Lesson 9 of 10 · 11 min

Deducing relations by dimensions

NCERT §1.6.2

Before learning any theory, Riya guesses that her pendulum's period could depend on the thread's length, on g and on the bob's mass. Dimensions alone can tell her exactly how.

The lesson in notes

In short

If you know which quantities (up to three independent ones) a result depends on, assume it is a product of their powers and match dimensions to find the powers.

For a simple pendulum take T = k lˣ gʸ mᶻ. Matching [T] = [L]ˣ [L T⁻²]ʸ [M]ᶻ gives x + y = 0, −2y = 1 and z = 0.

So x = ½, y = −½, z = 0 and T = k√(l/g): the period does not depend on the mass of the bob.

The dimensionless constant k cannot come from dimensions; experiment or theory gives k = 2π, so T = 2π√(l/g).

The method fails to give dimensionless constants, cannot tell apart quantities with the same dimensions (work and torque, for example), and cannot handle sums of terms or trigonometric, logarithmic and exponential dependence.