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Musical Animated Large-Angle Pendulums

The period of a pendulum depends on its amplitude in a complicated way. Here are four pendulums whose amplitudes are chosen so that their periods are commensurate. Plus, they're musical! Each pendulum plays a note when it passes through its lowest point. The musical frequencies are proportional to the pendulums' frequencies, so they play a happy tune. You can change which instrument they play.

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The Wolfram Functions page Jacobi elliptic function dn: Theorems gives an exact analytic solution for the position and 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.
,
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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 and . 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 , so .
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. So only depends on the amplitude. Working per unit mass, .
The kinetic energy is then , so .
The formula for kinetic energy is , so that , and . Notice that we only get the magnitude here.
And . Again, this is only the magnitude.
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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 is decreasing.
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We would particularly like to know the ratio of the period to the period of the small-angle pendulum:
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where is the elliptic integral of the first kind (Mathematica's EllipticF). Here is another formula for the ratio that uses the complete elliptic integral of the first kind (Mathematica's EllipticK) instead. They are the same, except that , which is the behavior we want, instead of :
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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.
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