# 60. Mystery Pendulum Explanation

- This information is from Bob Keolian.

The trick is in the suspension. At the "Mystery Spot," a tourist trap in Santa Cruz, they suspend the pendulum with a chain by casually looping the chain around a horizontal beam associated with the roof of a shed. The chain reattaches to itself, forming a "V" at the top. Your's must do something similar, forming a V at the top, with two fixed attachments at the top of the V and a knot or some other junction at the bottom of the V to a single chain or rope that goes down to the pendulum. The trick is in noticing that for motion perpendicular to the plane of the V, the effective length of the pendulum is from the two attachments at top to near the center of mass of the pendulum below, but for motion in the plane of the V, the junction or knot point remains fixed and the effective length of the pendulum starts from there. This gives a shorter length and slightly higher frequency for motion in the plane of the V than for motion perpendicular to the plane of the V.

At reasonably small amplitudes, say less than about 10 degrees of deflection, the motion of the pendulum can be considered to be a linear superposition of in-plane and out-of-plane motion, with two modes of slightly different frequencies. So you can get different Lissajous figures depending on how you start off the pendulum. If you start off with a circular motion, you are exciting both modes with the same amplitude but with an initial 90 degree phase shift between them. As the modes progress in time, the instantaneous phase between them (w_1t - w_2t - initial phase) changes with time. Eventually, the two modes are in phase and the pendulum move in a straight line 45 degrees from the plane of the V. Later the phase is such that the circle changes direction. If you let the pendulum go, the pattern repeats itself. But that isn't as much fun as stopping the pendulum after it has reversed direction once and cooking up some BS story about a gravitational or magnetic anomaly in the mantle below Los Angeles that causes the change in direction.

If you start the pendulum off in planer motion either in the plane of the V or out of the plane of the V, then you excite only one of the modes, and the motion stays in its plane. At larger amplitudes the linear superposition of modes picture breaks down, at around 30 degrees of deflection. The two modes couple parametrically by modulating the tension in the rope at twice the oscillation frequency, but that is another story.