Angular acceleration

Angular acceleration is the rate at which angular velocity changes with time, measured in radians per second squared. It is nonzero only when a rotation is speeding up or slowing down, not merely because something is going round.

Speeding up or slowing down a spin is what this measures, and nothing else. EK 5.1.A.3 defines average angular acceleration as the average rate at which the angular velocity changes with respect to time:

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

Units are $\text{rad/s}^2$ , and every point on a rigid system shares one value of $\alpha$ .

Tangential, not centripetal. This distinction is where marks go. The linear acceleration that matches $\alpha$ is the tangential component,

a_T = r\alpha

which the equation sheet prints. It points along the circular path and changes the object's speed. Centripetal acceleration, $a_c = v^2/r$ , points at the center and changes only the direction of the velocity. EK 2.9.A.3 and EK 2.9.A.4 keep them apart and state that the net acceleration of an object moving in a circle is the vector sum of the two.

So the two cases pull apart cleanly:

Uniform circular motion: $\alpha = 0$ and $a_T = 0$ , while $a_c$ can be large.
A wheel starting from rest: $\alpha \ne 0$ and $a_T \ne 0$ , while $a_c = 0$ at that first instant because $v = 0$ .

What produces angular acceleration is net torque, through $\alpha_{\text{sys}} = \tau_{\text{net}}/I_{\text{sys}}$ : the same force gives a different $\alpha$ on a body with different rotational inertia. The constant- $\alpha$ equations are in rotational kinematics.

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