19.11 Work and energy in rolling motion  (Page 22/3)

Work and energy consideration in rolling has certain interesting aspects. It gives a new functional meaning to friction, which otherwise has always been associated with negative work. Also, work and energy consideration in rolling has two parts : one that pertains to translation and other that pertains to rotation.

In this module, we shall also find that work – energy consideration provides a very useful frame work to deal with situation in which rolling is not along a straight line, but a curved path.

Work done in rolling

Work is associated with force and displacement. When a body rolls on a horizontal plane at a constant velocity, there is no external force. Irrespective of the nature of surface, friction is zero. No work, therefore, is done on the body. There is no corresponding change in the kinetic energy of the rolling body.

For an accelerated rolling, an external force is applied on the body. Application of external force in rolling, however, is not very straight forward as in the case of translation. Static friction comes into picture – whenever there is sliding tendency. Depending on the situation, friction plays specific role to maintain rolling.

We shall begin with the simplest case of rolling along a horizontal line to the case of rolling along an incline to clearly understand work by different forces in rolling.

Accelerated rolling along a straight horizontal line

For illustration purpose, we consider an external force, “F”, that acts through the COM and parallel to the surface as shown in the figure below. Friction acts in the backward direction. Let the disk rolls a linear distance “x” in the x-direction.

(i) Work by external force “F”

(ii) Work by static friction

The static friction has dual role here. It negates translation i.e. does negative work. Also, it accelerates rotation. As such, friction does the positive rotational work. If “T” and “R” denotes translation and rotation respectively, then :

and

where “θ” is total angle covered during the motion. For rolling motion,

Putting the expression of angle in equation – 3,

The results for the work done by friction in translation and rotation are very significant. They are equal, but opposite in sign. The net work by friction, therefore, is zero.

$$\begin{array}{l}\Rightarrow W_f=W_{\mathrm{fT}}+W_{\mathrm{fR}}=-f_{S}x+f_{S}x=0\end{array}$$

Thus, total work done by the external forces is :

Accelerated rolling along an incline

In this case also, friction does negative work in translation and positive work in rotation.

(i) Work by gravity

The component of gravity perpendicular to motion is perpendicular to displacement. Hence, it does not do work. The work by the component of gravity parallel to incline is :

(ii) Work by static friction