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A sphere of mass M and radius r shown in...

A sphere of mass `M` and radius `r` shown in figure slips on a rough horizontal plane. At some instant it has translational velocity `V_(0)` and rotational velocity about the centre `(v_(0))/(2r)`. Find the translational velocity after the sphere starts pure rolling.
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(i) A solid sphere of radius R is released on a rough horizontal surface with its top point having thrice the velocity of its bottom point A (V_(A) = V_(0)) as shown in figure. Calculate the linear velocity of the centre of the sphere when it starts pure rolling. (ii) Solid sphere of radius R is placed on a rough horizontal surface with its centre having velocity V_(0) towards right and its angular velocity being omega_(0) (in anticlockwise sense). Find the required relationship between V_(0) and omega_(0) so that - (a) the slipping ceases before the sphere loses all its linear momentum. (b) the sphere comes to a permanent rest after some time. (c) the velocity of centre becomes zero before the spinning ceases.