Friction in Physics – How Can Friction Do No Work in Pure Rolling?

Question

I have read various answers, on PSE and elsewhere, and most of them explain that the point of contact of the rolling object undergoes no instantaneous displacement in the direction of friction, I agree, but then there is a feeling that the friction does provide a torque, so how can it do no work> (The pure rolling here is under a constant external force on the object so friction does act ).

Answer 1

In a scenario of pure rolling of a rigid wheel on a flat plane, you don't need any friction. Once the wheel is rolling, it will continue to do so, even if the friction coefficient becomes zero. If this were not the case, you'd be violating conservation of angular momentum. There is no force, no torque and therefore no work done.

The more interesting case is that where the object is rolling under an external force, say down an inclined plane. See the diagram below

Now, you can analyze it in two ways. One is similar to Farcher's answer where the point of contact moves perpendicular to the frictional force and hence, there is no work done. But you were interested in it from the point of view of torques (where we consider the whole wheel, not just the point of contact) so let's do that.

Friction does two things to the wheel as a whole. It does negative work when you look at the linear motion of the wheel. Indeed,

$$W_1 = -f.dS,$$

where $f$ is the force of friction and the wheel has moved a linear distance $dS$. Next, the friction provides a torque about the center of the wheel and the wheel has angular displacement. Hence, it does positive rotational work i.e.

$$W_2 = \tau. d\alpha,$$

where $\tau$ is the torque and $d\alpha$ is the angular displacement of the wheel. But note that $\tau = fR$ and $R d\alpha = dS$. Hence $W_2 = f.dS$ and you get

$$W_{tot} = W_1 + W_2 = 0$$