19.8 Pulley in rolling

In our earlier treatment on "pulley" in this course, we had limited our consideration to "mass-less" pulley. Here, we shall consider pulley, which has finite mass and is characterized by rolling motion and presence of friction in certain cases.

We need to distinguish two situations (finite mass and mass-less cases) in order to analyze the motion correctly. Mass-less pulley is characterized by the fact that it does not affect the magnitude of tension in the string. It means that tensions in the string on either side of the pulley remains same. In general, a "mass-less" pulley changes the direction of force (tension) without any change in magnitude.

Analysis of the motion of pulley in rolling

A pulley of finite mass, on the other hand, may rotate, fulfilling the condition of rolling. In this case, the length of rope/string released from the pulley is equal to the distance covered by a point on the rim. If the rolling is accelerated, there is friction between the pulley surface and the string/ rope, passing over it, enabling the pulley to accelerate/decelerate in rotation.

Evidently, the acceleration of pulley of finite mass will be associated with a net force on the pulley. This, in turn, means that the tension in the strong are not same as in the case of "mass-less" pulley. The pulley, in the figure above, translates and rotates with acceleration, as the string wrapped over it unwinds.

The length of rope unwound is equal to the vertical distance traveled by the pulley/ disk as in the case of rolling and as shown in the figure. Hence,

$$\begin{array}{l}y=\theta R\end{array}$$

The motion of pulley may not exactly look like rolling. But, we can see that string/rope plays the role of a surface in rolling. This analogy is not very obscure as string provides a tangential surface like horizontal surface for the pulley to roll. This is an analogous situation to the rolling of a disk. Differentiating the relation, as given above, with respect to time :

In certain situation, the pulley may be fixed to the ceiling as shown in the figure below and hence incapable of translation. We can not say here that pulley is actually not rolling. But, the rope translates as much as a point on the rim of the pulley and as such the rope translates at the same velocity and acceleration as that of the center of mass of the pulley, if it were free to translate. We can see that pulley is executing the rotational part of the rolling motion, whereas string, along with attached blocks, is executing the translational part of the rolling motion. Thus, motions of pulley and string together are equivalent to rolling motion.

We analyze motion of pulley in same manner as that of a rolling body with the help of two Newton’s second laws – one for the linear motion and other for the angular motion. Such consideration of law of motion, however, is conditioned by the equation of rolling and equation of accelerated rolling.