Circular Motion: Angular Kinematics

Free notes · JEE Main Physics

Formulas

\omega = \frac{d\theta}{dt}, \qquad \alpha = \frac{d\omega}{dt}, \qquad v = \omega r, \qquad a_t = \alpha r

Angular quantities and their connection to linear motion

a_c = \frac{v^2}{r} = \omega^2 r, \qquad a = \sqrt{a_t^2 + a_c^2}

Centripetal (radial) acceleration and the total acceleration

\omega = 2\pi f = \frac{2\pi}{T}, \qquad \omega = \frac{2\pi \cdot \text{rpm}}{60}

Angular velocity from frequency, period or rpm

Key points

In uniform circular motion, speed is constant but acceleration is NOT zero — the direction of velocity keeps changing. Centripetal acceleration always points toward the centre, never along the tangent.

Worked example

A wheel rotates at 300 rpm. Find (a) its angular velocity in rad/s, (b) the linear speed and centripetal acceleration of a point 0.20 m from the axis.

(a) ω = 2π × 300 / 60 = 10π ≈ 31.4 rad/s
(b) v = ωr = 31.4 × 0.20 ≈ 6.28 m/s
a_c = ω²r = (31.4)² × 0.20 ≈ 197 m/s²

Answer: ω ≈ 31.4 rad/s; v ≈ 6.28 m/s; a_c ≈ 197 m/s²