Pipsqueakalchemist said:
But what about when rolling without slipping. I thought friction does no work when rolling without slipping and if it does no work then it doesn’t change the objects velocity meaning acceleration is zero so friction force equal zero because force = ma. Is there something wrong with my assumption?
Let's consider a ring being pulled by a horizontal force, ##F##, through its centre of gravity a) on a smooth surface and b) on a rough surface, where we have an unknown frictional force ##f##.
In case a) we have simply linear acceleration from ##F = ma## and no rotation.
In case b) we have linear acceleration given by ##F - f = ma## and an angular acceleration given by ##fR = I\alpha##. And, of course, we have ##a = R\alpha##.
We can, therefore, solve for ##f##: $$f(1 + \frac{mR^2}{I}) = F$$ For a ring we have ##I = mR^2##, hence ##f = \frac{F}{2}##, which is definitely not zero.
In fact, it's clear that the frictional force is instrumental is causing some of the linear motion to become rotational motion - if we compare with case a).
The linear acceleration is: $$a = \frac{F}{2m}$$ Now we can check the work done. The linear kinetic energy after moving a distance ##d## is half what we would have in case a): $$KE_l = \frac{Fd}{2}$$ But, we also have rotational KE of $$KE_r = \frac 1 2 I \omega^2 = \frac 1 2 I \frac{v^2}{R^2} =\frac 1 2 mv^2 (\frac{I}{mR^2}) = KE_l$$ And the total kinetic energy (linear plus rotational) is the same as in case a).
In conclusion, the frictional force did no work. But, this absolutely didn't mean that it was zero.
If you do the same calculations for your more complicated problem, then you should get the book answer.