I set up an experiment with a rigid rod pendulum swinging on a hinge and the hinge attached to a small PMDC generator.

The generator has no load attached and its output voltage is proportional to its speed of rotation.

I swung the pendulum from 90 degrees and using a data logger I recorded the voltage of the generator (at 100 samples / second).

I approximated the output voltage (and thus speed) to a decaying sine wave.

To calculate mechanical rotating power using torque and angular velocity, can I simply multiply the instantaneous equations for both, or do I need to somehow do a dot product?

The equations are shown below including the units. I've clearly gone wrong some where since I end up with watts*radians^2.

(The units on the graph are not accurate because I've compressed/stretched various values so they nicely fit onto the same scale)

Interesting, I ran another test with a resistive load attached to the generator. Max power occurs at max voltage which occurs at max speed of rotation.

This maximum output electrical power actually occurs when there is apparently zero mechanical input power.

How can this be? Obviously the pendulum has the most angular momentum at the base of the swing. Is it that I'm not taking into account the rotor inertia (which in this case is so small it seems to barely effect the dynamics of the swing).