Does the hinge reaction do any work? 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.
 A: 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 ${1\over 2}I\omega ^2 + Mgh$ 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.
A: *

*"Will the work done by hinge reaction be 0 somehow?"
Work is defined as force applied over a distance. When a body is constrained, like in the case of a hinge, the body is constrained not to move in the direction where constraint forces can develop.
For a hinge, any force vector $\boldsymbol{F}$ in any direction can be a constraint force, since no movement (translation) is allowed in any direction.
On the other hand, the reaction moment vector $\boldsymbol{M}$ on the hinge must be perpendicular to the pivot axis $\boldsymbol{z}$. This condition is often expressed as $$ \boldsymbol{M} \cdot \boldsymbol{z} = 0 $$ where $\cdot$ is the vector dot product.


*"Will the total mechanical Energy of the rod be conserved?"
Yes. Again, by definition reaction/constraint forces & moments do no work. You can check the power added/removed for a rigid body, by picking an arbitrary point A and evaluating the following $$ P = \boldsymbol{v}_A \cdot \boldsymbol{F} + \boldsymbol{\omega} \cdot \boldsymbol{M}_A$$
For the case of pivot, if we pick the pivot location to be the point of summation for power, then $\boldsymbol{v}_A = 0$  since the pivot is not moving, and as we saw above $\boldsymbol{\omega} \cdot \boldsymbol{M}_A =0$ since $\boldsymbol{\omega}$ is parallel to $\boldsymbol{z}$. So is the case for the pivot, and really for any constraint that power added/removed is zero.


*"If the total mechanical Energy is not conserved then how can we calculate the angular velocity of rod after it makes some angle theta?"
Now you are asking about the equations of motion. You can use either the so-called Newton-Euler equations of motion for rigid bodies or Lagrangian mechanics.
Either way, the first step is deciding on the degrees of freedom of the system. Next is describing the kinematics in terms of these degrees of freedom, and track the centers of mass using the DOF variables and their derivatives.
Finally apply the equations of motion given the formulation you choose and it will given you a system of differential equations for each DOF variable.
In the case of an inverted pendulum, you will end up with an ODE of the form
$$\ddot{\theta} = f(t, \theta, \dot{\theta})$$ where $t$ is time, $\theta$ is the rod angle and $\dot{\theta}$ is the rod angle speed.
