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I think there is an extra two in the final answer. I think that's just a typo. Regardless, there is one difficult thing for me to understand in this approach. The same force of friction causes both linear and angular acceleration/deceleration as if there are two copies of the same force. It is doing both things at the same time with just one force. Is this what actually happens? Or is this just mathematically convenient to derive and there is some underlying simplification? It seems like the reason it doesn't violate energy principles is that if one considers the whole track rather than the instantaneous force applied a faster linear deceleration would "shorten the work" for the angular acceleration. But, at any instant, the force applied can only do one of two things in full or both things with some apportionment between them. How does it do both things? If these underlying assumptions are answered, I agree this is a much more elegant way of doing this problem. Thank you for posting this alternate solution.Steve4Physics said:Since a solution is now posted (and since I'd already done one for my own entertainment) here’s an alternative, if of interest.
##\mu_k## is abbreviated to ##\mu## to reduce clutter.
Initially there is both rolling and slipping.
The linear acceleration is ##a = -\frac {\mu mg}m = -\mu g##. The linear velocity at time ##t## is ##v= v_0 -\mu g t##.
The angular acceleration is ##\alpha = \frac {\tau}I = \frac {\mu mgR}
{(2/5)mR^2}= \frac {5\mu g} {2R}##. The angular velocity at time ##t## is ##\omega = \frac {5\mu g} {2R} t##.
Rolling without slipping starts when ##v = \omega R##. That's the critical condition to understand. Say this happens at time ##t=T##.
##v_ 0 - \mu g T = \frac {5\mu g} {2R} T R##.
A bit of algebra gives:
##T = \frac {2v_0}{7\mu g}##
Distance covered in this time is:
##d = v_0 T + ½ aT^2##
##= v_0 \frac {2v_0}{7\mu g}- ½ (\mu g) (\frac {2v_0}{7\mu g})^2##
A bit of algebra gives:
##d = \frac {12 v_02}{49 \mu g}##
##\mu = \frac {12 v_02}{49 g d}##