This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.

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\frac{1}{3} m l^2$$
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 be the angular displacement of the bar from the vertical equilibrium position, with positive being counter-clockwise. The pivot is at .
Step 1: Determine the moment of inertia about the pivot . For a uniform slender bar rotating about one end, the moment of inertia is:
Step 2: Calculate the torques acting on the bar.
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2.1 Derive the governing differential equation of motion of the system using an appropriate analytical method.
This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.