A solid sphere is set into motion on a rough horizontal surface with a linear speed v in the forward direction and angular speed v/r in the anticlockwise direction. Find the linear speed of the sphere when:
a) When it stops rotating
b) when slipping ceases
Basic Angular Momentum based equations
The Attempt at a Solution
Now I imagined it moving as described. Since the floor is rough a torque will be acted by the floor which acts in the backward direction. Causing the angular velocity to reduce. The velocity of the centre of mass remains intact. So applying angular momentum for case (i) Where v' is the velocity of the centre of mass.
a) mvr + 2/5 mrv = mv'r (Iw = 2/5mvr)
v' = 7v/5
Now the friction acts to the torque to increase the angular velocity in the clockwise direction till pure rolling is achieved. So now when achieved the sphere is rolling clockwise, and the velocity remains in the positive direction.
b) mvr + 2/5 mrv = mv'r - 2/5mv'r (Iw = 2/5mvr)
v' = 7v/3
The answer for (a) is 3v/5 and (b) is 3v/7. If you could explain my mistakes it would be very helpful. Thanks in advance.