Physical Quantities
Different phenomenon includes different measurable quantities,
these are physical quantities.
In the diagram, a ball is falling
with some velocity. During its fall
it has some amount acceleration, it
takes some time to fall to the
ground, at each moment its
position can be described in terms
of its height from the ground, there
is also some energy stored in the
ball which changes form during the
fall of the ball, it also experiences
some force due to air resistance in
the direction opposite to the
velocity of the ball, all of these are
physical quantities.
Physical quantities are a feature of some object and don’t have
any physical existence.
A quantity is something that can be talked about. What makes a
quantity physical is that it can be measured. Therefore, a physical
quantity must have two attributes:
 A unit
 A numeric value
Unit:
In order to measure a physical quantity, it has to compared with
some agreed upon standard. This standard is called a unit. For
example, standard for measuring length, is the length travelled by
light in
1
299792458
seconds. Using this standard length of any size can
be measured quantitatively by using a numeric value. Choice of a
standard is arbitrary, in fact there are many standards for measuring
length but all of them are defined in terms of meter. While choosing a
standard two things are taken into account: first that it is easily
accessible and second that it is invariable. Often both of these
conditions are difficult to meet and compromises has to be made
between them.
Physical quantities have been divided into three types:
 Base Quantities
 Supplementary Quantities
 Derived Quantities
Base Quantities:
Base quantities are a set of quantities that are chosen as a
reference or starting point for defining all other quantities in a system
of measurement. The act as a building block for constructing a system
of measurement and all other quantities can be derived through them
with multiplication of division with each other or some constants.
There are different systems of measurements but the most common
and widely used is the SI system (International system of units). In SI
system, there are
seven base quantities:
Name
Unit
Symbol
Mass
Kilogram
kg
Length
Time
Temperature
Electric Current
Amount of Substance
Luminous intensity
Meter
Second
Kelvin
Ampere
Mole
candela
m
s
K
A
mol
cd
Derived Quantities:
The quantities which are derived from (by division or
multiplication of various base quantities) other quantities are called
derived quantities. In SI system of units, derived quantities are
derived from base quantities. In SI system of units, there are some
derived quantities which have a special name (22 in total).
Some examples of derived quantities are as follow:
Name
Formula
Unit
Velocity
𝑚⁄
𝑠2
Force
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒(𝑙𝑒𝑛𝑔𝑡ℎ)
𝑡𝑖𝑚𝑒
𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦⁄
𝑡𝑖𝑚𝑒
𝑀𝑎𝑠𝑠 × 𝐴𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛
Momentum
𝑀𝑎𝑠𝑠 × 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦
Pressure
Density
𝐹𝑜𝑟𝑐𝑒⁄𝐴𝑟𝑒𝑎
𝑀𝑎𝑠𝑠 ⁄𝑉𝑜𝑙𝑢𝑚𝑒
Acceleration
𝑚⁄
𝑠
𝐾𝑔 × 𝑚
𝑜𝑟 𝑛𝑒𝑤𝑡𝑜𝑛(𝑁)
𝑠2
𝐾𝑔 × 𝑚
𝑜𝑟 𝑁𝑠
𝑠
𝑁 ⁄𝑚 2
𝑘𝑔⁄𝑚3
Supplementary Quantities:
In SI system, there are two special quantities which are
dimensionless (as they are ratio of two same quantities), they are
called supplementary quantities (This category was abolished in
1995). It includes plane angle and solid angle. Plane angle is
measured is radians while solid angle is measured in steradians.
Radians:
1 radian is equal to the angle subtended at the center of
circle by an arc whose length is equal to the radius of the circle.
Steradian:
1 steradian is equal to the solid angle subtended by a
surface area at the center of a sphere whose area is equal to the square
of the radius.
System of units:
Besides the SI unit there are some other systems of units. They
are used for different fields in different parts of the world. Some field
of physics only require a particular number of base quantities and all
the other quantities can be derived from them. For example, in
mechanic only three base quantities (mass, length, time) are needed.
While in the study of electricity and magnetism, one additional
unit(ampere) is added. In thermodynamics, base quantity of
temperature is needed. There are some advance fields of physics that
require all the seven base quantities, for example quantum mechanics.
Some examples of other systems of units are:
 FPS system
 CGS system
 MKS System (a subset of SI system)
FPS system:
There are two variants of FPS system: British gravitational
system and American English system. These units were in use in
England before the adoption of metric system.
BG system:
In BGS system unit of mass, length and time are slug, foot
and second respectively. In BGS system, force is a base unit instead
of mass.
Name
Unit(FPS)
Symbol
Conversion(SI)
Force
Length
Time
Mass
Pound
Foot
Second
Slug
lbf
ft
s
slug
Power
Horsepower
hp
Pressure
pounds per
square inch
psi
4.448 N
0.3048 m
Same
1 lbf.(s2/ft) or
14.59 kg
550(ft.lbf)/s or
745.7 watt
6894.7 pascal
AE system:
It is also known as US customary units. In this system,
there are more than one unit for length and mass.
Units of Length
Unit
Symbol
Conversion(SI)
Inch
in
2.54 cm
foot
ft
12 in or 30.48 cm
yard
yd
3 ft or 91.44 cm
mile
mi
1760 yd or 1.6 k
Units of Mass
Unit
Symbol
Conversion(SI)
ounce
oz
28.35 grams
pound
lb
16 oz or 0.4536 kg
Units of Mass
Unit
Symbol
Conversion(SI)
Ton(short)
ton
2000 lb or 907.2 kg
Ton(long)
ton
2240 lb or 1016 kg
Other Units
Unit
Symbol
Name
Force
Poundal
pdl
Energy
Poundal-foot
Ft-pdl
Conversion(SI)
0.1383 N or
1 (lb.ft)/s2
0.042 joules
CGS system:
It is another system of unit which was previously in use but now
has been replace by the SI system. This system uses the smaller
counter-part of SI base units (of mass, length and time) i.e., gramcentimeter-second.
Name
Unit (FPS)
Symbol
Conversion(SI)
Mass
Length
Time
Acceleration
Force
Energy
Pressure
Viscosity
Gram
Centimeter
Second
Galileo
Dyne
Erg
Barye
Poise
g
Cm
s
gal
dyn
erg
ba
P
0.001 kg
0.01 m
Same
10-2 m/s2
10-5 N
10-7 joules
0.1 pascals
0.1 Pa-s
Prefixes:
In SI units, prefixes are used to represent small and large
quantities elegantly. Following are some prefixes used in SI system:
Name
Deca
Hecto
Kilo
Mega
Giga
Tera
Peta
Exa
Symbol
da
h
K
M
G
T
P
E
Factor
10
100
1000
106
109
1012
1015
1018
Name
Deci
Centi
Mili
Micro
nano
pico
femto
atto
Symbol
d
c
m
µ
n
p
f
a
Factor
0.1
0.01
0.0001
10-6
10-9
10-12
10-15
10-18
Conventions for Writing Units:
Following are some SI conventions for writing units:
 The full of the unit does not being with a capital letter
even if it is name after a scientist.
 The symbol of a unit named after a scientist is initial
capital letter i.e., N for newton
 Prefix is written before a unit without any space
 Combination of base unit is written each with one space
apart (in writing this is not apparent)
 Compound prefixes are not allowed
 When a base quantity is written with a prefix, it is
considered as one single word i.e., mm as millimeter and
not as milli meter.
 When a multiple of a base unit is raised to a power the
power applies to the whole multiple i.e., 1 cm2 = 1 (cm)2 =
10-4 m2
Other Useful units:
In physics, there are some other units which are used alongside
SI units, Such as
Name
Length
Volume
Pressure
Energy
Unit
Angstrom
lightyear
Astronomical unit
Micron
Fermi
Liter
Gallon
Bar
Standard atmosphere
Torr or
(~ mmHg)
Calorie
Kilowatt-hour
Electron-volt
British thermal unit
Symbol Conversion(SI)
Å
ly
Au
µm
fm
L
gal
bar
atm
torr
cal
kWh
eV
BTU
10-10 m
9.46 x 1015 m
1.496 x 1011 m
1.0 x 10-6
1.0 x 10-15
1 dm3
3.785 L
105 Pa
101325 Pa
133.32 Pa or
1/760 of atm
4.184 J
3.6 MJ
1.6 x 10-19 J
1055.06 J
Errors and uncertainty:
In any measurement, there is always a possibility of slight
imperfections. One might find that repeating a certain measurement
gives different results, it could be due to different reasons. In physics
errors and uncertainty are used interchangeably, but they are two
different concepts
Errors:
The deviation of the measured value from the exact value
is called error. They may be due to different reasons such as:
 Negligence or inexperience of the experimenter
 Faulty apparatus
 Inappropriate (less accurate) method (used for taking
measurement)
In all of the above causes, errors are easy to remove. Errors mainly
occur due to human negligence.
Uncertainty:
It is the range of value within which a measured value may
be. It refers to not being sure of a measured value. It is an inherent
feature of an experiment and is often very hard to remove. It is not
possible to measure a quantity with infinite precision, so there is
always some uncertainty in a measured value. It may be due to
following reasons:
 Limitation of an instrument
In the above diagram, a meter rule is used to measure the
length of a pencil, as a meter rule can only measure
minimum length of 0.1 cm so there is some uncertainty in
this measurement.
 Imperfection of human sense.
Types of Errors:
Random Error:
In repeated measurements, when there are random
fluctuations in the result, it is due to random error. The causes of
random error are hard to detect and it may be due variety of
reasons. Random error can be reduced by taking the mean value
of many measurements of the quantity.
Parallax error:
Wrong positioning of eye when taking a
reading is called parallax error. It is a type of random error.
Systema
tic Error:
In repeated
measurements,
systematic error
gives a consistent
shift of the
measured value in either direction. It may scale up the
measurement by some factor or scale down. For example, in a
meter rule, after zero point next marked value is 2 and 1 is
missing, then all the measurements (greater than 1 unit) taken by
that meter rule will be shifted up by 1 unit. It can also occur due
to poor calibration or zero error. Systematic errors can be
corrected by applying the known correction factor or comparing
the faulty instrument with the correctly calibrated one.
Significant figures:







Significant figures are used to represent accuracy in a
measurement. Significant figures contain all the accurately
known digits and the first doubtful digit. For example, 243.4
contains 4 significant figures, first 3 are accurately known while
the 1st decimal place is doubtful. Why is it doubtful? Well
because the original might have been 243.367 and while
rounding it to 4 significant figures, we rounded the first decimal
place to 4, that is why it is doubtful. There are some rules for
spotting significant figures:
All the non-zero digits are significant
Zeros between non-zero digits are significant
All the zeros to the left of non-zero digits are insignificant I.e.,
0023, 0.0023 in both of these numbers all the zeros are
insignificant.
Zeros to the right of non-zero digits after the decimal place are
significant i.e., 0.2300 here zeros after .23 are significant.
Zeros to the right of non-zero digits before the decimal place
may or may not be significant.
If a decimal place is place at the end of number all of its digits
are significant figures i.e., 4300. It has 4 sig figs.
Exact quantities have unlimited or infinite number of significant
figures. For example, if you count 12 eggs then there is no
uncertainty in this measurement, therefore it has infinite number
of significant figures
According to the textbook, measurement should only be taken up
to the least count of the measurement. For example, using a meter
rule measurement only up to 1st decimal place (in case of cm)
should be measured and 2nd decimal should not be guessed.
In the above diagram, since the reading of the pencil is closer to
26.1 cm so the recorded measurement will be 26.1 cm with some
uncertainty. But it is also helpful to guess the 2nd decimal place as
it would tell us that is measurement is surely less than 26.1 cm. I
would guess the above measurement as 26.06 cm(maybe).
Rounding off:
While taking measurements, often we have to round of the
measurement to some number of significant figures or decimal
places. Following rules apply while rounding off:
 If the digit to be dropped is less than five, the immediate digit
to the left is retained as is
 If the digit to be dropped is greater than five, the immediate
digit to the left is retained by incrementing it by 1
 If the digit to be dropped is five, then if the digit to the left is
even is retained as is and if it is odd, it is incremented by 1
Precision and accuracy:
In everyday life, word precision and accuracy are used
interchangeably, but in physics they have two different meanings:
Precision is how close are the measured values to each other, while
accuracy is how close the measured value is to the actual value.
The is represented in the following diagrams:
In actual measurements precision is represented by the absolute
uncertainty while accuracy is represented by percentage or
fractional uncertainty. This is explained by the following example:
Let’s say two measurements are taken, one with a meter rule and
one with a vernier caliper as:
23.4 cm ± 0.1 cm by a meter rule
1.23 cm ± 0.01 cm by a vernier caliper
Absolute uncertainty of 0.1 cm suggests that the actual
measurement taken by the meter rule lies anywhere between 23.3
to 23.5 cm as:
Anywhere between
23.30 cm
23.35 cm
23.40 cm
23.45 cm
23.50 cm
And absolute uncertainty of 0.01 cm suggests that measurement
taken by vernier caliper lies anywhere between 1.22 to 1.24 cm as:
1.220 cm
1.225 cm
1.230 cm
Anywhere between
1.235 cm
1.240 cm
As the values of possible measurements taken by vernier caliper
are closer together, therefore measurement of 1.23 ± 0.01 cm is
more precise than 23.4 cm ± 0.1 cm. As can be seen precision
depends on the instrument being used.
Fractional uncertainty of the measurement taken by meter rule is:
23.4 ± 0.43%
And that of (measurement taken by) vernier caliper is:
1.23 cm ± 0.81%
As percentage uncertainty of vernier caliper is higher than that of
meter rule, so the measurement taken by meter rule is more
accurate. Accuracy also depends on the measurement, if higher the
measurement (taken by meter rule), then uncertainty of 0.1 cm
doesn’t make the measurement deviates that must from the actual
measurement i.e., there is not much difference between 99.9 and
100.
Compound Error:
When quantities with some uncertainty are manipulated using
arithmetical operations, their uncertainties are also manipulated
according to the following rules:

Addition or subtracted:
When quantities are added or subtracted, their absolute
uncertainties are added. For example, lengths of two objects are
measured as 23.4 cm and 1.23 one with a meter rule and other
with a vernier caliper respectively, the difference between them
is calculated as
23.4 – 1.23 = 22.17 cm
Since absolute uncertainty of meter rule is 0.1 cm and that of
vernier caliper is 0.01 cm, so uncertainty of final result will be
0.11 cm. Hence final result is 22.17 ± 0.11 cm

Multiplication or division:
When quantities are multiplied or divided, their fractional
or percentage uncertainties are added. Force is applied on
an object of mass 5.0 ± 0.1 kg which produces an
acceleration of 2.3 ± 0.2 m/s2, find the magnitude of force
and its uncertainty:
As F=ma So,
F = (5.0 kg)(2.3 m/s2) = 12 N
Now for uncertainty:
% Uncertainty in mass= ∆m = 2%
% uncertainty in acceleration= ∆a = %8.7
% uncertainty in acceleration= ∆F = ∆m+∆a
= 10.7%
Uncertainty in Final force = 12 x 10.7% = 1.3 N
So final result is 12 ± 1.3 N

Power factor
When a quantity is raised to some exponent, uncertainty is
determined by multiplying the exponent with the fractional
or percentage uncertainty of the quantity. Example, a
sphere has a radius of 0.56 cm having an absolute
uncertainty of 0.01 cm, determine its volume (with
uncertainty)
Volume = V = 4⁄3πr3
So, V = 0.74 cm3
% Uncertainty in radius = 1.78%
% Uncertainty in volume = 1.78% x 3 = 5.34%
Uncertainty in Volume = 0.04 cm3
So final result is 0.74 ± 0.04 cm3

For uncertainty in the average value of many
measurements:
Find the average of the all the values. Calculate the
deviation of each value from the average value. Take the
mean of all the deviations, which is the uncertainty in the
average value.

Counting Experiments
The uncertainty in experiments in which some quantity of
many objects (having same magnitude of the quantity) is
measured simultaneously is determined by dividing the
least count of the measuring instrument by the number of
objects. For example, three objects are placed side by side
and their length is measured with the help of a meter rule
as (30 ± 0.1) cm, now the length of the single object is
given as (30 ± 0.1)/3 cm or (10 ± 0.03) cm. Another
example is of measuring the time period of a pendulum.
Usually, time period of a pendulum is measured by noting
the time for many oscillations of the pendulum and then
dividing the time by the number of oscillations. In this
case the uncertainty is also divided by the number of
oscillations.
Dimensions of physical quantities:
In physics, each base quantity has been given a dimension and is
represented by a capital letter enclosed in square brackets. Following
are dimensions of base quantities:
Base Quantity
Dimensions
[M]
Mass
[L]
Length
[T]
Time
[K], [θ]
Temperature
[A]
Electric Current
[n]
Amount of Substance
[J]
Luminous Intensity
There are many units for measuring a single base quantity but there is
only a single dimension for a base quantity. Dimensions are used for
analyzing different equations. Dimensions of all derived quantities are
written in terms of the dimensions of base quantities. Dimensions are
used for checking the homogeneity (correctness) of an equation. A
method called dimensional analysis utilizes dimension to find the
units(dimension) of an unknown variable in an equation and also the
convert between different units.
In an equation dimensions on both sides of the equation must be
equal. Using this fact, a possible formula for some physical quantity,
knowing all the factors that affect it, can be derived.
Imp MCQs(Entry Test)
Which one of the following statements is false:
a. Two different physical quantities of different units may have same
dimension
b. A dimensionless quantity may have units
c. A dimensionally consistent statement may be inconsistent
d. None of these
Values of some, time intervals(s)
Age of the universe:
Age of the Earth:
One year:
One day:
Time between normal
beats
Period of audible
sound waves
Period of typical
radio waves
Period of vibration of
an atom in a solid
Period of visible light
waves
Side
Information:
5 x 1017
1.4 x 1017
3.2 x 107
8.4 x 104
8 x 10-1
heart
1 x 10-3
1 x 10-6
1 x 10-13
2 x 10-15
areas of
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1 min 20 sec
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8 min 20 sec
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5 h 20 sec