Rotational Work done by Friction

Question

(Okay so I have asked this in a previous post but I think this should be a separate question)

Consider a cylinder performing accelerated pure rolling (Static friction is non-zero) on a sufficiently rough surface.

The point of contact keeps changing since the body is rolling. Now I understand rotational work to the work done by a torque to rotate a body by some angle. The torque is provided by a force which is applied to a PARTICULAR POINT of the body and stays in contact with that PARTICULAR POINT for the entire rotation (let us say by an angle A). However, this is what confuses me, In pure rolling the point of contact is at rest and is not displaced at that particular instant when it is in contact with the surface, the point of contact then changes when the body rolls. Now if the initial point of contact was at rest (Was not displaced by A) then how can friction do rotational work ?

Answer 1

The "momentarily at rest" fact indeed implies that no work is done by friction.

And it indeed is not the torque by friction which does the work; it is the torque that causes the acceleration, which does the work!

Since your cylinder is accelerating, there must be such a torque present. Maybe your cylinder is a wheel on a car. Then it is the engine that causes a torque on the wheel about the axle.

This, shall we call it engine torque, is doing rotational work on the cylinder. Was the cylinder hanging free with no contact to the ground, then this work done would add rotational kinetic energy. The cylinder would spin faster and faster - it would not move/translate, just spin/rotate.

You can think of the torque by friction as an "intermediate enabler" that causes this work, which otherwise would have turned into rotational kinetic energy, to be converted into translational kinetic energy of the cylinder. Friction doesn't do this work, it just acts as a means of changing from rotation to translation.

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You can be further convinced of this fact that friction does no work, by thinking of what would happen if the engine torque (the torque that causes the acceleration) stopped acting:

Then the cylinder would stop speeding up. But would it slow down? No! It would just continue rolling. Forever. At constant speed (constant translational and constant rotational speed). The torque by friction does no work on its own (neither positive nor negative, so it doesn't speed up nor slow down). (In fact, that static friction is not even present, since there are no other forces present to counteract.) It will continue rolling until some other forces/torques, which can do work appears, or until energy in other ways are removed/added to its motion.

Answer 2

Static friction fixes the location of the bottom of a rolling cylinder (or wheel). The wheel is undergoing an instantaneous rotation about this fixed point. As Steeven points out, an external source of torque will try to accelerate this rotation. This acceleration will exert a force on the center of mass (or axle) of the wheel (which is moving).