A wheel `A` starts rolling up a rough inclined plane and another identical wheel `B` starts rolling up a smooth plane having same inclination with the horizontal. If initial velocity of both the wheels is same then:

Options

A. `A` stops ascending earlier than `B`
B. kinetic energy of `B` never becomes zero
C. maximum height ascended by `A` is less than that, by `B`
D. friction acting on `A` remains constant during the round trip

Answer

B, D

Explanation

If we consider free body diagram of the wheel `B`, the retarding force on wheel `B` is equal to `mgsintheta`. Therefore, its retardation is equal to `mgsintheta` and no moment is produced by these forces about its cenrodial axis. Hence, the wheel `B` continues to rotate with a constant angular velocity and its translational velocity decreases during up the plane motion. Hence, at highest point it has a rotational `KE`

While during ascending motion friction acts on wheel `A`. That is the friction which is responsible for angular retardation of wheel `A` as shown in the figure.
The resultant retarding force on wheel `A` is equal to `(mgsintheta-` friction). Hence, net retarding force on wheel `A` is less than`mg sintheta`. Therefore, it continues ot move up the plane for longer time and comes to rest after moving move distance than wheel `B`.
Wheel `A` comes to rest when its total kinetic energy becomes equal to zero. It means, at highest points its potential energy will be equal to total initial `KE`, while at highest point, potential energy of wheel `B` is equal its initial translational `KE` up. It means, maximum heights ascended by wheel `A` is greater than that by `B`.
When wheel `A` descends, it tries to slide down the plane due to component `mgsintheta` of its weight. Hence the friction remains acting up the plane and produces a moment about centroidal axis. Hence, wheel experiences an angular acceleration in anticlockwise direction.
Since all the remaining forces remains same, therefore, magnitude of friction also remains same.