Angular Acceleration

Angular acceleration is how quickly a rotation rate is changing, $\alpha = (\omega - \omega_0)/t$. It is the exact rotational mirror of $a = (v - v_0)/t$, and that mirroring is not a coincidence or a mnemonic — the entire structure of linear kinematics carries across intact, with $\theta$ in place of $x$, $\omega$ in place of $v$, and $\alpha$ in place of $a$. Every uniform-acceleration technique you already know works unchanged in the rotational world.

A hard-disk platter spinning up from rest to 7200 rpm in 4 s: convert first, $\omega = 7200 \times 0.10472 \approx 754$ rad/s, so $\alpha = (754 - 0)/4 \approx 188$ rad/s². A large industrial fan coasting down from 1200 rpm (126 rad/s) to rest over 90 s has $\alpha = -1.4$ rad/s², and the negative sign is the whole of what distinguishes a spin-down from a spin-up.

What makes $\alpha$ worth computing is that it is the bridge to torque. Once you know the angular acceleration a duty cycle demands, $\tau = I\alpha$ tells you the torque the drive must supply, and that is how motors get sized for anything that has to start and stop repeatedly — a centrifuge, a machine-tool spindle, a conveyor, a robot joint. The starting torque is very often larger than the running torque, and this equation is where that requirement first appears.

The unit mistake here has an extra layer to it. As always, $\omega$ must be in rad/s before it goes anywhere near the formula, but the output also has to be read correctly: $\alpha$ comes out in rad/s², and a figure quoted as "rpm per second" is not the same quantity — it needs the same 0.10472 factor applied. A drive rated to accelerate at "500 rpm/s" is delivering 52.4 rad/s². The second issue is signs. Pick a positive direction of rotation and keep it: braking a shaft turning positively gives a negative $\alpha$, and the calculator's own answer will tell you if you have them mixed, because a spin-down entered with both velocities positive and a positive $\alpha$ is describing something that cannot happen. Finally, this is the average angular acceleration over the interval. A real motor's torque varies enormously with speed, so the instantaneous $\alpha$ during a start is nothing like constant; the average is the right number for a duty-cycle estimate and the wrong one for a stress calculation at the moment of highest load.