A solid 0.4750 kg ball rolls without slipping down a track toward a vertical loop of radius R = 0.7350 m. What minimum translational speed Umin must the ball have when it is a height H = 1.111 m above the bottom of the loop in order to complete the loop without falling off the track? Assume that the radius of the ball itself is much smaller 9.810 m/s? for the than the loop radius R. Use g = acceleration due to gravity.

Applying the conservation law of energy. Here, E, m, v, I, ω, and h represent the ball’s total energy, the ball’s mass, the ball’s velocity, the ball’s moment of inertia, the ball’s angular velocity, and the ball’s height from the loop’s bottom, respectively. The subscripts S, and T represent the ball’s starting position and the ball’s position at the top of the loop, respectively. The ball’s moment of inertia can be given as, Also, The angular velocity can be given as, Thus, Also, the ball’s height in the starting position is H and at the loop’s top, the height is 2R. Thus,Now, at the loop’s top, the ball’s weight acts as the centripetal force required to move the ball in the loop. Thus, the ball’s velocity at the loop’s top can be found as, Thus, Substitute the relevant values. Thus, the minimum starting velocity the ball must have to complete the loop is about 3.49 m/s.