A mass \(m\), connected to an elastic rubber band of unstretched length \(l\) and stiffness \(k\), is permitted to swing as a pendulum bob, as shown in Fig. 13.31. Derive the nonlinear equations of motion of the system using \(x\) and \(\theta\) as coordinates. Linearize the equations of motion and determine the natural frequencies of vibration of the system.

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Rubber band, stiffness k 1+x m FIGURE 13.31 Mass connected to a rubber band.