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Question

A sphere is rotating uniformly about a fixed axis passing though its centre. Then,
  1. The particles in the sphere do not have any angular acceleration.
  2. The particles on the axis do not have any linear acceleration.
  3. Different particles in the sphere have the same angular speeds.
  4. All the particles in the sphere have the same linear speed.


Solution

The correct options are
A The particles in the sphere do not have any angular acceleration.
B The particles on the axis do not have any linear acceleration.
C Different particles in the sphere have the same angular speeds.
a) Sphere is rotating uniformly, so there will be no angular acceleration. (constant angular speed).


b) As you can see in the figure, all the particles on the sphere are in circular motion about the points which are present on the axis of rotation, which is fixed. Hence, particles present on the axis of rotation do not have any centripetal acceleration i.e no linear acceleration.

c) Angular speed gives the rate at which angle at the centre is swept by a particle as it moves in a circular path.


All the particles present at different radii from the centre, cover the same angle θ from time t=0 to t.
Hence, all the particles present at different radii have the same angular speed.

d) Linear speed is the distance travelled per unit time.


In this diagram, in time t, particle 1, 2, 3, 4 all cover different distances. Hence distancetime=linear speed will be different for  particles present at different distances from the origin.
Hence, A,B,C options are correct.

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