## Equation of Motion of a Rotating Rod

If a rod AB of lengthand massis smoothly hinged at a point a distancefrom 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 velocityof the rod after it has turned through an angle

The moment of inertia of a rod of lengthand mass M about its centre isso 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 toin 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 anglethe 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

whereThis is the same equation as obtained by differentiating (1) with respect to t.

Divide byand putThe result is