A uniform ball of radius R rolls without slipping between two rails such that the horizontal distance is d between the two contact points of the rails to the hall.
(a) In a sketch, show that at any instant Vcm = CdYR2 - d 2/4. Discuss this expression in the limits d = 0 and d = 2R.
(b) For a uniform ball starting from rest and descending a vertical distance h while rolling without slipping down a ramp, Vcm = Y IOgh/7 €¢ Replacing the ramp with the two rails, show that

In each case, the work done by friction bas been ignored.
(c) Which speed in part (b) is smaller? Why? Answer in terms of how the loss of potential energy is shared between the gain in translational and rotational kinetic energies.
(d) For which value of the ratio d/ R do the two expressions for the speed in part (b) differ by 5.0% by 0.50%?

Transcribed Image Text:

10gh 5+ 2/(1 - d/4R?) Vem