Angular kinematic quantities

A rotating object changes its orientation as it turns about an axis. Angular displacement, $\theta$ , describes the change in its angular position. It is measured in radians: one complete revolution is $2\pi$ radians.

To give angular displacement a sign, choose one direction of rotation as positive. Rotation in that direction has a positive displacement; rotation in the opposite direction has a negative displacement. For example, if anticlockwise is positive, a quarter-turn clockwise has an angular displacement of $-\frac{\pi}{2}$ radians.

Angular speed and angular velocity

Angular speed describes how quickly an object rotates. It is a scalar: it has a magnitude but does not specify a direction. Angular velocity, $\omega$ , describes both the rate and the direction of rotation. For rotation about a fixed axis, its sign indicates the direction chosen as positive or negative.

The average angular velocity over a time interval is the angular displacement during that interval divided by the time taken:

\omega_{\text{avg}} = \frac{\Delta\theta}{\Delta t}

Here $\Delta\theta$ is the change in angular position and $\Delta t$ is the time interval. Instantaneous angular velocity is the rate of change of angular position at a particular instant. Angular velocity is measured in radians per second, $\mathrm{rad,s^{-1}}$ .

Angular acceleration

Angular acceleration, $\alpha$ , describes how quickly angular velocity changes. The average angular acceleration over a time interval is the change in angular velocity divided by the time taken:

\alpha_{\text{avg}} = \frac{\Delta\omega}{\Delta t}

Instantaneous angular acceleration is the rate of change of angular velocity at a particular instant. Angular acceleration is measured in radians per second squared, $\mathrm{rad,s^{-2}}$ .

The sign of angular acceleration describes the direction of the change in angular velocity. If angular acceleration has the same sign as angular velocity, the object speeds up; if they have opposite signs, it slows down.

Angular and translational quantities

Angular displacement, angular velocity and angular acceleration describe rotation in the same respective ways that translational displacement, velocity and acceleration describe motion along a line. Displacement describes a change in position, velocity describes its rate of change, and acceleration describes the rate of change of velocity.

Graphs of angular motion

Graphs of angular motion show how angular displacement, angular velocity or angular acceleration changes with time. Their gradients and the signed areas between their lines and the time axis connect these three quantities.

A sign is used to distinguish the chosen positive direction of rotation from the opposite direction. A negative value therefore has meaning: it is not simply ignored when interpreting a graph.

Angular displacement–time graphs

The gradient of an angular displacement–time graph gives angular velocity. A steeper gradient means a greater magnitude of angular velocity. A positive gradient indicates rotation in the chosen positive direction; a negative gradient indicates rotation in the opposite direction.

For a straight line, the gradient is constant, so the angular velocity is constant. For a curved graph, the gradient changes. To find the instantaneous angular velocity at a particular time, draw a tangent to the curve at that point and find the tangent’s gradient.

Angular velocity–time graphs

The gradient of an angular velocity–time graph gives angular acceleration. Uniform angular acceleration produces a straight line with a constant gradient. If angular acceleration is non-uniform, the gradient changes; the instantaneous angular acceleration at a particular time is the gradient of the tangent at that point.

The signed area between the graph and the time axis over an interval gives the angular displacement during that interval. Count area above the time axis as positive and area below it as negative. Adding these signed areas gives the net angular displacement.

Angular acceleration–time graphs

The signed area between an angular acceleration–time graph and the time axis gives the change in angular velocity over the chosen interval. Area above the axis represents a positive change; area below it represents a negative change. The result is the change in angular velocity, not necessarily the final angular velocity.

Equations of uniformly accelerated angular motion

When an object rotates with constant angular acceleration, its angular velocity changes by the same amount in each unit of time. This lets us relate its angular displacement, angular velocities and elapsed time using four equations.

Let $\omega_1$ be the initial angular velocity, $\omega_2$ the final angular velocity, $\alpha$ the constant angular acceleration, $$$t$$$ the elapsed time and $\theta$ the angular displacement during that time. Angular displacement is usually measured in radians, angular velocity in radians per second and angular acceleration in radians per second squared.

Because acceleration is the rate of change of angular velocity, the change in angular velocity over time $$$t$$$ is $\alpha t$ :

\omega_2 = \omega_1 + \alpha t

Angular velocity changes uniformly, so its mean value over the interval is $({\omega_1 + \omega_2})/2$ . Angular displacement is mean angular velocity multiplied by time:

\theta = \left(\frac{\omega_1 + \omega_2}{2}\right)t

Substituting $\omega_2 = \omega_1 + \alpha t$ into the displacement equation gives a form that does not require the final angular velocity:

\theta = \omega_1 t + \frac{\alpha t^2}{2}

Eliminating time from the equations gives a form that relates the angular velocities directly to acceleration and displacement:

\omega_2^2 = \omega_1^2 + 2\alpha\theta

Choosing signs

Choose one rotational direction as positive and use it consistently for $\theta$ , $\omega_1$ , $\omega_2$ and $\alpha$ . A quantity directed the opposite way is negative. For example, if an object rotates in the positive direction while slowing down, its angular velocity is positive but its angular acceleration is negative. The equations describe net angular displacement, including any change of direction.

These equations have the same form as the constant-acceleration equations for translational motion: angular displacement, angular velocity and angular acceleration take the places of displacement, velocity and acceleration.