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Question
A ball rolls down an incline plane and acquires a velocity vr 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 vs, then which of the following statements are not correct ?
A ball rolls down an incline plane and acquires a velocity vr 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 vs, then which of the following statements are not correct ?
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Solution
The correct options are
A vr=vs, because kinetic energy of the two balls is same at the bottom of the planes.
B vr<vs, because a work is done by the rolling ball against the frictional force.
C vr>vs. because the angular velocity acquired makes the rolling ball to travel faster.
When the ball moves along a smooth plane, the accelerating force on it is mg sin θ. Hence, its acceleration is equal to g sin θ.
When the ball rolls down the rough inclined plane, then mg sinθ acts down the plane but a friction comes into existence which acts up the plane. That friction produces angular acceleration on the ball and the net accelerating force on the ball becomes equal to (mg sin θ -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 vs>vr.
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 bottom of the plane while ball sliding down a smooth plane has translational KE only. Therefore, translational KE of the rolling ball will be less than the translational KE of the ball sliding down the smooth plane. Hence, option (d) is correct.
A vr=vs, because kinetic energy of the two balls is same at the bottom of the planes.
B vr<vs, because a work is done by the rolling ball against the frictional force.
C vr>vs. because the angular velocity acquired makes the rolling ball to travel faster.
When the ball moves along a smooth plane, the accelerating force on it is mg sin θ. Hence, its acceleration is equal to g sin θ.
When the ball rolls down the rough inclined plane, then mg sinθ acts down the plane but a friction comes into existence which acts up the plane. That friction produces angular acceleration on the ball and the net accelerating force on the ball becomes equal to (mg sin θ -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 vs>vr.
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 bottom of the plane while ball sliding down a smooth plane has translational KE only. Therefore, translational KE of the rolling ball will be less than the translational KE of the ball sliding down the smooth plane. Hence, option (d) is correct.
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