19.4 Understanding rolling motion  (Page 3/4)

$$\begin{array}{l}{v}_C=\omega R\end{array}$$

Looking at the motion of rolling (without sliding), it is easy to realize that it is not physically possible that the velocity of center of mass increases, but the angular velocity of the rolling body decreases. As a matter of fact, presence of friction shall contradict the physical reality of rolling itself!

According to the above relation, two velocities increase or decrease simultaneously. We can repeat this analysis for an opposite situation in which we assume that the friction operates in the direction of translation (not opposite) as shown in the figure above on the right. Even in this case, we shall find that the body is accelerated in translation, but decelerated in rotation – a contradiction of the condition of rolling. We, therefore, conclude that there is no friction when a body is rolling uniformly.

Absence of friction for rolling at constant velocity has a very significant implication as a disk in uniform rolling shall move indefinitely, if no net external force/ torque is acting. This is a slightly unrealistic deduction for we know that all rolling disk is brought to rest ultimately unless external force is applied to maintain the speed. This needs explanation.

As a matter of fact, it is not possible to realize an ideal pure rolling in the first place. All rolling motion in our real world involves contact which spreads beyond a point and there is some amount of deformation involved and, therefore, existence of normal force constituting a torque in the opposite direction to rotation of the object. As such, the rolling body decelerates.

We can have a direct feeling of the absence of friction in uniform rolling. We use a dumbbell that we often use for exercise. Just try to push across so that dumbbell slides without rolling at a constant speed. Then, push it to roll at a constant speed (approximately) without sliding. Experience the difference. We know that it is lot easier to roll than to slide the dumbbell. Had there been single point contact without deformation, the dumbbell would have continued rolling.

Condition of accelerated rolling

We shall discuss the implications of the external force soon in terms of Newton’s second law of motion. But, we first need to ascertain whether the body in pure rolling, when subjected to external force, shall retain the basic nature of the rolling motion or not? In simple words : can a rolling body shall continue rolling when external force or torque is applied?

Recall that rolling requires that linear and angular velocities are tied together by the equation of rolling motion :

$$\begin{array}{l}{v}_C=\omega R\end{array}$$

This means that if the motion retains the rolling character even after application of external force/ torque, then any change in velocities (i.e. linear and angular accelerations) should also be related (tied). We can use the above relation to obtain a conditional relation between linear and angular accelerations.