E1.1 Dimension of Physical quantity:

The dimension of a physical quantity is defined as the powers to be raised on fundamental units of length (L), Mass (M), & Time (T) to give the unit of that physical quantity. For example:
$Velocity = \frac{Displacement}{Time} = m/s  = \frac{ [L]}{ [T]} = [L][T^{-1}] = [M^0 L^1 T^{-1}]$

Dimensional Formula:
It is a relation that shows how & which fundamental quantities are involved into a physical quantities. For Example:
Dimensional Formula of force is [$MLT^{-2}$];  Dimensional Formula of acceleration is [$LT^{-2}$]

Dimensional equation:
An equation containing physical quantities with dimensional formula is known as dimensional equation.
Or, the dimensional formula of a physical quantity expressed in the form of an equation is called dimensional equation of that quantity. For example:
Dimensional equation of v = u + at is, 
$[M^0 L^1 T^{-1}] = [M^0 L^1 T^{-1}] + [M^0 L^1 T^{-2}] [M^0 L^0 T^1] = [M^0 L^1 T^{-1}]$


The following table shows the dimensional formulas of some Physical Quantities:
S. No
Physical Quantities
Formula
Dimensional Formula
SI Unit
1
Area
$l * b$
$[L]*[L] = [M^0\,L^2\,T^0]$
$m^2$
2
Volume
$l*b*h$
$[L]*[L]*[L] = [M^0\,L^3\,T^0]$
$m^3$
3
Speed or Velocity
$\frac{distance}{time}$
$\frac{[L]}{[T]} = [M^0\,L^1\,T^{-1}]$
$m/s$
4
Density
$\frac{mass}{volume}$
$\frac{[M]}{[L^3]}= [M^1\,L^{-3}\,T^0]$
$Kg/m^3$
5
Acceleration
$\frac{velocity}{time}$
$\frac{[L\,T^{-1}]}{[T]}= [M^0\,L^{1}\,T^{-2}]$
$m/s^{-2}$
6
Frequency
$\frac{no\;of\;vibrations}{time}$
$[M^0\,L^{0}\,T^{-1}]$
$hertz$
7
Momentum
(P = MV)
$mass * velocity$
$[M]*[L\;T^{-1}] = [M^1\,L^1\,T^{-1}]$
$kg\,m\,s^{-1}$
8
Force
$mass * acceleration$
$[M]*[L\,T^{-2}] = [M\,L\,T^{-2}]$
N (Newton)
9
Impulse
force * time
[M\,L\,T^{-2}] * [T] = [M\,L\,T^{-1}]
N s
10
Surface Tension
$\frac{force}{length}$
$\frac{[M\,L\,T^{-2}]}{[L]} = [M\,L^0\,T^{-2}] $
$N\,m^{-1}$
11
Pressure
$\frac{force}{area}$
$\frac{[M\,L\,T^{-2}]}{[L^2]}$
$N\,m^{-2}$ or Pa
12
Coefficient of Viscosity
$\frac{force}{area * velocity\;gradient}$
$[M\,L^{-1}\,T^{-1}]$
da P (decapoise)
13
Work
force * distance
$[M\,L\,T^{-2}]*[L] = [M\,L^2\,T^{-2}]$
J (Joule)
14
Energy
work = force * distance
$[M\,L\,T^{-2}]*[L] = [M\,L^2\,T^{-2}]$
J (Joule)
15
Power
$\frac{Work}{time}$
$\frac{[M\,L^2\,T^{-2}]}{[T]} = [M\,L^2\,T^{-3}]$
W (Watt)
16
Gravitational Constant (G)
$\frac{force\, *\, (distance)^2}{{mass}^2}$
$[M\,L^3\,T^{-2}]$
$N\,m^2\,kg^{-2}$
17
Gravitational Field Strength
$\frac{force}{mass}$
$[M\,L^1\,T^{-2}]$
$N\, kg^{-1}$
18
Gravitational Potential
$\frac{work}{mass}$
$[M^0\,L^2\,T^{-2}]$
$J\, kg^{-1}$
19
Force Constant (K)
$\frac{F}{L}$
$[M\,L^0\,T^{-2}]$
$N\, m^{-1}$
20
Angle
$\frac{arc}{radius}$
Dimensionless
rad
21
Moment of Inertia
$Mass * (distance)^2$
$[M\,L^2\,T^0]$
$Kg\,m^2$
22
Angular Momentum
Moment of inertia * angular velocity
$[M\,L^2]*[T^{-1}]=[M\,L^2\,T^{-1}]$
$kg\,m^2\,s^{-1}$
23
Torque or Couple
Force * perpendicular distance
$[M\,L\,T^{-2}]*[L] = [M\,L^2\,T^{-2}]$
N m
24
Kinetic Energy
$\frac{1}{2}mv^2$
$[M\,L^2\,T^{-2}]$
J (Joule)
25
Potential Energy
$mgh$
$[M\,L^2\,T^{-2}]$
J (Joule)
26
Stress
$\frac{force}{area}$
$[M\,L^{-1}\,T^{-2}]$
$N\,m^{-2}$ or Pa
27
Strain
$\frac{change\;in\;length}{original\;length}$
$[M^0\,L^{0}\,T^{0}]$
No unit
28
Modulus of Elasticity
$\frac{stress}{strain}$
$[M\,L^{-1}\,T^{-2}]$
$N\,m^{-2}$ or Pa
29
Angular Displacement
$\frac{arc}{radius}$
$[M^0\,L^{0}\,T^{0}]$
No unit
30
Angular Velocity ($\omega$)
$\frac{angular\;displacement}{time}$
$[M^0\,L^{0}\,T^{-1}]$
$rad/sec$
31
Angular Acceleration
$\frac{change\;in\;angular\;velocity}{time}$
$[M^0\,L^{0}\,T^{-2}]$
$rad/sec^{-2}$
32
Angular Momentum
$I \omega$
$[M^\,L^{2}\,T^{-1}]$
$kg\,m^2\,sec^{-1}$
33
Angular Impulse
$I \omega$
$[M^\,L^{2}\,T^{-1}]$
$kg\,m^2\,sec^{-1}$
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