# Angular Acceleration

## Angular Acceleration Formula - Equation & Solved Examples

In circular motion, angular acceleration is the rate with which the angular velocity changes with time. It is also referred to as the rotational acceleration. It is a vector quantity, that is, it has both magnitude and direction.

Angular acceleration is denoted by $\alpha$.
If $\theta$is the angular displacement, $\omega$is the angular velocity and $\alpha$, the angular acceleration, then;
$\alpha = \frac{{d\omega }}{{dt}} = \frac{{{d^2}\theta }}{{d{t^2}}}$ (as; $\omega = \frac{{d\theta }}{{dt}}$)
The above formula gives the instantaneous angular acceleration.
If $\Delta \omega$is the change in angular velocity over a time interval $\Delta t$, then average angular acceleration is given by:
$\alpha = \frac{{\Delta \omega }}{{\Delta t}}$
In the case of uniform rotation, the average and instantaneous values coincide.
It is expressed in the units of rad/s2 or radians per second squared.

Example 1
If the angular velocity of a body in rotational motion changes from $\frac{\pi }{2}$rad/s to $\frac{{3\pi }}{4}$ in 0.4 s. Find the angular acceleration.
Solution:
${\omega _1} = \frac{\pi }{2}rad/s,\,\,\,{\omega _2} = \frac{{3\pi }}{4}rad/s,\,\,\,\Delta t = 0.4\,s,\,\,\,\alpha = ?$
$\alpha = \frac{{\Delta \omega }}{{\Delta t}} = \frac{{{\omega _2}--{\omega _1}}}{{\Delta t}} = \frac{{\frac{{3\pi }}{4}--\frac{\pi }{2}}}{{0.4}} = \frac{{5\pi }}{8}rad/{s^2}$

Example 2
The angular displacement of an object in rotational motion depends on time t according to the relation
$\theta = 2\pi \,{t^3}--\pi {t^2} + 3\pi --6$, where $\theta$is in radians and t in seconds. Find its angular acceleration at t = 2 s.
Solution:
We have: $\theta = 2\pi \,{t^3}--\pi \,{t^2} + 3\pi \,t--6$ rad
Angular velocity: $\omega = \frac{{d\theta }}{{dt}} = 6\pi \,{t^2}--2\pi \,t + 3\pi$ rad/s
Angular acceleration: $\alpha = \frac{{d\omega }}{{dt}} = 12\pi \,t--2\pi$ rad/s2
At t = 2s, ${\alpha _{t = 2s}} = 12\pi \, \times 2--2\pi = 22\pi \,\,\,\,rad/{s^2}$

Question: A wheel rotating at 10 rad/s2 is imparted with a constant angular acceleration of 4 rad/s2 for 5 seconds. The number of rotations made by the wheel in this 5 s interval is:
Options:
(a) $\frac{{20}}{\pi }$
(b) $\frac{{40}}{\pi }$
(c) $\frac{{100}}{\pi }$
(d) $\frac{{50}}{\pi }$