First slide

Rotational motion

Question

The mass of a sphere placed on a smooth horizontal surface is distributed non-uniformly. A horizontal force equal to weight of sphere is applied at a distance 'a' from centre of the sphere for a short time. The sphere is rotated by an angle $θ$ but it behaves as if in stable equilibrium. Find the distance between centre and CG of the sphere:

Difficult

A

a tan $θ$
B

a cot $θ$
C

a sec $θ$
D

a cosec $θ$

Solution

The sphere is in stable equilibrium. It will come back to original position after rotation. Rotating Moment = Fa. For stable equilibrium,
Restoring moment = Rotating moment
Since $F = W, W a = F a$
Let O be the centre and C and C' be the positions of CG before and after rotation.
$O C = O C^{'} and C^{'} B = a . . . . . . . . (f r o m e q u a t i o n (i))$
$In Δ O B C^{'}, OC'= C^{'} B cosec θ = a cosec θ$

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