We knew that a after a bowling ball is threw out with a certain velocity to a non smooth track, it first rolls and skids as the translational velocity (decelerates due to friction) of the center of mass is greater than tangential velocity of the point of contact of ball with the floor which has a opposite direction. But after some time torque due to friction causes the angular velocity to increase and eventually the tangential velocity of point of contact achieve same value with translational velocity and have v=0, it starts rolling without slipping, and eventually come to stop.
My question here is after rolling without slipping is achieve how does the translational kinetic energy and rotational kinetic energy change? How does translational kinetic energy decrease while rotational kinetic energy is increase by torque due to friction?
Best Answer
In rolling without slipping motion, velocity of the centre of mass is related to angular velocity by $v=\omega r$ [1]. The term $\omega r$ represents the tangential velocity of a particular point on the object.
There is a maximum friction force that the floor can exert on the ball. Since the rotation of rolling motion is helped by this friction force, there is a pre-defined maximum angular velocity which the ball can have. Therefore there is a maximum tangential velocity too that one point can obtain.
Let's take the tangential velocity of the point of contact between ball and the track as $v'$. Then, if the initial velocity of the centre of the mass ($v$) is greater than $v'$, there cannot be a rolling without slipping motion, because $v\neq v'$ (or $v\neq \omega R$). Therefore the friction force acts against the translational motion, which results in slowing down the ball (because friction forces acts against the translational motion of a body most of the times causing deceleration) . When $v$ decreases to $v'$, the necessity for rolling without slipping motion is obtained. Thus it starts to roll with the aid of the torque provided by the friction force (quite ironical, because the same force caused the deceleration of the translational motion). Note that this rolling motion cannot happen if you place the ball on the track without an initial velocity. That is because, then there is no motion and therefore no resistance to motion by friction and therefore no torque provided by the friction. Thus it is obvious that rolling is helped by friction.
If the ball is perfectly rigid, it will roll forever due to that torque. Although in real life we cannot find perfectly rigid bodies, hence the ball will slow down due to phenomenon called rolling resistance [2], [3].
Hope this helps.