A uniform bar of mass 𝑚 and length 𝐿 is pivoted at point 𝑂, as shown in Figure 2 below. A point

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A uniform bar of mass 𝑚 and length 𝐿 is pivoted at point 𝑂, as
shown in Figure 2 below. A point mass 𝑀 is attached to the bar at a
distance 𝑑 from point 𝑂.
Part I. The bar is released from rest from the position shown.
Immediately after the release:
(a) Draw the free-body-diagram showing all the forces acting on the
bar.
(b) Obtain an expression for the angular acceleration of the bar in
terms of 𝑚, 𝑀, 𝐿, 𝑑, 𝑔, and 𝜃 (𝑔 is the acceleration due to
gravity) in the fixed frame 𝐴 (in terms of 𝑎⃗3 unit vector).
(c) Find the magnitude of 𝑑 for which the angular acceleration
maximises. Briefly explain the reasoning for your choice
Part II. Now assume that the bar has an angular velocity 𝜔𝐵 𝑎⃗3
when passing through the position shown in Figure 2. For this
system:
(d) Does the angular acceleration of the bar change from the
expression obtained in Part I(b)? Explain why.
(e) How do the magnitudes of the reaction forces at 𝑂 change
compared to Part I? You do not need to obtain expressions for the
forces; discuss whether each force is larger or smaller compared to
that of the system of Part I and explain why.