For a loss-less oscillating system the restoring force can be
described in terms of a potential energy function. In order for the
restoring force to be linear the potential energy must take the form
The total mechanical energy then reads
Thus harmonic oscillations is basically a phenomenon in which
mechanical energy is continuously converted back and forth
between potential and kinetic energy. The total mechanical energy
in the harmonic oscillations is proportional to the squared
amplitude and the the squared resonance frequency.
(note that there is a typo in Fig. 13-7 of Fishbane et al. At time t=T/2 the kinetic energy is zero).
For a physical pendulum the corresponding expression for the mechanical energy
is
where r is the distance from pivot point to center of mass, I
is the moment of inertia for oscillations about the pivot point,
is the time dependent
angular position of the pendulum, and
is the angular velocity of the pendulum as it oscillates.