Angular Acceleration
A turbine spinning up from rest, a flywheel braking to a stop, a drill bit reaching full speed in a fraction of a second — none of these rotate at constant angular velocity throughout. Angular acceleration describes how fast the spin rate itself is changing, and reading its behaviour off a graph is a skill worth mastering thoroughly.
Defining angular acceleration
Angular acceleration $\alpha$ (alpha) is the rate of change of angular velocity:
It is measured in radians per second squared (rad s$^{-2}$), and is the direct rotational analogue of linear acceleration $a = \Delta v/\Delta t$. A positive $\alpha$ means $\omega$ is increasing (the rotation is speeding up, assuming the standard sign convention), while a negative $\alpha$ means $\omega$ is decreasing — which could mean slowing down, or speeding up in the opposite sense of rotation, exactly as with linear acceleration and deceleration.
Engineering situations that call for angular acceleration are common: a motor's rotor accelerating up to its running speed after being switched on, a flywheel decelerating under a braking torque, a turbine's blades speeding up as steam pressure builds. In every case, $\alpha$ captures how quickly the angular velocity itself is changing.
Graphical treatment: reading rotation from graphs
Just as linear kinematics uses displacement-time and velocity-time graphs, rotational kinematics uses angular displacement-time ($\theta$–$t$) and angular velocity-time ($\omega$–$t$) graphs, and the same gradient and area rules apply directly.
- The gradient of a $\theta$–$t$ graph at any instant gives the angular velocity $\omega$ at that instant, since $\omega = d\theta/dt$.
- The gradient of an $\omega$–$t$ graph at any instant gives the angular acceleration $\alpha$ at that instant, since $\alpha = d\omega/dt$.
- The area under an $\omega$–$t$ graph, between two times, gives the angular displacement $\theta$ that occurred in that interval — the rotational equivalent of area under a velocity-time graph giving linear displacement.
A straight-line $\omega$–$t$ graph has a constant gradient, meaning constant (uniform) angular acceleration — this is the case the four rotational suvat equations, covered in the next lesson, apply to directly. A curved $\omega$–$t$ graph, by contrast, has a gradient that changes from point to point, meaning $\alpha$ itself is changing: this is non-uniform angular acceleration, and reading a value of $\alpha$ off such a graph means finding the gradient of a tangent to the curve at the instant of interest, rather than simply dividing rise by run over the whole graph.
Uniform angular acceleration means $\alpha$ is constant: the $\omega$–$t$ graph is a straight line, and the $\theta$–$t$ graph is a smooth curve (a parabola, for constant non-zero $\alpha$) with a steadily changing gradient. Non-uniform angular acceleration means $\alpha$ itself varies with time: the $\omega$–$t$ graph is curved, not straight. A common error is reading a changing gradient on an $\omega$–$t$ graph as if it represented uniform acceleration — always check whether the graph is straight or curved before deciding which suvat-style equations, if any, can be applied.
Key takeaways
- Angular acceleration $\alpha = \Delta\omega/\Delta t$ is the rate of change of angular velocity, measured in rad s$^{-2}$.
- It is the rotational analogue of linear acceleration, and the sign follows the same convention as $\theta$ and $\omega$.
- The gradient of a $\theta$–$t$ graph gives $\omega$; the gradient of an $\omega$–$t$ graph gives $\alpha$; the area under an $\omega$–$t$ graph gives $\theta$.
- A straight-line $\omega$–$t$ graph means uniform (constant) angular acceleration; a curved $\omega$–$t$ graph means non-uniform angular acceleration.
- For non-uniform acceleration, $\alpha$ at an instant is found from the gradient of the tangent to the curve at that point, not the overall rise-over-run.
- This completes the three core angular quantities — $\theta$, $\omega$, $\alpha$ — that feed directly into the rotational equations of motion covered next.