24.In the spring–mass system of Problem 23, suppose that the spring force is not given by Hooke's law but instead satisfies the relation Fs=−(ku+εu3),Fs=−(ku+εu3), where k > 0 and ε is small but may be of either sign. The spring is called a hardening spring if ε > 0 and a softening spring if ε < 0. Why are these terms appropriate? a.Show that the displacement u(t) of the mass from its equilibrium position satisfies the differential equation mu′′+γu′+ku+εu3=0 Suppose that the initial conditions are u(0)=0,u′(0)=1.u(0)=0,u′(0)=1. In the remainder of this problem, assume that m = 1, k = 1, and γ = 0.
24.In the spring–mass system of Problem 23, suppose that the spring force is not given by Hooke's law but instead satisfies the relation Fs=−(ku+εu3),Fs=−(ku+εu3), where k > 0 and ε is small but may be of either sign. The spring is called a hardening spring if ε > 0 and a softening spring if ε < 0. Why are these terms appropriate? a.Show that the displacement u(t) of the mass from its equilibrium position satisfies the differential equation mu′′+γu′+ku+εu3=0 Suppose that the initial conditions are u(0)=0,u′(0)=1.u(0)=0,u′(0)=1. In the remainder of this problem, assume that m = 1, k = 1, and γ = 0.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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24.In the spring–mass system of Problem 23, suppose that the spring force is not given by Hooke's law but instead satisfies the relation
Fs=−(ku+εu3),Fs=−(ku+εu3),
where k > 0 and ε is small but may be of either sign. The spring is called a hardening spring if ε > 0 and a softening spring if ε < 0. Why are these terms appropriate?
a.Show that the displacement u(t) of the mass from its equilibrium position satisfies the differential equation
mu′′+γu′+ku+εu3=0
Suppose that the initial conditions are
u(0)=0,u′(0)=1.u(0)=0,u′(0)=1.
In the remainder of this problem, assume that m = 1, k = 1, and γ = 0.
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