Angular Velocity — Earth's Rotation (Worked Example)

Medium

Problem

Earth completes one full rotation (2π radians) every 24 hours. What is Earth's angular velocity?

Given

Δθ = 2π rad
Δt = 24 h = 86,400 s

Find

Angular velocity ω.

Principle

Angular velocity is the rate of rotation: ω = Δθ/Δt, the rotational analogue of v = Δx/Δt.

Formula

\omega = \frac{\Delta\theta}{\Delta t}

ω = Δθ/Δt

Solution

Write the formula: $\omega = \Delta\theta / \Delta t$
Substitute: $\omega = 2\pi / 86400$
Calculate: $\omega = 7.27 \times 10^{-5}\ \mathrm{rad/s}$

7.27 × 10⁻⁵ rad/s

Unit check

rad ÷ s = rad/s — correct.

What this means physically

A tiny 7.27 × 10⁻⁵ rad/s — yet at the equator (r = 6371 km) that means v = ωr ≈ 463 m/s eastward. You are moving faster than a jet right now and feel nothing, because the motion is uniform.

Common mistake

Using 24 h without converting to seconds. ω in rad/s needs Δt in seconds — 86,400 s, not 24.