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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