Units and Dimensions - Consistency of Equations

Description:

Principle of homogeneity of dimensions

Let’s assume we have two physical quantities A and B −

• A ± B is only valid if and only if [A] = [B] (dim. of A = dim. of B)
• A = B is only valid if and only if [A] = [B] (dim. of A = dim. of B)

In simple words, only the quantities with similar dimensions could be added and subtracted from each other or equated to each other.

Example −

Force can’t be added or subtracted to velocity, distance, time, etc. It can only be added or subtracted to another force.

Force can’t be equal to anything else except force itself.

Checking Dimensional Consistency

Problem − Verify dimensional consistency of

x₀ = ut + 1/2 at²

Where, x₀ ≡ length; u ≡ Velocity; t ≡ time; a ≡ acceleration

Method

Dimensions of x₀ = [L]

Dimensions of ut = [L][T]^-1[T] = [L]

Dimensions of 1/2 at² = 1[L][T]^-2[T]² = [L] (Pure numbers like (1/2) are dimensionless)

Dimension of ut and 1/2 at² are equal hence, their addition is valid.

Dimensions of both L.H.S and R.H.S are equal to [L] hence, the equation is dimensionally consistent.

Important Note

Quantities in the argument of any Mathematical function must be dimensionless.

E.g

sin x, logy, e^z, etc.

Each x, y and z should be dimensionless.

• If an equation fails the dimensional consistency test, it is proved wrong.

• If an equation passes the dimensional consistency test, it is NOT proved right.

E.g. x₀ = ut + 1/2 at² and x₀ = ut + 4at², both are dimensionally consistent. But only the former is the correct equation.