A uniform rod of length $ L $ and mass $ M $ is free to rotate about one end, as shown in the diagram. The free end is released from rest at a horizontal position, as shown. The pivot point is supported by a stand so that only the free end can move. The moment of inertia of a rod about its end is $\tfrac{1}{3} M L^{2}$.

Question

A uniform rod of length $ L $ and mass $ M $ is free to rotate about one end, as shown in the diagram. The free end is released from rest at a horizontal position, as shown. The pivot point is supported by a stand so that only the free end can move. The moment of inertia of a rod about its end is $\tfrac{1}{3} M L^{2}$.(a) Draw a Free Body Diagram for the rod immediately after the free end is released.
, (b) Derive expressions for the angular and tangential accelerations of the center of mass of the rod immediately after the free end is released. Express your answers in terms of $ M $, $ L $, and $ g $.
, (c) Describe qualitatively how the angular acceleration of the rod changes as the rod rotates, from its starting position until it is vertical (assume there is a hole in the floor so the rod does not hit anything). Justify your answer.
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Jason · Accepted Answer

$\alpha = \dfrac{3g}{2L}$
$a_t = \dfrac{3g}{4}$

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