Question

A sphere is rotating about a diameter:
A. The particles on the surface of the sphere do not have any linear acceleration.
B. The particles on the diameter above do not have any linear acceleration.
C. Different particles on the surface have different angular speeds.
D. All the particles on the surface have the same linear speed.

Answer 1

Hint:If a body is moving in a straight line, the uniform acceleration caused by it is known as linear acceleration. It depends on some factors such as initial and final velocity, displacement and time.

Formula Used:
Since the sphere is rotating about a diameter, it is related to angular acceleration and it can be written by the formula:
\[a = r\alpha \]
Here, $a$ = Linear acceleration, $\alpha$ = angular acceleration and $r$= radius.

Complete step by step solution:
When the particles are rotating on the surface of the sphere, they will have centripetal acceleration due to the circular path. So, option A is incorrect.

The angular velocity for the particles is given by \[v = \omega r\]. When the particles are on the diameter, then r=0. Therefore, the particles on the diameter do not have any linear acceleration.

In completing one revolution, the time taken by different particles will be different. This results in different angular speed. All the particles rotating on the surface will have different linear speed due to different position from the axis of rotation.

Hence, Option B is the correct answer.

Note: It is to be noted that though linear and angular velocity are related to each other, there are some points of differences between them. The angular velocity for a particle around the axis of a circle will remain the same throughout its motion, but in linear velocity, since the particle is moving in a straight path, therefore, it is different for the particle as the particle changes its position.