In Section 3.8 we assumed that the restoring force F of the spring satisfied Hookes law

Chapter 11, Problem 18

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In Section 3.8 we assumed that the restoring force F of the spring satisfied Hooke's law F=ks, where s is the elongation of the spring and k is a positive constant of proportionality. If we replace this assumption with the nonlinear law \(F=k s^{3}\), then the new differential equation for damped motion becomes \(m x^{\prime \prime}=-\beta x^{\prime}-k(s+x)^{3}+m g\), where \(k s^{3}=m g\). The system is called overdamped when (0, 0) is a stable node and is called underdamped when (0, 0) is a stable spiral point. Find new conditions on m, k, and \(\beta\) that will lead to overdamping and underdamping.