In the treatment of the ordinary pendulum above, we just used Newton's Second Law directly to get the equation of motion. This was possible only because we could neglect the mass of the string and because we could treat the mass like a point mass at its end.
However, real grandfather clocks often have a large, massive pendulum like the one above - a long massive rod (of length and mass ) with a large round disk (of radius and mass ) at the end. The round weight rotates through an angle of in each oscillation, so it has angular momemtum. Newton's Law for forces no longer suffices. We must use torque and the moment of inertia to obtain the frequency of the oscillator.
To do this we go through the same steps (more or less) that we did
for the regular pendulum. First we compute the net gravitational torque
on the system at an arbitrary (small) angle :
Next we set this equal to , where is the total moment of
inertia for the system about the pivot of the pendulum and
Note that for this problem:
With in hand, we know everything. For example:
Note that the energy of this sort of pendulum can be tricky. Obviously
its potential energy is easy enough - it depends on the elevation of
the center of masses of the rod and the disk. The kinetic energy,