Question

A plank with a uniform sphere placed on it rests on a smooth horizontal plane. Plank is pulled to right by a constant force $F$, If sphere does not slip over the plank :

• A. both have same acceleration
• B. acceleration of centre of sphere is less then that of the plank
• C. work done by friction acting on the sphere is equal to its total KE
• D. total KE of the system is equal to work done by the force $F$

Answer 1

Answer: (B), (C), (D)

The correct options are
B acceleration of centre of sphere is less then that of the plank
C total KE of the system is equal to work done by the force $F$
D work done by friction acting on the sphere is equal to its total KE

When the force $F$ is applied to the plank, it has tendency to slide to right below the sphere, so friction acts on the plank to the left and that on the sphere to the right. The friction works as accelerating force on the sphere, and also produces Clockwise (CW) movement. So the sphere experiences a rightward, transitional acceleration and a CW angular acceleration simultaneously.
If $\alpha$ be the angular acceleration of sphere, then its COM has a backward acceleration $R\alpha$ relative to plank. So its net acceleration becomes less than that of plank.
Since there is no sliding between surfaces of sphere and plank, so no energy is lost against friction.
Because friction acting on the sphere, is the only force that provides motion to it, therefore KE of sphere at any instant is equal to work done by the friction acting on it. Since no energy is lost against friction, so kinetic energy of whole system is equal to work done by the force.