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CHAPTER 1: MEASUREMENT
•
Physical Quantities & SI Units
•
Scalars & Vectors
• Errors & Uncertainties
Learning Outcomes
Candidates should be able to:
(a)
recall the following base quantities and their units: mass (kg), length (m), time
(s), current (A), temperature (K), amount of substance (mol).
(b)
express derived units as products or quotients of the base units and use the
named units listed in ‘Summary of Key Quantities, Symbols and Units’ as
appropriate.
(c)
use SI base units to check the homogeneity of physical equations.
(d)
show an understanding of and use the conventions for labelling graph axes and
table columns as set out in the ASE publication Signs, Symbols and
Systematics (The ASE Companion to 16-19 Science, 2000).
(e)
use the following prefixes and their symbols to indicate decimal sub-multiples or
multiples of both base and derived units: pico (p), nano (n), micro (μ), milli (m),
centi (c), deci (d), kilo (k), mega (M), giga (G), tera (T).
(f)
make reasonable estimates of physical quantities included within the syllabus.
(g)
distinguish between scalar and vector quantities, and give examples of each.
(h)
add and subtract coplanar vectors.
(i)
represent a vector as two perpendicular components.
(j)
show an understanding of the distinction between systematic errors (including
zero errors) and random errors.
(k)
show an understanding of the distinction between precision and accuracy.
(l)
assess the uncertainty in a derived quantity by simple addition of actual, fractional
or percentage uncertainties (a rigorous statistical treatment is not required).
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CONCEPT 1: BASE & DERIVED QUANTITIES
INTRODUCTION
length = 23.0 m
physical
quantity
numerical
magnitude
unit
❖ 2 types of physical quantities: _________ and ______________ quantities.
BASE QUANTITIES
Base quantities are a set of __________ physical quantities of the SI system by
which all other physical quantities are defined.
❖ The six base quantities and its SI unit are:
base quantity
base unit
symbol for units
length
metre
m
time
second
s
mass
kilogram
kg
current
ampere
A
amount of substance
mole
mol
thermodynamic temperature
kelvin
K
luminous intensity
candela
cd
Table 1: Base Quantities and its SI units
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DERIVED QUANTITIES
Derived quantity is a quantity obtained from the combination of the __________
quantities via their ___________ or ___________.
❖ Examples of derived quantities:
Derived quantity
Formula
Derived SI base units
volume
V =l x h x b
m3
density
ρ=
m
V
kg m-3
velocity
v=
s
t
m s-1
acceleration
a=
Δv
Δt
m s-2
force
pressure
energy
torque (moment)
τ = Fd
power
P=
kg m2 s-2
E
t
Table 2: Derived Quantities and its SI base units
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WORKED EXAMPLE 1
a)
b)
Show that the SІ base units of power are kg m2 s–3.
The rate of heat flow
[2]
Q
in a solid is governed by
t
Q CAT
=
t
x
where A is the cross-sectional area of the solid,
T is the temperature difference across the thickness of the solid,
x is the thickness of the solid,
C is a constant.
Determine the SІ base units of C.
[2]
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WORKED EXAMPLE 2
a)
Intensity is power incident on a unit area.
Show that the SI base units of intensity are kg s -3.
b)
[2]
The intensity I of a mechanical wave is governed by
I = Wρvf 2 A02
where A0 is the amplitude of the wave,  is the density of the medium in which the
wave is passing, v is the speed of the mechanical wave, f is the frequency of the
mechanical wave, and W is a constant.
Show that W has no units.
[2]
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SELF-CHECK 1: BASE & DERIVED QUANTITIES
EASY
1
List the 6 SI base units.
2
The drag force F experienced by a steel sphere of radius r dropping at speed v
through a liquid is given by
F = arv
where a is a constant
What are the SI base units for a?
3
What are the SI base units for pressure?
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MEDIUM
4
5
Express the SI base units for work done.
If p is the momentum of an object of mass m, the expression
p2
has base units
m
identical to ___________. [Hint: momentum = mass x velocity]
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CONCEPT 2: DIMENSIONLESS & HOMOGENEITY
DIMENSIONLESS QUANTITY
Dimensionless quantity is a physical quantity that only has _________________,
but no ________.
❖ e.g. all numbers, e.g. 2,
1
, , e
2
trigonometrical functions, e.g. sine, cosine, tangent.
all logarithmic functions, e.g. logx, ln
x
exponents (powers), e.g. in 10 y , the ratio
x
must be unitless.
y
f x has a unit,
then the unit of y must have the same unit as that of x.
unitless physical constants, e.g. refractive index of glass.
NOTE:
•
Not all constants are unitless.
E.g.: gravitational acceleration is a constant = 9.81 m s-2 and it has unit of m s-2.
WORKED EXAMPLE 1
The terminal speed vT of a sphere falling through water of density  is given by
vT =
mg
W A
where m is the mass of the sphere, A is its surface area and W is a constant.
What is the unit of W?
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HOMOGENEOUS EQUATIONS
Homogenous equation is an equation that has the same ________ units for all its
terms on ______ side of the equation.
WORKED EXAMPLE 2
Velocity of a liquid is governed by its wavelength  , acceleration of free fall g, depth h
and density  . Which of the following is the correct expression for velocity of the liquid?
A
g / 
B
h / 
C
gh
D
g
WORKED EXAMPLE 3
The oscillation of a pendulum with period T is governed by equation T = 2
is the length of the pendulum and g is acceleration due to gravity.
Explain if the equation is homogenous.
L
where L
g
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SELF-CHECK 2: DIMENSIONLESS & HOMOGENEITY
EASY
1
Physical quantities A, B, C and D are related by the equation A = B – CD.
Which statement explain that the equation is homogenous?
A
A, B, C and D all have the same units.
B
A, B, C and D are all vector quantities.
C
The product CD has the same units as A.
D
The product CD is numerically equal to (B - A).
MEDIUM
2
Heat capacity C of a substance depends on temperature T, given by the expression
C = p/T + qT4
What are the SI units of p and q?
3
Current I across a thermistor due to p.d. V at temperature T is governed by
−
qV
I = I 0e kT where q is the electron charge, and k is a constant. Find the SI base units
of k.
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CONCEPT 3: PREFIXES
UNIT CONVERSIONS
❖ Prefixes simplify the way we write very big or very small numbers.
Sub – multiples & Multiples
Prefix
Symbol
Acronym
peta
P
Peter,
tera
T
The
giga
G
Great
mega
M
Mighty
kilo
k
king
deci
d
drinks
centi
c
chocolate
milli
m
milk
micro
μ
until
nano
n
night
pico
p
and pee
femto
f
the floor
Table 3: Prefixes and its symbols
NOTE
•
femto is commonly used in Nuclear Physics, the last chapter in Physics.
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WORKED EXAMPLE 1
50 nm = __________ km
WORKED EXAMPLE 2
5
2 Tb cm-2 = ____________ kb m-2
WORKED EXAMPLE 3
0.05 kb2 cm-3 = _______________ Mb2 mm-3
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SELF-CHECK 3: PREFIXES
EASY
1
The prefix ‘centi’ indicates x 10-2. That is, 1 centimetre is equal to 1 x 10 -2 metre.
Which line in the table correctly indicates the prefixes micro, nano and pico?
x 10-12
x 10-9
x 10-6
A
nano
micro
Pico
B
micro
pico
nano
C
pico
nano
micro
D
pico
micro
nano
MEDIUM
2
Convert 1 light year to metre.
3
2 Tb cm-2 = ____________ kb m-2
4
What is equivalent to 2000 picovolts?
A
0.002 J C
B
0.02 GV
C
0.2 x 104 TV
D
2 x 10-15 MJ C-1
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ADVANCED
5
400 μg2 dm-3 = _____________ mg2 cm-3
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CONCEPT 4: ESTIMATION OF PHYSICAL QUANTITIES
ESTIMATE SIZES OF PHYSICAL QUANTITIES
Fig. 1: Order of magnitude of various objects in the Universe
Quantity
Approx. value
Order of magnitude
mass of an electron
9.11 x 10-31 kg
10-31
mass of a raindrop
2.0 x 10-6 kg
10-6
mass of an ant
0.0025 kg
10-3
mass of a tennis ball
0.057 kg
10-2
mass of a water melon
5 kg
100
mass of a small size car
1000 kg
103
mass of the Earth
5.98 x 1024 kg
1024
mass of the Sun
2.0 x 1030 kg
1030
400 nm – 700 nm
10-7
diameter of a proton
2.0 x 10-15 m
10-15
diameter of an atom
1.0 x 10-10 m
10-10
thickness of an A4 paper
0.1 mm
10-4
area of Singapore
760 km2
102
diameter of the Earth
1.28 x 107 m
107
current in electric coil
8A
100
power of a filament light bulb
60 W
101
power of electric heater
1000 W
103
frequency of radio waves
100 MHz
108
resistance of a filament lamp
700 Ω
102
acceleration of MRT
2.0 m s-2
100
voltage of a CRO
1000 eV
103
wavelength of visible light
Table 4: Estimation of physical quantities ant its associated order of magnitude
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WORKED EXAMPLE 1
Which estimate is realistic?
A
The temperature in the domestic refrigerator is 200 K.
B
The surface area of a standard soccer ball is 0.15m2.
C
The current drawn by a laptop is 15 A.
D
The volume of a pen is 4.010-3 m3.
WORKED EXAMPLE 2
Estimate the following with appropriate reasoning.
(a)
The area of the island of Singapore.
(b)
The acceleration of a train on the Singapore rapid transit system.
(c)
The power of a car travelling along an expressway.
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SELF-CHECK 4: ESTIMATION OF PHYSICAL QUANTITIES
EASY
1
What is a reasonable estimate for the volume of a wooden metre rule found in a
school laboratory?
2
In an experiment, the width w and the thickness x of a metre rule are to be
measured as precisely as possible using normal laboratory apparatus.
Which combination of instruments is most appropriate for these measurements?
Measurement of w
Measurement of x
A
Half-metre rule
Half-metre rule
B
Half-metre rule
Vernier calipers
C
Vernier calipers
Half-metre rule
D
Vernier calipers
Micrometer screw gauge
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3
Estimate the kinetic energy of an Olympic gold medalist running in a 100 m race.
4
What is a reasonable estimate of the kinetic energy of a person walking at a
normal pace?
5
Make reasonable estimates of the following quantities.
(a)
Mass of an apple
(b)
Number of joules of energy in 1 kilowatt-hour
(c)
Wavelength of red light in a vacuum
(d)
Pressure due to a depth of 10 m of water
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MEDIUM
6
What is the order-of-magnitude of the mass of a single sheet of A4 paper?
7
State if the following statement is true.
(i)
The kinetic energy of a bus travelling on an expressway is
30 000 J.
8
(ii)
The power of a domestic light is 300 W
(iii)
The temperature of a hot oven is 300 K
(iv)
The volume of air in a car tyre is 0.03 m 3
What is the order of magnitude of the energy of an electron when it hits the screen
of a cathode-ray tube?
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9
The volume of the Earth is approximately 10 12 km3 and the volume of a grain of
sand is approximately 1 mm3. The order of magnitude of the number of grains of
sand that can fit in the volume of the Earth is ___________
10
The estimated number of ping pong balls that can completely fill in a classroom
with dimensions 20.0 m by 10.0 m by 3.50 m without being crushed is ______
11
You are using water to dilute small amounts of chemicals in the laboratory, drop by
drop. Estimate the number of drops of water in a 1 litre bottle of water.
(Hint: 1 litre = 1000 cm3)
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CONCEPT 5: SCALARS & VECTORS
BASIC OF SCALARS & VECTORS
❖ Scalars can be added or subtracted algebraically.
e.g. speed, distance, energy, work, power, pressure, mass, time, temperature,
electric current, potential difference (voltage), electric charge, density, volume.
❖ Vectors cannot be added or subtracted algebraically.
e.g. displacement, velocity, acceleration, force, weight, momentum, electric
field, gravitational field, magnetic field, moment, angular velocity.
VECTOR ADDITION
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WORKED EXAMPLE 1
Two forces of magnitudes 4 N and 6 N are applied to a point.
Which one of the following could not be the magnitude of their resultant?
A
0.5 N
B
4.2 N
C
9.9 N
D
10 N
WORKED EXAMPLE 2
Two forces of 10 N act on a point P, as shown in the diagram below.
The angle between the forces is 120 o. Determine the resultant force.
WORKED EXAMPLE 3
Each option below shows 3 vectors of equal length.
Which option shows that resultant vector different from the other three?
A
B
C
D
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VECTOR SUBTRACTION
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WORKED EXAMPLE 4
A car travelling in a circle changes its velocity from 50 km h -1 North to 30 km h-1 East.
Determine the change in velocity of the car.
WORKED EXAMPLE 5
A car has an initial velocity of 15 m s-1 to the right (in the x direction) as shown in the
diagram below. Later, its velocity is 15 m s-1 at an angle of 60o to x.
Calculate the change in velocity of the car.
15 m s-1
600
x
15 m s-1
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VECTOR RESOLUTION
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WORKED EXAMPLE 6
F1, F2, F3 act on a ball, as shown in the figure below. Find the resultant forces acting on
the ball.
y- axis
F1 = 15.0 N
F2 = 8.0 N
530
x- axis
320
F3 = 20.0 N
WORKED EXAMPLE 7
A ball of mass 4.9 kg is pulled by 3 forces on a horizontal table. The ball is in equilibrium.
Find the magnitude of tension T.
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SELF-CHECK 5: SCALARS & VECTORS
EASY
1
2
3
Which one of the following groups contains only vector quantities?
A
displacement, velocity, energy
B
displacement, velocity, momentum
C
velocity, acceleration, power
D
force, work, energy
Which of the following option is correct?
force
kinetic energy
moment
A
scalar
scalar
scalar
B
scalar
vector
vector
C
vector
scalar
scalar
D
vector
scalar
vector
E
vector
vector
vector
Diagram below shows the magnitude and direction of vectors X and Y.
Which diagram represents vector X – Y?
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4
Two forces act on a circular disc as shown in the diagram.
Which arrow best shows the line of action of the resultant force?
MEDIUM
5
A plane changes its velocity from 8 m s-1 due north to 6 m s-1 due east.
Determine its change in velocity.
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6
A picture frame is supported by 2 strings, with same angle of inclination θ , as shown in
the diagram below.
Draw a suitable vector triangle in order to determine the resultant force acting on the
frame.
7
3 forces of 20 N, 40 N and 50 N are acting on the same point, as shown in the figure
below. To maintain equilibrium, an additional force must be applied.
Determine the direction of this applied force.
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ADVANCED
9
A supersonic plane flies at 1000 km / h through a stream of wind of 300 km/h from the
west. The pilot wishes to go to north.
Find the speed in the north direction with respect to the ground.
\
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CONCEPT 6: PRECISION & ACCURACYRECISION &ACY
PRECISION
Precision is the ____________ of readings from each other.
❖ measure of ______________________.
❖ High precision = readings close to each other = small ___________ of data.
❖ Precision = maximum - ____________ readings.
❖ An instrument maybe precise but still be inaccurate. Why?
ACCURACY
Closeness of the _________ reading from its _________ value. High accuracy
implies low systematic error.
•
Accuracy = true value – mean value
PRECISION
ACCURACY
do not depend on __________________
depend on ____________________
high precision implies
high accuracy implies
_________________________________
_________________________________
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WORKED EXAMPLE 1
Which of the following option shows high precision but low accuracy?
A
B
C
D
WORKED EXAMPLE 2
A ball falls from rest through a vertical distance h under free fall acceleration g and the
time t for it to fall is measured. The results are plotted, as shown below.
Identify if the following statements are valid. Indicate True (T) or False (F).
(a)
The result is accurate as data is close to actual value.
(b)
The result is not accurate as the line does not pass through the origin
(c)
Data is precise as there are equal number of data points on both sides of the
line
(d)
Data is precise as the data points do not deviate from the line
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SELF-CHECK 6: PRECISION & ACCURACY
EASY
1
Diameter d of a uniform wire is measured with a micrometer and following are the
readings obtained.
1.02 mm
1.02 mm
1.01 mm
1.02 mm
1.02 mm
The zero error of the micrometer is -0.02 mm. Determine the value d.
2
A student takes the following readings of the diameter of a wire:
1.52 mm, 1.48 mm, 1.49 mm, 1.51 mm, 1.49 mm.
Express the diameter of the wire with its value and uncertainty.
MEDIUM
3
A student uses an ammeter to measure the current across a resistor. The ammeter is
marked every 0.02 A but has a zero error of 0.08 A. The student is not aware of this
zero error and writes down a reading of 2.16 A.
Is the reading accurate and is it precise?
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4
A student is conducting an experiment to determine the value of free fall
acceleration g. Which of the following is precise but not accurate?
results, g /m s-2
A
9.81
9.79
9.84
9.83
B
9.81
10.12
9.89
8.94
C
9.45
9.21
8.99
8.76
D
8.45
8.46
8.50
8.41
ADVANCED
6
A ball is falling for a distance s under constant gravitational acceleration g in time t.
The motion is plotted in the graph below.
The gradient of the graph is found to be 0.6 s2 m-1.
Deduce if the experiment is precise and accurate.
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CONCEPT 7: SYSTEMATIC & RANDOM ERRORS
Systematic errors are errors in which readings taken deviate from its true value by a
constant _________________ and constant ______________.
It can only be eliminated by correct lab practices, not by _________________.
Random errors are errors in which readings taken are _______________ about its
mean value with ________________ magnitudes and ________________ signs.
It can be reduced by _________________.
systematic errors
random errors
above or below its true value with a
scatter of readings about its mean
fixed pattern. OR same sign.
value with no fixed pattern. OR
different sign.
dependent on actual value.
independent of actual value.
cannot be reduced by taking repeated
can be reduced by repeated readings.
readings.
can be eliminated by making a
cannot be eliminated.
mathematical correction or using
calibration curves.
examples are zero errors, heat loss to
examples are reaction time, limitation
surroundings, incorrect experiment
of instrument’s precision, limitation of
technique and radioactive background
observer’s experimental skill and
count rate.
parallax error.
Table 5: Systematic and random errors
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❖ So, what are the relationships between systematic errors, random errors, precision,
and accuracy? Table below summarises their relationships.
illustration
precision
high
low
high
random error
low
high
low
accuracy
low
high
high
systematic error
high
low
low
explanations
•
•
readings close
•
readings far
•
readings
to each other,
from each
close to
so precise.
other, so
each other,
mean of
imprecise.
so precise.
readings far
•
mean of
•
mean of
from actual
readings close
readings
value, so
to actual value,
close to
inaccurate.
so accurate.
actual value,
so accurate.
conclusions
•
high systematic error, low accuracy, depends on actual
value.
•
high random error, low precision, independent of actual
value.
•
systematic has nothing to do with precision.
•
random error has nothing to do with accuracy.
Table 6: Precision, accuracy, systematic errors and random errors
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WORKED EXAMPLE 1
Errors in measurement may be either systematic or random.
Which of the following consists of random error?
A
zero error on the reading of length of a ruler.
B
stop a stopwatch in the end of a pendulum experiment.
C
subtract excess radiations from the surroundings.
D
use acceleration g = 10 m s-2 when calculating the speed of a falling object.
WORKED EXAMPLE 2
S1: error can possibly be eliminated
S2: error cannot possibly be eliminated
R1: error is of constant sign and magnitude
R2: error is of varying sign and magnitude
L1: error will be reduced by averaging repeated measurements
L2: error will not be reduced by averaging repeated measurements
Which statement(s) above applies to random errors?
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SELF-CHECK 7: SYSTEMATIC & RANDOM ERRORS
EASY
1
An object of mass 1.000 kg is measured multiple times by a weighing machine.
Which balance has the smallest systematic error but is not very precise?
Balance
Reading/kg
Mean/kg
1
2
3
4
5
A
1.000
1.000
1.002
1.001
1.002
1.001
B
1.011
0.999
1.001
0.989
0.995
0.999
C
1.012
1.013
1.012
1.014
1.014
1.014
D
0.993
0.987
1.002
1.000
0.983
0.993
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MEDIUM
2
A ball falls from rest through a vertical distance h under free fall acceleration g and
the time t for it to fall is measured. The results are plotted, as shown below.
Which of the following is an explanation for the intercept?
A
Air resistance was ignored.
B
Release and pressing of stopwatch is not synchronised.
C
Timer run consistently faster than usual.
D
graph h against t2 should be plotted.
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CONCEPT 8: UNCERTAINTIES
UNCERTAINTIES AND ABSOLUTE UNCERTAINTIES
Uncertainty is the range of values on both sides of a measurement in which the
actual value of the measurement is expected to lie.
Absolute uncertainty is the maximum ____________ in a reading, equals to one
___________ division of an instrument and is rounded off to ______ significant
figure. The calculated value is expressed as the same number of ______ as the
uncertainty.
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WORKED EXAMPLE 1
Given that a known value for x = 83568.7241, complete the following table with the
appropriate final answer (x ± ∆x).
∆x
∆x rounded off to 1 s.f.
final answer (x ± ∆x)
0.6245
0.657
5.4
18.45
584
2368.85
FRACTIONAL & PERCENTAGE UNCERTAINTIES
❖ The uncertainty of a measured value can also be presented as a simple fraction or a
percentage:
1. The fractional uncertainty of Q =
Q
Q
2. The percentage uncertainty of Q =
Q
 100%
Q
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WORKED EXAMPLE 2
The length of a rod is expressed as (25 ± 1) cm. Determine its absolute uncertainty,
fractional uncertainty and percentage uncertainty.
WORKED EXAMPLE 3
A student makes measurements from which he calculates the speed of sound to be
327.66 m s-1. He estimates that the percentage uncertainty is 3%. Round off the speed to
an appropriate number of significant figures.
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UNCERTAINTIES WITH VARIOUS FUNCTIONS
Addition
L=mP+nQ
 L = |m| P + |n| Q
Subtraction
L=mP–nQ
 L = |m| P + |n| Q
L=mP
 L = |m| P
Power
L = k Pm
L
P
= m
L
P
Product
L = k Pm x Qn
L
P
Q
=m
+n
L
P
Q
Quotient
L = k Pm ÷ Qn
L
P
Q
=m
+n
L
P
Q
Multiplication by
Constant
For example,
Special
L = tan x
Functions
L = cosine x
L = ½ (Lmax –Lmin)
Table 7: Formulae for uncertainties
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WORKED EXAMPLE 4
The measurements of the dimensions of a particular piece of rectangular cardboard are
(18.5 ± 0.5) mm and (12.5 ± 0.5) mm. Determine the (i) perimeter and (ii) area of the
cardboard with its associated uncertainty (in mm2).
WORKED EXAMPLE 5
The radius of a circle is r = (3.0 ± 0.2) cm. Find the circumference with its uncertainty.
WORKED EXAMPLE 6
Given that a sphere of radius r = (18.5 ± 0.5) mm, find the volume of the sphere
with its associated uncertainty (in mm3).
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WORKED EXAMPLE 7
The period of oscillation of a simple pendulum is given by
T = 2
L
.
g
A student conducts an experiment to find the acceleration of free fall, g. He measures
the length of the pendulum, L = 0.23 ± 0.01 m and the period of 20 oscillations,
t = (19.24 ± 0.01) s. Find g and its associated error.
WORKED EXAMPLE 8
A hollow rod has the following specifications:
Length L = (150.0  0.5 ) mm
External diameter Dext = (15.0  0.2 ) mm
Internal diameter Dint = (10.0  0.2 ) mm
Determine the volume of the hollow pipe with its associated uncertainty (in mm3).
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SELF-CHECK 8: UNCERTAINTIES
EASY
1
The diameter of a golf ball is measured, as shown in the diagram below.
What is the diameter of the ball?
3
The power loss P in a cable of resistance R during electrical transfer of voltage V is
governed by equation P = V2/R. The uncertainty in V is 3% and R is 2%.
What is the uncertainty in P?
4
The formula for the period T of a simple pendulum of length L is T = 2π
Such a pendulum is used to determine free fall acceleration g.
The fractional error of T is ±A and L is ±B.
In term of A and B, determine fractional error of g.
L
.
g
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5
Peter wishes to measure the density of a block. The mass, length, breadth, and
height of the material are measured using appropriate instruments.
mass = (25.0 ± 0.1) g
length = (5.00 ± 0.01) cm
breadth = (2.00 ± 0.01) cm
height = (1.00 ± 0.01) cm
The density was calculated to be 2.50 g cm-3.
What was the uncertainty in this measurement?
6
An unknown instrument reads 0.00160 ± 0.00005.
Determine the actual uncertainty, absolute uncertainty, fractional uncertainty and
percentage uncertainty of this reading.
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7
Rachel wants to measure the resistivity of a resistor using an electrical circuit.
Following are the readings obtained in the experiment.
Diameter d of the resistor
= 1.20 ± 0.01 cm
Current I flowing in the resistor = 1.50 ± 0.05 A
Length L of the wire
= 100 ± 1 cm
Voltage V across the resistor = 5.0 ± 0.1 V
d 2V
The resistivity of the resistor is governed by equation ρ =
.
4L I
Which variable gives rise to the least uncertainty in the value for the resistivity?
MEDIUM
8
4 balls are arranged in a row. The diameter of the balls is measured, as shown in
the diagram below.
Uncertainty of the reading is half of the smallest division of the scale.
What is the diameter of the ball together with its associated uncertainty?
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9
The current flows in a resistor is measured as (2.50 ± 0.05) mA.
The resistor is marked as having a value of 4.7 Ω ± 2%.
What is the percentage uncertainty in the power loss P of the resistor?
10
In an experiment, the external diameter d1 and internal diameter d2 of a metal tube
are found to be (64 ± 2) mm and (47 ± 1) mm respectively. The percentage error in
(d1 – d2) expected from these readings is at most
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11
The dimensions of a cube are measured with vernier callipers.
The measured length of each side is 30 mm. If the vernier callipers can be read with
an uncertainty of ± 0.1 mm, what does this give for the approximate uncertainty in
the value of its volume?
A
12
1
%
27
B
3
%
10
C
1
%
3
D
The specification of a digital voltmeter is written as
'accuracy ± 2% with an additional uncertainty of ± 5 mV'
The meter reads 14.074 V.
How should this reading be recorded, together with its uncertainty?
1%
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ADVANCED
13
The focal length f of a lens with object distance u and image distance v is governed
by equation
1 1 1
+ =
u v f
The values of u and v are as shown below:
u = 60 mm ± 3 mm
v = 180 mm ± 5 mm
What is the uncertainty in this value?
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DEFINITION LIST
CHAPTER 1: MEASUREMENT
1.
Define physical quantities.
[1]
Measurable quantities with magnitude and unit.
2.
Define base quantity.
[2]
One of the seven physical quantities in which other quantities are defined.
3.
Define derived quantities.
[2]
Physical quantities expressed in terms of the product and/or quotients of the
base quantities.
4.
State what is meant by a homogenous equation.
[2]
Equation that has the same base units for all its terms on each side of the
equation.
5.
Distinguish between scalar and vector quantities.
[2]
Scalar quantities have magnitude only, while vector quantities have both
magnitude and direction.
6.
State what is meant by a zero error.
[3]
A systematic error which cannot be eliminated by averaging but can be
corrected by calibration.
7.
State what is meant by systematic error.
[2]
Readings deviate from its true value by a constant magnitude and constant
sign. It can only be eliminated by correct lab practices, not by averaging.
8.
State what is meant by random error.
Readings are scattered about its mean value with varying magnitudes and
different signs. It can be reduced by averaging.
[2]
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9.
Describe what is meant by precision.
[3]
The closeness of readings from each other. High precision implies small
scatter of readings.
10.
Describe what is meant by accuracy.
Closeness of the mean reading from its true value.
[3]