A bar supported by a hinge at its base is held in place by a spring connected to a collar, as per Figure 2.69. The spring is unstretched when the bar is vertical. As the bar is displaced from equilibrium by an angle \(\theta\), the collar slides on the frictionless bar so that the spring remains horizontal. Derive the equation of motion using an energy method, and then linearize the model for small amplitude motion. Identify the frequency of oscillation.

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collar k 6000000 772 72 Figure 2.69: An inverted bar is held in place by a spring connected to a collar.