Serway & Jewett — Oscillatory Motion: Problem 42

Your thumb squeaks on a plate you have just washed. Your sneakers squeak on the gym floor. Car tires squeal when you start or stop abruptly. You can make a goblet sing by wiping your moistened finger around its rim. When chalk squeaks on a blackboard, you can see that it makes a row of regularly spaced dashes. As these examples suggest, vibration commonly results when friction acts on a moving elastic object. The oscillation is not simple harmonic motion, but is called stick-and-slip. This problem models stick-and-slip motion. A block of mass \(m\) is attached to a fixed support by a horizontal spring with force constant \(k\) and negligible mass (Fig. P15.42). Hooke’s law describes the spring both in extension and in compression. The block sits on a long horizontal board, with which it has coefficient of static friction \(\mu_s\) and a smaller coefficient of kinetic friction \(\mu_k\). The board moves to the right at constant speed \(v\). Assume the block spends most of its time sticking to the board and moving to the right with it, so the speed \(v\) is small in comparison to the average speed the block has as it slips back toward the left. (a) Show that the maximum extension of the spring from its unstressed position is very nearly given by \(\mu_s mg/k\). (b) Show that the block oscillates around an equilibrium position at which the spring is stretched by \(\mu_k mg/k\). (c) Graph the block’s position versus time. (d) Show that the amplitude of the block’s motion is \(A = \frac{(\mu_s - \mu_k)mg}{k}\). (e) Show that the period of the block’s motion is \(T = \frac{2(\mu_s - \mu_k)mg}{vk} + \pi \sqrt{\frac{m}{k}}\).