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.
\[{\omega _1} = \frac{\pi }{2}rad/s,\,\,\,{\omega _2} = \frac{{3\pi }}{4}rad/s,\,\,\,\Delta t = 0.4\,s,\,\,\,\alpha = ?\]
\[\alpha = \frac{{\Delta \omega }}{{\Delta t}} = \frac{{{\omega _2}--{\omega _1}}}{{\Delta t}} = \frac{{\frac{{3\pi }}{4}--\frac{\pi }{2}}}{{0.4}} = \frac{{5\pi }}{8}rad/{s^2}\]

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.
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:
(a) \[\frac{{20}}{\pi }\]
(b) \[\frac{{40}}{\pi }\]
(c) \[\frac{{100}}{\pi }\]
(d) \[\frac{{50}}{\pi }\]
Answer: (d)