A ball rolls down an inclined plane and acquires a velocity `v_(r)` when it reaches the bottom of the plane. If the same ball slides without friction and acquires rolling from the same height down an equally inclined smooth plane and acquires a velocity `v_(s)` (then which of the following statements are not correct?

Options

A. `v_(r)ltv_(s)` because a work is done by the rolling ball against the frictional force.
B. `v_(r)gtv_(s)`, because the angular velocity acquired makes the rolling ball to travel faster
C. `v_(r)=v_(s)` because kinetic energy of the two balls is same at the bottom of the planes.
D. `v_(r)gtv_(s)` because the rolling ball acquires rotational as well as translational kinetic energy.

Answer

A, B, C, D

Explanation

When the ball moves along a smooth plane, the accelerating force on it is `mgsintheta`. Hence its acceleration is equal to `g sintheta`.
When the ball rolls down the rough inclined plane, the `mg sintheta` acts down the plane but a frictioncomes into existence which acts up the plane. That friction prodces angular acceleration on the ball and the net accelerating force on the ball becomes equal to `(mgsintheta-"friction")`. Therefore, it has a smaller acceleration. Therefore its velocity at the bottom of the plane is less than that of the ball moving down a smooth plane or `v_(s)gtv_(r)`.
In fact, at the bottom of the planes both the balls have the same `KE` because loss of potential energy of both the balls is the same. But the ball rolling down a rough plane has translational as well as rotational kinetic energy at the botom of the plane while ball sliding down a smooth plane has translational `KE` only. Therefore, translation `KE` of the ball will be less than the translational `KE` of the ball sliding down the smooth plane. Hence, option d is correct.