Halliday, Resnick & Walker — Rolling, Torque, and Angular Momentum: Problem 83

When an object rolls over a surface, the contact area of both the object and the surface can continuously deform and recover. Energy is lost in that continuous motion and thus the kinetic energy of the object gradually decreases. A rolling friction is said to act on the object, with a magnitude \(f_r\) given by \(f_r = \mu_r F_N\), where \(\mu_r\) is the coefficient of rolling friction and \(F_N\) is the magnitude of the normal force. In Fig. 11.40, a pool ball rolls rightward over the felt of a pool table. The deformation of the ball is negligible but the deformation of the felt creates the rolling friction. The support forces along the contact area can then be represented by shifting the normal force \(\vec{F}_N\) rightward by distance \(h\) from being directly under the center of mass. The torque due to that force about the center of mass works against the rotation. The ball has mass \(m = 97.0 \text{ g}\) and radius \(r = 26.2 \text{ mm}\), and the shift distance is \(h = 0.330 \text{ mm}\). (a) What is the magnitude of the torque due to the normal force? How much energy is lost to the torque if the ball rolls through (b) one revolution and (c) distance \(L = 30.0 \text{ cm}\)? (d) What is the value of \(\mu_r\)?