(Solved): Consider the spring-loaded inverted pendulum shown in Figure. Assume that the spring force acting o ...

Consider the spring-loaded inverted pendulum shown in Figure. Assume that the spring force acting on the pendulum is zero when the pendulum is vertical \( (\theta=0) \). Assume also that the friction involved is negligible. Obtain a mathematical model of the system when the angle \( \theta \) is small, that is, when \( \sin \theta \cong \theta \) and \( \cos \theta \cong 1 \). Also, obtain the natural frequency \( \omega_{n} \) of the system.