Taylor Example 4.8. Is my reasoning sound? [closed]

Question

This problem has been giving me all sorts of fits. For one, Taylor states that because the frictional force and normal force are forces of constraint, they produce no work. I'm trying to figure out the right way to think about why friction does no work in this case.

My initial rationalization was that since the friction produces a constraint (i.e. no slipping), its functional form at each moment in time will exactly ensure that the constraint is satisfied, while avoiding any contribution to the kinetic energy.

While trying to couch this in analytical terms, I figured that since dW=F⋅dr, I can also say that dW=F⋅(dr/dt)dt=F⋅vdt. Since the instantaneous velocity at the point of contact is 0, this would give dW=0 in the case of friction. Is this a good argument though?

My second issue was how Taylor equated the length of CB with r$d\theta$. I wrote out my thoughts on why this is possible in the picture, but if someone has an easier way to see this, or if my rationale is flawed, I would love some input.

Answer 1

For your first question, you are completely correct. Anytime something is rolling without slipping, friction does zero work for the same reason: it does not contribute to changing the kinetic energy. I particularly like the way you reasoned it out using the pull-back of the work integral.

As for your second question, I believe you are thinking along the right lines but are having some trouble putting it into the correct words. Just because we can create an injective mapping between the points of the line segment $\overline{CB}$ and the points of the arc $r\theta$ doesn't mean that the lengths are equal. The physics of the situation is that a rolled object will cover the same ground as the exterior arclength it traverses. Equality comes from the fact that their measures are the same. If we were to "flatten" the arc, it would be a line segment congruent to $\overline{CB}$.

As a matter of fact, we can produce an injective map between any two segments (or arcs, or lines, or just any curves in general) regardless of length. Using horizontal lines as an example, let's parameterize one line of length $\ell_1$ as $x_1(t) = t\ell_1$ and another line of length $\ell_2$ as $x_2(t) = t\ell_2$ with $t\in[0,1]$. It follows that for any point on the line $x_1$, there will be a unique value of $t$. For each unique value of $t$ there will correspond a unique point on $x_2$. By symmetry, this argument is reversible and thus we have produced an injection between the two segments.