# Dimensions 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]

2. a) Above expression shows that dimensions of 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