19.7 Rolling with sliding  (Page 2/3)

• The body initially has greater angular velocity.
• The body initially has greater linear velocity.

We shall first consider a case, in which the body is given initial spin (angular velocity) without any linear velocity. In this case, kinetic friction appears such that the angular velocity is decreased and linear acceleration is increased and over a period of time so that the equation of rolling is satisfied.

Second, we consider a case in which the rigid body initially has greater linear velocity than required for rolling without sliding i.e. for rolling motion. In this case, kinetic friction appears such that the linear acceleration is decreased and angular velocity is increased over a period of time so that the equation of rolling is satisfied. The friction works to balance combination of linear angular motion such that equation of rolling is satisfied.

The solution in each case requires us to apply Newton’s second law for translational and rotational motion separately. Then, we use equation(s) of rolling as limiting condition when motion of rolling with sliding of the body is rendered as pure rolling. In the subsections below, we consider two examples to illustrate the concept discussed here.

The rigid body initially having greater angular velocity.

Problem : A hollow sphere of mass “M” and radius “R” is spun and released on a rough horizontal surface with angular velocity, “ ${\omega }_0$ ”. After some time, the sphere begins to move with pure rolling. What is linear and angular velocity when it starts moving with pure rolling ?

Solution : In the beginning, the sphere has only angular velocity about the axis of rotation. There is sliding tendency in the backward direction at the point of contact as it continues rotating. This is the natural tendency. This causes friction to come into picture. This friction, in the opposite direction of the sliding, does two things : (i) it works as a net force on the sphere and imparts translational acceleration and (ii) it works as torque to oppose angular velocity i.e. imparts angular deceleration to the rotating sphere. In simple words, the sphere acquires translation motion at the expense of angular motion. This process continues till the condition of pure rolling is met i.e.

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

It is therefore clear that we must strive to find a general expression of translational and rotational velocities at a given time “t”. Once we have these relations, we can use the condition as stated above and find out the required velocity, when sphere starts pure rolling. Now, we know that friction is causing acceleration. This force, incidentally, is a constant force for the given surfaces and allows us to use equation of motion for constant acceleration.

Let us first analyze translation motion. Here,

$$\begin{array}{l}v=v_0+at\end{array}$$

Initially, the sphere has only rotational motion i.e. $v_0=0$ . Hence,

The translational acceleration is obtained from the Newton’s law as force divided by mass :