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 ball. (a) In a sketch, show that at any instant . Discuss this expression in the limits ?d? = 0 and ?d? = 2?R?. (b) For a uniform ball starting from rest and descending a vertical distance ?h? while rolling without slipping down a ramp, . Replacing the ramp with the two rails, show that In each case, the work done by friction has 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? d0 the two expressions for the speed in part (b) differ by 5.0%? By 0.50%/?

Solution 100CP Step 1 of 4: a) The distance from the center of the ball to the midpoint of the line joining the points where the ball is in 2 d 2 2 d2 contact with the radius is R ( )2 so v cm = R 4. When d = 0, this reduces to v cm = R, the same as rolling on a flat surface. When d = 2R, the rolling radius approaches zero, and v 0 for any cm Step 2 of 4: b)K = mv + I 1 2 2 2 = [mv 2 + mR ( 2 vcm ) ] 2 cm 5 R (d2) 4 mv 2 = cm[5 + 22 ] 10 (14 R) The conservation of energy of the ball is K = U 2 mvcm[5 + 2 ] = mgh 10 (1 d2) 4 R Now we can find v cm using above relation