is
not
a
linear
one.
(It
is
also
the
spectral
class
of
OBAFGKM)
of
such
a
plot
is
shown
below.
6
10
4
10
L/ytisonimul
2
10
our Sun
0
10
2
10
4
10
right.
sequence
helium.
The
0
bottom
temperature/K
3
to
main
to
0
hydrogen
left
a
0
4
is
0
5
top
Sun
0
fusing
6
from
Our
0
0
goes
stars.
0
7
are
5
They
0
mass.
0
(roughly)
sequence
0
their
1
that
main
0
to
This
star.
stars
line
These
that
is
known
stars
are
not
are
on
the
main
sequence
‘normal’
stable
stars
the
as
main
sequence
–
the
can
and
only
also
be
categories.
the
lines
is
line
0
are
them
a
as
0
addition
These
into
on
known
0
between
put
fall
2
In
stars
are
5
broadly
it
0
difference
of
on
0
5
are
0
number
that
0
0
large
stars
0
A
surface
broad
going
regions,
from
top
lines
left
to
of
constant
bottom
radius
can
be
added
to
show
the
size
of
stars
in
comparison
to
our
Sun’s
radius.
right.
1
0
1
0 2
s
o
la
r
3
s
o
la
r
r
a
d
ii
ra
d
ii
red giants
instability
strip
main
Sun
sequence
1
0
2
s
o
la
r
ra
d
iu
s
white
dwarfs
0
0
0
0
0
0
0
0
0
0
0
0
eective
3
6
1
0
5
0
temperature/K
ma SS -luminoSity rel ation for main Sequence StarS
For
stars
the
main
on
the
main
sequence
sequence,
(i.e.
higher
there
up)
are
is
a
correlation
more
massive
between
and
the
the
star's
mass,
relationship
M,
and
its
luminosity,
L.
Stars
that
are
brighter
on
is:
3.5
L
∝
M
O p t i O n
D
–
A s t r O p h y s i c s
197
cd vs
principleS
Very
small
satellite
these
100
mathematicS
parallax
angles
observations
measurement
kpc
observe
know
away.
the
the
The
light
can
(e.g.
are
Gaia
difference
a
measured
mission)
limited
essential
from
be
to
stars
difculty
very
distant
between
a
but
that
is
The
even
are
that
star,
bright
using
away
and
principal
of
dimmer
problem
astronomical
When
we
that
away
this
of
then
stars
a
luminosity.
‘standard
to
other
do
source
•
Locate
•
Measure
What
make
their
that
we
in
all
the
a
of
that
variable
need
In
is
other
a
star
provide
a
•
Use
•
the
Use
light
we
the
had
other
words
of
Cepheid
variable
layers
contraction
its
the
variable
variation
in
in
the
a
galaxy
as
(in
which
the
follows:
galaxy.
brightness
luminosity–period
over
a
given
period
of
time.
relationship
for
Cepheids
to
estimate
we
known
such
star
is
undergo
quite
and
this
a
a
rare
periodic
a
type
of
the
average
law
to
luminosity,
estimate
the
the
average
distance
to
brightness
the
and
the
inverse
star.
star.
produces
a
compression
periodic
1
2
3
4
5
6
7
8
9
10
11
time/days
and
variation
Variation
in
of
apparent
magnitude
for
a
particular
Cepheid
variable
luminosity.
A Cepheid variable star undergoes
NUS
periodic compressions and
L/ytisonimul kaep
contractions.
increased
luminosity
6
10
5
10
4
10
3
10
2
10
10
lower
1
luminosity
1
General
These
stars
period
be
of
this
related
Cepheid.
are
to
by
to
variation
the
Thus
calculated
useful
astronomers
in
average
the
luminosity
absolute
luminosity
observing
of
because
turns
a
Cepheid
the
variations
has
a
in
2
5
luminosity–period
10
20
graph
the
out
magnitude
to
of
can
the
be
brightness.
e x ample
11
A
Cepheid
variable
star
period
of
10.0
days
and
apparent
peak
brightness
of
6.34
×
10
luminosity
Using
the
of
the
Sun
is
luminosity–period
3.8
×
graph
10
W
.
Calculate
the
distance
peak
luminosity
=
10
L
=
5012
×
3.8
×
sun
L
=
b
×
4πr
____
L
____
r
=
√
4πb
_________________
30
1.90 × 10
____________________
=
√
11
4
×
π
×
6.34
×
10
19
=
4.88
×
10
m
19
4.88 × 10
___________
=
pc
16
3.08
=
198
1590
×
10
pc
O p t i O n
D
–
Cepheid
26
×
2
∴
the
(above)
3.7
⇒
to
A s t r O p h y s i c s
10
30
=
1.90
×
10
W
variable
in
2
W
26
The
the
luminosity.
O
outer
to
be
candle’.
Cepheid
Its
might
stars
If
with
distance
the
distance
galaxy.
is
stars
the
the
imaged)
is
is
square
same
really
the
luminosities.
–
This
determination
comparisons
candle’
Cepheid
closer.
be
not
galaxies.
galaxy,
luminosity
‘standard
is
experimental
another
could
judge
that
approximately
Earth.
known
we
and
need
are
the
the
distances
observe
galaxy
from
source
in
source
can
ssenthgirb tnerappa
in
A
a
estimating
we
average
far
of
stars
about
when
we
process
individual
pc.
m
50
100
period / days
rd  ss
If
af ter the main Sequence
The
the
mass–luminosity
amount
fuel
is
used.
of
time
relation
different
Consider
a
star
(page
mass
that
is
197)
stars
10
can
take
times
be
used
before
more
to
the
can
hydrogen
massive
than
it
has
higher
compare
sufcient
and
mass,
higher
a
red
elements
giant
and
can
the
continue
process
of
to
fuse
nucleosynthesis
continue.
our
newly formed red giant star
3.5
Sun.
=
This
3,162
means
times
that
more
this
luminosity
star
effectively
is
the
luminosity
luminous
the
mass
of
that
of
the
our
larger
Sun.
hydrogen
in
star
Since
the
star,
will
the
be
(10)
source
then
the
of
dormant hydrogen-
400 million km
larger
burning shell
more
than
has
3000
10
times
times
the
more
rate.
‘fuel’
The
but
more
is
using
massive
the
star
fuel
will
at
nish
helium-burning
1
___
its
fuel
in
of
the
time.
A
star
that
has
more
mass
exists
for
a
shell
300
shorter
A
star
amount
cannot
of
time.
continue
in
its
main
sequence
state
forever.
It
is
carbon–oxygen core
fusing
hydrogen
into
helium
and
at
some
point
hydrogen
in
core of star
the
core
less
will
often.
and
the
become
This
rare.
means
gravitational
that
force
The
the
fusion
star
will,
is
once
reactions
no
longer
again,
will
in
cause
happen
nucleosynthesis
equilibrium
the
core
to
collapse.
This
collapse
increases
the
temperature
of
the
core
still
old, high-mass red giant star
further
the
and
star
that
the
to
helium
increase
outer
fusion
is
now
massively
layers
are
in
cooler.
possible.
size
It
–
this
becomes
The
net
result
expansion
a
red
is
for
hydrogen-burning shell
means
giant
700 million km
helium-burning shell
star.
carbon-burning shell
red
giant star
neon-burning shell
oxygen-burning shell
silicon-burning shell
iron core
star runs out
helium fusion possible
of hydrogen
due to increased
∴ collapses
temperature
fur ther
∴
This
end
of
process
with
the
other
of
the
greatest
words
fusion
as
a
source
nucleosynthesis
binding
the
of
energies
fusion
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iron
of
energy
iron.
per
to
The
must
iron
nucleon
form
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come
nucleus
of
higher
all
to
nuclei.
mass
an
has
one
In
nucleus
expansion
would
star
on
need
cannot
the
to
take
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continue
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to
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shine.
What
than
release
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energy.
next
is
The
outlined
page.
O p t i O n
D
–
A s t r O p h y s i c s
199
S v
poSSible fateS for a Star (af ter reD giant
h – r Diagram interpretation
All
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stars
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have
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pha SeS)
described
Page
199
showed
that
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red
giant
phase
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star
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essentially
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nal
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mistake
star
the
red
giant
star
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with
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sequence.
collapse
taking
of
the
place
gravitational
remnant.
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forces
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–
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examinations
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imply
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Chandrasekhar
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times
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electron
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be
of
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Below
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and
ytisonimul
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nebula
than
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In
and
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so
it
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ultimately
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n
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e
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e
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e
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ytisonimul
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red giant
phase
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star
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4
and
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masses,
Solar
electron
collapse.
supernova.
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In
its
remnant
masses.
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degeneracy
this
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becomes
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m
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pressure
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s
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energy
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neutron
star
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expected
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black hole
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40
their
It
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that
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and
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years
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mass
a
Volkoff
limit
limit
will
200
and
neutron
form
masses
star
and
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is
black
involved
have
2–3
is
Solar
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the
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above
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O p t i O n
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largest
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Remnants
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process
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black
holes.
that
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rely
energy
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but
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theoretical
existence
radiated
the
whole
temperatures
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of
these
super-massive
stars
W!).
They
models
must
10
stars
energy
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t b b d
e xpanSion of the uniVerSe
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us
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If
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Any galaxy would see all the other galaxies
Note
moving away from it.
explain
way
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to
think
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Universe
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sheet
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distance.
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galaxies
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If
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this
how
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masses
in
attraction
the
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model
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it
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expansion
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to
best
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way
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moves
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good
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be
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B (at rest)
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of
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Universe
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to
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The
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itself)
time
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Bang.
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egnar htgnelevaw tinu rep ytisnetnI
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Universe
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details).
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for
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210
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temperature
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peak wavelength ~1.1 mm (microwaves)
individual data
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theoretical
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theoretical
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wavelength
The
O p t i O n
D
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A s t r O p h y s i c s
201
g 
DiStributionS of gal axieS
Galaxies
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The
Virgo
in
region
a
are
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in
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grouped
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into
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cluster
million
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(50
light
huge
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example,
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million
million
years
together
throughout
region
light
light
across.
superclusters
arranged
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in
of
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away
even
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joined
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us)
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As
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galaxies
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mean
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at
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the
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Earth,
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speed
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wavelength
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_______________
___
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e xperimental obSerVationS
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galaxy
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distance
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it
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constant
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distance
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from
Earth.
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Hubble’s
law.
speed
has
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elapsed
beginning
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calculated
Universe,
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the
time
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from
1
10 000
s mk / yticolev lanoissecer
distance
________
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=
speed
8 000
x
____
=
H
x
0
6 000
1
___
=
H
4 000
0
This
is
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limit
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the
age
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the
Universe.
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all
galaxies
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predicts
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the
time.
0
0
20
40
60
80
100
120
140
size of
R
distance / Mpc
obser vable
Mathematically
v
∝
v
=
this
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universe
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d
or
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d
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1
T ≈
where
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Hubble
constant.
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H
0
0
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the
data
mean
that
the
value
of
H
is
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0
known
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precision.
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1
constant
are
s
SI
units
1
,
but
the
unit
of
km
of
the
Hubble
1
s
Mpc
is
often
used.
time
now
1
H
0
the coSmic Sc ale factor (R)
v
Δλ
___
Page
202
shows
how
the
Doppler
red
shift
equation,
=
≈
,
can
be
used
to
calculate
the
recessional
velocity,
v,
of
certain
c
λ
0
galaxies.
the
than
1.0.
quantity
As
This
speed
of
equation
light,
This
introduced
object
the
on
means
10
stretched
over
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cosmic
was
emitted
used
when
plenty
their
speed
of
scale
factor
(R).
page
time
10
be
however
that
a
years
scale
only
are
cosmic
that
million
can
There
implies
called
Universe
c.
201,
the
ago,
and
this
factor,
million
will
R,
be
can
is
a
years
the
and
considered
way
ago
of
undertaken
stretched
be
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≪
as
or
in
is
at
some
be
λ
words,
night
than
is
best
time
in
recorded
rescaling
the
wavelength
the
greater
will
a
other
in
Universe
quantifying
with
c
objects
recession
expansion
measurement
v
of
of
the
the
as
the
recessional
(e.g.
speed
pictured
the
expansion
when
the
sky
of
as
a
larger
Universe
that
scale
has
light.
the
distant
for
value
(the
was
place.
R
by
Δλ
to
a
larger
value
λ
(λ
=
λ
+
Δλ).
,
of
space
the
getting
the
to
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small
observed
it
itself.
is
now.
The
All
in
red
comparison
shift,
helpful
to
of
light
z,
is
of
a
the
emitted
measurement
to
greater
consider
expansion
wavelength
measured
In
the
has
the
situations
example
when
v,
which
these
Universe
0
increased
In
for
expansion
past,
taken
factor
velocity,
quasars)
will
by
an
be
bigger).
above
wavelength
example,
if
measured
the
wavelength
today
would
have
0
This
is
because
the
cosmic
scale
factor
has
increased
by
ΔR
(to
the
larger
value
0
λ
R
R
=
R
+
ΔR).
All
measurements
will
have
increased
by
the
ratio,
0
.
The
ratio
of
the
measured
wavelengths,
R
,
is
equal
to
the
λ
0
0
R
ratio
of
the
cosmic
scale
factors,
,
so
the
red
shift
ratio,
z
is
given
by:
R
0
λ
Δλ
____
z
=
λ
λ
___
0
_____
=
λ
=
1
=
0
0
0
1
R
λ
λ
0
R
___
R
___
or
z
=
-
1
R
0
R
So
a
measured
red
shift
of
4
means
that
=
5.
If
we
consider
R
to
be
the
present
‘size’
of
the
observable
Universe,
then
the
light
R
0
must
have
been
emitted
when
the
Universe
was
one
fth
of
its
current
size.
O p t i O n
D
–
A s t r O p h y s i c s
203
t  vs
SupernoVae anD the accelerating uniVerSe
Supernovae
(the
last
one
observed.
over
a
An
light
whereas
aspects
Type
be
Ia
is
of
type
at
which
Ia
a
no
and
can
its
be
single
sees
months.
into
light
in
a
a
I
a
in
1604)
increase
different
II
development
the
in
of
exceeds
type
the
but
amounts
often
two
occur
in
rapid
Huge
from
type
can
place
supernova
The
hydrogen
that
took
Earth
or
categorized
many
main
types
There
stars
(see
further
the
stars
stars
the
are
or
page
indicates
are
some
of
(hence
energy
local
of
number
brightness
radiated
supernova
supernova.
large
in
word
emitted
for
subdivisions
of
of
a
200).
=
new
short
Supernovae
means
star)
period
of
that
which
time
are
many
then
and,
rare
have
events
been
diminishes
at
its
peak,
the
galaxies.
more
presence
page
Universe
‘nova’
in
individual
207
(see
the
details)
according
hydrogen
these
(from
types
(Ia,
to
the
Ib,
a
spectral
absorption
etc.)
based
analysis
spectra)
on
different
spectrum.
are
explosions
and
apparent
used
with
Universe
gravitational
supernovae
it
explosions
galaxy
the
emit.
accurately
expanding
of
on
been
light
our
weeks
of
they
supernovae
technique
result
some
have
the
supernova
a
of
there
of
in
observer
that
predicted
The
catastrophic
occur
brightness
Supernovae
the
to
period
apparent
of
are
increases
involving
brightness
galaxies
(which
is
as
up
larger
to
dwarf
be
in
with
a
provided
time
the
expect
as
galaxy,
1,000
Big
Bang
rate
at
evidence
In
When
other
these
‘standard
given
the
strong
passes.
stars.
used
approximately
might
as
can
observed
consistent
we
however,
getting
white
supernovae
attraction,
has,
is
these
a
Mpc
distance
model)
that
take
By
place,
the
comparing
measurement
amount
the
to
known
that
of
energy
released
luminosity
galaxy
can
be
of
a
can
type
calcuated.
Ia
This
away.
which
means
R
not
words
events
candles’.
the
that
increases
only
is
that
to
the
expansion
be
the
cosmic
of
cosmic
slowing
the
scale
scale
down.
factor,
Universe
is
factor,
Analysis
R,
R,
of
is
a
increasing
accelerating.
increasing.
large
but
The
As
number
the
rate
evidence
2
from
type
Ia
supernovae
identies
this
effect
from
a
time
when
the
universe
was
approximately
of
its
current
size.
Note
that
this
3
acceleration
The
different
mechanisms
counteract
(see
is
page
the
that
to
the
cause
inward
very
an
rapid
period
accelerating
gravitational
pull.
of
expansion
Universe
There
must
are
not
also
be
of
the
fully
a
early
Universe
understood
source
of
but
energy
which
must
which
is
called
involve
has
been
an
ination.
outward
given
the
accelerating
name
212).
dark energy
accelerated expansion
development of
galaxies
dark ages
ination
st
1
stars
13.7 billion years
Source:
204
NASA/WMAP
Science
O p t i O n
D
Team
–
A s t r O p h y s i c s
dark
force
energy
to
n s –  Js 
HL
the Je anS criterion
As
seen
on
hydrogen,
in
stable
(e.g.
a
can
such
with
as
time,
a
the
thought
stars
and
equilibrium
in
be
196,
helium
collision
incident
point
page
nucle ar fuSion
form
other
for
many
another
total
as
a
of
interstellar
materials.
years
cloud
supernova)
of
out
energy
until
or
starts
Such
the
can
external
inuence
collapse.
associated
combination
clouds
an
the
clouds
with
the
A
exist
gas
is
another
any
star
the
the
given
cloud
The
negative
gravitational
mass
process
energy,
E
,
main
sequence
nuclei.
proton–proton
stars
alternative
of:
potential
the
helium
predominant
small
In
•
on
produce
event
of
At
of
this
(up
place
reaction,
chain
method
process,
takes
One
to
called
at
as
for
just
fusing
the
hydrogen
by
outlined
nuclear
above
higher
carbon,
is
process
the
CNO
which
on
page
fusion
mass
to
of
is
196.
take
our
to
achieved
This
place
Sun).
is
in
An
(carbon–nitrogen–oxygen)
temperatures
nitrogen
nuclei
this
and
in
larger
oxygen
are
mass
used
stars.
as
which
P
catalysts
the
cloud
possesses
as
a
result
of
its
mass
and
how
it
and
the
in
space.
density
of
Important
the
factors
are
thus
the
to
aid
the
fusion
of
protons
into
cycle
is
shown
The
positive
random
cloud.
kinetic
START
4
energy,
E
,
that
the
particles
1
H
H
the
cloud
possess.
temperature
The
cloud
will
of
the
An
important
factor
is
thus
One
1
K
in
nuclei.
below:
mass
He
•
helium
is
possible
distributed
γ
the
cloud.
remain
gravitationally
bound
together
if
12
C
E
+
E
P
that
is
<
zero.
Using
this
information
allows
us
to
15
predict
K
13
N
the
collapse
greater
than
of
an
interstellar
a
certain
a
given
critical
cloud
mass,
may
M
.
begin
This
is
if
the
its
Jeans
ν
J
criterion.
For
cloud
of
gas,
M
N
mass
is
dependent
on
the
J
cloud’s
to
density
collapse
if
and
it
temperature
and
the
cloud
is
more
likely
has:
15
13
O
C
•
large
mass
14
N
•
small
•
low
size
temperature.
γ
γ
In
symbols,
the
Jeans
criterion
is
that
collapse
can
start
1
H
1
if
H
M
>
M
J
γ
proton
ν
neutron
gamma ray
neutrino
positron
time Spent on the main Sequence
For
so
long
as
hydrostatic
a
star
remains
equilibrium
(see
on
the
page
main
192)
sequence,
and
have
a
hydrogen
constant
‘burning’
luminosity
is
L.
the
A
source
star
that
of
energy
exists
on
that
the
allows
main
the
star
sequence
to
for
remain
a
time
in
T
MS
must
E
in
=
total
L
×
radiate
an
energy
E
given
by:
T
MS
This
star
energy
M
has
release
been
comes
converted
from
into
the
nuclear
energy
synthesis
according
to
that
has
Einstein’s
taken
place
famous
over
its
lifetime.
A
certain
fraction
f
of
the
mass
of
the
relationship:
2
E
=
f
×
Mc
L
×
T
2
∴
=
f
×
Mc
MS
2
f
×
Mc
_______
T
=
MS
L
3.5
But
the
mass–luminosity
relationship
applies,
L
∝
M
M
____
∴
T
∝
MS
3.5
M
2.5
∴
T
∝
M
MS
Thus
the
higher
the
mass
of
a
star,
the
shorter
the
lifetime
that
it
spends
on
the
main
2.5
Time
on
main
sequence
for
star
A
Mass
____________________________
on
main
sequence
for
star
of
star
A
Mass
____________
=
Time
B
(
Mass
of
star
B
sequence
2.5
of
star
B
____________
=
)
(
Mass
of
star
A
)
10
For
example
times
its
our
mass
be
Sun
is
expected
expected
to
to
have
a
main
sequence
lifetime
of
approximate
10
years.
How
long
would
a
star
with
100
last?
2.5
1
____
10
Time
on
MS
for
100
solar
mass
star
=
10
×
(
100
)
5
=
10
years
O p t i O n
D
–
A s t r O p h y s i c s
205
nsss    s
HL
nucleoSyntheSiS off the main Sequence
For
so
long
‘burning’
emitting
more
as
is
star
remains
source
energy
helium
helium
a
the
of
whilst
exists
(helium
in
on
the
energy
remaining
the
core.
‘burning’)
in
A
does
main
that
sequence,
allows
a
the
stable
nuclear
release
state.
to
continue
More
synthesis
energy
hydrogen
star
the
high
that
and
the
involve
very
high
temperature
massive
and
involving
(since
In
nuclei
the
can
mass
of
stars,
the
core
continue
release
of
to
energy.
gravitational
can
be
produced.
Typical
binding
of
neon:
nucleon
of
the
products
is
greater
than
that
of
Production
but
can
only
take
place
at
high
+
high
mass
stars,
the
helium
burning
of
magnesium:
can
+
and
spread
throughout
the
core
whereas
begin
in
Production
stars
this
process
starts
suddenly.
Whatever
the
mass
star,
a
new
equilibrium
of
oxygen:
state
is
created:
the
red
→
C
page
common
nuclear
process
reactions
by
C
→
addition
giant
if
the
or
burning
temperatures
can
the
helium
triple
is
converted
alpha
is
process
a
in
are
high
series
enough,
Two
neon
γ
→
helium
nuclei
ray),
+
He
2
4
+
of
24
He
→
Mg
2
16
fuse
releasing
sulfur:
O
+
+
γ
12
16
32
O
→
S
8
+
γ
16
into
a
beryllium
nucleus
(and
reactions
4
He
+
and
→
Be
2
are
and
also
other
heavy
produced.
nuclei
Some
of
such
as
these
+
nuclear
reactions
also
produce
neutrons,
which
can
γ
4
easily
beryllium
produce
possible
phosphorus
8
He
2
are
a
energy.
alternative
4
to
4
O
8
10
silicon
The
and
produced.
gamma
2.
He
2
16
+
Ne
of
which
Many
1.
2
occur:
8
is
+
red
Production
carbon
4
O
8
200).
which
called
γ
16
6
20
A
+
12
+
Ne
(see
He
2
Mg
6
12
10
phase
+
24
20
supergiant
4
Ne
of
oxygen
the
all
small
In
mass
→
10
C
6
6
gradually
reactions
20
C
12
C
temperatures.
process
means
more
include:
6
12
For
These
and
the
12
reactants)
rise
12
C
6
per
to
reactions
12
Production
energy
contraction
continue
a
nucleus
carbon
fuses
with
nucleus
another
(and
a
helium
gamma
ray),
be
captured
by
other
nuclei
to
form
new
isotopes.
This
nucleus
process
of
In
high
neutron
capture
is
explored
further
below.
releasing
very
mass
stars,
silicon
burning
can
also
take
place
energy.
56
which
results
in
the
formation
of
iron,
Fe.
As
explained
on
26
8
4
Be
+
3.
→
C
2
Some
can
12
He
4
of
go
the
on
oxygen.
fuse
+
produced
with
this
4
C
in
another
process
the
triple
helium
releases
alpha
nucleus
nucleon
process
to
199,
produce
in
a
and
fusion
acquired,
energy:
iron
has
one
of
represents
process
but
the
the
the
that
highest
largest
releases
reactions
binding
nucleus
energy.
require
an
energies
that
can
Heavier
energy
per
be
nuclei
created
can
be
input.
16
He
6
γ
page
carbon
to
Again
12
+
6
→
O
2
+
γ
8
nucle ar SyntheSiS of he aV y elementS – neutron c apture
Many
of
involve
any
are
the
the
charge,
present
reactions
release
it
is
in
of
easy
the
that
take
place
neutrons.
for
star.
them
When
to
a
in
Since
the
core
neutrons
interact
nucleus
with
of
are
stars
other
captures
a
also
nuclei
that
neutron,
s-process
giant
without
than
nucleus
is
said
to
be
neutron
rich.
Given
helium
most
of
these
neutron-rich
nuclei
would
alternative
undergo
In
emitting
1
this
an
process,
electron
1
n
→
0
+
X
+
changes
into
a
vast
β
A
n
+
+
206
process,
takes
place
numbers
rapid
when
that
neutron-rich
nuclei
neutrons
captured.
to
A
X
+
1
are
be
created.
→
Z
Y
Z
+
catastrophic
0
+
1
able
to
be
of
a
red
heavier
created.
neutron
the
capture
neutrons
are
or
the
present
there
is
not
sufcient
time
for
in
the
to
undergo
The
beta
result
is
decay
for
before
very
several
heavy
more
nuclei
Typically
β
+
known
as
slow
result
of
the
neutron
s-process
O p t i O n
D
explosion
the
that
r-process
is
a
takes
supernova.
place
during
Elements
the
that
are
v
1
capture
or
the
s-process.
is
–
a
new
element.
Typically
A s t r O p h y s i c s
than
iron,
such
as
uranium
and
thorium,
can
only
be
The
created
overall
are
are
proton,
v
1
→
0
is
iron
stage
that
antineutrino:
heavier
This
than
burning
elements
1
1
Z
neutron
an
lighter
helium
that
0
p
1
A
the
and
the
means
beta
such
decay.
but
during
this
enough
r-process,
time,
place
Typically
the
The
resulting
takes
star.
the
in
this
way
at
very
high
temperatures
and
densities.
ts  sv
HL
SupernoVae
Supernovae
on
their
are
light
among
curves
–
the
a
most
plot
gigantic
of
how
explosions
their
in
brightness
the
Universe
varies
with
(see
time
page
and
a
200).
The
spectral
two
categories
analysis
of
the
of
light
supernova
that
they
are
emit.
based
Type
I
10
supernovae
quickly
reach
a
maximum
brightness
(and
an
equivalent
luminosity
of
10
Suns)
which
then
gradually
decreases
over
9
time.
Type
II
supernovae
often
have
lower
peak
luminosities
(equivalent
to,
say,
10
Type II
Type I
ytisonimuL
ytisonimuL
0
100
200
300
0
100
days after maximum brightness
Supernovae
the
types
elements
•
Type
Ia
•
Type
Ib
•
Type
Ic
All
Suns).
type
II
presence,
The
are
distinguish
identied
and
the
by
analysis
different
of
the
presence
of
singly
shows
the
presence
of
non-ionized
not
show
supernovae
or
reasons
not,
for
of
the
show
the
different
these
presence
Supernova
Does
Cause
White
Context
Binary
ionized
of
300
400
light
(Ia,
spectra.
Ib
and
All
Ic)
type
are
I
supernovae
based
on
a
do
more
not
include
detailed
the
spectral
hydrogen
spectrum
in
analysis:
silicon.
helium.
helium.
presence
of
hydrogen.
The
different
subdivisions
(IIP
,
IIL,
IIn
and
IIb)
again
depend
on
the
elements.
differences
Spectra
their
subdivisions
shows
does
200
days after maximum brightness
not
are
the
Type
show
different
mechanisms
that
are
taking
Ia
place:
Supernova
hydrogen
but
does
show
singly
ionized
Shows
Type
II
hydrogen.
silicon.
dwarf
star
orbiting
exploding.
system
each
with
white
dwarf
and
red
giant
other.
Large
mass
Large
star
of
its
The
gravity
material
mass
of
eld
from
the
of
the
white
the
red
white
giant
dwarf
star,
star
thus
attracts
increasing
When
the
dwarf.
the
cannot
The
The
extra
mass
total
mass
limit
(1.4
of
Solar
degeneracy
the
gained
the
release
matter
of
(up
to
is
the
for
no
collapse.
iron)
energy
being
by
beyond
masses)
pressure
gravitational
elements
star
a
distributed
white
the
star
takes
dwarf.
the
fusion
the
to
to
of
heavier
resulting
explode
throughout
halt
sudden
with
the
star
any
and
material
moving
Solar
out
of
masses)
elements
fuel,
further
under
degeneracy
degeneracy
be
runs
gravitational
stable
8
lighter
at
up
the
to
end
the
its
the
energy
own
iron
by
centre
nuclear
gravity
core
fusion.
forming
a
star.
Electron
the
Electron
sufcient
Nuclear
and
dwarf
Chandrasekhar
longer
starts
causes
white
the
collapsing.
iron.
collapses
neutron
Explosion
of
star
than
fusing
release
star
giant
(greater
lifetime,
production
Process
red
pressure
rigid
pressure
collapse
is
and
neutron
bounces
off
outwards.
the
This
of
is
the
the
star.
core
not
sufcient
core,
core
The
but
becomes
rest
creating
causes
all
of
of
a
the
to
halt
neutron
the
a
infalling
shock
outer
wave
layers
to
ejected.
space.
O p t i O n
D
–
A s t r O p h y s i c s
207
t s  d  ds
HL
section
the coSmologic al principle
The
cosmological
structure
The
two
of
An
isotropic
large
–
but
upon
are
scale
that
a
pair
which
the
of
assumptions
current
Universe,
structures
in
the
about
models
are
providing
Universe,
is
the
based.
one
only
isotropic
statement
the
Earth.
in
an
the
this
is
one
the
of
galaxies
in
not
Earth,
all
same
this
to
see
other.
of
the
the
same
we
on
will
the
universe.
structures
the
in
validity
used
be
that
homogeneous
expansion
universe
is
one
where
the
local
galaxies
and
galaxy
clusters
that
exists
in
one
region
of
the
turns
universe.
out
to
be
Provided
the
one
same
is
distribution
considering
a
in
all
regions
stars
mass.
in
a
galaxy
Different
rotate
models
around
can
be
their
used
to
reasonably
common
predict
based
if
the
the
of
the
develop
Hubble’s
Universe
and
now
Big
He
in
large
to
scale
question
of
of
by
general
existence
that
model
this
galaxies
is
observational
the
model
cosmological
principle
agree
Bang
a
did
between
equations
cosmological
the
very
static.
attraction
physicists
around
space
look
of
the
of
an
correct,
the
discovery
CMB
Universe
is
expanding
cosmological
for
the
principle
future
of
the
is
also
linked
Universe
to
(see
three
page
211).
large
The
centre
how
The
models
of
rotation curVeS – mathematic al moDelS
The
of
to
was
yet-to-be-discovered
analysis
to
hundreds
of
astrophysicists
principle
Universe
non-static.
many
choose
apparently
some
several
the
possible
universe
that
we
to
volume
distribution
universe.
of
a
that
principle.
gravitational
by
of
equal
in
wherever
cause
the
that,
be
radius
discoveries
Subsequent
meant
of
galaxies
cosmological
the
must
the
same
which
Universe
of
cosmological
balanced
non-static
A
the
showed
has
the
the
in
sphere
Universe
relativity
of
Earth
observe.
of
a
number
Recent
the
Universe
would
(e.g.
the
effectively
repulsion.
the
random
on
be
space
then
proposing
a
they
basic
do
be
From
universe,
wherever
as
they
to
of
Mpc),
Einstein
every
observers
clusters
direction
in
any
appears
observers,
to
to
structure
apply
galaxy
whatever
the
different
scale
expected
and
is
only
universe
are
looks
on
large
does
isotropic
true
that
direction
observer
about
universe,
distribution
and
an
assumption
In
is
particular
of
of
the
universe
no
perspective
true
are
the
is
homogeneous
direction
the
Universe
assumptions
considers
and
the
principle
the
star
circular
of
at
a
given
motion
gravitational
speed
distance
because
its
r
from
the
centre
centripetal
force
will
is
orbit
in
provided
by
the
attraction:
2
varies
with
distance
from
the
galactic
GMm
_
centre.
mv
_
=
2
r
r
1.
Near
the
galactic
centre
GM
_
2
∴
v
=
r
A
simple
model
to
explain
the
different
speeds
of
rotation
of
stars
The
near
the
galactic
centre
assumes
that
density
of
the
galaxy
total
centre,
ρ,
is
constant.
A
star
of
mass
m
feels
a
resultant
of
stars
that
orbit
closer
resultant
force
attraction
is
the
in
same
towards
as
if
the
the
total
centre.
mass
M
The
of
value
all
the
of
=
volume
to
the
galactic
centre
were
concentrated
in
the
density
stars
that
4
centre.
An
v
point
to
note
is
that
the
net
effect
of
all
the
M,
is
stars
=
that
3
πr
×
ρ
3
πr
ρ
4πGρ
3
_
_
=
=
2
r
r
important
star,
3
G
closer
×
this
2
are
this
force
4
gravitational
of
by
M
of
than
near
given
its
mass
3
_____
are
4πGρ
_
orbiting
at
radius
that
is
greater
than
r
sums
to
zero.
∴
v
=
r
√
3
i.e.
v
α
r
density of stars
in galactic centre = ρ
2.
Far
away
Far
away
from
the
galactic
centre
speed ν
of
visible
reduced
mass of
star, m
be
from
stars
so
much
by
galactic
show
considered
unaffected
the
to
that
that
be
their
centre,
the
individual
freely
observations
effective
stars
orbiting
neighbouring
density
at
the
these
central
stars.
In
this
of
of
the
the
number
galaxy
distances
mass
and
to
be
situation,
radius r
GM
_
2
v
=
where
M
is
the
mass
of
the
galaxy
r
1
i.e.
v
α
√
Comparisons
agreement
r
with
with
observations
mathematical
withmathematical
model
ν
discussed
stars outside r have
overall no net eect
M
m
r
stars inside r have
eect of total mass M
at centre
208
O p t i O n
D
–
A s t r O p h y s i c s
on
page
209.
(2).
of
real
model
The
galaxies
(1)
but
proposed
show
no
good
agreement
solution
is
has
can
r  vs d d 
HL
rotation curVeS
350
Galaxies
this
rotation
analysis
show
an
a
this
initial
at
speed
be
Most
centre
calculated
speed
A
galactic
for
mass
and
individual
with
the
speeds
stars
curve
for
distance
from
a
an
galaxy
from
the
show:
in
orbital
velocity
with
distance
centre
increasing
rotation
of
rotation
varies
galaxies
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slightly
of
their
spectra.
orbital
linear
the
or
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star’s
centre.
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•
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roughly
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NGC 4378
NGC 3145
250
NGC 1620
200
NGC 7664
150
100
constant
50
centre.
5
10
15
20
25
distance from centre of galaxy (kpc)
that
e ViDence for Dark matter
As
shown
agree
but
above,
with
the
theoretical
orbital
distance
observed
away
from
of
the
orbital
velocity
is
the
orbital
velocity
v
radius,
models
velocity
the
rotation
within
stars
centre
roughly
of
a
star
curves
is
the
not
as
real
galactic
observed
would
constant
is
for
be
at
(v
α
decrease
expected.
whatever
constant
centre
to
the
different
we
can
matter.
galaxies
outside
r)
with
matter
Instead,
radius.
values
Further
In
the
Dark
so
the
galactic
could
be
that
suggestion
situation
evidence
neutrons
If
of
see
this
it
centre
constitute
is
that
to
a
only
be
that
have
forming
suggests
imagined
would
be
halo
a
made
ordinary,
there
to
or
around
very
up
must
small
of
be
dark
concentrated
the
baryonic,
the
galaxy.
amount
protons
of
this
and
matter.
matter:
then
•
gravitationally
•
does
attracts
ordinary
matter
GM
_
2
since
v
=
r
not
emit
radiation
and
cannot
be
inferred
from
its
M
_
=
constant
or
M
α
r
interactions
r
Thus
the
must
be
total
mass
increasing
certainly
not
true
that
is
with
of
keeping
distance
the
visible
the
from
mass
star
the
(the
orbiting
galactic
stars
in
its
galaxy
centre.
emitting
This
•
is
unknown
•
makes
•
machoS, wimpS anD other theorieS
explain
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why
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The
are
there
are
a
matter
could
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evidence
that
or
could
lots
of
of
account
or
be
in
for
Massive
matter
of
could
light.
small
with
and
theories
what
it
to
short.
does
as
There
exist
in
low-mass
even
be
black
Evidence
is
some
could
These
are
‘failed’
holes.
Perhaps
correct.
simply
these
suggests
the
There
so-called
•
Astronomical
for
up
Universe
majority
stars
made
be
the
of
into
new
of
the
Universe
ordinary
with
particles
that
Interacting
around
the
world
baryonic
less
than
5%
of
we
do
not
Massive
are
matter.
know
about.
Particles.
searching
for
Many
these
WIMPs
our
current
Some
a
up
Weakly
experimenters
consists
theories:
thought
no
a
up
matter
MACHOs
They
or
come
possible
ordinary
can
little
to
dark
found
planets.
produce
only
be
much
Objects
These
high-mass
would
so
number
Compact
groupings.
attempting
is
structure
light)
the
Astrophysicists
in
is
theories
theories
failure
of
our
try
to
of
gravity
explain
current
are
the
theories
not
completely
missing
to
take
matter
as
everything
account.
These
that
these
proportion.
O p t i O n
D
–
A s t r O p h y s i c s
209
t s   uvs
HL
projection
fluctuationS in cmb
The
cosmic
isotropic
in
the
space
(the
early
with
precisely
no
to
be
any
uniformly
Further
distributed
(anisotropies)
are
in
typically
a
as
the
few
identical
know,
is
the
temperature
with
the
If
this
projection
in
the
Just
2.73
which
CMB
like
of
a
The
diagram
highlights
(with
map
K.
the
the
effects
–
it
includes
all
the
our
in
received
CMB
from
the
would
stars
is
and
be
would
be
not
galaxies.
uctuations
of
the
temperature
the
right
minor
of
variation
was
would
early
variations
background
Variation
effective
in
Anisotropy
temperature
the
matter
matter
matter
distribution
These
compared
all.
and
that
into
tiny
that
shows
wholeUniverse.
essentially
Universe
Universe
however,
is
throughout
at
the
everywhere
reveals
directions.
µK
of
concentrated
CMB
implies
variations
throughout
it
This
(CMB)
distributed
development
We
the
different
directions).
uniformly
the
structure.
of
radiation
temperature
then
distributed
in
all
was
absolutely
analysis
Universe
background
in
random
case
uniformly
without
same
Universe
the
expected
be
microwave
is
an
observed
own
of
the
The
variations
galaxy
countries
CMB
as
Probe
observed
by
the
Wilkinson
Microwave
(WMAP)
enhanced
in
removed).
world,
minute
Universe
this
differences
densities,
which
in
allow
temperature
structures
to
imply
be
minor
differences
developed
as
the
expands.
the hiStory of the uniVerSe
We
can
‘work
•
Very
•
As
•
The
soon
the
backwards’
after
the
Universe
and
Big
imagine
Bang,
expanded
it
the
the
process
Universe
cooled.
It
had
that
must
to
took
have
cool
to
place
been
a
soon
very
certain
after
the
Big
Bang.
hot.
temperature
before
atoms
and
molecules
could
be
formed.
36
Universe
underwent
a
short
period
of
huge
expansion
(Ination)
that
would
have
taken
place
from
about
10
s
after
the
32
Big
Bang
to
10
s.
Time
What
45
is
happening
Comments
36
10
s
→
10
s
→
10
36
s
Unication
s
Ination
of
forces
This
is
the
starting
point.
32
10
A
rapid
rapid
32
period
of
expansion
expansion
are
not
–
fully
the
so-called
inationary
epoch.
The
reasons
for
this
understood.
5
10
s
→
10
Quark–lepton
s
era
Matter
and
slightly
5
antimatter
more
matter
(quarks
than
and
leptons)
are
interacting
all
the
time.
There
is
antimatter.
2
10
s
→
10
Hadron
s
era
At
the
beginning
protons
2
and
of
this
neutrons)
short
to
be
period
it
is
just
cool
enough
for
hadrons
(e.g.
stable.
3
10
s
→
10
Nucleosynthesis
s
During
nuclei.
3
this
period
some
The
matter
that
and
antimatter
The
formation
have
of
the
now
protons
exists
is
and
the
neutrons
‘small
have
amount’
combined
that
is
left
to
form
over
helium
when
matter
interacted.
5
10
s
→
3
×
Plasma
10
era
(radiation
years
era)
plasma
with
of
light
nuclei
electrons,
has
protons,
now
nished
neutrons,
and
helium
the
Universe
nuclei
and
is
in
the
photons
all
form
of
a
interacting.
5
3
×
10
years
Formation
→
of
atoms
At
the
beginning
of
this
period,
the
Universe
has
become
cool
enough
for
the
rst
atoms
9
10
to
years
exist.
the
Under
matter.
It
microwave
these
is
conditions,
these
photons
radiation.
The
the
that
photons
are
Universe
is
now
that
exist
being
essentially
stop
received
75%
having
as
part
hydrogen
to
of
and
interact
the
with
background
25%
helium.
9
10
years
→
Formation
now
galaxies
of
and
stars,
Some
galactic
of
matter
clusters
are
the
is
matter
dense
expansion
any
radiation
over
time
very
(see
short
microwaves
The
the
that
12
Universe
has
page
approximately
at
of
been
202).
billion
–
spectrum
the
of
CMB
CMB
emitted
Thus
years
wavelengths
is
means
the
ago
now
that
in
the
the
radiation
(shortly
being
will
that
after
received
was
as
hot
‘stretched’
emitted
Big
smaller
enough,
longer
radiation
was
and
corresponds
to
black-
much
λ
the
change
radiation
at
a
temperature
of
2.73
K.
The
at
two
different
calculation
λ
α
so
R
uses
∴
law
to
link
the
peak
wavelength,
λ
,
of
the
radiation
max
the
temperature,
T,
of
the
black
body
in
kelvins:
3
2.9 × 10
__________
λ
=
max
T
1
λ
α
max
210
T
O p t i O n
D
–
A s t r O p h y s i c s
to
was
reactions
emitted
also
1
α
T
of
times
1
Wien’s
gravitational
nuclear
wavelength,
max
body
by
hotter,
was
stretching
in
wavelengths
received
together
interactions.
can
take
place
If
this
and
stars
the
the
temperature
cosmic
scale
proportionally
of
factor,
much
the
R,
was
much
smaller.
max
Since
Bang)
much
the
universe
of
radiation.
radiation
and
When
be
the
brought
formed.
wavelength
past
be
enough
coSmic Sc ale factor anD temperature
The
can
R
or
T
α
R
the
Universe
then
must
the
be
is
ratio
the
the
of
same
cause
cosmic
as
the
of
the
scale
ratio
factors
of
peak
t    uvs
HL
rotcaf elacs cimsoc
future of the uniVerSe (without Dark
energy)
If
the
to
do
Universe
in
the
is
expanding
future?
are
moving
away
the
galaxies,
As
from
then
this
a
at
result
us.
If
the
of
the
there
expansion
moment,
Big
were
could
what
Bang,
no
be
is
it
other
forces
going
galaxies
between
thought
of
as
being
R
open
at
closed
constant.
now
rotcaf elacs cimsoc
An
open
current rate
force
of expansion
a
of
Universe
gravity
is
slows
one
the
that
rate
continues
of
to
recession
expand
of
the
forever.
galaxies
The
down
1
little
halt.
A
bit
The
expansion
uniform.
means
there
force
is
The
that
is
a
of
if
two
force
must
going
of
have
to
expansion
the
force
do
and
gravity
masses
are
attraction
slowed
in
the
the
it
is
would
closed
not
Universe
back
on
the
expansion
the
Universe
strong
happen
to
is
one
itself.
an
if
is
bring
in
the
brought
force
This
to
density
that
The
end.
enough
the
of
to
gravity
would
the
expansion
Universe
a
is
happen
stop
and
enough
if
were
the
to
a
low.
then
to
bring
density
in
time
Universe
of
but
This
collapses
now
it
time
R
however,
between
moving
pulling
the
depends
of
matter
all
apart
them
expansion
future
density
cannot,
acts
from
back
down
on
in
the
the
have
masses.
one
been
the
This
past.
current
What
rate
at
and
another
together.
in
A
This
of
down
but
The
it
is
takes
happen
if
a
little
make
bit
the
high.
the
force
an
the
electron-positron
be
Universe.
Universe
closed.
only
were
mathematical
of
gravity
innite
Universe
pair
bigger.
Universe
time
were
more,
Just
possibility
keeps
and
enough
to
on
get
to
exactly
the
to
between
slowing
rest.
the
the
This
right
gravitational
start
the
open
expansion
would
density.
force
One
would
contraction
and
closed.
critic al DenSity, ρ
c
The
theoretical
Universe
is
value
called
of
the
density
critical
that
would
density,
ρ .
create
Its
a
value
at
is
not
c
radius r
certain
because
the
current
rate
of
expansion
is
26
measure.
Its
order
of
magnitude
is
10
not
easy
to
3
kg
m
or
a
few
proton
total mass in
masses
every
cubic
metre.
If
this
sounds
very
small
recessional
remember
sphere, M
that
enormous
mass
at
amounts
space
exist
that
contain
little
or
velocity = ν
no
all.
The
density
It
reasonably
is
of
of
the
Universe
easy
to
is
not
estimate
an
the
easy
mass
quantity
in
a
to
galaxy
measure.
by
The
total
energy
E
of
the
galaxy
is
the
addition
of
its
kinetic
T
estimating
the
number
of
stars
and
their
average
mass
but
the
energy
E
and
gravitational
potential
energy,
E
K
majority
of
the
mass
in
the
Universe
is
dark
= E
T
value
of
ρ
can
be
estimated
using
Newtonian
+
E
K
P
gravitation.
1
c
E
a
consider
recessional
a
galaxy
at
a
distance
velocity
of
v
with
r
away
respect
to
from
the
an
observer
2
=
K
We
by:
matter.
E
The
given
P
mv
but
Hubble’s
law
gives
v
=
H
r
0
2
with
observer.
1
∴
E
2
=
m(H
K
r)
0
2
average density of universe
GMm
_
E
inside sphere = ρ
=
4
-
but
M
=
volume
×
density
=
r
P
3
πr
ρ
3
radius r
3
G4πr
2
ρm
4Gπr
_
E
=
ρm
_
-
=
P
-
3r
3
obser ver
recessional velocity = ν
If
E
is
positive,
the
galaxy
will
escape
the
inward
attraction
–
T
the
If
universe
E
is
is
open.
negative,
the
galaxy
will
eventually
fall
back
in
–
the
T
universe
If
The
net
effect
of
all
the
masses
in
the
Universe
outside
the
E
on
the
galaxy
is
zero
(see
page
208
for
an
is
exactly
The
galaxy
is
thus
gravitationally
to
zero,
rest
–
the
the
galaxy
universe
will
is
take
at.
an
This
innite
will
time
occur
to
when
be
the
analogous
density
situation).
closed.
T
brought
sphere
is
attracted
in
by
of
the
universe
ρ
is
equal
to
the
critical
density
a
ρ
c
2
total
mass
M
which
acts
as
though
it
was
located
at
the
4Gπr
observer
1
∴
as
shown
m(H
r)
m
_
2
=
0
2
(above).
ρ
3
2
8Gπr
2
∴
mH
r
ρ
m
_
2
=
0
3
2
3H
0
_
∴
ρ
=
c
8πG
O p t i O n
D
–
A s t r O p h y s i c s
211
D 
HL
coSmic DenSity parameter
The
cosmic
density
parameter,
Ω
is
the
ratio
of
the
average
If
Ω
If
Ω
0
density
of
matter
and
energy
in
>
1,
the
universe
is
closed.
<
1,
the
universe
is
open.
=
1,
the
universe
is
at.
0
the
Universe,
ρ,
to
the
critical
0
density,
ρ
c
If
Ω
ρ
0
___
Ω
=
0
ρ
C
Dark energy
Gravitational
of
expansion
with
time.
candles
has
in
Currently
fact,
are
on
cause,
been
both
and
the
the
slowing
brink
of
the
single
discovery
for
the
dark
of
yet
either
down
means
that
that
expected
Ia
to
the
the
time
the
•
rate
as
Universe’s
page
rate
to
is
of
need
of
to
new
Universe’s
name
matter
‘dark
are
a
that
be
our
gravity
204).
mass
of
for
physics.
•
this
theories
modied.
concepts.
their
basis
existence
that
had
to
matter
within
been
observation
source
different
theoretical
than
to
galaxies
be
explain
for
able
adds
to
galaxies
previously
the
to
the
‘missing
known
explain
the
a
laws
galaxy’s
attractive
implying
expected,
matter’
of
more
hence
rate
force
of
unseen
the
name
mass
galaxies.
energy’.
implies
acting
opposes
has
within
attraction
of
that
means
counteracting
we
the
expansion
The
hypothesized
Dark
accelerating
of
Perhaps
Whatever
accelerating
two
evidence
agree
explanation
is
exist
rotation.
dark
possible
matter
must
gravitational
standard
expansion
(see
Dark
that
decrease
supernovae
that
over
accepted
it
relativity
experimental
have
type
evidence
indicate
course,
general
and
be
increasing.
no
of
general
energy
existence
is
and,
reason
cases
physicists
been
there
given
Dark
been
masses
would
using
strong
currently
has
observation
gravity
between
Universe
provided
Observations
expansion
the
Measurements
have
not,
attraction
of
the
The
the
expansion
that
then
attractive
attractive
resulting
energy,
of
force
force
the
be
gravity.
gravity
in
name
Universe
must
of
of
increase
hence
the
there
energy
dark
a
is
force
Dark
that
is
energy
between
implies
an
unseen
energy
In
but
explains
the
concept.
effect of Dark energy on the coSmic Sc ale factor
The
existence
The
graph
of
below
dark
energy
compares
counteracts
how
a
at
the
attractive
Universe
is
force
predicted
of
to
gravity.
develop
This
with
will
and
cause
cosmic scale
at Universe with dark energy
factor, R
(accelerating expansion)
at Universe without dark energy
(approaches maximum size)
now
212
O p t i O n
D
–
A s t r O p h y s i c s
time
the
without
cosmic
dark
scale
energy.
factor
to
increase
over
time.
asss s
HL
•
a StrophySicS reSe arch
Much
of
the
undertaken
current
in
collaboration
proud
at
the
of
time
astrophysics
and
their
fundamental
the
that
the
involves
sharing
record
of
research
of
close
previous
edition
is
Scientists
collaboration.
the
Cassini
spacecraft
of
had
this
been
book
in
can
For
was
orbit
is
be
be
•
example,
several
years
sending
information
about
the
and
is
currently
(2014)
continuing
to
Cassini–Huygens
spacecraft
was
funded
Space
Agency),
NASA
(the
planet
back
Administration
of
the
United
States
of
Life
as
Spaziale
general
mission
was
the
the
23
information
was
moon
and
The
scientic
Dark
countries
research
and
against,
nature
Future
of
Survey
by
ESA
they
Arguments
the
•
Is
as
and
budget
signicant
this
focus
descended
120
resources
for
such
dark
as
the
Euclid
Universe
mission
continues
to
for
towards
•
on
Earth
future,
will,
we
the
new
The
rise
to
of
technology
life
for
many
that
people.
at
If
some
time
in
humankind’s
the
distant
descendents
future,
are
to
become
exist
must
be
able
to
travel
to
distant
stars
in
and
planets.
against:
money
and
could
be
medical
are
suffering
the
world.
If
money
is
from
to
worthwhile
the
This
example
It
is
better
rather
scientic
both
offers
improving
for
•
map
be
give
quality
more
care
usefully
to
the
spent
many
providing
millions
of
food,
people
hunger,
homelessness
and
disease
who
around
the
researching
to
will
the
probe
scientists
arguments
into
to
well
process.
are
important
deserves
ASI
of
among
projects,
than
available
There
As
Huygens
shared
current
continue
undertake.
it
is
more
The
and
therefore
area.
for,
be
to
allocated
invest
the
immediate
the
quality
to
than
fund
Sending
a
life
great
rocket
research
•
Is
information
•
calculated
•
showed
the
of
for
deal
all
into
should
research,
resources
possibility
concentrating
space
the
a
on
limited
not
gained
funding
a
is
is
much
more
medical
saving
lives
research.
and
sufferers.
small
space
be
of
some
of
it
into
diverse
into
one
expensive,
research
expensive
thus
funding
priority.
really
worth
the
cost?
the
planned.
for:
fundamental
Why
Many
Agency).
important
it
and
Universe.
Understanding
•
an
Titan.
discovered
(involving
limited
investing
the
of
a
can
called
interesting
and
(the
Aeronautics
America)
Space
information
worldwide),
research,
geometry
back
Italian
Saturn,
Saturn
community.
have
that
the
about
of
sent
–
information
Energy
institutions
All
•
a
released
surface.
entire
Italiana
research
improve
impossibility.
shelter
(Agenzia
whole
to
•
Space
a
data.
jointly
National
as
Saturn
Arguments
European
fundamental,
researched.
eventually
colonize
The
most
being
around
produce
the
humankind
fundamental
this
Earth
of
for
properly
All
an
for
one
may
•
published,
It
areas
being
international
resources.
international
that
are
there
the
nature
of
philosophical
we
the
Universe
questions
sheds
light
on
like:
here?
(intelligent)
life
elsewhere
in
the
Universe?
current obSerVationS
1
Three
recent
in
detail
of
the
scientic
have
experiments
together
Universe.
added
Particular
a
that
great
have
deal
experiments
to
of
studied
our
note
the
CMB
NASA
’s
Cosmic
•
NASA
’s
Wilkinson
Background
•
ESA
’s
Explorer
Hubble
constant
to
be
67.15
km
s
1
Mpc
that
their
results
were
consistent
with
the
Big
Bang
include:
and
•
the
understanding
specic
ination
theories
(COBE)
•
showed
the
Universe
to
be
at,
Ω
=
1
0
Microwave
Anisotropy
Probe
(WMAP)
•
Together
Planck
these
space
observatory.
experiments
have:
In
calculated
the
23%
matter
summary,
matter
•
mapped
the
anisotropies
of
the
CMB
in
great
detail
and
discovered
200
many
•
that
million
scientists
calculated
years
the
years
the
had
age
rst
after
generation
the
Big
previously
of
the
of
Bang,
stars
much
to
shine
earlier
did
and
Universe
current
dark
and
to
scientic
energy
be
71.4%
are
composed
dark
evidence
taken
into
of
4.6%
atoms,
energy.
suggests
that,
consideration,
when
the
dark
Universe:
with
precision
•
dark
so
•
is
•
has
a
the
critical
•
has
an
•
is
than
at
density
that
is,
within
experimental
error,
very
close
to
density
expected
Universe
as
13.75
±
0.14
accelerating
expansion
billion
composed
mainly
of
dark
matter
and
dark
energy.
old
O p t i O n
D
–
A s t r O p h y s i c s
213
ib qss – sss
1.
This
question
Wolf
about
determining
some
properties
of
the
star
3.
a)
The
spectrum
(i)
(ii)
star
Wolf
359
Describe
how
Calculate
toWolf
has
a
this
the
parallax
parallax
distance
in
angle
angle
of
is
0.419
seconds.
measured.
light-years
from
evitaler
The
ytisnetni
a)
is
of
light
from
the
Sun
is
shown
below.
359.
[4]
1.0
0.8
Earth
359.
[2]
0.6
(iii)
State
used
why
for
the
stars
method
at
a
of
parallax
distance
of
less
can
only
than
a
be
few
0.4
hundred
b)
The
parsecs
from
Earth.
[1]
ratio
[4]
apparent
brightness
of
Wolf
_____________________________
15
is
apparent
brightness
of
the
0.2
359
3.7
×
10
Sun
0
0
Show
that
the
500
1000
1500
2000
2500
3000
ratio
wavelength / nm
luminosity
of
Wolf
359
_____________________
4
is
8.9
×
10
4
.
(1ly
=
6.3
×
10
AU)
Use
luminosity
of
the
of
c)
The
surface
temperature
of
Wolf
359
is
2800
K
is
3.5
×
b)
10
W
.
Calculate
the
the
radius
Outline
be
By
2.
The
S
=
to
white
diagram
onwhich
(L
a
estimate
the
surface
temperature
[2]
how
the
following
quantities
can,
in
principle,
determined
from
the
spectrum
of
a
star.
[2]
reference
neither
to
of
Wolf359.
d)
spectrum
Sun.
and
23
itsluminosity
this
Sun
the
below
the
data
dwarf
shows
positions
luminosity
of
of
the
in
nor
(c),
a
the
suggest
red
grid
selected
why
Wolf
359
of
an
stars
[2]
HR
are
(i)
The
(ii)
Its
elements
present
in
its
outer
layers.
[2]
is
giant.
diagram,
4.
a)
shown.
speed
Explain
model
how
of
relative
to
Hubble’s
the
the
law
Earth.
supports
[2]
the
Big
Bang
Universe.
[2]
Sun.)
b)
Outline
saying
one
how
other
it
piece
supports
of
evidence
the
Big
for
the
model,
Bang.
[3]
B
A
5
c)
1.0 × 10
The
Andromeda
about
700
kpc
Virgo
nebula
away
at
galaxy
from
is
2.3
is
a
relatively
the
Milky
Way,
Mpc
away.
If
close
galaxy,
whereas
Virgo
is
the
moving
1
3
1200
km
s
,
show
that
Hubble’s
law
1.0 × 10
predicts
that
Andromeda
should
be
moving
away
1
at
roughly
400
km
s
.
[1]
luminosity L/L
s
1
d)
1.0 × 10
Andromeda
is
in
fact
moving
towards
the
Milky
Way,
1
with
a
speed
discrepancy
1
and
of
about
from
direction,
the
be
100
km
s
.
prediction,
How
in
can
both
this
magnitude
explained?
[3]
1.0 × 10
e)
If
light
of
wavelength
Andromeda,
what
500
would
nm
be
is
emitted
the
from
wavelength
observed
3
from
1.0 × 10
4
4
3.0 × 10
Earth?
[3]
3
1.2 × 10
3.0 × 10
5.
A
quasar
has
a
redshift
of
6.4.
Calculate
the
ratio
of
the
surface temperature T/K
a)
(i)
(ii)
Draw
a
circle
Label
this
Draw
a
around
circle
circle
(iii)
Draw
a
line
stars
that
are
red
this
Explain,
the
circle
through
without
stars
that
are
astronomerscan
diameterthan
the
c)
Using
the
show
emitted
the
light
stars
that
are
6.
main
the
following:
calculation,
that
star
B
has
how
a
Explain
a)
Why
b)
The
more
jeans
from
data
that
and
star
information
A
is
at
a
from
Mean
being
size
when
the
quasar
detected.
[3]
massive
stars
have
shorter
lifetimes
[2]
criterion
[2]
c)
How
d)
How
e)
The
elements
distance
type
1a
heavier
than
supernovae
can
iron
be
are
used
produced
as
by
standard
stars
candles
[2]
[2]
the
of
signicance
of
observed
anisotropies
in
the
about
Microwave
background
[2]
Earth.
brightness
of
the
Sun
=
1.4
×
10
brightness
distance
of
of
Sun
star
from
A
=
4.9
×
Earth
=
1.0
AU
10
=
2.1
×
f)
The
signicance
g)
The
evidence
h)
What
of
the
critical
density
of
universe
[2]
2
W
m
9
Apparent
is
its
[3]
3
Apparent
that
to
larger
Cosmic
800pc
universe
[1]
A.
following
HRdiagram,
any
deduce
star
the
HL
white
W
.
[1]
doing
of
[1]
sequencestars.
b)
size
giants.
R.
around
dwarfs.Label
the
current
2
W
m
is
for
meant
dark
by
matter
dark
[2]
energy
[2]
5
1
pc
10
AU
[4]
7.
d)
Explain
why
determined
214
i B
the
by
distance
the
of
method
star
of
Q u E s t i O n s
A
from
stellar
–
Earth
cannot
be
parallax.
A s t r O p h y s i c s
[1]
Calculate
the
critical
density
for
of
1
Hubble
constant
of
71
km
s
the
universe
using
the
1
Mpc
[3]
17
a P P e n d i x
gp
Plotting graPhs – axes and best fit
The
us
f
reasn
t
fr
identify
ltting
trends.
reresenting
the
a
T
grah
be
in
the
recise,
variatin
f
it
ne
rst
lace
allws
us
quantity
is
a
that
it
visual
with
allws
•
All
•
Errr
the
•
A
the
When
ltting
fllwing
ints
grahs,
have
yu
been
need
t
make
best-t
sure
that
–
it
The
grah
shuld
•
The
scales
have
a
title.
the
axes
shuld
If
the
Smetimes
they
be
–
als
need
a
curse,
be
any
sudden
r
suitable
there
shuld
The
inclusin
f
the
rigin
uneven
has
If
best-t
the
‘jums’
in
the
be
shuld
have
imrved
by
the
this
rigin
being
been
thught
abut.
can
always
The
nal
draw
a
included
missed
secnd
As
–
ut.
it
If
is
rare
in
fr
dubt
a
grah
shuld,
if
in
either
grah
withut
it
if
ssible,
cver
mre
best-t
a
general
The
axes
AND
the
are
The
(e.g.
crsses
are
are
‘jins
the
is
a
curve,
this
line
is
a
straight
has
been
drawn
as
a
single
line,
this
has
been
added
WITH
rule,
there
the
line
shuld
as
be
belw
rughly
the
the
same
number
line.
Check
that
the
ints
Smetimes
are
ele
randmly
try
t
t
a
abve
best-t
and
belw
straight
the
line
necessary.
than
half
ints
that
shuld
be
reresented
by
a
gentle
curve.
If
the
was
dne
then
ints
belw
the
line
wuld
be
at
the
directin.
labelled
units
ints
just
it.
with
bth
the
quantity
(e.g.
current)
wuld
clear.
better
be
f
at
the
the
curve
end,
and
r
all
vice
the
ints
abve
the
line
versa.
ams).
•
•
line
abve
beginning
•
NEVER
trend.
line.
ints
this
aer
line
verall
grah
include
t
•
This
the
Mst
line.
Yu
added.
shw
numbers.
•
t
is
t
RULER.
f
grahs
line
there
nt,
•
•
trend
is
key.
A
f
crrectly.
arriate.
all
•
f
ltted
if
remembered:
smth
•
been
included
t
•
f
have
are
way
resect
dts’
anther.
ints
bars
Vertical
than
45
and
hrizntal
degree
crsses
r
lines
t
Any
make
ints
that
d
nt
agree
with
the
best-t
line
have
been
identied.
dts.
The
Me a suring intercePt, gradient and are a
the
gradient
tangent
f
t
a
curve
the
at
curve
any
at
articular
that
int
is
the
gradient
f
int.
under the graPh
Grahs
fr
be
can
be
used
straight-line
used
the
fr
curves
intercept,
1.
t
analyse
grahs,
as
the
the
thugh
well.
Three
gradient
data.
many
This
f
things
and
the
the
are
area
is
articularly
same
articularly
under
easy
rinciles
the
can
useful:
∆y
graph.
P
Intercept
In
general,
times.
A
a
grah
can
straight-line
intercet
grah
can
(cut)
nly
either
cut
axis
each
any
axis
number
nce
and
f
ften
x
it
is
the
y-intercept
that
has
articular
imrtance.
y-intercet
is
referred
t
as
simly
‘the
intercet’.)
If
at point P on the cur ve,
a
∆y
∆y
line =
grah
has
an
intercet
f
zer
it
ges
thrugh
the
gradient =
rigin.
∆x
∆x
Proportional
–
nte
that
tw
quantities
are
rrtinal
if
the
3.
grah
is
a
straight
line
THAT
pASSES
THRoUGH
THE
Area
a
grah
has
t
be
‘cntinued
n’
(utside
under
the
range
area
under
readings)
in
rder
fr
the
intercet
t
be
fund.
This
rcess
as
extrapolation.
The
rcess
f
assuming
that
the
the
alies
between
tw
ints
is
knwn
as
the
x-axis.
average
grah
quantity
n
is
the
the
rduct
y-axis
by
f
the
This
des
nt
always
reresent
a
useful
quantity
hysical
trend
quantity.
line
graph
straight-line
is
n
knwn
a
f
multilying
the
a
oRIGIN.
The
Smetimes
x
∆x
(Smetimes
gradient of straight
the
∆y
∆x
When
wrking
ut
the
area
under
the
grah:
interpolation.
•
If
the
grah
cnsists
f
straight-line
sectins,
the
area
can
be
ex trapolation
wrked
•
If
the
ut
grah
by
is
a
dividing
curve,
the
the
shae
area
u
can
int
be
simle
calculated
shaes.
by
‘cunting
y-intercept
the
•
squares’
The
units
and
fr
wrking
the
area
ut
under
what
the
ne
grah
square
are
the
reresents.
units
n
the
The line is interpolated
The ex trapolation of the
y-axis
multilied
by
the
units
n
the
x-axis.
between the points.
graph continues the
•
If
the
mathematical
equatin
f
the
line
is
knwn,
the
area
f
trend line.
the
grah
can
be
calculated
y
2.
gradient
value
The
f
divided
A
•
The
a
by
fllwing
•
rcess
called
integration
y
straight-line
the
increase
ints
straight-line
triangle
shuld
grah
used
t
has
grah
in
be
a
the
is
the
x-axis
increase
in
the
y-axis
value.
remembered:
cnstant
calculate
the
gradient.
gradient
shuld
be
as
large
x
area under graph
as
x
area under graph
ssible.
The
gradient
divided
•
a
Gradient
The
•
using
only
if
by
the
reresent
has
the
x-axis
the
units.
units
is
RATE
n
a
They
the
are
the
measurement
at
units
n
the
y-axis
x-axis.
which
the
f
time
quantity
des
n
the
the
gradient
y-axis
increases.
A P P E N D I X
215
gp y  m  p
Yu
equation of a straight-line graPh
All
straight-line
grahs
can
be
described
using
ne
the
general
been
equatin
y
y
=
and
the
m
mx
x
+
are
c
the
tw
variables
(t
match
with
the
y-axis
and
x-axis).
and
c
shuld
same
are
bth
cnstants
–
they
have
ne
xed
=
u
y
=
c
if
x-axis
•
c
reresents
In
is
sme
give
a
x
straight
we
t
intercet
t
Fr
examle,
a
the
y-axis
(the
value
y
we
see
that
the
hysics
mathematical
s
as
t
equatin
equatin.
emhasize
the
The
trlley
this
as
is
it
v
In
a
lt
sme
in
f
the
ther
rder
straight
calculate
simle
rlls
=
situatin
u
v
has
at
these
are
the
tw
equatins,
velcity
ging
t
get
n
a
the
yu
y-axis
straight-line
shuld
and
be
the
able
time
t
n
see
the
grah.
t
situatins
get
line,
ther
measured
a
we
we
straight
then
variable
have
line.
use
the
t
In
will
chse
either
gradient
and
20
15
10
20
gradient =
+
values.
exeriment
dwn
at
and
a
sle.
where
t
are
u
ur
is
2
= 4 ms
5
5
1
mtin
exactly
link.
might
measure
the
velcity
2
3
4
5
f
= 0
a
has
rder
mx
lt
we
t
the
takes
grah.
direct
lt
have
the
f
a
line.
what
nce
the
0)
gradient
situatins,
carefully
case,
=
the
n
s m / v yticolev
m
intercet
+
+
as
1
when
•
the
able
belw
cmaring
that
value.
changed
v
By
be
frm
The
the
equatin
initial
variables,
a
that
velcity
and
u
describes
f
are
the
the
time t / s
the
bject.
In
cnstants.
The
cmarisn
•
c
•
m
(the
In
this
als
wrks
y-intercet)
(the
gradient)
must
must
fr
be
be
the
cnstants.
equal
equal
t
t
the
the
initial
velcity
acceleratin
u
a
choosing what to Plot to get a
examle
the
grah
tells
us
that
the
trlley
must
straight line
have
With
a
little
rearrangement
we
can
ften
end
u
with
started
frm
rest
(intercet
zer)
and
it
had
a
cnstant
the
2
acceleratin
hysics
equatin
equatin
•
f
Identify
a
in
which
symbls
the
straight
same
line.
symbls
reresent
frm
as
the
Imrtant
reresent
The
the
•
If
symbls
symbls
yu
take
that
that
a
variables
and
which
Example
crresnd
crresnd
t
t
x
and
m
reading
and
and
y
must
be
variables
c
must
be
cnstants.
square
it
(r
cube,
and
an
image
etc.)
–
the
result
is
still
a
variable
f
1
square
and
t
lt
this
n
ne
f
the
Yu
can
lt
any
mathematical
the
1
n
ne
axis
–
this
is
a
is
lens
are
f
yur
int
f by
the
hysical
quantities
mass)
seed
symbls
f
invlved
fr
light).
gradient
frce
F
that
acts
n
we
get
an
distance
image.
u
and
The
the
fcal
equatin.
ssible
straight–line
ways
frm.
Yu
t
rearrange
shuld
check
this
in
that
rder
all
t
these
get
are
riginal
use
the
symbls
the
same.
+
u
=
v
r
Be
careful
r
nt
t
an
u
at
a
1
1
r
1
=
u
f
1
-
v
f
cnfuse
intercet.
bject
v
=
m
1
gravitatinal
fllwing
distance
v
The
the
(e.g.
lens
bject
u
with
c
a
the
)u + v(
these
Example
r
f
f
many
f
(e.g.
the
t
1
uv
_
Smetimes
frnt
variable.
v
•
in
related
axes.
cmbinatin
still
v
=
v
algebraically
readings
laced
yu
it
•
is
distance
length
There
chse
2
bject
+
culd
s
cnstants.
variable
recircal
m
include
u
rt,
4.0
mathematical
ints
If
•
f
r
1
gradient =
away
frm
the
centre
f
a
lanet
is
given
by
the
1
equatin
gradient =
f
GMm
_
F
f
=
where
M
is
the
mass
f
the
lanet
and
the
2
r
m
is
mass
f
the
bject.
uv
v
= 0
If
we
lt
frce
against
distance
we
get
a
curve
(grah
1).
intercept = –1
GMm
____
We
can
restate
the
equatin
as
F
=
+
0
and
if
we
lt
F
2
r
1
y-axis
and
n
the
x-axis
we
will
get
a
straight-line
2
r
u
the
1
n
(grah2).
1
intercept =
F
F ecrof
1
A
2
f
gradient = –1
A
gradient = GMm
B
1
B
C
v
C
distance r
I
intercept = 0
2
r
216
A P P E N D I X
gp y – m 
hl
logs – ba se ten and ba se 
Power l aws and logs
Mathematically,
Thus
if
ln
n
(l)
we
lt
the
ln
(T)
x-axis
n
we
the
will
y-axis
get
a
and
straight-
(log – log)
line
grah.
b
If
a
=
10
When
a
Then
lg
(a)
=
an
wer
exerimental
law
be
abslutely
is
ften
nly
invlves
ssible
recise
lg
(a)
=
it
int
straight-line
frm
a
them.
But
we
dn’t
lgs.
the
base.
We
can
have
use
any
Fr
examle,
the
time
t
will
be
equal
use
be
t
equal
p
t
a
simle
endulum,
T,
is
(k)
[s
k
=
e
]
erid
buttn
related
t
its
gradient = p
10
length,
as
will
(intercet)
ln
‘lg’
T nI
have
f
n
intercet
b]
taking
calculatrs
gradient
The
by
10
Mst
The
t
b
transfrm
[t
it
situatin
l,
by
the
fllwing
equatin.
number
p
T
that
we
like.
Fr
examle
we
k
use
2.0,
563.2,
17.5,
42
r
=
k
l
culd
and
p
are
cnstants.
even
intercept = ln (k)
2.7182818284590452353602874714.
A
Fr
but
it
the
values
IS
cmlex
the
mst
reasns
useful
this
last
number
number
t
use!
It
lt
f
is
the
nt
variables
clear
f
k
will
frm
and
p
give
this
a
curve,
curve
wrk
ut
what
t
be.
In l
on
plt
is
given
the
symbl
e
and
lgarithms
t
f
this,
if
we
d
is,
we
can
nt
knw
what
this
base
are
called
logarithms.
The
lgarithms
ln
is
natural
symbl
(x).
This
fr
is
value
natural
als
n
values
=
The
e
ln
(p)
werful
means
that
=
q
nature
we
lt
a
calculate
straight-line
have
f
fllwing
erid)
versus
(length)
gives
a
straight-line
grah
the
grah.
Bth
the
gravity
frce
are
inverse-square
frce
and
the
electrstatic
relatinshis.
This
2
means
The
that
same
generate
lgarithms
the
(time
the
frce
technique
a
∝
(distance
can
straight-line
be
aart)
used
.
t
grah.
ecrof
Then
t
nt
sdnoces / T
q
p
p
mst
calculatrs.
If
f
ln
the
ln
t
f
k
force =
rules
2
(distance apar t)
l / metres
ln
(c
ln
(c
ln
(c
×
d)
=
ln
(c)
÷
d)
=
ln
(c)
+
ln
(d)
ln
(d)
Time
n
)
=
n
ln
(c)
)
=
-ln
(c)
erid
versus
length
fr
a
simle
endulum
distance apar t
The
‘trick’
is
t
take
lgs
f
bth
sides
f
1
These
c
natural
all
have
been
lgarithms,
lgarithms
The
be
the
rules
int
used
f
t
exressed
but
they
whatever
lgarithms
exress
the
is
sme
fr
wrk
equatin.
used
fr
fr
natural
all
The
lgarithms
belw
but
whatever
base.
that
equatins
lgarithms,
have
wuld
the
)ecrof( gol
(
ln
wrk
base.
p
they
ln
(T)
=
ln
(k l
)
ln
(T)
=
ln
(k)
+
ln
(T)
=
ln
(k)
+
can
intercept = log (k)
gradient = -2
p
ln
(l
)
relatinshis
log (distance apar t)
(articularly
wer
exnentials)
means
with
A
that
amunt
2
scale
each
3
be
frm.
ltting
This
grahs
This
is
scales.
y
increases
by
the
nw
equatin
(l)
=
c
in
fr
+
the
a
same
straight
frm
as
the
line
mx
Inverse
and
same
square
lg-lg
relatinshi
–
direct
lt
lt
time.
4
5
6
7
8
9
10
11
lgarithmic
scale
increases
by
examle,
the
cunt
rate
R
at
R
Fr
A
ln
and
straight-line
will
lgarithmic
nrmal
1
in
we
laws
p
any
the
given
time
t
is
given
by
the
equatin
R
0
same
rati
all
the
λt
time.
R = R
λt
R
=
R
0
e
e
0
0
10
1
2
10
3
10
10
R
and
λ
are
cnstants.
0
If
1
10
100
we
take
lgs,
we
get
1000
λt
ln
(R)
=
ln
(R
e
)
0
t
λt
(R)
=
ln
(R
ln
(R)
=
ln
(R
ln
(e
λt
ln
)
)
(e)
0
(log – line ar)
ln
lgarithms
+
0
e xPonentials and logs
Natural
)
)R( nl
ln
are
very
(R)
=
ln
(R
imrtant
)
λt
[ln
(e)
=
intercept = ln (R
gradient =
because
many
exnential.
imrtant
natural
rcesses
Radiactive
examle.
In
decay
this
are
is
case,
an
This
fr
the
taking
f
lgarithms
can
a
equatin
fr
t
a
be
cmared
straight
with
the
λ
equatin
c
+
grah
mx
t
allws
with
if
we
lt
ln
(R)
n
the
y-axis
and
t
the
n
equatin
cmared
straight-line
=
Thus
the
be
nce
y
again
)
0
1]
0
the
x-axis,
we
will
get
a
straight
line.
line.
Gradient
Intercet
=
=
- λ
ln
(R
)
0
A P P E N D I X
217
Answers
Topic
1
(Page
8):
Measurements
and
uncertainties
Topic
9
(Page
104):
Wave
2
1.
(a)(i)
0.5
×
acceleration
down
the
slope
(a)(iv)
0.36
phenomena
λ
1
ms
1.
B
2.
(b)
27.5
m
s
3.
(a)
0.2°;
4.
(a)
(i)
zero;
(ii)
π
or
;
2
2
2.
C
3.
6.
(b)(i)
D
4.
D
5.
(b)
2.4
±
4
Topic
3;
2
(b)(ii)
(Page
2.6
24):
0.1
s
(c)
2.6
±
0.2
10
ms
(iii)
zero;
(b)
110
3
10
nm;
5.
19
Nm
(d)
Mechanics
(ii)
Topic
5.0
10
×
(i)
1.5
×
10
m;
J;
(a)(ii)
2
10
(Page
(b)
m
s
111):
Fields
11
1.
C
2.
D
3.
B
4.
B
5.
(a)
520
N;
(b)(i)
1.2
MJ;
(b)(ii)
270
W
1.
A
2.
C
1
6.
(a)
equal;
7.
(c)
3.50
(b)
left;
(c)
20
km
hr
3.
C
4.
(a)(i)
1.9
×
-1
10
12
;
(e)
car
driver;
(f)
No
2.2
×
7.7
km
hr
(a)(iii)
2
10
J;
(c)
2.6
hr
5.
(b)(i)
2.5
m
s
N
Topic
11
(Page
120):
Electromagnetic
induction
4
Topic
3
(Page
32):
Thermal
Physics
1.
D
3.
B
4.
B
5.
D
6.
(b)
0.7
v
7.
(a)
7.2
×
10
C;
(b)
2.9
3
1.
B
2.
width
D
=
3.
5m,
D
4.
D
5.
temp
=
25
(a)
°C;
(i)
length
(a)(ii)
$464;
=
20
(b)(i)
m,
depth
84
=
2
m,
×
10
s;
(c)(ii)
5CR
=
3.5
s;
(c)(iii)
No
days
Topic
12
(Page
130):
Quantum
and
nuclear
1
6.
(a)(i)
7.8
J
K
mol
1
1.
C
2.
B
3.
(b)
ln
R
&
t;
(c)
Yes;
(e)
34
Topic
1.
C
4
2.
(Page
C
3.
50):
(a)
Waves
5.
longitudinal
(b)
(i)
0.5
m;
(ii)
0.5
(b)(i)
6.9
Option
330
m
0.510
×10
Js;
(b)(ii)
3.3
;
(h)
1.85
hr
4
×10
J
6.
4.5
×10
Bq
A
(Page
151):
relativity
7
s
4.
(c)
(i)
2.0
Hz;
(ii)
1.25
(1.3)
cm;
(f)
(i)
4.73
×10
m;
19
2.
(ii)
hr
19
mm;
1
(iii)
0.375
mm
5.
(a)(i)
1.40c;
(a)(ii)
0.95c;
(c)
6.0
×
10
J
3.
a)
2
yrs;
45°
b)
4
yrs
;
c)
x
=
5
ly;
d)
0.5
c
4.
(c)
front;
(d)
T:100
m,
S:87
m;
2
Topic
5
(Page
64):
Electricity
and
magnetism
(e)
T:75m,
S:87
m;
5.
(a)(i)
zero;
(a)(ii)
2.7
m
c
;
(b)(i)
0.923
c;
0
15
1.
C
2.
A
3.
(c)
(ii)
7.2
×
2
10
m
2
s
(c)
(iv)
100
v
4.
(d)
B;
(b)(ii)
2.4
m
c
2
;
(b)(iii)
3.6
m
0
(e)
(i)
Equal;
(ii)
approx.
0.4A;
(iii)
lamp
A
will
have
6
(Page
68):
Circular
motion
and
gravitation
A
2.
A
3.
C
4.
(a)(ii)
No;
(b)
1.4
m
a)
300
agree.
B
0
(Page
170):
Engineering
0.95
J;
N
(d)
m;
b)
500
J;
25.2
(e)
J;
c)
500
J;
13.4
(f)
N
physics
3.
150
J;
(a)
(g)
No;
(b)
16%
Equal;
4.
(b)
(c)
990K;
(c)
M
_
1
1.
(c)
dissipation;
2.
Topic
;
greater
Option
power
c
s
5.
(b)
(i)
g
=
G
;
(i)
1;
(c)
(ii)
2
&
3;
(c)
(iii)
3;
6.
Laminar
(R=1200)
7.
(a)
2Hz;
2
1.9
×
R
M
_
27
(b)(ii)
10
kg
6.
(a)
g
=
G
24
;
(b)
6.0
×
10
(b)
kg;
21
mW
2
R
Topic
1.
B
7
(Page
2.
D
81):
3.
fundamental;
A
Atomic
4.
(iii)
3
D
5.
quarks
nuclear
B
6.
or
3
7.
uu[π
(a)(i)
2
]
8.
(b)(i)
12
(a)
12
C
→
(ii)
(iv)
+
electron
a
quark
24
Mg
→
is
1.
and
an
(a)
C
14
(b)
24
(c)
4.5
(Page
cm
cm
189):
behind
behind
Imaging
mirror,
diverging
virtual,
lens,
upright,
real,
magnied
inverted,
(×2);
magnied
(×3);
4
Na
12
Option
physics
+
11
He
cm
behind
second
lens,
real,
upright
&
diminished
(0.25)
2
0
N
6
uud;
26
H
1
9.
particle
antiquarks;
0
antiquark;
and
+
+ν ;
β
7
(b)(ii)
11600
years;
10.
(a)(i)
3;
2.
(d)
upside
down;
5.
(a)
1
MHz;
6.
(b)(ii)
(e)
60
cm;
4.
(a)
–
10
dB;
(b)
0.5
mW
1
19
(b)(i)
1.72
Topic
8
×
10
(Page
11.
94):
A
Energy
(c)
15
MW
(d)(i)
20%
4
mm;
9.3
mm;
(b)(iii)
d
=
38mm,
l
=
130mm
production
(b)(iii)
1.
→20
4.
(a)
1000
(b)(iv)
18.6
mm
MW;
1
(b)
1200
MW;
(c)
17%;
(d)
43
kg
s
5.
(c)
1.8
Option
MW
D
(Page
214):
Astrophysics
7
1.
(ii)
d
=
7.78
4.
(e)
499.83
ly,
(c)
r
=
8.9
×
10
m;
3.
(a)
5800
K
27
nm
5.
14%
of
current
size
7.
9.5
×
10
3
kg
m
Origin of individual questions
The
questions
Topic
1:
detailed
Measurement
1
N99S2(S2)
5
M98SpH2(A2)
Topic
2:
below
2
and
M98H1(5)
6
are
all
taken
from
past
IB
examination
Topic
uncertainties
3
N98H1(5)
4
M99H1(3)
N98H2(A1)
M98S1(2)
6
N00H2(B2)
2
M98S1(4)
7
3
M98S1(8)
5
1
3:
2
1
4:
(16)
3
N99H1(17)
5
M98
M98S3(C2)
6
M122H2(B2.1)
5:
4
HL2
N03
7
2
10:
11:
6:
N10H1(15)
Electricity
N10S1(7)
5
M08
N98H2(A5)
3
M111H3(G3)
5
N09
HL3
G4
4
N98H2(B4)
5
N01H2(A3)
induction
3
N03S2(A3)
4
5
N00H1(31)
N98H2(A4)
and
3
M98H1(33)
4
M112H1(24)
5
N99H1(34)
N04H2(B4.1)
1
magnetism
12:
Quantum
N10H1(34)
2
and
nuclear
M01H1(35)
3
physics
N00H2(A1)
2
motion
and
M111H1(4)
3
2
gravitation
M101S1(8)
4
SpS3(A3)
6
N05H2(B2.1)
–
part
A
relativity
M111H3(2)
4
M00H3(G1)
5
N01H3(G2)
6
M092H3(3)
M1112(A5)
question
–
sections
B
Engineering
physics
(d)
N01H2(B2)
4
N98H2(A2)
(g)
Option
Topic
7:
Atomic,
nuclear
and
particle
N98S1(29)
2
M99S1(29)
3
8
M99S1(30
M98S2(A3)
9
4
M99S2(A3)
10
t o
q u e s t i o n s
D
M03H3(D2)
Astrophysics
M99H2(B4)
1
A n s w e r s
5
M98SpS1(29)
M122H1(32)
218
imaging
N00H3(H1)
Option
M98SpS1(30)
C
physics
2
11
phenomena
2
Electromagnetic
6
3
5
M98SpS3(C2)
Fields
1
Option
1
4
Q2.2
Circular
1
to
M98SpS3(C3)
M091S2(A2)
Option
Topic
Wave
N10H1(24)
Topic
Topic
9:
N01H1(24)
3
Sp2(B2)
Waves
M01H1(14)
production
5
3
Physics
N99H1
M112H2(A5)
Topic
IB.
M99S3(C1)
Topic
6
Energy
©
M091S2(A2)
Thermal
N99H1(15)
all
2
Topic
Topic
8:
are
NO1S3(C1)
1
M101S2(A2)
and
1
Topic
Mechanics
1
papers
M101H3(E1)
2
M111H3(E2)
3
N01H3(F2)
4
N98H3(F2)
Index
Page
numbers
in
italics
refer
to
question
sections.
A
galactic
absolute
magnitude
194
absolute
uncertainties
absolute
zero
absorption
69
gases
9,
14,
acceleration,
during
equations
34,
experiment
uniformly
acoustic
addition
air
alpha
free-fall
motion
11
178
187
90
alternating
coil
red
72
72,
current
rotating
ac
in
bridges
losses
in
the
eld
transformer
of
energy
atomic
physics
atoms
26
mass
power
115
183
39,
impulse
magnication
angular
momentum
conservation
157
of
177
157,
165
angular
series
barium
meals
momentum
77
spectra
69
77
124
base
becquerels
73
antineutrinos
units
78
194
beta
164
scale
25
variables
198
reaction
76
Chandrasekhar
charge
coefcient
185
30
196
198
limit
3
Big
capacity
60
law
51
54
acceleration
Charles’s
law
chemical
energy
22
units
190,
(AUs)
[RC]
capacitor
charging
investigating
circuit
Universe
astrophysics
research
model
213
194
parallel
series
165
of
circular
the
Bernoulli
198
equation
and
208
213
212
Universe
72,
shift
Bohr
Boyle’s
201
211
Brackett
motion
rectication
116
circuits
57
smoothing
circuits
motion
67,
nucleon
76
41
196,
194
200
radius
150
of
the
atom
31
series
49
124
a
time
period
vertical
66
plane
66
65
of
of
law
circular
circular
of
motion
motion
universal
65
65
gravitation
66
23
190
communications
coaxial
wire
184
cables
bres
pairs
complex
184
184
184
numbers
composite
123
in
79
comets
optical
conditions
68
and
68
radians
colour
90,
65–7,
circular
Newton’s
per
57
56
velocity
collisions
150,
circuits
circuits
mechanics
195
law
and
mathematics
radiation
model
boundary
118
119
56
angular
75
energy
holes
circuits
circuits
diode-bridge
divider
examples
129
45
birefringence
blue
165
72
Schwarzchild
models
observations
equation
model
black-body
discharge
circuits
potential
sensor
energy
binding
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principle
mathematical
the
204
201
variables
energy
193,
92
experimentally
rectication
effect
binoculars
accelerating
cosmological
191,
214
(CFCs)
55
capacitor
209
electric
31
73
178
and
145
73
73
decay
Bang
binding
200
51
116
radiation
beta
190
of
163
60
2
Bernoulli
principle
117
capacitor
163
charge
166
magnitude
current
engine
185
79
applications
48
parallel
92
theorem
Coulomb’s
60
Bernoulli
129
and
charged
118
119
163
Carnot
186
matter
batteries
Cepheid
in
circuits
circuits
92
Carnot
charge
123
78,
210
future
model
123
count
baryons
177
Bang
cycles
principles
coefcient
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size
astrophysics
Carnot
particle
distortion
anisotropies
dark
69,
xation
chain
radiation
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angular
astronomical
discharge
series
mathematics
81
atomic
constant
motion
Big
77,
atomic
of
dioxide
carbon
current
angular
assumptions
in
stored
carbon
circuits
barrel
157
Archimedes’
energy
Cepheid
129
matter
background
angular
antiparticles
charging
Celsius
148
183
background
48
angular
asteroids
207
shifts
B
amplitude
antinodes
[RC]
capacitor
chlorouorocarbons
ampliers
antimatter
117
116
capacitor
cells
77
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209
56
63
apparent
69,
of
spectra
matter
200
levels
for
190–1
195
model
atomic
114
electrical
sequence
193
frequency
attenuation
operation
main
dark
supernovae
atomic
114
transmission
amperes
of
attenuation
values
ammeters
spectra
explanation
power
and
stellar
structure
of
criterion
199
parallax
clock
114
72
capacitors
the
Universe
stars
atomic
–
115
transmission
the
evolution
evidence
magnetic
Jeans
196
off
stellar
54
115
RMS
C
capacitors
the
stellar
atomic
generator
diode
–
curves
127
a
factor
119
in
giant
types
decay
scale
192
fusion
rotation
16
radiation
stars
41
210
cosmic
206
objects
5
alpha
164
capacitance
nucleosynthesis
angular
accelerated
resistance
albedo
motion
13
impedance
law
buoyancy
194
nucleosynthesis
determine
doublets
Universe
and
Brewster’s
197
205
152
acceleration
of
nuclear
displacement
harmonic
uniform
to
10
and
95
acceleration
the
law
diagram
cancer
nature
graphs
velocity
of
of
luminosity
108
simple
[SHM]
202
203
92
acceleration-time
achromatic
history
Hubble’s
spectra
acceleration
5
29
greenhouse
motion
Hertzsprung–Russell
125
particles
compression
14,
78
16
I n d e x
219
compression
concentration
conduction
conductors
quantities
of
temperature
constant
volume
constellations
constructive
29,
190–1
interference
continuous
waves
convection
40,
47
division
69
35
converging
lenses
lenses
Copenhagen
173
173
density
cosmic
microwave
parameter
radiation
uctuations
125
212
background
(CMB)
factor
scale
210
dark
scale
and
Coulomb’s
coulombs
couples
temperature
on
the
cosmic
208
speed
drift
velocity
density
and
45
211
tomography)
scans
186
54
of
the
168
energy
of
scale
dark
204,
gravity
212
on
the
cosmic
212
WIMPs
and
other
theories
209
Davisson
De
and
Broglie
destructive
basic
122
waves
35
of
elastic
collisions
elastic
potential
elds
light
energy
52,
Davisson
Germer
diffraction
grating
explanation
multiple-slit
single-slit
diffraction
of
uses
of
of
diffraction
gratings
with
white
220
gratings
115
I n d e x
99
in
of
46
101
light
types
and
an
and
electric
energy
electrical
energy
22
electrical
meters
56
51–60,
for
work
stars
22
22
192
22
energy
143
momentum
energy
and
energy
of
in
eld
elds
53,
53,
a
52
energy
production
power
warming
and
primary
109
metal
wind
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law
51
55
54
sources
87,
energy
power
83
sources
88
91
transfer
and
other
89
technologies
83
energy
difference
of
and
53,
Kirchoff’s
and
electric
109
circuits
59
60
and
and
parallels
electromagnetic
force
electromagnetic
induction
current
56
71
112–19,
120
power
engineering
charge
119
capacitor
discharge
law
of
force
112
83
22
152,
170
166
examples
155
thermodynamics
uids
at
rest
uids
in
motion
161
164
–
oscillations
Bernoulli
and
effect
165
resonance
168–9
engines
rotational
(emf)
density
88
examples
equilibrium
second
and
second
inertia
118
electromotive
–
83
83
89
physics
Newton’s
capacitor
energy
transformations
heat
117
and
35,
83
sources
sources
energy
forced
114–15
energy
energy
sources
energy
transfer
Bernoulli
sensors
energy
energy
wind
laws
cells
of
non-renewable
specic
58
capacitance
88
85–6
energy
sources
rst
alternating
87
technologies
90
renewable
52
series
92
power
energy
comparison
resistance
in
84
88
energy
and
divider
production
developing
power
thermal
54
82
82
93
effect
power
94
production
conversions
fuel
solar
162
82–93,
power
secondary
109
64
use
48
degradation
new
57
induced
and
energy
energy
nuclear
electric
conduction
resistivity
99
ow
energy
greenhouse
132
potential
example
1
of
hydroelectric
electric
electric
resistors
diffraction
diffraction
bridges
122
46
diffraction
and
110
difference
internal
97
diffraction
electric
potential
potential
experiment
97
diode
122
101
99
signicance
resolvance
experiment
22
78
energy
energy
fossil
electric
potential
resolution
diffraction
of
97
51
69
of
of
radiation
potential
and
practical
difference
electric
and
examples
elds
elds
diffraction
electron
between
current
46
range
global
elds
electric
46,
22
between
electric
observations
16,
energy
22
electrical
61
circuits
47
128
90
energy
electric
40,
121
23
2
117
14,
particles
spectra
wave
22
charge
interference
200
experiment
potential
conservation
115
model
electricity
113
144
electric
material
diffraction
122
relative
53
relativistic
14
units
dielectric
experiment
hypothesis
deformation
derived
Germer
emf
electrons
electrostatic
mass
currents
electrical
by
125
concepts
191
emf
pressure
force
energies,
20
representation
212
induced
112
electrostatic
emission
54
89
60
112
of
scattering
elementary
53
209
MACHOs,
102
191
energy
energy
factor
matter
102
190
magnetic
dark
nuclear
energy
friction
comparison
effect
effect
54
comparison
D
dark
Doppler
190
electric
113
circuits
77
electronvolts
103
102
gravitational
damping
103
applications
equation
planets
Einstein
(computed
175
E
eddy
154
current
drift
efciency
critical
broadening
earthquake
51
53
angle
lens
electrons
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year
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37,
(emf)
degeneracy
orbital
source
day
212
principle
law
critical
CT
energy
factor
cosmological
diverging
175
16
Earth
of
rays
203
factor
induced
production
102
moving
dynamic
210
effect
by
observer
dwarf
CMB
motion
electron
important
moving
drag
73
in
displacement
waves
involving
mathematics
interpretation
cosmic
scale
54
172,
and
harmonic
175
and
effect
law
smoothing
force
motion
created
examples
current
Faraday’s
and
transformer-induced
Doppler
89
conventional
velocity
5
Doppler
and
electromotive
lenses
images
165
law
electromagnetic
34,
denitions
spectrum
60
95
diverging
continuous
rectication
116
simple
[SHM]
29
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116
183
during
160
rectication
54
characteristics
acceleration,
22
29
equation
diode-bridge
experimentally
current
discharge
126
energy
constant
cosmic
54
dispersion
pressure
cosmic
circuit
direct
electrons
constant
convex
investigating
42
51
conservation
continuity
35
solutions
89
conduction
conjugate
waves
of
heat
law
pumps
–
163
moment
of
156
dynamics
law
of
157
thermodynamics
and
entropy
solving
162
ideal
rotational
thermodynamic
problems
systems
158
and
concepts
159
translational
and
equilibrium
translational
rotational
154
and
rotational
motion
density
112
ux
linkage
113
ux
losses
forces
entropy
an
gas
160
162
equilibrium
hydrostatic
equilibrium
translational
and
equilibrium
equipotentials
of
bending
to
eld
principle
of
106
light
lines
estimation
146
6
free-fall
exchange
particles
excited
state
exponential
of
78
78
point
129
113
application
and
Feynman
bre
Faraday’s
coils
diagrams
105,
moving
105
elds
lines
106
energy
and
107
108
and
propagation
uniform
electric]
eld
strength
Galilean
dynamic
affecting
110
gases
49
friction
of
of
uid
resistance
rest
13
curves
of
buoyancy
and
Archimedes’
principle
164
denitions
of
density
and
pressure
164
of
equilibrium
principle
variation
of
uid
motion
Bernoulli
165–6
164
107
107
shift
support
22,
107
108
gravitational
red
148
of
gravity
dark
energy
dark
matter
of
155
212
212
gravity
greenhouse
effect
greenhouse
gases
on
92,
spacetime
149
93
92
simulation
half-value
30
heat
investigations
gases
model
gases
of
an
30
ideal
gas
heat
ow
heat
pumps
and
146
as
a
whole
to
the
positioning
163
measuring
heat
matter
heat
capacity
from
systems
lenses
148
latent
capacities
27
heat
28
27
principle
uncertainty
126
principle
126
Hertzsprung–Russell
and
heat
capacities
and
uncertainty
estimates
149
43
mirrors
of
of
Heisenberg
relativity
163
25
specic
specic
universe
of
159
engines
31
73
optics
26,
185
49
heat
phases
general
experimentally
74
thickness
methods
78
relativity
geometry
29
92
real
half-life
74
72
and
74
investigating
29–30
gases
geometric
78
74
example
131
transformation
of
global
148
71
half-life
209
applications
164
pressure
165–6
effect
164
67
147
hadrons
131
state
counters
general
hydrostatic
and
16
red
harmonics
bosons
Geiger
34
H
203
202
208,
Galilean
molecular
gauge
202
26
laws
ideal
galaxies
galaxies
greenhouse
164
strength
14,
energy
effect
observations
radiation
25,
gas
eld
gradient
205
215
110
potential
2
transformations
equation
16
electric
potential
gravity
t
[SHM]
gravitational
76
experimental
uid
best
gravitational
centre
fusion
and
elds
potential
shift
area
11
lensing
to
and
10
between
gravitational
20
units
graphs
speed
217
158
gravitational
20
factors
uncertainty
217
motion
force
gravitational
friction
216
215
axes
motion
elds
evidence
equations
gamma
76
in
168
121
of
gradient
logs
harmonic
escape
resonance
–
gravitational
188
and
10
graph
40
base
graph
and
gravitational
148
straight
217
waves
gravitational
13
shifts
logs
intercept,
velocity-time
free-fall
168
graphs
and
graphs
gravitational
187
experimental
109
132
elds
harmonic
uids
142
of
ten
laws
simple
16
of
and
the
rotational
determine
a
191
failure
potential
Pascal’s
28,
rotation
and
105
at
5
frequency
and
motion
[gravitational
under
131
frequency
distributions
potential
motion
potential
uids
14,
galaxies
base
power
potential
106
105,
potential
ssion
stations
G
gravitational
rst
fusion
–
measuring
14
frequency
nuclear
and
84
133,
10
get
straight-line
representation
interference
84
reference
frequency
ultrasound
110
109
orbital
to
clock
fundamental
energy
power
to
4
71
48
coefcient
gravitational
equipotentials
eld
of
natural
static
potential
9,
a
comparison
friction
and
fuel
frequency
friction
182–3
compared
electric
to
113
80
elds
fossil
Larmor
111
describing
electric
law
forces
disadvantages
diagrams
threshold
rotating
optics
elds
of
of
of
graphical
mediate
11
atomic
177
law
and
frame
frequency
F
Faraday’s
154
and
logs
acceleration
72
73,
(torque)
plot
216
exponentials
experience
graphs
to
displacement-time
14
uncertainties
driving
far
force
experiment
processes
78
what
equation
reference
free-body
89
bosons
71,
transformations
inertial
exchange
choosing
plotting
fractional
3
evaporation
16
84
energy
frames
bars
14,
84
146
73
10
line
forces
that
efciency
93
79
tubes
14
fundamental
fuels
106
forces
forces
advantages
106
of
105
of
particles
causes
acceleration-time
vectors
moment
4
error
154
as
the
surfaces
93
GM
forces
fossil
93
mechanisms
gluons
types
154
equipotentials
relationship
equivalence
164
rotational
165
93
for
possible
different
measuring
106
equipotential
examples
155
warming
evidence
graphs
magnitude
examples
the
115
fundamental
16
equilibrium
errors
ideal
equation
and
14
couples
by
global
streamlines
ux
167
done
165
ow,
continuity
152–3
viscosity
work
uid
laminar
176
interpretation
Higgs
bosons
78,
diagram
197
200
79
I n d e x
221
Hubble
constant
Hubble’s
law
Huygen’s
203
kinetic
principle
hydraulic
97
systems
hydroelectric
164
power
advantages
hydrogen
and
hysteresis
87
gas
laws
gas
processes
done
29,
of
192
constant
an
ideal
gas
pressure
convex
model
of
aberrations
of
compound
images
image
diverging
bre
X-ray
171
in
convex
intensity
magnifying
glass
equation
177
174
imaging
187–8
techniques
186
165
23
39,
90,
of
two-source
188
lms
equation
lens
law
22,
energy
26,
of
47,
98
14,
ideal
159
60
experimentally
quantities
square
law
resistance
60
of
light
bending
of
star
circularly
of
radiation
light
clock
70
light
curves
light
energy
light
gates
light
waves
criterion
205
light
light
moving
glass
magnication
of
of
mass
mass
and
mass
defect
years
(lys)
191
of
monochromatic
light
of
light
liquid-crystal
light
57
41
mass
41
effect
duality
displays
122
[LCDs]
functions
waves
Lorentz
217
35
sound
waves
transformations
of
effect
derivation
of
from
length
Lorentz
transformation
123
49
scale
222
25
I n d e x
1–7,
9–23,
and
and
power
138
weight
and
second
Newton’s
third
friction
free-fall
13
diagrams
14
19
impulse
23
law
of
motion
law
law
of
of
15
motion
motion
17
18
20
21
78,
79
pathways
meteorites
methane
134
and
free-body
9–12
72
190
92
micrometers
example
22
16
rst
metabolic
134
8
24
Newton’s
solid
132
33
Newton’s
mesons
from
77
122
resistance
work
Lorentz
transformation
194
pipe
contraction
factor
series
a
137
Lorentz
luminosity
in
134
47
88
momentum
217
logarithms
12
193
equations
motion
147
202
interference
position
and
129
shift
motion
waves
mass
42
red
galaxies
structure
forces
198
102
decay
matter
uid
26
natural
of
58
microscopes
compound
scanning
microscopic
microscopes
tunnelling
travelling
Kelvin
5
equilibrium
41
75
183
variables
power
energy
132
light
1
units
parallax
mechanics
41
143
67
masses
measurement
light
3
75
matter
mean
47
plane-polarized
light
(LDRs)
1,
152
dispersion
Maxwell’s
resistors
177
194
energy
masses
wind
40
177
19
two-source
22
62
41
parabolic
207
35,
point
105,
magnitude
law
62
charge
177
far
1,
to
177
size
and
due
61
angular
motion
41
lines
angular
stellar
light-dependent
Lyman
K
148
11
Lorentz
J
a
gravitational
137
transformation
39
on
Cepheid
146
polarized
light
derivation
136
force
exponential
bending
longitudinal
internal
magnetic
Doppler
longitudinal
resistance
current
unied
78
eld
62
a
magnifying
63
64
on
mathematics
number
magnetic
force
range
174
the
eld
point
178
16,
of
and
63
14,
magnetic
material
family
logarithmic
monatomic
174
solenoid
magnetic
centre
16
liquids
159
an
172,
176
a
wires
force
orders
lenses
in
electric
132
63
currents
mass
and
elds
eld
parallel
Malus’s
172
wave–particle
26
178
78
lepton
61
wire
magnitude
113
unpolarized
100
forces
determining
invariant
thin
speed
40
interference
energy
gas
185,
138
172
polarized
waves
from
175
mirrors
magnication
partially
89
parallel
internal
172,
of
plane-polarized
51,
intermolecular
Jeans
173
185
interference
isotopes
lenses
181
89
insulators
inverse
in
contraction
175
aberration
light
collisions
infra-red
internal
noise
172,
axis
leptons
and
23
inelastic
internal
172
length
172
spherical
Lenz’s
and
1
principal
175
incompressibility
thin
176
182
imaging
X-rays
impulse
mirrors
length
aberration
point
lenses
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156
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190
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128
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156
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176
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30
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89
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188
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223
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31
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law
60
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136
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196
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200
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168
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quantum
physics
atomic
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121–7,
spectra
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and
126
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model
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potential
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atom
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microwave
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123
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35
46
electromagnetic
graphs
183
48
26
and
work
work
done
work
function
by
work
done?
an
done
160
21
ideal
gas
160
121
X
X-rays
185
basic
X-ray
imaging
display
techniques
techniques
intensity,
quality
185
186
and
attenuation
185
Y
Young’s
226
double-slit
experiment
I n d e x
47,
98
O X
F O
R
D
I B
S
Physics
T
U
D
Y
G
U
I D
E
S
2014 edition
Author
f o r
T h e
I B
d I p l o m a
Ti Kik
Csy su ting t pysics Cus Bk , tis cnsiv stuy
gui eectively reinforces  t ky cncts  t tst sybus t Sl
n hl (st xin 2016). pck wit ti
assessment guidance,
it su ts t igst civnt in xs.
O xford IB study guides build unrivalled assessment potential.
Yu cn tust t t:
●
Comprehensively cv t sybus, tcing IB scictins
●
rinc all t ky tics in  cncis, us-iny t,
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●
Eectively prepare stunts  ssssnt wit visin su t n
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Us c n
stigtw ngug t
suppor t E AL learners
mti is snt in cncis cunks,
helping students focus
Su ting Cus Bk ,
v wit t IB
978 0 19 839213 2
digtic t bks wn
cnging cncts, building
understanding
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