# Angular acceleration and angular velocity relationship

It is given that angular velocity is the rate of change in the angular position of a body while the angular acceleration is the rate of change in angular velocity.

Other relationships include that when angular velocity is constant, angular acceleration is zero.

Given a situation where it is definite that angular velocity is zero, does it also mean that angular acceleration is zero as well?

Picture this. You throw a ball into the air while gravity is pulling the ball down, back to Earth. The acceleration of the ball at all times is $$-9.8~\rm\frac{m}{s^2}$$ Assuming that the up direction is positive. The ball will rise until it reaches that certain point, and then fall back down. At the exact moment the ball reaches its highest point, the velocity is zero, but the acceleration is still $$-9.8~\rm\frac{m}{s^2}$$ (Actually, you need a difference in time to compute a velocity but we'll ignore that in this case.) So if the "instantaneous velocity", that is, the velocity at a point in time is zero, the acceleration can be other than zero, but it doesn't have to be. It is also possible that the ball is at rest with no forces acting on it.

In real life, however, this isn't possible. You need to compute the displacement between two separate points in time to compute a velocity. If we assume that there is a force that causes acceleration, the velocity of the two points we measure would be different and thus lead to different displacements.

Angular motion is very similar to linear motion. We can measure displacement, velocity, acceleration, and inertia in both. The relationships between the 4 is the same for both.

Angular acceleration is the change in angular velocity, as you said. So if the angular velocity is zero and constant, then yes, the angular acceleration is also zero.

However, in general, the angular velocity can be instantaneously zero, but changing. In that case, the angular acceleration is not zero.

It is analogous to linear velocity and acceleration. If you throw a ball straight up in the air, it will slow down until it instantaneously has zero velocity at its highest height. Then it starts moving downward. But the ball is always accelerating downward. Before the top, at the top, and after the top.

Same for angular velocity.