Dimensional Formula

A dimensional formula is an expression that shows how and which base quantities represent the dimensions of a physical quantity.

General form: [MaLbTcAdKemolfcdg][M^a L^b T^c A^d K^e mol^f cd^g] where a,b,c,…a, b, c, \dots are integers.

Examples:

  • Velocity → [M0L1T−1][M^0 L^1 T^{-1}]
  • Force → [M1L1T−2][M^1 L^1 T^{-2}]
  • Work → [M1L2T−2][M^1 L^2 T^{-2}]
  • Pressure → [M1L−1T−2][M^1 L^{-1} T^{-2}]

Dimensional Equations

When a physical quantity is equated with its dimensional formula, the result is a dimensional equation.

Examples:

  • Volume: V=[M0L3T0]V = [M^0 L^3 T^0]
  • Acceleration: a=[M0L1T−2]a = [M^0 L^1 T^{-2}]
  • Force: F=[M1L1T−2]F = [M^1 L^1 T^{-2}]

Application of Dimensional Equations

They are useful for:

  1. Checking dimensional consistency of equations.
  2. Converting units from one system to another.
  3. Deriving relationships between physical quantities (if form of dependence is known).

Dimensional equations reveal how a quantity depends on basic dimensions, but not on dimensionless constants like π\pi or numerical factors like 1/21/2.

Example 1:

Find the dimensional formula of power.

Solution: Power = Work / Time Work = [M1L2T−2][M^1 L^2 T^{-2}] Time = [T][T] Power=[ML2T−2][T]=[ML2T−3]\text{Power} = \frac{[M L^2 T^{-2}]}{[T]} = [M L^2 T^{-3}]

Example 2:

Find the dimensional formula of surface tension. Surface tension = Force / Length Force = [MLT−2][M L T^{-2}], Length = [L][L] Surface tension=[ML0T−2]\text{Surface tension} = [M L^0 T^{-2}]