Dimensions of a Physical Quantity

What is dimension of a physical quantity?

1. In all the systems of units, the derived units of all the physical quantities can be expressed in terms of the fundamental units of mass, length and time, raised to some power. The powers to witch fundamentals units must be raised in order to express a physical quantity, are called its dimensions. To make it clear, consider the physical quantity density which is defined as mass per unit volume.
Hence, density=mass/volume=M/L³=[M1L³T⁰].

Dimensions of a Physical Quantity

2. a) Above expression shows that dimensions of a physical quantity (density) are 1 in mass, -3 in lengths and 0 in time.
c) Dimensional formula for density is [ML^-3] or [ Ml^-3T⁰].
d) Dimensional equation for density is [d] =[ML^-3T⁰].

Dimensional formula of some physical quantities in mechanics and their units in different systems

Physical Quantity	Formula	Dimensional Formulae	S.I Unit
Area	Length × breadth	[L2]	m2
Volume	Length × breadth × height	[L3]	m3
Density	Mass/vol.	[ML-3]	Kg/m3
Speed or velocity	Distance/time	[LT-1]	m/s
Acceleration	Velocity/time	[LT-2]	m/s2
Force	Mass × acc	[MLT-2]	Newton
Work	Force × distance(FsCosθ)	[ML2T-2]	Kg.m2/s2 or jule
Energy(Mechanical heat, light etc)	1/2mv2 or mc2	[ML2T-2]	N ×m
Power	dW/dt	[ML2T-3]	J/s or watt
Momentum or Impulse	mV	[MLT-1]	Kg-m/sec
Momentum of force or Torque	Force × distance	[ML2T-2]	N ×m
Angle or solid angle	Arc/radius	[M0L0T0]	Radians
Stress	Force/Area	[ML-1T-2]	N/m2
Strain	Change in dimensions/original dimensions	[M0L0T0]	No unit
Coefficient of Elasticity	Stress/Strain	[ML-1T-2]	N/m2
Surface Tension	Energy/length	[ML0T-2]	N/m
Surface Energy	Energy/Area	[ML0T-2]	J/m2
Velocity gradient	Velocity/distance	[ML0T-1]	s-1
Coefficient of viscosity	Force/Area × velocity gradient	[ML-1T-1]	Kgm/m/sec
Radius of gyration	Distance	[M0LT0]	m
Moment of Inertia	Mass ×(Radius of gyration)2	[ML2]	Kg × m2
Angular velocity	Angle/Time	[T-1]	Radian/sec
Frequency	1/T	[T-1]]	Cps or Hertz
Angular acceleration	Angular velocity/time	[M0L0T-2]	Radian/sec2
Angular momentum	Moment of inertia × velocity (Iω)	[ML2T-1]	Kgm2s-1
Pressure	Force/Area	[ML-1T-2]	N/m2

If anyone ask you:

What do you mean by the dimension of a physical quantity?

You can simply tell him:

The dimensions of a physical quantity are the powers to which the fundamental units are raised to represent a physical quantity.