KinexiaLEARN PHYSICS

Pendulum Period — Finding Length from Period (Worked Example)

Medium
Problem

In a lab, students time a pendulum and find its period is 1.42 s. What is the pendulum's length? (g = 9.8 m/s²)

Given

  • T = 1.42 s

  • g = 9.8 m/s²

Find

The length L.

Principle

Rearranging T = 2π√(L/g) gives L = g(T/2π)² — the same measurement that historically determined g itself.

Formula

Solution

  1. Rearrange: L=g(T/2π)2L = g(T/2\pi)^2

  2. Compute T/2π=1.42/6.283=0.226T/2\pi = 1.42/6.283 = 0.226

  3. Square: 0.2262=0.05110.226^2 = 0.0511

  4. Multiply: L=9.8×0.0511=0.501 mL = 9.8 \times 0.0511 = 0.501\ \mathrm{m}

0.501 m

Unit check

m/s² × s² = m — a length, correct.

What this means physically

0.501 m — about half a metre of string. This inverse use of the formula is exactly how 17th-century scientists measured g: time the swing, solve for gravity.

Common mistake

Forgetting to square the (T/2π) term. Length depends on the square of the period — small timing errors grow quickly.