Consider the mass \(m\) suspended by a nonlinear hardening spring as shown in Figure 2.34. The spring obeys \(F=k x^{3}\) and has an original length of \(L\). The coordinate \(x\) is the stretch of the spring, and \(y\) is the displacement of the mass measured from its static equilibrium position. Derive the linearized equation of motion about its equilibrium position in terms of \(x\) and also in terms of \(y\). Find the natural frequency of this system.

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original length, L eeeee static equilibrium position Yo xo y X m Figure 2.34: A mass-spring system with a hardening spring.