(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.

Answer

(i) `(12V_(0))/(7)`
(ii) (a) `V_(0) gt (2)/(5) omega_(0) R`
(b) `V_(0) = (2)/(5) omega_(0) R`
(c) `V_(0) lt (2)/(5) omega_(0) R`