IB Physics
Summer
Packet
1. Read Chapter 1.
2. On a separate paper to be turned in on the second day of
school, Answer the questions at the end of chapter 1 and at the
end of the study guide of chapter 1
1
MEASUREMENTS
AND
U N C E R TA I N T I E S
Introduction
This
topic
course
used
You
is
different
book.
in
most
will
The
aspects
come
from
content
across
of
other
topics
discussed
your
many
studies
aspects
in
here
in
of
the
will
in
be
physics.
this
work
the
you
context
may
expected
would
wish
to
of
read
return
other
to
to
do
this
it
subject
so,
as
you
topic
and
matter.
would
in
one
when
it
Although
not
go,
is
be
rather
you
relevant.
1.1 Measurements in physics
Understanding
Applications and skills
➔
Fundamental and derived SI units
➔
Scientic notation and metric multipliers
➔
Signicant gures
➔
Orders of magnitude
➔
Estimation
➔
Using SI units in the correct format for all
required measurements, nal answers to
calculations and presentation of raw and
processed data
➔
Using scientic notation and metric multipliers
➔
Quoting and comparing ratios, values, and
approximations to the nearest order of
magnitude
➔
Estimating quantities to an appropriate number
of signicant gures
Nature of science
In physics you will deal with the qualitative and the
with words than symbols and vice-versa. It is
quantitative, that is, descriptions of phenomena
impossible to avoid either methodology on the IB
using words and descriptions using numbers. When
Diploma course and you must learn to be careful with
we use words we need to interpret the meaning and
both your numbers and your words. In examinations
one person's interpretation will not necessarily be
you are likely to be penalized by writing contradictory
the same as another's. When we deal with numbers
statements or mathematically incorrect ones. At the
(or equations), providing we have learned the rules,
outset of the course you should make sure that you
there is no mistaking someone else's meaning. It is
understand the mathematical skills that will make
likely that some readers will be more comfor table
you into a good physicist.
1
1
M E A S U R E M E N TS
A N D
U N C E R TA I N T I E S
Quantities and units
Physicists
that
are
etc.
deal
with
physical
measureable
Quantities
are
such
as
r e l a ted
quantiti e s ,
mass,
to
one
which
leng th,
time ,
a n ot h e r
by
are
those
e le ctr ic al
e quat i on s
thi ng s
c u r r en t ,
such
as
m
__
ρ
=
which
is
the
symb o l i c
f or m
of
s ay ing
that
d e ns i t y
is
t he
ra ti o
V
of
the
mass
equation
can
be
but
are
sure
written
same
of
in
all
o b j e ct
to
its
w r i tte n
in
i ta l ic
that
t he
Roman
symbol
“m”
an
with
a
represents
throughout
the
s y m bol s
( up r ig ht)
vo lume .
the
( s lop in g )
r e pr es e nt
font
q ua ntity.
co ur s e
uni t
Not e
“ m”
“me tre ”.
b ook ,
fon t s
a nd
it
–
th ey
We
wi ll
a l so
s y m bols
this
is
Units
the
use
the
in
ho w
ar e
so m e t im e s
represents
is
the
qu a n t it i e s.
bec a u s e
So
t h at
a l wa ys
sh a r e
qua nti ty
t h is
t he
we
the
“ma s s”
con ven t i on
co n ve n t io n
us ed by
theIB.
Nature of science
Greek
The use of symbols
The
use
of
common
that,
and
Greek
in
even
capitals),
uses,
one
physics.
using
Sometimes
of
52
soon
symbols
that
trying
such
There
the
we
meaning
way
letters
are
Arabic
run
such
Greek
to
as
tie
a
out
as
d
rho
so
( ρ)
many
letters
of
x
symbol
a
case
symbols.
have
to
ν
β
beta
ξ
ksi
γ
gamma
ο
omicron
δ
delta
π
pi
ε
epsilon
ρ
rho
ζ
zeta
σ
sigma
η
eta
τ
tau
θ
theta
υ
upsilon
iota
φ
phi
κ
kappa
χ
chi
λ
lambda
ψ
psi

mu
ω
omega
very
(lower
have
α
nu
quantities
unique
and
letters
is
multiple
become
just
quantity
ι
uniquely.
happens
then
use
Of
course,
when
we
Russian
we
run
ones
must
out
of
from
consider
Greek
the
Russian
alpha
what
letters
Cyrillic
too
–
we
alphabet.
Fundamental
quantities
so
basic
other
In
the
that
all
are
those
quantities
quantities
need
to
be
that
are
expressed
considered
in
terms
of
to
be
them.
m
__
density
ρ
equation
=
only
mass
is
chosen
to
be
fundamental
V
(volume
said
It
is
by
to
being
be
essential
To
that
Système
metric
quantities
form
and
are
of
them
unit
are
only
these
rst.
2
The
The
use
and
multiples
or
as
This
density
means
both
by
that
and
volume
that,
either
for
has
and
of
are
units
been
when
people,
units
person
understood
standard
sub-multiples
being
one
are
system
between
denes
classied
made
units
worldwide
d’unités .
units
SI
of
angular
angle
two
supplementary
during
The
angle”.The
the
lengths),
are
there
the
is
of
–
never
letters
units.
The
(or
as
from
scientic
should
fundamental
have
known
developed
values
prexes
understood
to
be
used
units
base),
derived,
supplementary.
There
one
of
we
communicated
decimal
themselves
this
international
are
three
measurements
meaning.
anyconfusion.
to
all
system
of
quantities.
achieve
unambiguous
the
product
derived
others.
SI–
the
two
the
Diploma
subtended
in
SI
course,
so
we
supplementary
measurement
radian
units
is
by
a
useful
an
arc
of
and
units
the
circle
to
having
the
the
you
might
the
steradian
alternative
a
are
and
will
as
radian
(sr)
–
same
well
(rad)
the
degree
meet
length
mention
–
unit
and
is
as
only
the
of
“solid
dened
as
theradius,
1 . 1
as
shown
in
gure
Sub-topic6.1.
the
a
radian
1.
The
and
We
will
steradian
uses
the
look
is
idea
at
the
of
the
radian
in
more
three-dimensional
mapping
a
circle
on
M E A S U R E M E N T S
detail
I N
P H Y S I C S
in
equivalentof
to
the
surface
r
of
r
sphere.
1 rad
Fundamental and derived units
In
SI
on
there
the
are
seven
Diploma
completeness).
electric
and
are
that
precisely
any
standards
standard
You
will
to
for
metre
not
quantities,
but
precise
they
metre
(m):
can,
in
a
example,
given
they
are
the
to
are
be
the
right
made
will
provided
the
with
be
allow
is
easily
light
to
▲
Figure 1 Denition of the radian.
▲
Figure 2 The international prototype kilogram.
means
fundamental
the
you
and
denitions
This
compared
by
for
time,
quantity
of
these
substance,
exact
more
travelled
to
of
of
here
mass,
the
that
against
six
included
have
denitions
here
use
equipment.
of
usually
distance
know
is
amount
compared
are
will
length,
quantities
measurement
length
than
expected
these
you
candela,
temperature,
theory,
that
the
r
and
quantities
for
measurements
rather
be
units
ensure
most
so,
The
units
seventh,
reproducible,
quantity
practice,
(the
fundamental
thermodynamic
intensity.
measurement
In
course
The
current,
luminous
fundamental
accurate.
achieved
with
in
a
a
vacuum.
fundamental
see
just
how
are.
the
length
of
the
path
travelled
by
light
in
a
vacuum
during
1
_________
a
time
interval
of
299
kilogram
of
the
(kg):
kilogram
Sèvres,
near
second
(s):
kept
second.
to
the
458
equal
at
a
the
mass
Bureau
of
the
international
International
des
Poids
prototype
et
Mesures
at
Paris.
the
corresponding
ground
792
mass
of
state
duration
to
of
the
the
of
9
192
transition
631
770
between
caesium-133
periods
the
two
of
the
radiation
hyperne
levels
of
the
atom.
TOK
ampere
(A):
that
straightparallel
constant
current
conductors
of
which,
innite
if
maintained
length,
negligible
in
two
circular
Deciding on what is
cross-section,
and
placed
1
m
apart
in
vacuum,
would
produce
between
fundamental
7
these
conductors
a
force
equal
to
2
×
of
the
10
newtons
per
metre
of
length.
Who has made the decision
1
_____
kelvin
(K):
the
fraction
thermodynamic
temperature
of
the
273.16
triple
point
of
that the fundamental
water.
quantities are those of
mole
(mol):
elementary
the
mole
atoms,
such
is
the
amount
entities
used,
as
the
molecules,
of
there
substance
are
atoms
elementary
ions,
of
in
entities
electrons,
other
a
system
0.012
must
that
kg
be
particles,
of
contains
carbon–12.
specied
or
and
specied
as
many
When
may
be
groups
of
mass, length, time, electrical
current, temperature,
luminous intensity, and
amount of substance? In an
alternative universe it may
particles.
be that the fundamental
candela
(cd):
the
luminous
intensity,
in
a
given
direction,
of
a
source
quantities are based on force,
12
that
emits
that
has
monochromatic
radiation
of
frequency
540
×
10
hertz
and
volume, frequency, potential
1
___
a
radiant
intensity
in
that
direction
of
watt
per
steradian.
683
dierence, specic heat
All
quantities
can
always
relevant
be
that
are
not
expressed
equation.
For
fundamental
in
terms
example,
of
the
speed
are
known
as
fundamental
is
the
rate
of
derived
and
quantities
change
of
these
capacity, and brightness.
through
distance
a
with
Would that be a drawback
or would it have meant that
s
___
respect
to
time
or
in
equation
form
v
=
(where
s
means
the
change
in
“humanity” would have
t
distance
and
and
time
are
t
means
the
fundamental
change
in
quantities,
time).
speed
As
is
a
both
distance
derived
(and
length)
progressed at a faster rate?
quantity.
3
1
M E A S U R E M E N TS
A N D
U N C E R TA I N T I E S
The
units
used
fundamental
Note
units.
It
is
a
for
fundamental
units
and
those
straightforward
quantities
for
derived
approach
are
unsurprisingly
quantities
to
be
able
to
are
known
known
express
as
the
as
derived
unit
of
If you are reading this at the
any
quantity
in
terms
of
its
fundamental
units,
provided
you
know
the
star t of the course, it may seem
equation
relating
the
quantities.
Nineteen
fundamental
quantities
have
that there are so many things
their
own
unit
but
it
is
also
valid,
if
cumbersome,
to
express
this
in
terms
that you might not know; but,
of
fundamental
units.
For
example,
the
SI
unit
of
pressure
is
the
pascal
take hear t, “Rome was not
1
(Pa),
which
is
expressed
in
fundamental
units
as
m
2
kg s
built in a day” and soon much
will come as second nature.
1
When we write units as m s
Nature of science
2
and m s
it is a more eective
and preferable way to writing
what you may have written in
Capitals or lower case?
Notice
that
when
we
write
the
unit
newton
in
full,
we
use
a
lower
case
2
the past as m/s and m/s
; both
forms are still read as “metres
per second” and “metres per
second squared.”
n
but
we
some
All
is
a
the
have
the
capital
been
in
In
the
in
this
is
a
volt
refers
to
are
the
a
is
Sir
to
a
form
the
ampère
farad,
will
this
have
but
the
take
a
but
care!
those
symbol
the
is
that
scientist
“newton”
scientist’s
anyway)
Faraday,
so
letter,
between
Newton
of
unfortunately
case
confusion
Isaac
–
correct
scientist
no
of
unit
lower
abbreviations
shortened
Volta,
with
of
there
for
setting
start
honour
units
after
symbol
default
way
“Newton”
(which
the
should
derived
letter.
for
have
full
Sometimes
Ampère,
N
processors
unit:
unit.
amp
a
written
capital
and
so
word
units
that
use
means
surname,
named
after
etc.
Example of how to relate fundamental and derived units
The
unit
of
force
is
the
newton
(N).
This
is
a
derived
unit
and
can
2
be
expressed
for
this
is
in
that
acceleration
terms
force
or
F
=
of
can
ma.
fundamental
be
dened
Mass
is
a
as
units
as
being
kg
the
fundamental
m
s
.
The
product
of
quantity
reason
mass
but
and
acceleration
v
___
is
not.
Acceleration
is
the
rate
of
change
of
velocity
or
a
=
where
v
t
represents
time
is
a
the
fundamental
another
step
the
of
in
rate
the
change
in
of
velocity
quantity,
dening
change
topic
in
and
velocity
velocity
in
displacement
but,
for
now,
equation
for
velocity
it
t
is
change
not
so
fundamental
(a
simply
the
quantity
means
in
we
time.
need
to
quantities.
that
distance
we
in
will
a
Although
take
Velocity
discuss
given
is
later
direction).
s
___
So
the
is
v
=
with
s
being
the
change
in
t
displacement
length)
and
and
time
t
are
again
both
being
the
change
fundamental,
so
in
we
time.
are
Displacement
now
in
a
(a
position
to
1
put
N
into
already
fundamental
fundamental–
acceleration
will
units.
there
therefore
The
is
be
no
unit
of
velocity
shortened
those
form
ofvelocity
is
m s
of
and
this.
divided
by
The
these
units
time
and
are
of
so
1
m
s
____
will
be
2
which
is
written
as
m s
.
So
the
unit
of
force
will
be
the
unit
s
2
of
mass
This
is
multiplied
such
a
by
the
common
unit
unit
of
that
acceleration
it
has
its
and,
own
therefore,
name,
the
be
kg m s
newton,
2
(N≡kg m s
–
identical).
So
if
you
always
a
mathematical
you
are
in
an
way
of
expressing
examination
and
that
forget
the
the
two
unit
units
of
2
▲
could
write
kg m s
(if
you
have
time
to
work
it
out!).
Figure 3 Choosing fundamental units in
an alternative universe.
Signicant gures
Calculators
decide
4
how
usually
many
give
digits
you
to
many
write
digits
down
in
for
an
the
answer.
nal
How
answer?
do
are
force
you
.
1 . 1
Scientists
gures
use
a
(often
Consider
the
it
that
tells
the
us
next
most
we
Even
ten
we
a
digit
round
84
decimal
4
express
telling
it
8
is
eighty
is
a
the
us
that
zero,
is
number
it
tells
the
and
this
such
us
next
most
number
two
0.00246
we
would
have
rounded
up
digits
–
to
all
you
you
help
rounded
to
case
really
can
so
is
need
choose
you,
two
there
if
to
the
to
to
down
so
also
digit,
digit,
four
the
the
number
is
signicant,
rounded
gures
signicant
0,
because
The
4
is
thousand
is
the
and
third
2
is
two
the
most
thousandth
showing
that
there
are
something.
three
down,
are
P H Y S I C S
meaningful.
something.
0.00245,
the
have
this
of
means
signicant
and
next
as
that
from
In
number
here
most
there
the
would
do?
certain
thousand
we
we
number
the
a
I N
here.
because
The
is
to
“Signicant”
0.00244
do
the
to
072,
though
thousandths
wish
rounding
tos.f.).
number
digit
something.
four
of
signicant
face
signicant
If
the
signicant
When
and
number
most
something.
method
abbreviated
M E A S U R E M E N T S
two
equal
to
be
is
round
digits.
to
gures
to
the
0.0025.
consistent
or
had
it
If
last
0.0024
been
if
your
Often
is
rounded
need
had
had
it
a
5
up
will
and
up
been
so
what
and
for
have
you
to
to
been
choice
you
0.002451
be
it
rounding
with
down.
would
for
we
number
and
However,
justication
up
number
signicant
signicantgures
a
set
of
further
wanted
it
0.0025.
Some rules for using signicant gures
●
A
digit
that
signicant
●
Zeros
that
signicant
●
a
occur
–
Zeros
that
1.00
(3
When
there
–
400
(4
s.f.);
zeros
0.345
to
are
0.34500
is
(1s.f.);
always
be
signicant
no
that
(5
s.f.);
decimal
(3
s.f.)
to
the
right
–
345
is
three
point,
but
non-zero
the
(2
are
always
left
a
(1
a
non-zero
digit
are
point
non-zero
are
signicant,
digit
–
1.034
(4
s.f.);
s.f.).
trailing
needs
is
of
s.f.).
decimal
this
digits
s.f.).
of
0.003
there
–
(6
0.034
of
the
signicant
400.
occur
s.f);
right
to
between
10.3405
(3
the
they
them
will
sandwiched
occur
that
s.f.);
(tomake
–
zero
(3s.f.).
3405
signicant
provided
●
not
Non-sandwiched
not
●
is
gures
zeros
to
be
a
rarely
are
not
signicant
decimal
point)
written.
Scientic notation
One
of
(e.g.
the
the
constants
small.
fascinations
universe)
and
(quantities
This
presents
The
answer
The
speed
is
to
that
a
use
for
the
physicists
very
do
not
problem:
scientic
is
small
dealing
(e.g.
change)
how
can
with
the
electrons).
are
also
writing
very
Many
very
many
large
digits
large
physical
or
be
very
avoided?
notation.
1
of
light
has
a
value
of
299
792
458m s
.
This
can
be
rounded
to
1
three
and
signicant
it
would
be
gures
easy
to
as
300
miss
000
one
000
out
or
m s
8
this
Let
number
us
mass
000
is
analyse
of
the
000
written
writing
Sun
000
as
to
000
another
four
kg
3.00×
10
(that
is
1989
There
are
another.
In
a
lot
of
zeros
scientic
in
this
notation
1
m
large
signicant
.
add
s
(to
three
number
gures
and
is
in
1
signicant
scientic
989
000
twenty-seven
gures).
notation.
000
zeros).
000
To
The
000
000
convert
5
1
M E A S U R E M E N TS
A N D
U N C E R TA I N T I E S
this
into
moving
as
scientic
the
many
it).
This
notation
decimal
trailing
brings
point
zeros
our
as
we
30
we
number
write
places
it
as
to
1.989
the
like
to
a
back
to
the
left
decimal
and
then
we
(remember
number
original
imagine
we
can
without
number
and
write
changing
so
it
gives
30
the
A
mass
of
similar
the
idea
electron,
000
000
and
we
bring
10
the
and
is
as
000
must
000
×
10
very
an
small
decimal
000
kg.
accepted
coulombs.
the
000
our
to
has
1602
move
moving
1.989
applied
which
000
0.000
Sun
000
decimal
Again
point
000
point
numbers
value
to
we
19
1602
of
such
write
places
into
the
as
the
charge
approximately
to
this
right
the
coefcient
the
right
form.
means
The
the
on
0.000
in
000
as
1.602
order
base
is
to
always
exponent
is
19
negative.
Apart
We
from
scientic
when
can
write
avoiding
notation
weare
is
this
number
making
with
1.602
mistakes,
preferable
dealing
as
to
there
writing
several
×
is
10
a
second
numbers
numbers
in
an
8
the
value
of
the
“coefcient”
and10.
There
●
The
are
the
10
is
some
When
same
of
speed
simple
or
made
to
multiplying
●
When
dividing
●
When
raising
a
and
the
the
to
the
will
equation.
m s
the
why
This
is
Inwriting
1
10
always
and
reason
longhand.
8
,
be
is
3.00
a
is
called
number
the
between
1
the“exponent”.
numbers
the
exponent
must
be
the
same.
numbers
number
it
in
apply:
numbers
multiply
3.00 ×
“base”
rules
be
When
and
as
subtracting
●
power
light
number
called
adding
or
of
C.
we
we
to
add
the
subtract
a
power
exponent
exponents.
one
we
by
exponent
raise
the
the
from
the
coefcient
other.
to
the
power.
Worked examples
In
of
these
the
examples
we
are
going
evaluate
each
Here
1.40
×
10
the
exponents
negative)
6
1
to
calculations.
to
give:
are
5
subtracted
8
=
(since
5
+
3.5
×
the
8
is
−3
3
10
So
we
write
this
product
as:
7.8
×
10
5
Solution
4.8 × 10
_
4
2
6
These
so
must
that
be
both
written
numbers
as
1.40
have
×
the
10
same
0.35
×
can
now
be
added
directly
to
1.75
×
10
Solution
exponents.
give
×
10
6
They
3.1
6
+
The
coefcients
subtracted
5
2
3.7
×
10
are
divided
and
the
exponents
so
we
have:
4.8
÷
3.1
=
1.548
(which
8
×
2.1
×
10
we
round
to
1.5)
Solution
The
coefcients
are
multiplied
and
the
And
5
2
=
This
makes
3
exponents
3
are
added,
so
we
have:
3.7
×
2.1
=
7.77
(which
the
we
to
will
7.8
to
discuss
be
in
later
line
in
with
this
the
topic)
data
and:
–
5
something
+
8
=
result
of
the
division
1.5
×
10
we
7
round
5
(3.6
×
10
3
)
13
Solution
13
So
we
write
this
product
as:
5
3
3.7
×
10
7.8
×
3
10
We
8
×
2.1
×
cube
And
10
3.6
and
multiply
7
3.6
by
3
=
to
46.7
give
21
21
This
Solution
gives
46.7
×
10
,
which
should
22
4.7
Again
the
coefcients
exponents
6
are
added,
are
so
multiplied
we
have:
and
3.7
×
the
2.1
=
are
10
7.8
×
10
in
scientic
notation.
become
1 . 1
M E A S U R E M E N T S
I N
P H Y S I C S
Factor
Metric multipliers (prexes)
24
10
Scientists
have
a
second
way
of
abbreviating
units:
by
using
21
(usually
called
“prexes”).
An
SI
prex
is
a
name
18
symbol
that
is
written
before
a
unit
to
indicate
Υ
zetta
Ζ
exa
Ε
or
10
associated
Symbol
yotta
metric
10
multipliers
Name
the
15
10
peta
Ρ
tera
Τ
12
appropriate
power
of10.
So
instead
of
writing
2.5
×
10
J
we
could
12
10
alternatively
write
this
as
2.5
TJ
(terajoule).
Figure
4
gives
the
20
SI
9
giga
10
prexes
in
–
these
are
provided
for
you
as
part
of
the
data
booklet
6
used
mega
10
3
examinations.
2
hecto
10
1
deka
10
Orders of magnitude
1
10
2
An
important
skill
for
physicists
is
to
understand
whether
or
not
10
the
3
physics
being
considered
is
sensible.
When
performing
a
calculation
10
in
6
10
which
someone’s
mass
was
calculated
to
be
5000 kg,
this
should
10
bells.
Since
average
adult
masses
(“weights”)
will
usually
12
a
value
of
5000 kg
is
an
10
18
number
rounded
to
the
nearest
power
of
10
is
called
an
order
of
10
21
magnitude.
For
example,
when
considering
the
average
adult
10
human
24
mass:
60–80
kg
is
closer
to
2
magnitude
humans
to
100
than
c
m
micro

nano
n
pico
p
kg
than
10
kg,
making
the
order
10
of
femto
atto
f
a
zepto
z
yocto
y
1
10
have
100
milli
d
impossibility.
15
A
centi
be
10
60–90 kg,
deci
ring
9
alarm
Μ
kilo
10
and
a
not
mass
10.
In
a
10
of
.
Of
100
similar
course,
kg,
we
simply
way,
the
are
that
mass
not
their
of
a
saying
that
average
sheet
of
all
mass
A4
adult
is
▲
Figure 4 SI metric multipliers.
closer
paper
may
be
3
3.8
g
which,
expressed
in
kg,
will
be
3.8
×
10
kg.
Since
3.8
is
closer
to
3
1
than
to
suggests
10,
that
this
the
makes
ratio
the
of
order
adult
of
mass
magnitude
to
the
mass
of
of
its
a
mass
piece
10
of
kg.
paper
This
(should
2
10
____
you
wish
to
make
this
comparison)=
2
=
−(
3)
5
10
=
10
=
100
000.
In
3
10
other
words,
heavier
than
an
a
adult
sheet
human
of
A4
is
5
orders
of
magnitude
(5powers
of
10)
paper.
Estimation
Estimation
produce
a
is
Estimation
result
in
a
can
skill
is
that
that
is
closely
value
Whenever
you
a
value
you
is
a
related
that
is
see
to
more
measure
usually
used
the
by
useable
a
scientists
nding
precise
length
whole
and
approximation
an
order
than
with
a
number
the
of
a
in
true
order
power
calibrated
millimetres
to
value.
magnitude,
nearest
ruler
of
others
to
in
but
but
of
may
10.
millimetres
will
need
to
1
__
estimate
to
the
next
mm
–
you
may
need
a
magnifying
glass
to
help
10
you
to
do
this.
The
same
thing
is
true
with
most
non-digital
measuring
instruments.
Similarly,
you
area
of
shows
The
a
a
curve
curve
under
26
and
in
has
about
of
the
all.
60
This
area
N s
or
gives
gives
the
how
the
total
30
to
under
area
2
to
×
s
object
(as
you
yellow
=
2
to
there
varies
will
the
This
curve,
nd
are.
with
see
about
under
N s.
need
squares
totalling
squares
1
non-regular
will
rectangles
impulse
full
a
you
an
squares
N
so
many
complete
partial
about
area
applied
nearly
further
the
actual
force
equivalent
for
nd
the
estimate
the
graph
are
to
out
and
how
complete
there
an
need
work
rectangle
are
squares
you
truly
graph
area
There
of
when
cannot
in
Figure5
time.
Topic2).
under
four
the
full
curve.
gives
the
an
Each
estimate
impulse.
7
1
M E A S U R E M E N TS
A N D
U N C E R TA I N T I E S
In
an
examination,
tolerance
(60
±
2)
given
estimation
with
the
questions
accepted
will
answer,
so
always
in
this
have
case
a
it
might
be
N s.
12
These two par tial
10
N/ecrof
squares may be
8
combined to approximate
6
to a whole square, etc.
4
2
0
0
1
2
3
4
5
6
7
8
time/s
▲
Figure 5
1.2 Uncer tainties and errors
Understanding
Applications and skills
➔
Random and systematic errors
➔
Absolute, fractional, and percentage uncer tainties
➔
Error bars
➔
Uncer tainty of gradient and intercepts
➔
Explaining how random and systematic errors
can be identied and reduced
➔
Collecting data that include absolute and/or
fractional uncer tainties and stating these as an
uncer tainty range (using ±)
➔
Nature of science
Propagating uncer tainties through calculations
involving addition, subtraction, multiplication,
division, and raising to a power
In Sub-topic 1.1 we looked at the how we dene
the fundamental physical quantities. Each of these
is measured on a scale by comparing the quantity
➔
Using error bars to calculate the uncer tainty in
gradients and intercepts
with something that is “precisely reproducible”. By
Equations
precisely reproducible do we mean “exact”? The
answer to this is no. If we think about the denition
Propagation of uncer tainties:
7
of the ampere, we will measure a force of 2 × 10
N.
If: y = a ± b
7
If we measure it to be 2.1 × 10
it doesn’t invalidate
then: y = a + b
the measurement since the denition is given to just
_____
ab
one signicant gure. All measurements have their
If: y =
c
y
______
a
______
limitations or uncer tainties and it is impor tant that
then:
=
______
b
+
b
both the measurer and the person working with the
n
If: y = a
measurement understand what the limitations are.
y
_______
then:
This is why we must always consider the uncer tainty
in any measurement of a physical quantity.
8
______
a
=
y
⎜n
a
______
c
+
a
y
⎟
c
1 . 2
U N C E R T A I N T I E S
A N D
E R R O R S
Uncertainties in measurement
Introduction
No
it
experimental
is
always
reasons
There
for
are
reading
this
two
–
quantity
subject
in
to
this
types
be
absolutely
degree
of
accurate
uncertainty.
when
We
will
measured–
look
at
the
section.
of
systematic
can
some
error
and
that
contribute
to
our
uncertainty
about
a
random.
Systematic errors
As
the
used
name
to
example,
of
the
scale
warp
timer
can
because
▲
a
the
the
may
and,
run
these
types
measurement.
equipment,
Rulers
A
suggests,
make
or
as
be
result,
slowly
battery
incorrectly
because
a
of
if
its
voltage
it
has
the
may
are
be
to
to
no
a
being
apparatus.
during
longer
the
system
period
becomes
when
the
faulty
either
over
are
crystal
fallen–
due
due
calibrated
changed
divisions
quartz
has
errors
This
For
manufacture
of
time.
symmetrical.
damaged
timer
(not
simply
stops).
Figure 1 Zero error on digital calliper.
When
measuring
dependent
actually
The
setting
zero
Figure
0.000
resistors
positioned
zero
reads
1
shows
mm
but
readings
be
reset
to
it
is
to
accept
in
making
standard
The
real
them
sets
by
give
well;
In
on
will
zero
not
the
that
same
if
are
a
deal
is
is
be
the
of
usage,
0.01
subtracted
never
errors
with
that
is
in
that
the
it
no
it
should
This
The
any
longer
is
calliper
that
can
made.
experimenters
the
signicant
the
scale
only
possible
are
results
a
If
both
third
have
effort
against
systematic
apparatus.
they
read
means
readings
spend
checking
another
likelihood
disagreement
be.
and
else
removes
is
is
is.
that
This
mm.
from
error
by
light-
source
LDR
so
or
error
should
or
the
the
closed.
reads
systematic
task
to
they
sources
where
zero
jaws
it
than
a
re-calibrated
to
the
a
error.
check
two
performing
set
may
be
difference.
with
zero
errors
experimentation.
measurement
the
and
reading
same
results,
due
instruments,
systematic
the
any
could
know
surface
called
with
a
to
drift,
is
bigger
spot
radioactive
active
error
their
they
there
resolve
we
to
on
with
performing
our
mm
Repeating
problem
to
mm
hard
can
this
zero
0.01
0.01
–
calliper
a
possible
sure
the
is
reading
scale.
general
with
or
is
sealed
the
apparatus
digital
be
it
where
should
there
however,
needed
a
it
from
(LDRs),
or
on
when
all
Often
distances
said
to
be
as
When
well
as
we
can
systematic
and
errors
then
are
move
small,
a
accurate
9
1
M E A S U R E M E N TS
A N D
U N C E R TA I N T I E S
Nature of science
Systematic errors
Uncertainty
quoted
view
To
of
be
larger
the
▲
to
when
±0.5
how
on
the
or
precisely
safe
side
uncertainty
reading
using
mm
lies
a
300
±1
mm
you
you
and
within
can
mm
gauge
might
then
your
ruler
may
depending
you
the
wish
will
to
be
be
on
The
your
reading.
use
meter
with
a
giving
this
gure
large
4
shows
scale
reading
as
–
an
there
being
analogue
is
(40
ammeter
justication
±
5)
in
A.
the
sure
that
bounds.
Figure 2 Millimeter (mm) scale on ruler.
Figure 4 Analogue scale.
▲
You
should
make
sure
you
observe
the
scale
above
and
at
right
angles
to
the
plane
ruler
in
order
to
avoid
parallax
digital
0.27
A
In
±
0.01)
each
of
quoted
to
decimal
to
9
8
7
6
4
3
2
which
in
gure
should
be
5
gives
recorded
a
value
as
errors.
(0.27
01
ammeter
of
of
the
Figure5 Digital scale.
▲
from
The
directly
do
A.
these
the
places)
this
as
examples
same
as
the
indicative
energy
value
as
the
uncertainty
precision
the
of
(number
reading
number
always
of
–
8
J
is
essential
decimal
precision.
being
it
we
places
When
are
is
of
we
is
write
implying
an
that
it
1
is
(8
±
1)
precision
▲
in
fairly
J
and
of
if
we
write
it
as
8.0
J
it
implies
a
±0.1J.
Figure 3 Parallax error.
Random errors
Random
errors
frequently
gure
be
can
when
when
difcult
the
to
judge
For
smallest
the
maximum
use
this
calibrated.
is
a
single
you
Choosing
is
measuring
likely
to
be
oscillations
being
(16.3
precision
than
0.1
of
s
as
of
±
far
a
then
is
than
pendulum
s.
This
timer.
you
If
no
but,
to
is
it
it
a
this.
take
idea
quote
is
a
is
the
your
that
this
value
to
be
to
the
scales
are
severely
manufacturer’s
do.
the
precision
time
twenty
recorded
dominates
time
of
better
–
may
use
be
may
determining
timed
time
reaction
instead
it
different
reaction
you
should
reaction
your
well
the
can
your
if
in
problem
the
you
since
then
precaution
a
know
best
example,
s
signicant
are
scale
inappropriate
16.27
that
you
precisely
on
you
the
could
sensible
digit
last
most
changed
since
scales
how
reading,
For
know
is
unless
because
you
should
in
the
up
insensitive
have
But,
digital
probably
manually
uncertainty
greater
0.1)
the
time
with
is
crop
uncertainty
signicant
uncertainty
a
the
the
but
estimate
would
values,
really
it
to
available.
of
least
calibration,
timer
reading
have
the
has
instrument
reading
Dealing
the
the
a
range
data
of
measurement,
an
division
underestimate
When
If
whether
precision.
regarding
any
scale.
scale
that
in
experimenter
a
possible
larger
likelihood
10
the
reading
circumstances.
than
occur
0.1
is
s.
as
the
greater
1 . 2
The
best
way
readings
values
value
will
that
largest
a
more
value
not
handling
nd
give
Readings
does
of
and
the
value
that
advanced
minus
with
mean
random
average
the
small
they
of
is
a
error
is
set
good
to
of
take
data.
a
series
Half
approximation
analysis
smallest
provides.
the
to
The
of
E R R O R S
repeat
range
the
A N D
of
the
statistical
range
is
the
value.
random
are
errors
each
U N C E R T A I N T I E S
errors
accurate,
are
said
to
be
precise
(this
however)
Worked examples
1
In
measuring
glass
the
the
interface
following
angle
for
a
of
refraction
constant
results
were
angle
at
of
obtained
an
air-
incidence
(using
a
T/ °C
0
1
2
3
4
5
6
7
8
9
10
o
protractor
with
a
precision
of
± 1
):
Express
45°,
47°,
46°,
45°,
an
How
should
we
the
temperature
and
its
uncertainty
to
44°
express
the
angle
appropriate
number
of
signicant
gures.
of
Solution
refraction?
The
scale
is
calibrated
in
degrees
but
they
are
Solution
quite
The
mean
(47°
−
of
44°)
Half
these
=
the
values
is
45.4 °
and
the
range
is
a
3°
is
6
then
do
to
decision)
so
angle
of
we
record
our
overall
value
0.5)
the
of
the
protractor
is
(although
the
values
°C.
quote
value
should
our
mean
to
a
whole
This
a
be
is
closer
judgement
the
recorded
as
measurement
and
should
be
to
the
same
number
of
and
it
will
round
places.
A
student
down
a
takes
certain
not
and
minimize
so
we
our
should
means
that
the
angle
a
series
quantity.
of
He
measurements
then
averages
his
to
What
aspects
of
systematic
uncertainty
round
this
random
of
uncertainties
is
he
addressing
by
up
taking
2°.
should
is
expect
number
and
unrealistically
to
to
meniscus
that
Remember
measurements.
We
The
6.5
uncertainty
of
45°.
°C.
reasonable
±1°,
3
(integral)
is
refraction?
precision
weshould
±0.5
it
for
decimal
Since
so
1.5°
the
the
of
than
(6.0±
How
here,
precision
to
range
clear
repeats
and
averages?
refraction
Solution
should
be
recorded
as
45 ±
2°
Systematic
2
The
diagram
below
shows
the
position
of
repeat
meniscus
of
the
mercury
in
a
errors
are
not
dealt
with
by
means
of
the
readings,
but
taking
repeat
readings
and
mercury-in-glass
averaging
them
should
cause
the
average
value
thermometer.
to
be
closer
chosen
to
the
individual
true
value
than
a
randomly
measurement.
Absolute and fractional uncer tainties
The
values
called
as
of
uncertainties
absolute
thequantity
that
we
uncertainties .
and
should
be
have
These
written
been
values
to
the
looking
have
same
the
at
are
same
number
units
of
decimalplaces.
Dividing
(one
the
the
with
uncertainty
no
fractional
units)
and
by
the
gives
uncertainty
value
us
gives
the
the
itself
leaves
fractional
percentage
a
dimensionless
uncertainty.
quantity
Percentaging
uncertainty
11
1
M E A S U R E M E N TS
A N D
U N C E R TA I N T I E S
Worked example
Calculate
the
uncertainties
force,
2.5
N,
absolute,
for
the
fractional,
following
and
percentage
measurements
of
range
a
F:
up
2.8
N,
2.6
=
(2.8
absolute
N
to
2.5)
N
uncertainty
0.2
=
of
0.3
N,
0.15N
giving
that
an
rounds
N
We
would
write
the
fractional
our
value
for
F
as
(2.6
±
0.2)
N
0.2
___
uncertainty
is
=
0.077
and
the
2.6
Solution
percentage
2.5
N
+
2.8
N
+
2.6
uncertainty
will
be
0.077
×
100%
=
N
________________
mean
value
=
=
2.63
N,
this
is
7.7%
3
rounded
down
to
2.6
N
Propagation of uncer tainties
Often
we
calculate
the
measure
other
calculated
uncertainties
from.
This
There
are
is
value
in
and
if
want
further
the
11
and
b,
of
you
uncertainty
a,
c,
be
then
an
determined
that
rules
that
advanced
rules
these
can
use
we
we
can
here
them
a
combination
used
apply
of
to
to
in
of
calculate
in
a
when
this
but
(since
up
uncertainty
the
the
value
uncertainties
developed,
rules
look
have
of
measurements
The
from
treatment
are
our
equation.
propagation
these
application
them
you
as
more
how
and
with
quantities
simple
In
prove
In
we
you
text
we
topic
are
will
book
are
we
propagating
would
going
never
or
on
to
be
the
focus
asked
to
Internet
information).
equations
etc.
are
the
discussed
absolute
next,
a,
b,
c,
uncertainties
etc.
in
are
the
these
quantities
quantities.
Addition and subtraction
This
21
the
some
demonstrate
your
will
known
uncertainties.
on
quantities
quantities
is
the
quantities
a
=
b
of
the
always
+
c
rules
add
or
a
because
their
=
when
absolute
b
c
we
add
or
subtract
uncertainties.
then
a
=
b
+
c
31
When
easiest
we
In
order
41
added
So
if
to
or
we
use
these
subtracted
are
relationships
must
combining
have
two
don’t
the
forget
same
masses
m
and
m
1
51
be
m
the
=
sum
(200
of
±
the
10)
other
g
and
two
m
1
61
m
7
1
81
91
use
are
means
(300
±
When
that
ruler
is
found
is
the
used
by
Larger
write
20)
we
a
quantities
being
then
the
total
m
mass
will
2
masses.
(100
this
±
10)
g
so
m
=
300
g
and
m
=
20
g
more
set
often
the
zero
judgement
to
is
subtracting
=
measurement
a
the
each
measurement
±
we
our
0.5)
realise
ruler
where
the
when
against
zero
is
we
one
are
end
measuring
of
positioned
an
and
object
this
really
mm.
metal
smaller
rod
as
shown
in
measurement
measurement
195.0
=
than
of
of
(0.0
measure
for
as:
g
value
uncertainty
is
± 0.5
gure
from
6.
the
The
larger
length
one.
mm.
mm
118.5
mm
Figure 6 Measuring a length.
Length
12
should
making
Smaller
▲
we
subtraction
A
The
02
=
lengths.
we
the
2
meaning
We
=
that
units.
=
(76.5
±
1.0)
mm
as
the
uncertainty
is
0.5
mm
+
0.5
mm
1 . 2
U N C E R T A I N T I E S
A N D
E R R O R S
Nature of science
Subtracting values
When
you
subtraction
need
quantity
to
be
is
involved
particularly
becomes
smaller
in
a
careful.
in
size
If
relationship
The
=
then
resulting
(because
a
while
the
absolute
c
a
=
uncertainty
subtracted:
of
b
addition).
=
4.0
±
Imagine
0.1
and
c
two
=3.0
values
±
won’t
concern
quantities
ourselves
actually
with
what
1.0
and
since
a
=
b
+
c
have
gone
from
two
uncertainty
values
is
2.5%
in
which
and
the
3.3%
that
to
a
calculated
value
with
uncertainty
0.1.
of
We
=
0.2
respectively
are
a
becomes
percentage
larger(because
then
of
We
subtraction),
b
20%.
Now
that
really
is
propagation
of
uncertainties!
these
arehere.
Multiplication and division
When
we
multiply
percentage
or
divide
uncertainties,
quantities
a
=
bc
or
a
=
a
b
___
form
or
a
=
b
___
+
c
b
are
of
fractional
c
___
=
a
There
their
c
or
c
then
add
so:
b
when
we
very
few
relationships
multiplication
or
in
physics
that
do
not
include
some
division.
m
__
We
have
seen
that
density
ρ
is
given
by
the
expression
ρ
=
where
V
m
is
the
mass
particular
the
the
of
is
be
sides
uncertainty
the
of
the
substance
percentage
and
uncertainty
V
in
is
its
the
volume.
mass
is
For
5%
a
and
for
uncertainty
in
the
calculated
value
of
the
density
will
±17%
sample
the
sample
12%
percentage
therefore
If
a
sample,
volume
The
of
had
was
of
been
4%
we
cubical
can
see
in
shape
how
and
this
the
brings
uncertainty
about
a
in
each
volume
with
12%:
3
For
a
cube
l
__
V
___
so
=
V
This
the
volume
l
__
+
example
the
cube
of
the
side
length
(V
=
l
=
l
×
l
×
l)
l
__
+
l
l
is
=
4%
+
4%
+
4%
=
12%
l
leads
us
to:
Raising a quantity to a power
V
___
From
the
cube
example
you
might
have
spotted
that
l
__
=
3
V
l
n
This
result
positive
or
can
a
The
whole,
so
that
integral,
when
or
a
=
decimal
b
(where
n
can
be
a
number)
___
=
a
generalized
b
___
then
be
negative
⎜
n
b
modulus
uncertainty
⎟
sign
can
is
be
included
either
as
an
positive
alternative
or
way
of
telling
us
that
the
negative.
13
1
M E A S U R E M E N TS
A N D
U N C E R TA I N T I E S
Worked example
The
period
T
of
oscillation
of
a
mass
m
on
a
spring,
Using
the
division
and
power
relationships:
m
__
having
spring
constant
k
is
T
=
2π
√
a
b
___
k
=
at
worry
this
about
what
these
quantities
actually
+
uncertainty
m
5%.
so
the
in
m
in
k
is
11%
and
the
uncertainty
m
1
here
___
=
k
1
___
2
k
+
T
m
2
percentage
+
uncertainty
in
Calculate
the
approximate
uncertainty
in
a
=
1.20
that
in
T
will
be
half
that
k
means
that
the
percentage
0.5
×
0.5
×
5%
+
11%
=
error
in
8%
value
If
of
half
in
T
T
or
c
mean
This
for
n
b
stage.
The
is
T
___
___
n
a
Don’t
c
___
the
measured
value
of
T
is
1.20
s.
s
then
the
8
___
absolute
uncertainty
is
1.20
×
=
0.096
This
100
rounds
Solution
(1.20
First
let’s
write
it
adjust
the
equation
a
little
–
we
up
±
to
0.10
0.10)
and
so
we
quote
T
as
being
s.
can
Remember
as
that
the
quantity
and
the
uncertainty
n
1
m
T
=
2π
(
k
Although
to
as
must
b
__
be
to
the
same
number
of
decimal
places
2
which
)
is
of
the
form
a
=
2π
(
)
c
and
we
many
percentage
will
truncate
signicant
π,
gures
uncertainty
in
π,
we
as
as
can
we
in
really
wish
2
will
write
and
so
it
so
the
precision
zeros
in
the
are
important,
as
they
give
us
the
value.
the
bezero.
Drawing graphs
3.0
An
important
justication
for
experimental
work
is
to
investigate
the
1
relationship
s m/deeps
2.0
revealing
dividing
resistor
between
even
the
to
if
it
physical
can
potential
nd
the
be
quantities.
used
to
difference
resistance.
One
calculate
across
a
Although
a
set
values
physical
resistor
the
of
by
is
rarely
constant,
the
current
calculation
does
very
such
in
tell
as
the
you
the
1.0
resistance
for
resistance
depends
one
value
you
resistance
of
upon
current,
the
it
says
current.
nothing
Taking
a
about
series
of
whether
values
the
would
tell
0
0
0.1
0.2
0.3
if
the
was
constant
but,
with
the
expected
random
errors,
0.4
it
would
still
not
be
denitive.
By
plotting
a
graph
and
drawing
the
line
time/s
▲
of
Figure 7 Error bars.
best
some
t
the
other
pattern
of
results
is
far
easier
to
spot,
whether
it
is
linear
or
relationship.
Error bars
In
The value could lie
anywhere inside this
plotting
error
bars.
a
point
These
on
a
graph,
are
vertical
the
quantity
uncertainties
and
are
horizontal
recognized
lines
that
by
adding
indicate
the
rectangle
possible
range
of
being
measured.
Suppose
at
a
time
of
1
(0.2±
0.05)
plotted
as
s
the
shown
speed
in
of
gure
an
object
was
(1.2
±
0.2)
m
s
this
would
be
7.
3.0
This
means
that
the
value
could
possibly
be
within
the
rectangle
that
1
s m/deeps
touches
the
ends
of
the
error
for
the
data
bars
as
shown
in
gure
8.
This
is
the
zone
2.0
of
uncertainty
spreads
1.0
line
0
the
and
points
also
so
passes
that
point.
they
through
are
the
A
line
evenly
error
of
best
t
should
distributed
on
be
both
one
that
sides
of
the
bars.
Uncertainties with gradients
0
0.1
0.2
0.3
0.4
Using
a
plot
graph
computer
application,
such
as
a
spreadsheet,
can
allow
you
to
time/s
▲
a
with
data
points
and
error
bars.
You
can
then
read
off
the
Figure 8 Zone of uncer tainty.
gradient
14
and
the
intercepts
from
a
linear
graph
directly.
The
application
1 . 2
will
automatically
lines
with
while
but
on
still
the
an
currents
graph
line
of
of
to
=
the
This
can
So
graph
a
The
a
potential
be
ε
table
of
V
on
The
that
to
graph
are
V
+
against
(the
emf
the
the
of
just
of
of
right
and
error
be
of
are
bars
by
see
the
force
the
it
Topic5
are
is
E R R O R S
trend
possible
–
commonly,
furthest
essential
(emf)
across
are
should
the
the
A N D
apart
that
all
bars.
resistors
I,
add
just
quite
error
current,
in
are
that
connected
across
will
then
appropriate,
all
against
r
can
that
Students
electromotive
V
,
you
You
bars.
the
could
the
emf
and
the
internal
cell.
then
give
of
a
The
measured.
a
straight
cell
is
theequation:
Ir
to
I
is
the
give
of
V
=
ε
Ir
gradient
r
(the
internal
resistance)
and
cell).
shows
a
voltmeter
9
of
resistors
shows
set
of
measurements
gure
error
differences
As
line.
gradients
through
resistance
rearranged
milliammeter
identical
=
pass
difference,
gradient.
r)
trend
the
these
series
internal
+
all
measure
a
potential
best
shallowest
extremes
draw
to
cell,
and
I(R
intercept
the
you
negative
related
ε
in
a
through
use
experiment
of
the
and
Although
lines
resistance
A
passing
graphs.
trend
draw
steepest
incorrectly,
the
In
the
U N C E R T A I N T I E S
of
results
low
given
the
in
line
from
precision
of
the
best
this
the
experiment.
repeat
values
With
are
table.
t
together
with
two
lines
possible.
EMF and Internal Resistance
1.8
I ±
1.6
5/mA
V ±
0.1/V
V = −0.013I + 1.68
15
1.5
20
1.4
25
1.4
30
1.3
35
1.2
0.6
50
1.1
0.4
55
0.9
70
0.8
85
0.6
90
0.5
V = −0.0127I + 1.64
1.4
V = −0.0153I + 1.78
1.2
1.0
V/V
0.8
0.2
0.0
0
20
40
60
80
100
120
140
I/mA
▲
Figure 9 A graph of potential dierence, V, against current, I, for a cell.
Converting
the
to
from
internal
15.3
This
The

(=
leads
to
2.6
a
intercept
to
1.7
V
of
the
just
)
the
the
(the
to
for
V
gures).
is
lines
This
that
=
axis
data
possible
r
amps,
gradient)
meaning
value
on
(since
signicant
milliamps
resistance
is
half
(13.0
of
the
the
±
1.6
means
the
to
that
of
to
ε
=
and
=
of
these
the
1.3
lines
range
is
suggest
from
that
12.7


.
best
2
1.8

range
1.3)
line
essentially
gives
equations
13.0
t
=
1.68
signicant
V
(when
(1.6
±
which
gures).
rounded
0.1)
to
rounds
The
range
two
V
.
15
1
M E A S U R E M E N TS
A N D
U N C E R TA I N T I E S
Nature of science
Drawing graphs manually
●
One
of
draw
the
skills
graphs
tested
on
by
this
expected
hand
in
the
and
data
of
physicists
you
may
analysis
is
well
to
include
be
question
origin
3
of
the
examination.
IB
Diploma
You
are
Programme
also
likely
to
graphs
for
your
internal
need
Try
to
look
at
your
extreme
Use
sensible
and
your
need
an
a
idea
of
what
minimum
reasonable
of
chance
scales
six
of
Use
scales
that
will
values
to
use.
points
to
drawing
allow
you
so
that
out
as
much
as
multiples
Try
to
plot
a
You
your
sheet
and
as
lose
page,
that
but
give
you
valid
to
possible
not
would
you
intercept
marks).
if
you
You
need
can
the
10
graph
shows
is
very
some
of
useful
the
key
when
that
3,
your
will
make
clear-cut
4,
or
a
line
always
when
graph
–
if
7
–
both
(avoid
stick
as
you
apparently
up,
you
plotting
scales
to
2,
5,
will
they
are
correct.
are
doing
the
see
them
unusual
values
and
check
can
Before
you
consider
draw
your
whether
line
or
of
not
it
best
is
t,
may
that
well
you
be
a
trend
can
anomalous
ignore,
but
if
points
there
denite
to
the
curve
then
you
for
a
smooth
curve
drawn
with
a
single
don’t
of
a
good
and
hand-drawn
not
“sketched”
graph.
artistically!
Calculating
the
gradients
values.
6
line that is just possible
3
m
10
Second “just possible”
line
(398 − 288) K
7
3
m
should be added
for a real investigation –
1
K
it has been missed out
56
here so that you can
gradient of just possible line
6
(58.6 − 42.0) ×
10
clearly see the values
3
m
=
on the two lines
(398 − 286) K
7
1.48 × 10
3
m
1
K
52
m
3
6
Use a large gradient
01/V
50
triangle to reduce
uncer tainties
All values on V axis
48
have been divided by
6
10
3
and are in m
46
outlier ignored
Uncer tainties in
for best
temperature are too
straight line
small to draw error bars
44
False origin – neither
42
line has been forced
through this point
40
260
280
300
320
340
T/K
▲
16
Figure 10 Hand-drawn graph.
is
should
calculate
(59.6 − 42.0) ×
=
a
quality
you
checking
There
(outliers)
=
54
need
or
second
elements
1.60 × 10
that
you
straight
gradient of best straight line
=
10).
your
best straight line
58
that
and
a
line
Figure
false
should
onto
your
one;
of
crop
opt
an
a
line.
spread
(you
overspill
damage
you
will
curve.
ll
give
you
to
points
axes
assessment.
●
●
scales
calculations
experiment
have
your
ne).
to
●
●
is
physics
are
draw
origin,
in
●
Paper
the
(which
360
380
400
on
1 . 2
U N C E R T A I N T I E S
A N D
E R R O R S
Linearizing graphs
Note
Many
relationships
proportional
one
and
quantity
with
a
between
straight
against
the
non-proportional
relationship
and
physical
line
quantities
cannot
other.
There
relationships:
when
we
do
be
are
are
not
obtained
two
when
directly
simply
approaches
we
know
the
by
to
Relationships can never be
plotting
“indirectly propor tional” – this
dealing
form
of
is a meaningless term since it
the
is too vague. Consequently the
not.
term “propor tional” means the
1
__
If
we
do
know
the
form
of
the
relationship
such
as
p
∝
(for
a
gas
held
V
at
constant
of
temperature)
a
graph
a
straight-line
one
quantity
origin
or
T
against
graph.
√
∝
l
the
An
(for
a
simple
power
of
alternative
the
for
same as “directly propor tional”.
pendulum)
other
the
we
quantity
simple
can
to
plot
obtain
pendulum
is
to
2
plot
a
graph
of
T
against
l
which
will
give
the
same
result.
This topic is dealt with in more
detail and with many fur ther
You
should
think
about
the
propagation
of
errors
when
you
examples on the website.
2
consider
The
the
following
technique
discussion
that
undertake
an
relative
SL
merits
applies
students
Extended
may
Essay
of
to
HL
wish
in
a
T
plotting
examinations,
to
utilize
science
it
√
against
but
when
l
T
or
offers
against
such
completing
a
IAs
l.
useful
or
if
they
subject.
expontential shape
If
we
don’t
that
one
the
actual
quantity
power
is
involved
related
to
the
in
a
relationship,
other,
we
can
but
write
a
45
we
stinu y rartibra/N
suspect
know
general
n
relationship
By
taking
can
logs
arrange
means
have
that
an
in
the
of
into
a
form
this
log
graph
intercept
y
=
kx
equation
y
=
of
on
n log
log
the
y
where
we
x
+
obtain
log
against
log
y
k
k
of
log
and
log
axis
and
x
is
will
log
y
n
are
=
log
of
be
the
constants.
k
+
n log
form
linear
of
y
=
x
which
m x
+
gradient
n
c.
we
This
and
0.039t
30
N = 32e
25
20
15
10
5
0
k
0
This
technique
relationship
is
also
logs
a
to
very
between
useful
base
is
e,
way
useful
two
of
instead
in
quantities
dealing
of
carrying
base
with
10.
out
really
investigations
is
not
exponential
For
example,
known.
when
The
relationships
radioactive
40
60
80
100
t/arbitrary units
technique
by
20
a
▲
Figure 11
taking
nuclides
linear form of same data
decay
so
that
either
the
activity
or
the
number
of
nuclei
remaining
falls
4.0
according
to
the
same
general
form.
Writing
the
decay
equation
for
the
3.5
number
of
nuclei
remaining
gives:
3.0
In N = −0.0385t + 3.4657
λt
N
=
N
e
by
taking
logs
to
the
base
e
we
get
ln N
=
lnN
0
is
(where
2.5
0
the
usual
way
of
writing
N nI
lnN
λt
N).
log
e
2.0
1.5
By
plotting
a
graph
of
ln N
against
t
the
gradient
will
be
λt
and
the
1.0
intercept
on
the
lnN
axis
will
be
lnN
.
This
linearizes
the
graph
shown
0
0.5
in
gure
11
producing
the
graph
of
gure
12.
A
linear
graph
is
easier
to
0.0
analyse
than
a
curve.
0
Capacitors
also
mathematical
discharge
through
relationship
as
that
resistors
used
for
using
the
same
radioactive
10
general
decay.
20
30
40
50
60
70
80
90
t/arbitrary units
▲
Figure 12
17
100
1
M E A S U R E M E N TS
A N D
U N C E R TA I N T I E S
1.3 Vectors and scalars
Understanding
Applications and skills
➔
Vector and scalar quantities
➔
Combination and resolution of vectors
Solving vector problems graphically and
➔
algebraically.
Equations
The horizontal and ver tical components of vector A:
= A cos θ
A
➔
H
= A sin θ
A
➔
V
A
A
V
u
A
H
Nature of science
All
physical
course
It
a
is
are
classified
impor tant
vector
quantity
the
quantities
or
a
is
used
by
aspect
scalar
of
of
it
is
a
being
this
will
forces
that
any
will
millennia,
recent
is
has
meet
vectors
whether
since
adding
for
you
or
an
the
scalars.
how
published
1687,
now
the
been
in
call
with
19th
the
for
complex
a
mat ter
of
quantities
generalized
vectors.
became
At
the
an
course
Vectors
by
are
now
physicists
which
this
star t
to
of
indispensible
three-dimensional
numbers.
of
Mathematica,
used
never
vectors
representing
mathematicians
the
but
concepts
century
and
Principia
Newton
vectors,
tool
as
analytical
In
Naturalis
deal
the
intuitive
development .
Philosophiæ
we
is
Although
probably
the
on
quantity
affect
mathematically.
vectors
sailors
of
as
know
treated
concept
application
to
that
space
used
and
alike.
Vector and scalar quantities
Scalar
quantities
direction.
use
the
both
We
rules
scalars,
are
treat
of
as
those
scalar
algebra
is
that
when
speed.
have
quantities
The
magnitude
as
dealing
average
numbers
with
speed
them.
is
(or
size)
(albeit
but
with
Distance
simply
the
no
units)
and
and
time
distance
are
divided
1
by
the
There
time,
are
Vector
We
no
use
take
called
vector
18
you
are
vector
into
those
algebra
account
equivalent
being
travel
80 m
in
10 s
the
speed
will
always
be
8 ms
surprises.
displacement
direction).
as
if
quantities
must
must
so
Time,
direction.
(i.e.,
of
as
it
speed
we
displacement
which
when
is
is
have
a
have
The
velocity
divided
by
is
in
(i.e.,
a
a
it
vectors
is
the
of
direction.
we
distance
direction).
speed
Average
and
since
equivalent
specied
scalar.
time.
magnitude
with
vector
distance
seen,
both
dealing
in
a
is
The
specied
velocity
is
dened
.
1 . 3
Dividing
do
a
vector
involving
with
scalars,
was,
again,
a
by
a
vector.
suppose
scalar
To
is
the
continue
the
easiest
with
displacement
operation
the
was
that
example
80 m
we
that
due
need
we
north
V E C T O R S
S C A L A R S
to
looked
and
A N D
the
at
time
1
10 s.
The
generalize,
when
vector
has
that
magnitude
we
the
equal
average
divide
a
direction
to
that
of
velocity
vector
of
the
the
would
by
a
scalar
original
vector
be
8 ms
we
one,
divided
end
but
by
due
up
north.
with
which
that
of
will
the
a
So,
to
new
be
of
scalar.
Commonly used vectors and scalars
Vectors
Scalars
Comments
force (F)
mass (m)
displacement (s)
length/distance (s, d, etc.)
F
displacement used to be called “space” – now that
means something else!
velocity (v or u)
time (t)
momentum (p)
volume (V)
acceleration (a)
temperature (T)
gravitational eld strength (g)
speed (v or u)
electric eld strength (E)
velocity and speed often have the same symbol
the symbol for density is the Greek “rho” not the letter “p”
density (ρ)
magnetic eld strength (B)
pressure (p)
area (A)
energy/work (W, etc.)
the direction of an area is taken as being at right angles to
the surface
power (P)
current (I)
with current having direction you might think that it
should be a vector but it is not (it is the ratio of two
scalars, charge and time, so it cannot be a vector). In
more advanced work you might come across current
density which is a vector.
resistance (R)
gravitational potential (V
)
G
the subscripts tell us whether it is gravitational or electrical
electric potential (V
)
E
ux is often thought as having a direction – it doesn’t!
magnetic ux (Φ)
Representing
A
vector
quantity
is
vector quantities
represented
●
The
direction
the
arrow
●
The
length
the
line
chosen
When
simple
we
of
by
points
a
line
with
represents
represents
the
an
the
arrow.
direction
magnitude
of
the
of
the
vector
vector.
to
a
scale.
are
matter
dealing
to
with
assign
vectors
one
that
direction
act
as
in
one
being
dimension
positive
and
it
the
is
a
5.0 N
this vector can represent a force
opposite
of 5.0 N in the given direction
direction
as
being
negative.
Which
direction
is
positive
and
which
(using a scale of 1 cm representing
negative
really
doesn’t
matter
as
long
as
you
are
consistent.
So,
if
one
1 N, it will be 5 cm long)
force
acts
simple
upwards
matter
to
on
nd
an
the
object
and
resultant
another
by
force
subtracting
acts
one
downwards,
from
the
it
other.
is
a
▲
Figure 1 Representing a vector.
19
1
M E A S U R E M E N TS
A N D
U N C E R TA I N T I E S
Figure
2
object
–
object
is
shows
the
an
upward
upward
line
tension
is
longer
and
than
downward
the
weight
downward
acting
line
since
on
an
the
tension = 20 N upwards
not
in
equilibrium
and
has
an
upward
resultant.
Adding and subtracting vectors
resultant = 5 N upwards
When
adding
direction.
subtracting
can
be
done
vectors,
either
by
account
a
scale
has
to
drawing
be
taken
of
their
(graphically)
or
algebraically.
weight = 15 N downwards
▲
and
This
Figure 2 Two vectors acting on an object.
V
2
V
V
1
V
1
V
2
▲
Figure 3 Two vectors to be added.
Figure 4 Adding the vectors.
▲
Scale drawing (graphical) approach
Adding
two
vectors
V
and
V
1
done
●
by
forming
Make
give
●
a
rough
you
an
the
space
the
vectors
Having
a
sketch
idea
of
available
of
how
how
to
of
V
a
you.
suitable
and
1
V
to
are
not
in
the
same
direction
can
be
vectors
your
This
is
a
are
scale
going
needs
good
idea
to
to
be
when
add
in
together
order
you
are
to
to
ll
adding
too.
scale,
(so
scale.
the
large
mathematically
chosen
direction
which
2
parallelogram
that
draw
they
the
scaled
form
two
lines
in
adjacent
the
sides
of
the
2
parallelogram).
●
Complete
●
The
blue
the
parallelogram
diagonal
magnitude
and
by
represents
drawing
the
in
the
resultant
remaining
vector
in
two
sides.
both
direction.
Worked example
Two
forces
of
magnitude
4.0
N
and
6.0
N
act
on
a
Scale 10 mm represents 1.0 N
8.7
single
point.
The
forces
make
an
angle
of
60 °
6
each
other.
Using
a
scale
diagram,
determine
N
the
90
10
90
80
80
0
7
resultant
N
with
force.
0
6
10
0
resultant
0
1
1
0
7
0
1
6
0
0
5
3
0
0
0
4
4
1
6
0
0
3
Solution
5
1
the
vector
must
have
a
6
that
0
forget
0
2
Don’t
means
that
the
0
is
just
as
important
as
the
size
of
the
081
angle
07
this
01
direction;
1
a
07
01
and
1
1
magnitude
36
force.
4 N
length of resultant = 87 mm so the force = 8.7 N
angle resultant makes with 4 N force = 36°
20
N
1 . 3
V E C T O R S
A N D
S C A L A R S
Algebraic approach
Vectors
can
situation
to
each
act
that
any
angle
to
each
you
are
going
to
deal
We
will
other.
at
deal
with
other
with
this
but
is
the
when
most
they
common
are
at
right
angles
rst.
v
2
Adding vector quantities at right angles
Pythagoras’
theorem
two
perpendicular
two
vector
can
be
vectors
used
are
to
calculate
added
(or
a
resultant
subtracted).
vector
when
Assuming
that
the
v
same
as
quantities
long
as
they
are
are
horizontal
and
vertical
but
the
principle
is
the
1
perpendicular .
v
1
Figure
5
shows
two
perpendicular
velocities
v
and
v
1
parallelogram
that
is
a
;
they
form
a
2
rectangle.
______
2
The
magnitude
of
the
resultant
velocity
=
√v
2
+
v
1
●
The
resultant
velocity
makes
an
angle
θ
to
2
the
horizontal
given
by
v
2
v
v
1
_
tan
θ
=
1
_
1
so
(v )
that
θ
=
tan
(
)
v
2
2
θ
●
Notice
to
the
that
the
length
order
or
the
of
adding
direction
the
of
two
the
vectors
makes
no
difference
resultant.
▲
Worked example
A
walker
point.
walks
He
north.
then
At
walker
the
from
sketch
4.0
km
stops
end
his
Figure 5 Adding two perpendicular vectors.
due
before
of
his
starting
west
from
walking
journey,
his
3.0
how
starting
km
far
due
is
the
point?
3 km
Solution
______
2
2
resultant
√
=
4
+
3
=
5
km
θ
3
1
angle
Before
see
θ
=
we
how
tan
(
look
to
4
at
)
=
36.9°
4 km
adding
resolve
a
vectors
vector
–
i.e.
that
split
are
it
not
into
perpendicular,
two
we
need
to
components.
Resolving vectors
We
have
seen
that
adding
two
vectors
together
produces
a
resultant
Fsin θ
vector.
It
resultant
true
for
sensible,
into
any
together,
of
is
the
two
vector
make
vectors
that
therefore,
–
the
can
vectors
it
can
added
imagine
from
be
resultant
be
to
which
divided
vector.
it
into
and,
we
was
could
formed.
components
There
together
that
is
no
limit
to
split
In
fact
which,
the
consequently,
F
the
this
is
added
θ
number
there
is
no
Fcos θ
limit
to
the
number
of
components
that
a
vector
can
be
divided
into.
▲
However,
that
are
we
most
perpendicular
we
look
The
at
force
equal
to
commonly
perpendicular
projectiles
F
F
vectors
in
gure
cos θ
component
and
opposite
one
have
in
6
a
to
no
Topic
has
affect
the
a
vector
The
on
into
reason
each
two
for
other
as
components
doing
we
this
will
is
see
that
when
2.
been
vertical
to
divide
another.
Figure 6 Resolving a force.
resolved
into
component
angle
used
is
the
equal
always
horizontal
to
Fsin θ.
the
component
(The
sinecomponent.)
21
1
M E A S U R E M E N TS
A N D
U N C E R TA I N T I E S
Worked example
1
An
ice-hockey
puck
is
struck
at
a
constant
speed
=
v
40
cos
60°
=
20
m
s
x
1
of
40
m
s
at
an
angle
of
60°
to
the
longer
side
of
distance
travelled
(x)
=
v
t
=
20
×
=
10
m
0.5
x
an
in
ice
rink.
How
directions
the
long
a)
side
far
will
parallel
after
0.5
the
and
puck
b)
have
travelled
perpendicular
to
b)
s?
Resolving
v
=
v
sin
=
40
parallel
to
shorter
side:
60°
x
Solution
1
v
cos
60°
=
34.6
m
s
y
distance
travelled
(y)
=
v
t
°06 nis 04 =
=
34.6
=
17
×
0.5
y
1
40 ms
Just
of
to
demonstrate
adding
the
that
resolving
components
we
can
m
is
the
use
reverse
Pythagoras’
V
y
theorem
to
add
together
our
two
components
giving:
_________
60°
2
total
V
speed
2
√ 20
=
+
1
34.6
=
39.96
m s
long side of rink
= 40 cos 60°
x
as
the
value
for
v
was
rounded
this
gives
the
y
1
expected
a)
Resolving
v
=
v
cos
parallel
to
longer
40
m
s
side:
60°
x
Adding vector quantities that are not at right angles
You
●
are
now
Resolve
1
will
in
a
each
often
be
position
of
the
to
add
vectors
horizontally
θ nis
perpendicular
to
a
any
in
and
vectors.
two
directions
vertically,
but
at
right
may
be
angles
parallel
–
this
and
surface.
1
V =
V
1
Add
all
●
Add
all
the
components
in
one
the
components
in
the
direction
to
give
a
single
component.
1
●
V
second
single
perpendicular
direction
to
give
a
component.
θ
1
●
V
Combine
the
two
components
using
Pythagoras’
theorem,
1
V
and
V
1
quantities
are
the
at
right
two
vectors
to
angles.
be
added.
2
2
Each
vector
Note
that
is
resolved
since
the
x
into
components
component
of
V
in
is
to
the
the
x
and
left
it
y
is
directions.
treated
as
being
θ
2
nis
2
V =
negative
(the
upwards
is
y
component
of
each
vector
is
in
the
same
direction . . . so
V
2
V
y2
Total
x
treated
as
component
positive).
V
=
V
x
+
V
1x
=
V
2x
cos
θ
1
V
1
cos
θ
2
2
θ
2
Total
y
component
V
=
V
y
= V
V
2x
2
+
V
1y
=
2y
V
sin
θ
1
+
1
V
sin
θ
2
2
cosθ
2
Having
calculated
V
and
V
x
we
can
nd
the
resultant
y
______
Figure 7 Finding the resultant
2
Pythagoras
so
V
=
√
V
2
+
V
x
y
of two vectors that are not
–1
perpendicular.
and
the
angle
θ
made
with
the
horizontal
=
tan
(
V
y
__
)
V
x
22
for
1
vector
▲
as
cos θ
= V
1x
by
using
1 . 3
V E C T O R S
A N D
S C A L A R S
Worked example
Magnetic
elds
respectively.
have
The
strength
elds
act
at
200
27 °
mT
to
and
one
150
Solution
mT
another
as
The
shown
in
the
150
mT
eld
is
horizontal
and
so
has
no
diagram.
vertical
component.
Vertical
200
component
sin
This
27°
=
makes
90.8
the
of
the
200
mT
eld
=
mT
total
vertical
component
of
the
200 mT
(not drawn
resultant
eld.
to scale)
Horizontal
component
of
the
200
mT
eld
=
27°
200
cos
27°
=
178.2
mT
150 mT
Total
horizontal
(150.0
+
component
178.2)
mT
=
of
328.2
resultant
eld
=
mT
____________
Calculate
the
resultant
magnetic
eld
strength.
2
Resultant
The
eld
resultant
eld
√ 90.8
makes
an
2
+
328.2
angle
=
340
mT
of:
90.8
_____
1
tan
strength
(
328.2
)
=
15°
with
the150
mT
eld.
Subtraction of vectors
Subtracting
negative
The
of
one
the
negative
direction.
Suppose
of
Let’s
we
vector
vector
a
vector
look
wish
from
to
to
at
be
has
an
nd
another
is
subtracted
the
same
example
the
to
very
and
simple
add
this
magnitude
see
difference
how
you
to
but
this
between
–
just
the
the
form
other
the
vector.
opposite
works:
two
velocities
v
and
v
1
shown
in
gure
2
8.
V
2
V
1
V
2
−
V
1
V
2
V
1
V
1
▲
Figure 8 Positive and negative vectors.
V
1
In
nding
from
the
the
difference
second;
so
we
between
need
v
.
two
We
values
then
we
add
subtract
v
1
gure
The
9
to
order
give
of
the
red
to
v
1
the
as
rst
value
shown
V
in
2
−
V
1
2
resultant.
combining
the
two
vectors
doesn’t
matter
as
can
be
seen
V
from
it
the
doesn’t
same
two
versions
matter
length
and
in
where
gure
the
direction
9.
In
resultant
it
is
the
each
is
case
the
positioned
same
vector.
resultant
as
long
as
is
it
the
has
2
same–
the
▲
Figure 9 Subtracting vectors.
23
1
M E A S U R E M E N TS
A N D
U N C E R TA I N T I E S
Questions
1
Express
the
following
fundamental
a)
newton
b)
watt
c)
pascal
d)
coulomb
e)
volt
units
in
terms
of
the
SI
5
units.
Write
down
following
(N)
(W)
(Pa)
(C)
(V)
the
(you
a)
the
length
b)
the
mass
c)
the
charge
d)
the
age
of
e)
the
speed
of
of
order
may
a
a
magnitude
to
human
do
of
some
the
research).
foot
y
on
a
the
of
of
need
proton
universe
electromagnetic
waves
in
a
vacuum
(5
marks)
(5
2
Express
the
signicant
following
numbers
to
marks)
three
gures.
6
a)
Without
using
a
calculator
estimate
to
one
2π4.9
_
signicant
a)
gure
the
value
of
.
257.52
480
b)
0.002
b)
347
c)
0.1783
d)
7873
When
a
under
the
wire
extension
potential
e)
of
stretched,
of
the
energy
a
graph
wire
the
of
gives
stored
in
area
force
the
the
against
elastic
wire.
1.997
Estimate
(5
3
is
line
Complete
the
following
calculations
marks)
the
the
energy
following
stored
in
the
wire
with
characteristic:
and
12
express
your
answers
to
the
most
appropriate
10
of
signicant
a)
×
3.2
1.34
gures.
N/ecrof
number
2
8
6
1.34 × 10
_
b)
3
2.1
×
10
4
2
×
3
+
c)
1.87
10
d)
(1.97
×
10
e)
(9.47
×
10
1.97
×
10
2
5
4
)
×
(1.0
×
10
)
0
2
)
3
×
(4.0
×
10
)
0
1
2
3
4
5
6
7
8
extension/mm
(5
marks)
(4
4
Use
the
appropriate
metric
multiplier
instead
of
7
a
power
of
ten
in
marks)
The
grid
below
sho w s
o ne
d ata
p oi n t
and
it s
thefollowing.
associated
errorba r
on
a
gr a ph.
Th e
x -axis
is
4
a)
1.1
×
b)
4.22
10
not
V
together
4
×
10
shown.
State
with
its
the
y-value
a b s o l ute
of
a nd
the
data
p o i nt
pe r c en t a g e
m
uncertainty.
10
c)
8.5
×
d)
4.22
10
W
7
×
10
m
5.0
13
e)
3.5
×
10
C
(5
marks)
4.0
3.0
2.0
1.0
24
(3
marks)
Q U E S T I O N S
8
A
ball
The
falls
freely
variation
from
with
1
is
given
by
s
=
rest
time
with
t
of
an
its
acceleration
displacement
c)
g.
s
p
was
V
measured
was
to
measured
an
to
accuracy
an
of
accuracy
5%
of
and
2 %.
2
gt
.
The
percentage
Determine
uncertainty
the
absolute
error
in
the
value
2
in
g
the
is
value
±2%.
of
t
is
±3%
Calculate
and
the
that
in
percentage
the
value
of
of
the
constant
A
uncertainty
(6
in
the
value
of
(2
marks)
11
(IB)
An
9
The
volume
V
marks)
s
of
a
cylinder
of
height
h
and
experiment
extension
x
of
was
a
carried
thread
of
out
a
to
measure
spider’s
web
the
when
2
radius
a
r
is
given
particular
from
by
theexpression
experiment,
measurements
of
r
V
is
to
and
be
h.
V
πr
=
h.
In
a
load
F
is
applied
to
it.
determined
The
percentage
9.0
thread breaks
uncertainty
in
V
is
±5%
and
that
in
h
is
±2%.
8.0
at this point
Calculate
the
percentage
uncertainty
in
7
.0
r
6.0
(3
marks)
2
N 5.0
F/10
4.0
3.0
10
2.0
(IB)
1.0
At
high
pressures,
a
real
gas
does
not
behave
0.0
0.0
as
an
ideal
gas.
For
a
certain
range
of
1.0
2.0
3.0
4.0
5.0
6.0
pressures,
2
x/10
it
is
suggested
that
for
constant
temperature
pressure
p
one
the
mole
of
relation
a
real
gas
between
at
the
a)
and
volume
V
is
given
by
the
Copy
=
A
+
Bp
where
A
and
B
are
the
graph
and
draw
a
best-t
between
F
and
line
for
equation
the
pV
m
constants.
b)
data
The
points.
relationship
x
is
of
the
form
In
an
experiment,
1
mole
of
nitrogen
gas
n
was
compressed
150 K.
for
of
The
volume
different
the
at
values
product
pV
a
V
constant
of
of
the
the
against
temperature
gas
was
pressure
p
is
F
of
measured
p.
shown
A
in
State
graph
in
kx
and
order
explain
to
the
graph
determine
the
you
value
would
plot
n
the
c)
diagram
=
When
a
load
is
applied
to
a
material,
it
is
below.
said
the
to
be
under
stress
is
stress.
given
The
magnitude
p
of
by
F
_
13
p
=
A
where
12
A
sample
is
of
the
the
cross-sectional
area
of
the
material.
2
pV/10
Nm
Use
the
that
the
graph
and
thread
the
used
data
in
the
below
to
deduce
experiment
has
11
a
greater
breaking
stress
than
steel.
9
Breaking
stress
of
steel
=
1.0
×
10
2
N
m
10
6
Radius
0
5
10
15
d)
6
p/ × 10
of
spider
web
thread
=
4.5
×
10
m
20
The
uncertainty
in
the
measurement
of
Pa
6
the
a)
Copy
for
b)
the
Use
the
pV
the
graph
data
your
A
+
draw
a
line
of
best
t
A
to
determine
and
Determine
the
points.
graph
constants
=
and
B
in
the
the
values
of
radius
value
of
the
the
of
thread
is
percentage
the
area
of
±0.1
×
10
m.
uncertainty
the
in
thread.
(9
marks)
equation
Bp
25
1
M E A S U R E M E N TS
12
A
cyclist
travels
a
A N D
U N C E R TA I N T I E S
distance
of
1200
m
14
due
A
boat,
starting
on
one
bank
of
a
river,
heads
1
north
before
going
2000
m
due
east
due
followed
south
with
a
speed
of
1.5
m
s
.
The
river
1
by
500
m
calculate
south-west.
the
cyclist’s
Draw
nal
a
scale
diagram
displacement
to
ows
initial
east
at
0.8
(4
Calculate
the
to
resultant
the
The
river
is
50
displacement
the
diagram
shows
three
forces
P,
Q,
bank
of
velocity
the
of
the
boat
river.
marks)
b)
The
s
position.
relative
13
m
from
a)
her
due
and
R
boat
m
wide.
from
reaches
its
the
Calculate
initial
the
position
opposite
when
bank.
in
(7
equilibrium.
P
acts
horizontally
and
Q
marks)
vertically.
15
A
to
car
of
the
mass
850
horizontal.
kg
rests
on
Calculate
a
the
slope
at
25 °
magnitude
of
R
P
the
component
parallel
to
the
of
the
car’s
weight
which
slope.
(3
Q
When
P
=
magnitude
5.0 N
and
and
Q
=
direction
3.0 N,
of
calculate
R
(3
26
the
marks)
acts
marks)
E N D - O F -TO P I C Q U E S T I O N S
Solutions for Topic 1 – Measurements and uncertainties
1. a) kg ms–2
d) A s
b) kg m2s–3
e) kg m2s–1A–1
c) kg m–1s–2
2. a) 258
d) 7870
b) 0.00235
e) 2.00
c) 0.178
d) 2.0 × 109
3. a) 4.3
b) 6.4 × 10–2
e) 3.8 × 102
c) 2.16 × 103
4. a) 11 kV
b) 0.422 mm or 422 μm
d) 0.422 μm or 422 nm
e) 0.35 pC or 350 fC
c) 85 GW
5. a) 10–1 m (a few 10s of cm)
b) 10–4 – 10–2 kg (flies come in many different shapes and sizes!)
c) 10–19 C (1.6 × 10–19 C)
d) 1010 year (13.7 billion years)
e) 108 ms–1 (3.0 × 108 ms–1)
2 × 3 × ​
5 =_
6. a)​ _
​  6  ​ = 0.06
500
100
b) between 70 and 74 mJ
7. 3.0 ± 0.4; ±13.3% (or rounding up ±14%)
∆g
∆s ​ = _
8.​ _
​  g ​ + 2 _
​  ∆t ​ = 0.02 + (2 × 0.03) = 0.08 = ±8%
s
t
∆r ​ = _
9.​ _
​  1 ​ ​ _
​  ∆h ​ + _
​  ∆V ​​= 0.5 × 0.02 × 0.05 = 0.035 = ±4%
r
V
2 h
(
)
10. a) check line of best fit reasonable
b) A = y-intercept; B = gradient
c) use difference between steepest and shallowest gradients
11. a) check line of best fit reasonable
b) log F = n log x + log k; plot log F against log x and find gradient of graph which is n
8.5 × ​10 ​–2​  ​ = 1.3 × 1​0 ​9​ N​m ​–2​
__
c) p = ​
π × 4.5 × 1​0 ​–6​
0.1 × 1​0 ​–6​ ​ = 0.04 = ±4%
∆A ​= 2 _
_
d)​ _
​  ∆r
​
=
2
×
​
r
A
4.5 × 1​0 ​–6​
12. (1850 ± 50) m at angle of N (60 ± 5) °E from scale starting point
______
13. magnitude = √
​ ​5 ​2​ + ​3 ​2​ ​ = 5.8 N
tan θ = _
​  5 ​; θ = 31° to horizontal
3
_________
14. a) magnitude of velocity = √
​ 1.​
5 ​2​ + 0.​8 ​2​ ​ = 1.7 m ​s ​–1​
0.8 ​; θ = 28° with original direction
tan θ = ​ _
1.5
b) s cos 28 = 50; displacement s = 57 m
15. component parallel to slope = mg sin θ = 850 × 9.8 sin 25 = 3500 N
© Oxford University Press 2014: this may be reproduced for class use solely for the purchaser’s institute
839213_Solutions_Ch01.indd 1
1
12/17/14 4:10 PM
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
Topic 1 (Page 8): Measurements and
uncertainties
1. (a)(i) 0.5 × acceleration down the slope
(a)(iv) 0.36 ms 2
2. C
3. D
4. D
5. (b) 2.4 ± 0.1 s
(c) 2.6 ± 0.2 ms 2
HL
2
6.
1
This
question
forces
In
0
0.0
0.1
0.2
0.3
0.4
0.5
2
(time taken)
2
/ s
an
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
possible
the
plate
values
with
graph.
are
[2]
measured,
their
and
about
experiment,
two
poles
and
separation
the
are
nding
magnets
seeking
results
Add
is
between
facing
shown
Separation
in
the
their
magnets
one
of
and
the
d/m
placed
The
magnets,
table
between
the
separations.
were
another.
the
relationship
d,
force
were
with
of
their
North-
repulsion,
measured
f,
and
the
below.
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]