Units and Dimensions.pdf

Units and Dimensions notes for physics. Here is the complete notes for unit and dimensions. Mechanics, physics notes for students. All abou unit and dimension. units and measurements class 11. dimensions of physical quantities.

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Physical Quantities
The physical phenomena which can be measured are called Physical Quantities.
1.  Length, Time, Speed, and Magnetic Field are Physical Quantities as they can be measured.
 Smell, Taste are not Physical Quantities as they can not be measured in a general sense.
 There are two types of physical quantities: Fundamental Quantities and Derived Quantities.
2. Fundamental Quantities:
 The quantities that are independent of other quantities are called fundamental quantities.
 There are seven fundamental quantities. They are Length, Mass, Time, Temperature, Electric Current, Luminous
Intensity, and Amount of Substance.
3. Derived Quantities:
 The quantities that are dependent on other quantities are called derived quantities.
 Area = Length  Length and Speed =  . Here, area depends on length and speed depends on length and
time hence they are derived quantities.
 Other derived quantities are Volume, Force, Power, Intensity of sound, Magnetic Flux, etc.
Units
1. Unit is the standard reference in which physical quantity is expressed.
 Length is expressed in meters and speed is expressed in meters/second. The meter and meter/second are the units.
2. System of Units
 The standard references can vary based on which unit Mass, Length, Time, and other fundamental quantities are
measured. On the basis of that, there are four systems of units. They are CGS, FPS, MKS, and SI systems.
Quantity CGS System FPS System MKS System
1. Length Centimeter Foot Meter
2. Mass Gram Pound Kilogram
3. Time Second Second Second
 SI system is an extension of the MKS system in which units of all fundamental quantities are included.
Quantity SI Unit Symbol
1. Length meter m
2. Mass Kilogram kg
Units and Dimensions
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→
→
→
→
→
→ ×​
time
Distance
→
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2. Quantity SI Unit Symbol
3. Time Second s
4. Electric Current Ampere A
5. Temperature Kelvin K
6. Luminous Intensity Candela Cd
7. Amount of Substance Mole mol
3. Types of Units
 There are three types of units. Namely, Fundamental units, Derived Units, and Supplementary Units.
 Fundamental Units: The units of fundamental quantities which are independent of other units are called fundamental
units. There are seven fundamental units. eg. m,kg, s, K, Cd, Mol, A.
 Supplementary Units: The units of plane angle and solid angle which are two purely geometric angles are called
supplementary units.
Quantity Unit Symbol
1. Plane Angle Radian Rad
2. Solid Angle Steradian Sr
 Derived Units: The units of derived quantities which are dependent on fundamental units and supplementary units are
called derived units. For eg. m/s, kgms, m/s etc.
4. To cover all the scientific measurements in terms of SI unit, the prefixes are used for the powers of ten.
Powers of Ten Prefixes Symbol Powers of Ten Prefixes symbol
Yotta Y Yocto y
Zetta Z zepto z
Exa E Atto a
Peta P femto f
Tera T Pico p
Giga G nano n
Mega M micro
Kilo K Milli m
Hecto h centi c
→
→
→
→
−2 2
10
24
10
−24
10
21
10
−21
10
18
10
−18
10
15
10
−15
10
12
10
−12
10
9
10
−9
10
6
10
−6
μ
10
3
10
−3
10
2
10
−2
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3. Powers of Ten Prefixes Symbol Powers of Ten Prefixes symbol
deca da deci d
The size of the nucleus is in order of  m and the size of an atom is in order of 
Common Non-SI Units 
1. SI units are commonly used worldwide and cover all scientific measurements. Due to historical, political, and situational
significance, there are common Non-SI units in use.
2. Length
Common Non-
SI unit
Symbol Value in SI unit Significance
1 Astronomical
Unit
1 AU 1.496 10 m The distance from Earth to the Sun
1 Light Year 1 Light
Year
9.46 10 m The distance covered by light in one year
1 Parallactic
Second
1 Parsec 3.084 10
m/(3.26 Light Year)
the distance at which the length of one
astronomical unit subtends an angle of one second
of an arc.
1 Angstrom 1 1 10m named after 19th-century Swedish physicist Anders
Jonas Ångström
1 Fermi 1 Fermi 1 10m The smallest unit of distance used in nuclear
physics/ named after Enrico Fermi
1 X-ray Unit xu 1 10m used to quote the wavelength of X-rays and
gamma rays.
1 Inch 1 " 0.0254 m Imperial System
1 Foot 1 ft/ 1' 0.3048 m Imperial System
1 Yard 1 yd 0.9144 m Imperial System
1 mile 1 mile 1609.344 m Imperial System
1 Nautical mile 1 Nm 1852 m Used in air, marine, and space navigation
3. Mass
Common Non-SI unit Symbol Value in SI unit Significance
1 Pound 1 lb 0.4536 kg Imperial System
1 Slug 1 slug 14.59 kg British system based on standard gravity
10
1
10
−1
10
−14
10.
−10
×
11
×
15
×
16
A
˚ ×
−10
×
−15
×
−13

4. Common Non-SI unit Symbol Value in SI unit Significance
1 Quintal 1 Q 100 kg historical unit of mass
1 Metric tonne 1 t 1000 kg historical unit of mass
1 amu 1 amu 1.66 10 kg precisely 1/12 the mass of an atom of carbon-12
4. Time
Common Non-SI unit Symbol Value in SI unit Significance
1 Minute 1 min 60 s Convenience
1 Hour 1 hr = 60 min = 3600 s Convenience
1 Day 1 day = 24 hr = 86,400 s Convenience
1 year 1 yr 1= 365.25 day = 3.156 s Convenience
1 shake 1 Shake 1 10 s Used in nuclear physics
5. Other Conversion
Quantities Non SI values Value in SI unit
Pressure 1 Pascal 1 Nm
Pressure 1 atm = 760 mmHg 1.01 Nm
Pressure 1 bar= 1 atm 1.01 Nm
Pressure 1 torr= 1 mmHg 133.322 Nm
Volume 1 liter = 1 cm
Energy 1 eV 1.6 J
Energy 1 erg 1 J
×
−27
×10
7
×
−8
−2
×10
5 −2
×10
5 −2
−2
3
1×10m
−33
×10
−19
×10
−7

5. Quantities Non SI values Value in SI unit
Energy 1 KwHr 3.6 J
Power 1 Hp 746 W
Power of lens 1 D 1 m
Plane Angle 1 Degree
Measurement
1. The comparison of the amount of physical quantity in terms of standard reference of the same quantity is called
measurement.
2.  The standard reference amount is a unit (u).
 The scale of the amount with reference to the unit is the numerical value (n).
Hence,
Measured Quantity(Q) = Numerical value (n)  unit (u)
 Eg. 10 Kg where, numerical value = 10 and unit = 1 Kg.
3. 1. For a given measurement, the measured quantity is constant. i.e. Q = nu = constant.
Dimensions and Dimensional Formula
1. The dimensions of a physical quantity may be defined as the powers to which the fundamental units of mass, length, and
time must be raised to represent the physical quantity.
E.g
2.
The dimension of velocity is 0 in mass, 1 in length, and -1 in time.
×10
6
−1
​
rad
180
π
→
→
×
→
n
​u
​=11n
​u
​
For two systems of units. 
 
22
velocity=​=
time
displacement
​=
T
L
[MLT]
01−1

6. 3.
The dimension of velocity is 0 in mass, 1 in length, and -2 in time.
4.
=
= 
The dimension of force is 1 in mass, 1 in length, and -2 in time.
5. Different Types of Variables and Constants
 Dimensional Variable: Quantities having dimension and variable values. eg. Acceleration, Volume, Force, etc.
 Dimensional Constant: Quantities having dimension but constant values. eg. gravitational constant, Planck's
constant, Stefan's constant, etc.
 Non-dimensional Variable: Quantities not having dimension with variable values. eg. strain, angle, relative density,
etc.
 Non-dimensional Constant: Quantities not having dimension with constant values. eg.  , ,  etc.
6. Non-Dimensional Quantities
 Relative density
 Angle and solid angle
 Strain
 Poisson’s ratio
 Refractive Index
 Mechanical Equivalent of Heat
 Emissivity
 Magnetic Susceptibility
 Electric Susceptibility
 Relative Permittivity
 Relative Permeability
 Coefficient of Friction
 Loudness ( Decibel is the Unit of Intensity Level)
 Dielectric Constant
7. Physical Quantities Dimension SI unit
Area (A = ) []
m
Volume (V = ) []
m
Density ()
[]
kg/m
Acceleration ()
[]
m/s
acceleration=
​=
time
velocity
​=
T
LT
−1
[MLT]
01−2
Force=mass×acceleration
M×LT
−2
[MLT]
−2
→
→
→
→ πe1,,
→
→
→
→
→
→
→
→
→
→
→
→
→
→
L
2
L
2
2
L
3
L
3
3
ρ
=
​
v
m
ML
−3
3
a
=
​
t
v
LT
−2
2

7. Physical Quantities Dimension SI unit
Speed ()
m/s
Momentum (p = mv ) kgm/s
Acceleration ()
[]
m/s
Force ()
[]
N
Impulse ()
[]
Ns
Work () []
J
Power( )
[]
W
Pressure( ) []
N/m
Use of Dimensional Formula
1. Conversion of one system of the unit to another.
If  are the fundamental units on one system of units,  are the fundamental units on the second
system of units.
and, the dimensional formula of the physical quantity is  then,
Where  and  are the numerical values in each system.
 Example: Convert 1 Joule into erg.
Joule and erg are the units of energy. So, dimensional formula = 
and Joule is the unit of the SI system and erg is the unit of the CGS system.
v
=
​
t
d
[LT
]
−1
MLT
−1
a
=
​
t
v
LT
−2
2
F
=
ma
MLT
−2
I
=
Ft
MLT
−1
W
=
Fd
ML
T
2
−2
P
=
W/t
ML
T
2
−3
P
=
F/A
ML
T
−1
−2
2
M
​,L
​,T
​111 M
​,L
​,T
​222
MLT
abc
n
​=2n
​(
​)(
​)(
​)1
M​2
M
​
1
a
L​2
L
​
1
b
T​2
T
​
1
c
n​1n​2
→
[MLT]
2−2
n​=2n
​(
​)(
​)(
​)1
M
​
2
M​1
a
L
​
2
L​1
b
T
​
2
T​1
c
n
​=21(
​)(
​)(
​)
1gm
1kg
1
1cm
1m
2
1s
1s
−2
n​=21(​)(​)(​)
1gm
1000gm
1
1cm
100cm
2
1s
1s
−2
n
​=210
7

8. Hence, 
2. Checking the consistency of the equation
The principle of homogeneity states that "dimensions of each of the terms of a dimensional equation on both sides
should be the same. "
 Example: check the consistency of the equation 
Dimensional formula of 
Dimensional formula of 
Dimensional formula of 
Dimensional formula of 
Dimensional formula of 
Hence,
Dimension of 
Dimension of 
Dimension of 
Hence, the given equation is dimensionally consistent.
Note: The dimensionally consistent equations are not necessarily correct physical equations. But, dimensionally
inconsistent equations are always incorrect physical equations.
3. Deriving the equation
 Example: The viscous force acting on a sphere is directly proportional to the following parameters:
the radius of the sphere(r)
coefficient of viscosity()
the terminal velocity of the object 
Find the expression for the viscous force on the sphere if the proportional constant is 
Solution:
Using homogeneity of dimension:
Then,
solving we get, 
Dimensional Formula of Important Physical Quantities
1. Physical Quantities Dimension SI unit
Angular displacement( ) [ ] rad
Angular velocity( ) [ ] rad/s
Frequency( ) [ ] Hz
Angular acceleration( ) [ ] rad/s^
1J=10erg
7
→ v=
2
u+
2
2as
v=[MLT]
01−1
u=[MLT]
01−1
2=[MLT]
000
a=[MLT]
01−2
s=[MLT]
010
v=
2
[LT]=
−12
[LT]
2−2
u=
2
[LT]=
−12
[LT]
2−2
2as=[MLT][MLT][MLT]=
000 01−2 010
[LT]
2−2
→
η
v
6π
F∝ηrv
abc
[MLT]=
−2
[MLT][L][LT]
−1−1ab −1c
[MLT]=
−2
[ML T]
a−a+b+c−a−c
a=1
−a+b+c=1
−a−c=−2
a=b=c=1
F=6πηrv
θ=l/r MLT
000
ω=θ/t MLT
00−1
η=1/T MLT
00−1
α=ω/t MLT
00−2
2

9. Physical Quantities Dimension SI unit
Moment of Inertia( ) [ ]
Torque( ) [ ] Nm
Angular Momentum( ) [ ] Js
Force constant( ) [ ] Nm
Surface tension( ) [ ] Nm
Gravitational Constant(
[ ]
velocity gradient(
[]
Coefficient of viscosity (
[ ]
Gravitational Potential (
[ ]
Planck's Constant (
[ ]
Gravitational Intensity(
[ ]
Charge( []
Electric Potential (
[ ]
Electric Dipole ( [ ]
Current Density (
[ ]
Capacitance (
[ ]
Permittivity (
[ ]
Resistance (
[ ]
Conductance (
[ ]
2. Combined Dimension:
The dimension of  is the same as that of velocity. (E stands for Electric Field Intensity and B stands for magnetic field
intensity.)
I=mr
2
ML
2
kgm
2
τ=Iα MLT
2−2
J=Iω MLT
2−1
k=F/x MT
−2 −1
k=F/l MT
−2 −1
G=
​
)
m
​
m
​
12
Fr
2
MLT
−13−2
Nmkg
2−2
​
)
δx
δv T
−1
s
−1
η=
​
)
A×dv/dx
F MLT
−1−1
kgms
−1−1
=
​
)
m
W LT
2−2
mkg
2−2
h=
​
)
ν
E MLT
2−1
Js
=
​
)
m
F LT
−2
Nkg
−1
Q=It) AT As
=
​
)
Q
W MLTA
2−3−1
JC
−1
=q×2l LTA Cm
J=
​
)
A
I AL
−2
Am
−2
C=
​
)
V
Q MLTA
−1−242
Farad
=
​
)
A
Cd MLTA
−1−342
Farad/m
R=
​
=
I
V
​
)
QI
W MLTA
2−3−2
Ohm
=
​
R
1
MLTA
−1−232
mho
​
B
E

10.  has the dimension of Time.
The dimensions of solar constant are
3. Physical Quantities having the same dimensions
 Inertia and mass have the same dimensional formula.
 acceleration, gravitational field strength, and acceleration due to gravity. 
 Linear momentum and impulse.
 Work, energy, torque, couple, moment, internal energy, heat, work function .
 Angular velocity, frequency, velocity gradient, radioactive decay constant  .
 Spring constant, force gradient, surface energy. 
 Pressure, stress, modulus of elasticity, energy density . 
 Gravitational Potential, specific latent heat. 
 Electric potential, EMF, and electric potential difference. 
 Intensity of radiation and solar constant. 
 Angular momentum and Planck’s Constant.
 Magnetising force and intensity of magnetization. 
 Rydberg’s constant and wavenumber.  .
 Never memorize the dimensional formula of any physical quantity. Instead, memorize the formula which contains
that physical quantity and try to obtain the dimensional formula of the designated physical quantity.
For eg; For calculating the dimensional formula of permittivity, we first have to know the Formula which uses the
Permittivity. The simplest one which I come across is the following relation obtained from Coulomb's Law:
For vacuum 
For medium
From the above relation, we know the dimensional formula of Force(F), Charge(q), and distance(r).  is a dimensionless
quantity.
Through simplification, we can easily obtain the dimensional formula of  .
Where  Is the permittivity of free space.
Non-Standard Bases of Dimensional Formula
1. Taking force, length, and time to be the fundamental quantities. Find the density, momentum, and energy.
Density
From homogeneity of dimension:
Solving, we get, 
2. Momentum
From homogeneity of dimension:
RC=​=
R
L
​LC
​
(area×time)
energy
→
→ [MLT]
0−2
→ [MLT]
−1
→ [MLT2]
2−
→ [MLT]
00−1
→ [MLT]
0−2
→ [MLT]
−1−2
→ [MLT]
02−2
→ [MLTA]
2−3−1∣
→ [MLT]
0−3
→ [MLTA]
2−1
→ [MLTA]
0−10
→ [MLT]
0−10
P.S.
F​=v
​​
4πϵ
​0
1
r
2
q​q​12
F​=m
​​
4πϵ​ϵ​0r
1
r
2
q
​q
​
12
∴
​=
F
​
v
F​m
ϵ
​r
Π
ϵ
​
0
ϵ
​
0
[ML]=
−3
[F][L][T]
abc
[ML]=
−3
[MLT][L][T]
−2abc
[ML]=
−3
[MLT ]
aa+b−2a+c
a=1,a+b=−3,−2a+c=0
a=1,b=−4,c=2
[Density]=[FLT]
1−42
[MLT]=
−1
[F][L][T]
abc
[MLT]=
−1
[MLT][L][T]
−2abc
[MLT]=
−1
[MLT ]
aa+b−2a+c

11. Solving, we get, 
3. Energy
From homogeneity of dimension:
Solving, we get, 
Percentage Change
1. For percentage change 
Equation type 
Taking log in both sides,
differentiating,
2. Due to properties of logarithm:
The power is changed into multiplication.
The multiplication is changed into addition and
The division is changed into subtraction.
3. For percentage change 
Equation type 
 If the percentage increase in x is  , 
 If the percentage decrease in x is  , 
Errors in Measurement
1. There is a certain difference between the measured value and the actual value during measurement. It is called an error
in measurement.
2. Error in Equation type 
Taking log in both sides,
differentiating,
a=1,a+b=1,−2a+c=−1
a=1,b=0,c=1
[Density]=[FLT]
101
[MLT]=
2−2
[F][L][T]
abc
[MLT]=
2−2
[MLT][L][T]
−2abc
[MLT]=
2−2
[MLT ]
aa+b−2a+c
a=1,a+b=2,−2a+c=−2
a=1,b=1,c=0
[Density]=[FLT]
110
≤10%
A=kx
n
logA=logk+nlogx
​=
A
ΔA
n×
​
x
Δx
​×
A
ΔA
100%=n×​×
x
Δx
100%
y=x
n
logy=nlogx
y=ab
logy=loga+logb
y=
​
b
a
logy=loga−logb
>10%
A=kx
n
→ a%%change in A=[(​)−
100
100+a
n
1]×100%
→ a%%change in A=[(​)−
100
100−a
n
1]×100%
f=kx
n
logf=logk+nlogx

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REACTIONS
3. Error in Equation type 
Relative Error: 
4. There is no error of measurement in constant value.
5. The power is changed into multiplication.
The multiplication and division are changed into addition.
​=
f
Δf
n×
​
x
Δx
%error in f=​×
f
Δf
100%=n×​×
x
Δx
100%
f=k
​
z
c
xy
ab
​=
f
Δf
a×
​+
x
Δx
b×
​+
y
Δy
c×
​
z
Δz
Percentage Error:​×
f
Δf
100%=(a×
​+
x
Δx
b×​+
y
Δy
c×​)×
z
Δz
100%
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