Does the hinge reaction do any work?

Question

I was doing some questions in rotational mechanics and I am not able to understand the concept behind hinge reaction which acts on a rod suspended at a point. 
I am taking a simple example here. Let's say that a uniform rod of some mass is free to rotate. Initially it's at rest in an unstable equilibrium vertically upwards. The hinge reaction acting initially would be -mg upwards. If the rod is slightly displaced and then makes some angle, hinge reaction will change but will not be 0.

My query is-

1 .Will the work done by hinge reaction be 0 somehow?

2 .Will the total mechanical Energy of the rod be conserved?

3 .If the total mechanical Energy is not conserved then how can we calculate the angular velocity of rod after it makes some angle theta?

I have done some similar questions in past and as far as I remember, I used conservation of energy. I was not familiar with hinge reaction at that time.

John Darby · Answer

Assume a frictionless hinge. Work is force acting through a distance. There is no relative displacement of the rod relative to the hinge point; therefore, the force of the hinge on the rod does no work regardless of the direction of this force. See  Conservation of Kinetic Energy?
The change in kinetic energy is the net work done.  The work done by gravity is considered as the negative of the change in the gravitational potential energy.  With no other force doing work except gravity, the kinetic energy plus potential energy is conserved. Therefore, considering rotation about the hinged end ${1over 2}Iomega ^2 + Mgh$ is constant where I is the inertia about the hinge, $omega$ is the angular velocity of the rod, M is the total mass and h is the elevation of the center of mass. Your problem is the "compound pendulum" discussed in most physics mechanics textbooks; for example Symon Mechanics.

Does the hinge reaction do any work?

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