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Imagine a rod that is held at angle to the vertical by some hook. The rod is in equilibrium. There would be three forces on this rod - Normal contact, Friction, and the weight of the rod.

I struggle to see how rotational equilibrium can be achieved. If we define the pivot to be the point of contact between the hook and the rod, that means that the normal contact force and Friction would not have a torque on the rod since it passes through the defined pivot. Hence, there would be a net torque about the pivot due to the weight of the rod.

This applies to any system where there are only 2 points which forces act. If we define pivot to be any one of the point of action, there would definitely be a net torque due to the other force.

How do I resolve this?

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The rod cannot achieve rotational equilibrium. Your thinking is right. The rod will have to rotate about the hinge because of the torque due to its weight. mgsinθL/2 = Iα

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