# 19.7 Rolling with sliding

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The equation of rolling does not hold when a body rolls with sliding.

We encounter many real time examples, when circular body in motion rolls with sliding. When we drive the car and are required to decelerate quickly, we need to apply sudden brake. The car in response may skid before coming to a stop. The length of slide depends on the friction between the surfaces. On a wet road, we may find that car slides longer than anticipated (reduced friction). For this reason, we experience almost uncontrolled sliding on a surface covered with ice, when we apply brake hard.

On the other extreme of above example, we may encounter a situation when we may not be able to move a car on a wet land at all. As we press the gas (paddle), the wheel rotates in its place without translating. Only, the wheel keeps rotating. The car may be stuck getting deeper into a groove created there. In such situations, we are required to push the vehicle to translate, keeping some hard matter like bricks or wooden plate near stuck wheel.

Evidently, rolling (pure) is not taking place in these examples. Clearly, rolling with sliding is not pure rolling.

## Implication of rolling with sliding

In this section, we outline differentiating aspects of rolling with sliding (impure rolling) via-a-vis pure rolling as :

1: In the case of applying sudden brake, we apply brake pad. The friction between brake pad and braking disk constitute a circumferential force. This force constitutes a torque that opposes angular motion of the rolling wheel. The net effect is that we disproportionately try to reduce angular velocity – not maintaining the relation between angular and linear velocity as given by equation of rolling.

When we accelerate or initiate a motion on a wet land without being able to translate, we apply torque and induce angular acceleration. But, friction between wheel and land is not sufficient to convert angular acceleration into linear acceleration as required for rolling.

2: For rolling with sliding, the distance covered in translation is not equal to the distance covered by a point on the rim of the body in rotation. This means that

$\begin{array}{l}s\ne \theta R\end{array}$

In the case of applying “sudden brake”, the car moves a greater distance than 2πR before the wheel completes one revolution i.e. there are fewer revolutions than in the rolling motion.

$\begin{array}{l}s>\theta R\end{array}$

In the case of driving on a wet land, the car, in one revolution of wheel, moves a lesser distance than 2πR. There are more revolutions than in the rolling motion.

$\begin{array}{l}s<\theta R\end{array}$

3: The friction involved in the sliding is kinetic friction – not static friction.

4: In the case of rolling with sliding, the motion is still analyzed, using two forms of Newton’s second law (linear and angular). However, equations of rolling for velocity and acceleration are not valid :

$\begin{array}{l}{v}_{C}\ne \omega R\end{array}$

$\begin{array}{l}{a}_{C}\ne \alpha R\end{array}$

5 : When left to itself, motion of a body initially rolling with sliding ultimately changes to rolling due to friction.

## Illustrations

We noted in earlier section that, when left to itself, a body initially rolling with sliding is rendered to roll without sliding due to friction. Here, we shall work out two examples to illustrate this assertion for two situations :

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