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# Role of friction in rolling

Module by: Sunil Kumar Singh. E-mail the author

Summary: Friction maintains accelerated rolling.

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The motion of a rolling body is altered due to external force/ torque. This can happen in following ways :

1. The line of action of external force passes through center of mass.
2. The line of action of external force does not pass through center of mass.

The effect of external force or torque on rolling motion is moderated by friction force between rolling body and surface. However, there is usually a bit of confusion with respect to its role and direction. This module, therefore, focuses on aspects of rolling that aims to instill definite clarity with regard to friction. The friction in rolling, however, is characteristically different and surprising in its manifestation with respect to the common negative perception (always negates motion) about it.

We have already learnt in the previous module that no friction is involved in uniform rolling. We shall find in this module that friction is actually the agent, which enforces the condition for accelerated rolling. In doing so, friction causes linear and angular accelerations or decelerations, depending on the requirement of rolling motion.

## Role of friction

### The line of action of external force passes through center of mass.

When the external force is passing through center of mass, it only produces linear acceleration as there is no moment arm and, thus, there is no torque on the body. Linear acceleration means linear velocity tends to increase. This, in turn, induces tendency of the rolling body to slide in the forward direction (i.e. in the direction of motion). Force of friction, therefore, appears in the backward direction of external force to check the sliding tendency.

We need to emphasize here that friction acts tangentially at the point of contact. Its direction is “backward” with respect to the component of force parallel to this tangential direction. This clarification is required as the external force may not be parallel to the surface in contact.

So long the condition of pure rolling is met as given by the equation of rolling motion, there is no actual sliding – rather there is only a tendency to slide. As such the friction involved in accelerated rolling is static friction. We may recall our discussion of friction in translation. Friction is a self adjusting force. It adjusts with respect to external force. Importantly, the static friction is any intermediate value less than the maximum static friction ( μ s N μ s N ).

The friction acts to balance the changes in a manner so that the condition as imposed by the equation of accelerated rolling is met. First, it reduces the net external force and hence the translational acceleration. Second, it constitutes a torque in clockwise direction inducing angular acceleration. In the nutshell, an increase in linear acceleration due to net external force acting through center of mass is moderated by friction by a two pronged actions and the rolling is maintained even when the body is accelerated. Corresponding to linear acceleration, there is a corresponding angular acceleration such that :

a C = α R a C = α R

In plain words, it means that if there is an increase in linear velocity, then there shall be an increase in angular velocity as well. So is the correspondence for a decrease in either of two velocities.

We can understand the situation from yet another perspective. Since external force induces linear acceleration, there should be a mechanism to induce angular acceleration so that condition as imposed by the equation of accelerated rolling is met. In other words, the friction appears in magnitude and direction such that above relation is held for rolling.

Static friction in rolling differs to its counterpart in translation in one very important manner. In translation, friction adjusts to the external force parallel to the contact surface completely till the body is initiated. What it means that intermediate static friction is equal in magnitude to the external force in opposite direction. Such is not the case in rolling. There is friction in opposite direction, but it is not equal. As a matter of fact, this is expected also as we are considering static friction for the body, which is already in motion (not required to be initiated as in pure translation). The situation is different.

The linear acceleration of the center of mass is given by :

a C = F M = F - f S M a C = F M = F - f S M

The angular acceleration of the center of mass is given by (note that external force causes clockwise rotation and hence negative torque) :

α = τ I = - R f S I α = τ I = - R f S I

The two accelerations are such that they are related by the equation of accelerated rolling (negative sign as linear and angular accelerations are in opposite directions) as :

a C = - α R a C = - α R

F - f S M = R 2 f S I F - f S M = R 2 f S I

Solving for “ f S f S “, we have :

f S = - I F ( I + M R 2 ) f S = - I F ( I + M R 2 )
(1)

This relation can be used to determine friction in this case. Since all factors are positive scalars, the friction is negative and is in the opposite direction to the applied force. Evidently, friction is different for rigid body of different moments of inertia. It is an important result in the sense that it indicates that friction does not only depend on the mass of the body or its constitution, but also on the relative distribution of mass about the axis of rotation.

### The line of action of external force does not pass through center of mass.

Now, we consider the second case. The external force that does not pass through center of mass causes angular acceleration apart from causing linear acceleration. Angular acceleration means angular velocity tends to increase. This, in turn, induces tendency of the rolling body to slide in the backward direction (i.e. in the opposite direction of motion). Force of friction, therefore, appears in the direction of external force to check the sliding tendency.

The friction acts to balance the changes in a manner so that condition of rolling is met. First, it enhances the net external force and hence the translational acceleration. Second, it constitutes a torque in anticlockwise direction inducing angular deceleration. In the nutshell, an increase in angular acceleration due to net torque is moderated by friction by a two pronged actions and the rolling is maintained even when the body is accelerated in rotation.

We can understand the situation from yet another perspective. Since external torque induces angular acceleration, there should be a mechanism to induce linear acceleration so that condition of accelerated rolling as given hereunder is met.

Friction appears in magnitude and direction such that above relation is held for rolling. The linear acceleration of the center of mass is given by :

a C = F M = F + f S M a C = F M = F + f S M

The angular acceleration of the center of mass is given by (note that external force causes clockwise rotation and hence negative torque) :

α = τ I = R ( f S - F ) I α = τ I = R ( f S - F ) I

The two accelerations are such that they are related by the equation of accelerated rolling (negative sign as linear and angular accelerations are in opposite directions) as :

a C = - α R a C = - α R

F + f S M = R ( f S - F ) I F + f S M = R ( f S - F ) I

Solving for “ f S f S “, we have :

f S = ( M R 2 - I ) ( M R 2 + I ) x F f S = ( M R 2 - I ) ( M R 2 + I ) x F
(2)

This relation emphasizes following aspects of friction force :

• Since I > 0 and I M R 2 I > 0 and I M R 2 , friction force, “ f S f S ”, is positive and is in the direction of external force. We should note here that for all rigid body M R 2 M R 2 represents the maximum moment of inertia which corresponds to a ring or hollow cylinder. MIs of all other bodies like sphere, disk etc. are less than this maximum value.
• For ring and hollow cylinder, I = M R 2 I = M R 2 . Thus, friction is zero even for accelerated rolling in the case of these two rigid body. This is one of the reasons that wheels are made to carry more mass on the circumference.

We conclude from the discussion as above that friction plays the role of maintaining the rolling motion in acceleration. As a matter of fact, had it not been the friction, it would have been possible to have accelerated rolling – it would not have been possible to accelerate bicycle, car, motor, rail etc!

The important aspect of the response of the body in rolling to external stimuli is that the “effect” takes place in both translation and rotation “together”– not selectively. For example, a force through center of mass is expected to produce translation alone. However, such is not the case. Friction ensures that external stimuli like force through center of mass works to affect both translation and rotation so that rolling continues.

We will strengthen our understanding of the role of friction from the perspective of energy in subsequent module. We shall find that friction by virtue of being capable to accelerate is actually capable of even doing positive work i.e. capable to impart kinetic energy in certain situation. Indeed, it is a totally different friction.

In case, the rolling is not maintained, the friction involved is kinetic friction as the body rotates with sliding. There can be many such situations in real life like applying a sudden brake to a moving car. We shall discuss these cases in a separate module.

## Direction of friction in real time motion

The discussion on the role of friction in rolling motion gives us definite clue about the direction of friction. Evidently it depends on the external force .vs. external torque situation.

We can accelerate rolling by applying force through its center of mass. As discussed earlier, friction in such case acts in the backward direction to counteract sliding in the forward direction. It is difficult to visualize a mechanical force that can be applied through center of mass of a rolling body. Incidentally, gravity provides a ready made arrangement in which force acts through center of mass.

A ball rolling along an incline is one such example, in which external force is gravity. Its component along the plane “mgsinθ” acts through COM of the rolling body.

The friction, in this case, acts in the backward direction as shown in the figure above.

Another way to accelerate rolling is to impart a force such that its line of action does not pass through center of mass. In this case, force constitutes a torque that imparts angular acceleration in addition to translational acceleration.

Such is the case with all transporting vehicles having wheels. The internal drive rotates the shaft, which in turn tends to rotate the wheel. We can visualize the situation better for the case of bicycle. When a bicycle is peddled, torque is imparted to the wheel. The chain – socket arrangement rotates the wheel. As discussed earlier, the friction, in this case, is in the direction of the acceleration of COM.

What if we apply the brake? The sole purpose of applying the brake is to apply a torque in the opposite direction to the direction of rotation. The brake pad jams on the rim. The tangential friction force constitutes the torque opposing the rotation. There is a corresponding linear deceleration of the wheel. A linear deceleration is equivalent to an acceleration in the backward direction. The friction between the wheel and surface, therefore, also acts backward along with acceleration as shown in the figure above.

## Summary

1: The friction between surface and rolling body is self adjusting static friction for accelerated rolling.

2: The friction between surface and rolling body is less than that in the case of sliding.

3: Friction facilitates simultaneous occurrence of two types of acceleration as a response to external stimuli (force or torque or both).

4: Direction of friction :

(i) If the force passes through center of mass, then there is sliding tendency in the forward direction of applied force. In turn, friction acts in backward direction of the external force.

(ii) If the force does not pass through center of mass, then it constitutes torque. There is sliding tendency due to torque in the backward direction of applied force. In turn, friction acts in forward direction of the external force. Apart from constituting a torque, the force also acts as “force”. There is sliding tendency due to force as “force” in the forward direction of applied force. In turn, friction acts in backward direction of the external force. The net result is that there is either no net friction (in special circumstance) or there is friction in the forward direction.

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