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Derive the governing differential equation of motion of the system using an appropriate analytical method.

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2.1 Derive the governing differential equation of motion of the system using an appropriate analytical method.

We will use Newton's second law for rotational motion. Let θ\theta be the angular displacement of the bar from the vertical equilibrium position, with positive θ\theta being counter-clockwise. The pivot is at OO.

  1. Determine the moment of inertia about the pivot OO.

    For a uniform slender bar rotating about one end, the moment of inertia is:

    IO=13ml2I_O = \frac{1}{3} m l^2

  2. Calculate the torques acting on the bar.

    • Torque due to gravity (τg\tau_g): The gravitational force $
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