Lesson 8 of 10 · 7 min
Checking dimensional consistency
NCERT §1.6.1
Kabir says the period of a pendulum is T = l/g. Riya says T = 2π√(l/g). Before either of them times a single swing, dimensions can already settle part of the argument.
The lesson in notes
In short
Only quantities with the same dimensions can be added, subtracted or equated; this principle of homogeneity of dimensions is a quick test of an equation.
In a correct equation every term on both sides has the same dimensions. In x = x₀ + v₀t + ½at², each term, x, x₀, v₀t and ½at², has dimension [L].
Units and dimensions are handled like algebraic symbols: identical ones in the numerator and denominator cancel.
The argument of a trigonometric, logarithmic or exponential function must be dimensionless, and pure numbers and ratios of like quantities (an angle, refractive index) have no dimensions.
An equation that fails the test is certainly wrong; one that passes is not thereby proved right, because dimensionless factors cannot be checked.
Of the candidates K = m²v³, ½mv², ma, (3/16)mv² and ½mv² + ma for kinetic energy, dimensions rule out m²v³, ma and the mixed sum, but cannot choose between ½mv² and (3/16)mv².
½mv² = mgh passes: both sides are [M L² T⁻²].