Angular Displacement, Velocity & Acceleration

Angular displacement is the angle through which a body rotates about a fixed axis. It is measured in radians (rad). One full revolution corresponds to $\theta = 2\pi$ rad ≈ 6.283 rad. The radian is defined geometrically: $\theta = s / r$, where $s$ is the arc length traced by a point on the body and $r$ is its distance from the rotation axis. Angular displacement is a signed quantity — positive values conventionally indicate counter-clockwise rotation and negative values indicate clockwise rotation, following the right-hand rule.

Angular velocity describes how fast the angular displacement is changing with time. The average angular velocity over a time interval $\Delta t$ is $\bar{\omega} = \Delta\theta / \Delta t$. The instantaneous angular velocity is the derivative: $\omega = d\theta/dt$. Its SI unit is radians per second (rad/s). Angular velocity is related to the ordinary frequency $f$ (in Hz) and the period $T$ (in s) by $\omega = 2\pi f = 2\pi / T$. A faster spin means a larger $|\omega|$. The direction of $\omega$ (as a vector) lies along the rotation axis, determined by the right-hand rule: curl the fingers of the right hand in the direction of rotation, and the thumb points in the direction of $\vec{\omega}$.

Angular acceleration describes how quickly angular velocity changes. The average angular acceleration is $\bar{\alpha} = \Delta\omega / \Delta t$, and the instantaneous value is $\alpha = d\omega/dt = d^2\theta/dt^2$. Its SI unit is radians per second squared (rad/s²). If $\alpha$ and $\omega$ have the same sign, the rotation is speeding up; if they have opposite signs, the rotation is slowing down. This is directly analogous to linear acceleration either speeding up or slowing down linear motion.

When $\alpha$ is constant, the rotational kinematic equations are perfectly parallel to the linear equations. Starting from $\alpha = d\omega/dt$ and integrating: $\omega = \omega_0 + \alpha t$. Integrating again: $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$. Eliminating $t$ gives the velocity-displacement relation: $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$. These three equations allow us to solve any constant-$\alpha$ rotational problem given sufficient initial conditions.

For a point at radius $r$ from the axis, the linear (tangential) speed is $v = r\omega$, the tangential acceleration is $a_t = r\alpha$, and the centripetal acceleration directed toward the axis is $a_c = v^2/r = r\omega^2$. These relations are crucial: they allow conversion between the rotational description of the whole body and the linear motion of any specific point on it. Note that all points share the same $\omega$ and $\alpha$, but points farther from the axis have larger $v$ and $a_t$.

\alpha = 18 \text{ rad/s}^2,\quad t \approx 6.67 \text{ s}

• Torque and Newton's Second Law for Rotation — Torque (τ = Iα) directly uses angular acceleration α; understanding α as defined here is essential before studying how forces cause rotational motion.
• Moment of Inertia — Moment of inertia connects angular acceleration to torque (τ = Iα) and depends on how mass is distributed about the rotation axis, building directly on the definition of α.
• Linear Kinematics — The angular kinematic equations are direct analogues of the linear kinematic equations; understanding linear displacement, velocity, and acceleration makes angular variables much easier to grasp.
• Uniform Circular Motion — Uniform circular motion is the special case where ω is constant (α = 0); it introduces centripetal acceleration a_c = rω², linking rotational and linear variables.
• Rotational Kinetic Energy — Rotational KE = ½Iω² uses angular velocity ω defined in this article; energy methods for rotating bodies require a solid understanding of all three angular variables.