1.
Physical Quantities, Units and Measurement
Content
1.1 Scalars and vectors
1.2 Measurement techniques
1.3 Units and symbols
Learning outcomes
Candidates should be able to:
(a)
define the terms scalar and vector.
(b)
determine the resultant of two vectors by a graphical method.
*** This will be covered in Topic-03 with Forces.
(c)
list the vectors and scalars from distance, displacement, length, speed, velocity,
time, acceleration, mass and force.
(d)
describe how to measure a variety of lengths with appropriate accuracy using
tapes, rules, micrometers and calipers. (The use of a vernier scale is not
required.)
(e)
describe how to measure a variety of time intervals using clocks and
stopwatches.
(f)
recognise and use the conventions and symbols contained in ‘Signs, Symbols and
Systematics’, Association for Science Education, 2000.
1
1.1 Scalars and vectors
Physical Quantities:
To analyse and study matter physically, there are quantities which can be measured
and they are known as Physical Quantities.
These quantities are divided into two major categories based on their Origin and
Nature.
Fig.1.1
Base Physical Quantities:
There are 7 base physical quantities.
S.No
1
2
3
4
5
6
7
Base Physical Quantities
Mass
Length
Time
Temperature
Electric Current
Amount of Substance
Intensity of Light
Symbol
m
l
t
T
I
n
L
S.I.Unit
Kilograms
Meters
Seconds
Kelvin
Amperes
Moles
Candela
Symbol
kg
m
s
K
A
mol
Cd
Derived Physical Quantities:
There are many derived physical quantities and more are yet to be discovered. They are
derived from basic physical quantities. Here only few familiar ones are mentioned.
S.No
1
2
Symbol
A
V
S.I.Unit
Meters Squared
Meters Cubed
Symbol
m2
m3
3
Derived Physical Quantities
Area (length х length)
Volume (length х length х
length)
Density (mass/Volume)
ρ
kg/m3
4
…
Speed (distance/time)
…
v
…
Kilograms per Meters
Cubed
Meters per Second
…
m/s
…
2
Unit:

Unit is a pneumonic used to measure physical quantities.

Such as for length we use meters, inches, feet, kilometres, miles and etcetera.

Students are expected to memorize and remember Units of Physical Quantities
used in this course.

Not all units are mentioned in this chapter.
S.I. Unit:

System International (S.I) Unit is one specific unit throughout the world for one
physical quantity.

For example, meters (m) is S.I Unit for Length.

S.I Units have great importance and students are preferred to use them primarily
in solution unless other units have been used in questions.
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In a test, four students linked the quantities on the left with their units on the right.
Which student matched them all correctly?
[P11-N14-Q2]
3
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What is the correct unit for the quantity shown?
[P11-N19-Q4]
Magnitude:

Magnitude is the size or value of any Physical Quantity.

Magnitude comprises of a Number and a Unit.
For Example:
10 meters

Now in this example 10 is a Number & meters is a Unit.

Separately they don’t mean anything but together they are giving value or size of
length.
Scalar:

Scalars are those physical quantities which have magnitude only.

Example for scalars will be mass, time, temperature, volume, and etcetera.
mass = 10 kg
Volume = 5 m3
Vectors:
While on the other hand some physical quantities are more than just magnitude, they
also have direction which is necessary to define them.
Example for vectors are Force, Velocity, Acceleration, and etcetera.
Force = 10 N towards North
Or
Force = 10 N at 45ᵒ
4
List of Scalars & Vectors:
Scalars
Vectors
Distance
Speed
Mass
Time
Length
Energy
Displacement
Velocity
Force
Acceleration
** Vector Addition is covered in Topic-3.
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The following statements are about motion.
1.
2.
3.
4.
A
A
A
A
plane flies due East for 600 km.
runner’s average speed in a race around a track is 5 m/s.
snail crawls at 3 mm/s in a straight line towards a lettuce.
tourist travels 500 km on a journey.
Which statements describe vector quantities?
A. 1 and 2
B. 1 and 3
C. 2 and 3
D. 2 and 4
[P11-J10-Q2]
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Which list contains only scalar quantities?
A. acceleration, displacement, velocity
B. distance, force, speed
C. force, length, time
D. length, mass, speed
[P11-N10-Q2]
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Velocity is given by the change in displacement divided by the change in time.
How many vector quantities appear in this statement?
A. 0
B. 1
C. 2
D. 3
[P11-N12-Q2]
5
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Is mass a scalar or a vector, and is acceleration a scalar or a vector?
[P11-J15-Q1]
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A car’s acceleration and maximum speed are improved by using an engine of smaller
mass and greater driving force.
How many of the underlined quantities are vectors?
A. 1
B. 2
C. 3
D. 4
[P11-J17-Q2]
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1.2 Measurement techniques
Length:


Length is one of the very base physical quantities and we see it all around us in
this world.
It is measured in S.I Unit called Meters (m).

There are different instruments we use to measure length.
1.
2.
3.
4.
Meter Rule
Tape Measure
Caliper (Vernier Caliper)
Micrometer
These instruments have certain advantages and limitations over one another. Choice of
instrument must be appropriate.
Least Count:



Least Count is the smallest value an instrument can measure.
The smaller the value is the more sensitive (accurate) the measuring instrument.
It gives the detailed value with more and more decimal places.
Instruments for Length Measurement:
1-Meter Rule:
Fig.1.2



Meter Rule is a solid rigid stick of 100 centimetre = 1 meter long.
The least count of meter rule is 1 millimetre = 0.1 centimetre.
It is used to measure flat and straight lengths between 0.1 cm and 100 cm.
7
2-Tape Measure:
Fig.1.3

Tape Measure is a flexible tape of various lengths based on common lengths


are measured.
The least count of Tape Measure is 1 millimetre = 0.1 centimetre as well.
It is used to measure round and curved lengths between 0.1 cm and 100 m.
3-Vernier Calliper:
Fig.1.4



Vernier Caliper is a solid rigid mechanical instrument to measure small lengths.
The least count of Vernier Caliper is 0.05 millimetre = 0.005 centimetre.
It is used to measure internal and external diameters.
***Note***
Vernier Caliper’s reading procedure is not part of O-Level’s Syllabus
Past-Paper Alert|||||||||||||| Past-Paper Alert|||||||||||||| Past-Paper Alert
Vernier calipers read to one tenth of a millimetre.
Which reading is given to this precision?
A. 3.3 cm
B. 3.31 cm
C. 3.310 cm
D. 3.312 cm
[P11-N12-Q1]
8
Micrometer (Screw Gauge):
Fig.1.5




Micrometer is a solid rigid mechanical instrument to measure small lengths.
The least count of Micrometer is 0.01 millimetre = 0.001centimetre.
It is used to measure external diameters.
It is used to measure smaller lengths than Vernier Caliper.
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A plumber measures, as accurately as possible, the length and internal diameter of a
straight copper pipe.
The length is approximately 80 cm and the internal diameter is approximately 2 cm.
What is the best combination of instruments for the plumber to use?
[P11-J11-Q1]
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A manufacturer measures accurately the dimensions of a wooden floor tile.
The approximate dimensions of the tile are shown.
Which instruments are used to measure accurately each of these dimensions?
[P11-J12-Q1]
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A length of copper pipe, of uniform cross-section and several metres long, carries water
to a tap.
Measurements are taken to determine accurately the volume of copper in the pipe.
Which instruments are used?
A. calipers and micrometer
B. micrometer and rule
C. rule and tape
D. tape and calipers
[P11-N13-Q2]
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Which device can be used to measure the thickness of a single sheet of paper?
A. a metre rule
B. a micrometer
C. a plastic ruler
D. a measuring tape
[P11-N14-Q1]
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A student determines the circumference of a golf ball.
Which instrument gives a reading that is the circumference of the golf ball?
A. calipers
B. micrometer
C. rule
D. tape
[P11-J16-Q3]
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A length of copper wire is labelled ‘length 30 m’ and ‘diameter 0.50 mm’.
Which instruments are most suitable to measure accurately the length and the diameter
of the wire?
[P11-N16-Q1]
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Reading Micrometer:



Value is a combination of Main Scale (M.S) & Circular Scale (C.S) readings.
Last value visible on Main Scale will be considered as M.S reading.
The value on Circular Scale which is coinciding with Datum Line on Main Scale
will be considered as C.S reading.
Fig 1.6

So the formula for Final Reading goes like this:
Final Reading = M.S + (C.S x L.C)

L.C stands for Least Count and for Micrometer, it is 0.01 mm.
Now READ the value of Micrometer given in fig.1.6 and check with your teacher or else
it is solved on next page with other examples.
ZERO ERROR & Other Errors will be discussed in ATP Addendum.
Micrometer Reading Examples:
Example-1:
Fig. 1.7
Solution:



M.S = 22.00 mm
C.S = 33
L.C = 0.01 mm
Formula: Final Reading = M.S + (C.S x L.C)
Final Reading = 22.00 mm + (33 x 0.01 mm)
Final Reading = 22.00 mm + 0.33 mm
Final Reading = 22.33 mm
12
Example-2:
Fig 1.6
Solution:



M.S = 3.50 mm
C.S = 16
L.C = 0.01 mm
Formula: Final Reading = M.S + (C.S x L.C)
Final Reading = 3.50 mm + (16 x 0.01 mm)
Final Reading = 3.50 mm + 0.16 mm
Final Reading = 3.66 mm
Past-Paper Alert|||||||||||||| Past-Paper Alert|||||||||||||| Past-Paper Alert
The diagram shows a micrometer scale.
Which reading is shown?
A. 5.64 mm
B. 7.14 mm
C. 7.16 mm
D. 7.64 mm
[P11-N15-Q2]
13
Time:



Time is also one of the very key Physical Quantity.
It is always there since the beginning of Universe.
It is measure with S.I unit called seconds (s).
Instruments for Time Measurements:
1234-
Clocks
Stop Watches
Digital Light Sensors
Pendulum
In this course we will discuss Pendulum in detail with its terminologies.
1- Clocks:



Digital and Analog Clocks are used to measure and display time.
Least Count of analog clock is 1 second.
Least Count of digital clock is 0.01 second.
2- Stop Watches:



Digital and Analog Stop Watches are used to measure and time intervals.
Least Count of analog stopwatch is 0.2 second.
Least Count of digital clock is 0.01 second.
14
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The diagram shows a stopwatch.
What is the reading on the stopwatch?
A. 30.6 s
B. 33.0 s
C. 36.0 s
D. 36.6 s
[P11-J14-Q2]
3- Digital Light Sensors:






Light Sensors are used in devices to measure time.
These light sensors work on principle of speed of light.
When light reflects back or sensors receive light it measures the time delay.
This measurement is very accurate.
Human Error ~ Reaction Time Delay can be avoided using this method.
International Sports and Olympics use these methods.
Fig.1.8
15
4- Pendulum:
Pendulum, a body suspended from a fixed point so that it can swing back and
forth under the influence of gravity.
Historically and even now pendulum mechanism is used for time measurement
in clocks.
Fig.1.9
Amplitude:
The maximum displacement of the object from Mean Position (Origin) is called
the Amplitude of a Pendulum.
Fig.1.10
Oscillation:
One complete cycle of back and forth swing of suspended object is called Oscillation of
the Pendulum.
The complete cycle from A-O-B-O-A
16
Time Period:
The amount of time taken to complete one oscillation is called Time Period or Period.
Frequency:
Number of Oscillations per second is known as Frequency of the Pendulum.
*** These characteristics are very much related to WAVES and will be discussed again
in detail in Topic-13: General Properties of Waves.
1.3 Units and symbols
Scientific Notation:
Scientific Notation is the method used to write very large or small numbers in a simpler
manner with a number between 1 & 10 with power of 10.
For example
0.0005 can be written as 5.0 x 10
-4
Or
785000 = 7.85 x 10
5
Prefixes:





Prefix is a pneumonic which has a predefined value in power of 10.
For example Kilo (k) has a value of 1000 which is 10 3
These prefixes are used for convenience and standardization.
Commonly used prefixes in O-Levels are given in the table below.
Students are expected to memorize them and remember their values and
symbols by heart.
Negative Prefixes
Nano
Micro
Milli
Centi
Deci
Symbol
n
μ (Mu)
m
c
d
Value
10-9
10-6
10-3
10-2
10-1
Positive Prefixes
Kilo
Mega
Giga
Symbol
k
M
G
Value
103
106
109
Fig.1.11
17
Past-Paper Alert|||||||||||||| Past-Paper Alert|||||||||||||| Past-Paper Alert
Power is measured in watts.
What is the correct symbol for millions of watts?
A. mw
B. mW
C. Mw
D. MW
[P11-J10-Q1]
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A quantity is quoted as having a value of 6.2 ms.
In what units is it measured?
A. metres
B. metres per second
C. microseconds
D. milliseconds
[P11-N11-Q2]
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The magnitudes of three different electric charges are given below.
What is the correct order of size, from largest to smallest?
A. 1 mC → 1 MC → 1 kC
B. 1 MC → 1 mC → 1 kC
C. 1 MC → 1 kC → 1 mC
D. 1 kC → 1 mC → 1 MC
[P11-J19-Q4]
18
Prefixes Conversions:
There are 3 possible scenarios in prefixes conversions normally.
1- Removing a Prefix (e.g. cm to m)
2- Introducing a Prefix (e.g. m to km)
3- Interconversion of Prefix (e.g. cm to km)
*** Students are allowed with Scientific Calculator in Physics Exam.
1- Removing a Prefix:


Removing a prefix is the simplest one.
We just simply multiply the number with the value of prefix being removed.
For Example:



15 cm =
m??
Now in this example we want to remove centi (c).
First of all student needs to remember the value of centi (c) = 10
Now simply multiply this 10 -2 with 15 and answer will be 0.15.
-2
.
Therefore 15 cm = 0.15 m
Solve some of the questions by yourself.

78 mg =
g

9.5 km =
m

838 μs =
s

512 MB =
B
2- Introducing a Prefix:

We just simply divide the number with the value of prefix being introduced.
For Example:
km??
90 m =


Now in this example we want to introduce kilo (k).
Here also student needs to remember the value of kilo (k) = 10 3.

Now simply divide
90 by 10 3 with and answer will be 0.09.
Therefore 90 m = 0.09 km
Solve some of the questions by yourself.

678 g =
mg

9523 m =
km

605 s =
μs

512 B =
kB
19
3- Interconversion of Prefixes:

This is tricky but if you have understood previous 2 methods simply just combine
them here.
For Example:
85 cm =
km??

kilo (k) = 10 3 & centi (c) = 10

Now here we have to Remove centi (c) and Introduce kilo (k).

Here we will do both of them together.
=
-2
85 × 𝟏𝟎−𝟐
𝟏𝟎𝟑
85 cm = 0.00085 km
Solve some of the questions by yourself.

342 mg =
kg

23 km =
mm

783 ms =
μs

512 kB =
GB
20