## Equation of Motion of a Rotating Rod

If a rod AB of length and mass is smoothly hinged at a point a distance from the end A and, held with the end B above A and then given a small push so that it is free to rotate, we can apply conservation of energy to obtain the angular velocity of the rod after it has turned through an angle  The moment of inertia of a rod of length and mass M about its centre is so the moment of inertia of the rod above about its centre is The parallel axis theorem states that if it is pivoted about a parallel axis a distance (equal to in this case) from the first, the moment of inertia about that point is The kinetic energy of the rad at an time is When the rod has turned through an angle the centre of mass has fallen a distance (see the diagram which shows the changing position of the centre of mass). The increase in kinetic energy is equal to the loss in gravitational potential energy. (1)

Taking moments about the hinge give where This is the same equation as obtained by differentiating (1) with respect to t.  Divide by and put The result is  