### The Ping Pong Pendulum

Galileo noticed the regular swinging of a candelabra in the cathedral in Pisa and speculated that the swing period was constant. This led him to use a pendulum to measure intervals of time for his experiments in dynamics. Bu not all pendulums behave like clock pendulums.

The ping pong pendulum.

The Ping Pong Pendulum

We consider a pendulum with two pivots. The motion is confined to a plane and the pendulum pivots alternately about each pivot. The pendulum bob moves on a circular arc centered at the left pivot when it is swinging to the right. When swinging to the left, it rotates about the right pivot, following another circular arc. Thus, the motion of the bob is along two circular arcs, intersecting at the point of equilibrium.

Bob moves on a circular arc centered at the left pivot when swinging to the right, and an arc centered at the right pivot when swinging left.

The origin of coordinates is taken to be at the point mid-way between the two pivot points. The distance between the pivots is ${2\delta}$, the distance from pivot to bob is ${\ell}$ and the point of equilibrium is at ${(0,-h)}$. The configuration of the system is determined by ${\theta}$, the angle between the vertical through the origin and the line from origin to bob.

Small Amplitude Motion

For small swing amplitudes, we may approximate the two circular arcs by line segments

$\displaystyle x = h \theta \,, \qquad y = - h + \delta\theta \,.$

The tangents to the circular arcs at the point ${(0,-h)}$ are the lines

${y=-h+(\delta/h)x}$ and ${y=-h-(\delta/h)x}$.

The potential energy becomes a V-shaped potential well, ${V(\theta) = \kappa|\theta|}$, where ${\kappa=\delta/h}$. The motion in such a potential well is described by a sequence of parabolic arcs and the period is ${\tau = 4\sqrt{{2x_0}/{\kappa}} = 4\sqrt{{2 h\theta_0}/{\kappa}}}$.

Damping and Frequency Growth

Now we add damping to the system, and model the motion by the equation

$\displaystyle \ddot\theta + k\dot\theta + \kappa \theta/|\theta| = 0$

This equation can be solved piecewise in analytical terms but it is inconvenient to have separate solutions for separate segments, so we solve the equation numerically. The result is shown below. It is clear that the frequency increases strongly as the amplitude decreases.

This pattern of frequency increasing as energy decreases is similar to the behaviour of a range of physical systems. We may mention the Euler Disk, but there are many others.

Sources

A more detailed article on the ping pong pendulum is available on ArXiv.