In the springmass system of 31, suppose that the spring force is not given byHookes law

Chapter 3, Problem 32

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In the springmass system of 31, suppose that the spring force is not given byHookes law but instead satisfies the relationFs = (ku + u3),where k > 0 and is small but may be of either sign.The spring is called a hardening springif > 0 and a softening spring if < 0. Why are these terms appropriate?(a) Show that the displacement u(t) of the mass from its equilibrium position satisfies thedifferential equationmu + u + ku + u3 = 0.Suppose that the initial conditions areu(0) = 0, u(0) = 1.In the remainder of this problem, assume that m = 1, k = 1, and = 0.(b) Find u(t) when = 0 and also determine the amplitude and period of the motion.(c) Let = 0.1. Plot a numerical approximation to the solution. Does the motion appearto be periodic? Estimate the amplitude and period.(d) Repeat part (c) for = 0.2 and = 0.3.(e) Plot your estimated values of the amplitude A and the period T versus . Describethe way in which A and T, respectively, depend on .(f) Repeat parts (c), (d), and (e) for negative values of .