Angular Displacement, Velocity and Acceleration

The need for angular quantities

When a rigid body rotates, every point on it moves in a circle. Different points have different speeds and different accelerations. But one thing is the same for all points: they all rotate through the same angle in the same time.

Angular quantities — angle, angular velocity, angular acceleration — describe this shared rotational behaviour, independent of which point you look at.

Angular displacement $\theta$

Angular displacement is the angle through which a body has rotated from a reference position.

Unit: radian (rad)

One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius.

$2\pi \text{ rad} = 360° \implies 1 \text{ rad} = \frac{180°}{\pi} \approx 57.3°$

Angular displacement is a scalar for small angles but behaves as a vector (along the rotation axis) for finite rotations.

Angular velocity $\omega$

Angular velocity is the rate of change of angular displacement:

$\omega = \frac{d\theta}{dt}$

Unit: rad/s

For uniform rotation (constant $\omega$):

$\omega = \frac{\theta}{t} = \frac{2\pi}{T} = 2\pi f$

where $T$ is the period and $f$ is the frequency.

Direction: By the right-hand rule — curl the fingers of the right hand in the direction of rotation, the thumb points along $\vec{\omega}$.

Angular acceleration $\alpha$

Angular acceleration is the rate of change of angular velocity:

$\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}$

Unit: rad/s²

Positive $\alpha$: angular velocity increasing (speeding up in positive direction) Negative $\alpha$: angular velocity decreasing (slowing down, or speeding up in negative direction)

The complete analogy with linear motion

Every linear quantity has a rotational analogue. Every linear equation has a rotational counterpart.

Instantaneous vs average

Average angular velocity over interval $\Delta t$:

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

Instantaneous angular velocity:

$\omega = \lim_{\Delta t \to 0}\frac{\Delta\theta}{\Delta t} = \frac{d\theta}{dt}$

Same distinction as linear velocity — average over an interval vs value at an instant.