Concept explainers
Coulomb Friction Revlslted In Problem 27 in Chapter 5 in Review we examined a spring/mass system in which a mass m slides over a dry horizontal surface whose coefficient of kinetic friction is a constant μ. The constant retarding force fk = μmg of the dry surface that acts opposite to the direction of motion is called Coulomb friction after the French physicist Charles-Augustin de Coulomb (1736–1806). You were asked to show that the piecewise-linear differential equation for the displacement x(t) of the mass is given by
- (a) Suppose that the mass is released from rest from a point x(0) = x0 > 0 and that there are no other external forces. Then the differential equations describing the motion of the mass m are
x″ + ω2x = F, 0 < t < T/2
x″ + ω2x = −F, T/2 < t < T
x″ + ω2x = F, T < t < 3T/2,
and so on, where ω2 = k/m, F = fk/m = μg, g = 32, and T = 2π/ω. Show that the times 0, T/2, T, 3T/2, ... correspond to x′(t) = 0.
- (b) Explain why, in general, the initial displacement must satisfy ω2 |x0| > F.
- (c) Explain why the interval −F/ω2 ≤ x ≤ F/ω2 is appropriately called the “dead zone” of the system.
- (d) Use the Laplace transform and the concept of the meander function to solve for the displacement x(t) for t ≥ 0.
- (e) Show that in the case m = 1, k = 1, fk = 1, and x0 = 5.5 that on the interval [0, 2π) your solution agrees with parts (a) and (b) of Problem 28 in Chapter 5 in Review.
- (f) Show that each successive oscillation is 2F/ω2 shorter than the preceding one.
- (g) Predict the long-term behavior of the system.
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Check out a sample textbook solutionChapter 7 Solutions
A First Course in Differential Equations with Modeling Applications (MindTap Course List)
- Trigonometry (MindTap Course List)TrigonometryISBN:9781305652224Author:Charles P. McKeague, Mark D. TurnerPublisher:Cengage Learning