Your thumb squeaks on a plate you have just washed. Your sneakers often 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. As you slide it across the table, a Styrofoam cup may not make much sound, but it makes the surface of some water inside it dance in a complicated resonance vibration. 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. P15.74). Hookes 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 μs and a smaller coefficient of kinetic friction μk.
The board moves to the right at constant speed v. Assume that the block spends most of its time sticking to the board and moving to the right, so that 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 μsmg/k.
(b) Show that the block oscillates around an equilibrium position at which the spring is stretched by 5kmg/k.
(c) Graph the blocks position versus time.
(d) Show that the amplitude of the blocks motion is
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