Question

A particle is moving along a circular path with uniform speed. Through what angle does its angular velocity change when it completes half of the circular path?
(A) $0^\circ $
(B) $45^\circ $
(C) $180^\circ $
(D) $360^\circ $

Answer 1

Hint: In this question, we will use the equation for the linear velocity of a particle moving in a circular path. Then, we identify the direction of the angular velocity to find its change.

Complete step by step answer:
When a particle moves around a circular path with a constant speed, we say that the particle is in uniform circular motion. If the particle moves along a circle of radius $r$ with angular velocity $\omega $ and uniform linear velocity $v$, then we can write the linear velocity of the particle as

$\overrightarrow v = \overrightarrow \omega \times \overrightarrow r $

Since it is a cross product, we can say that all three vectors will be perpendicular to each other.

The direction of the linear velocity is along the tangent of the circle along which the particle moves. The tangent to the circle is perpendicular to the radius of the circle. Since the radius vector and the velocity vector are perpendicular to each other, the third vector, which is the angular velocity vector, will be perpendicular to the plane containing $\overrightarrow v $ and $\overrightarrow r $. Hence, the angular velocity vector will be in a direction perpendicular to the plane containing $\overrightarrow v $ and $\overrightarrow r $ or the plane of the circle. So, even if the magnitude of the angular velocity changes, the direction of the angular velocity remains the same. It confirms that the change in the angular velocity will be zero.

Since the angular velocity will be zero even after the particle moved through half of the circular path.

So, the correct answer is “Option A”.

Note:
If the particle moves along a circle with uniform speed, the magnitude of the linear velocity will be the same throughout the motion. But the direction of the velocity vector changes at every point along the circle.