The Wolfram Functions page Jacobi elliptic function dn: Theorems
gives an exact analytic solution
for the position
of a pendulum, assuming that the clock starts when the pendulum is at its lowest point. Here
is the angular amplitude,
is the length of the simple pendulum, and
is the acceleration due to gravity.
To make the pendulum animation loop properly, we need to know how the period of a pendulum depends on its amplitude, so we can choose proper amplitudes. Now, a small-angle pendulum is approximately a harmonic oscillator, and its period is
, which does not depend on the angular amplitude. But for a large-angle pendulum, we have to figure out just how the period does depend on the amplitude. If we knew the angular velocity
as a function of the angle
, then we could integrate 1/ω
for one complete oscillation. Or even better, integrate it for 1/4 of an oscillation, that is, from
down to 0; then we do not have to worry about the changing signs for
. We can figure out
by using conservation of mechanical energy. (Actually, we use mechanical energy per unit mass to remove the mass from consideration.)
The potential energy is
, so the potential energy per unit mass is
. For convenience, we measure the height from the pivot point, not the lowest point (it does not matter). This means that the height
The total mechanical energy
is constant in this system. The pendulum has zero kinetic energy when it is at its highest point, that is, when θ=α
only depends on the amplitude. Working per unit mass,
The kinetic energy is then
The formula for kinetic energy is
, so that
. Notice that we only get the magnitude here.
. Again, this is only the magnitude.
Now we are ready to integrate. We look at only 1/4 of a complete oscillation, starting from rest when θ=α
, and ending when θ=0
. During this part of the motion,
is negative because
We would particularly like to know the ratio of the period to the period of the small-angle pendulum:
The amplitudes in the Demonstration are chosen so that their
's are in the ratios (1, 5/4, 5/3, 5/2); this means that they will complete 5, 4, 3, and 2 oscillations in one complete cycle.