Consider the system below; the pendulum (with mass m and length L) is connected to a freely moving block (with mass m). Assume that the surface where the block slides horizontally has negligible friction and the block is connected to a wall by a spring with spring constant k. Determine the (a) frequencies of the normal modes for small oscillations around the equilibrium positions and (b) the motion for each of the normal modes for special case where ??= √??=√??. Use Lagrangian formulations to solve the problem.

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Free Damped Vibrations

Free vibration is a type of vibration in which the external force is removed after giving the initial displacement to a body/system. In other words, in this type of vibration, force is applied at the initial displacement of the system, and after the initial displacement, force is removed, and the system vibrates at the natural frequency.