Wilberforce pendulum

A Wilberforce pendulum, invented by British physicist Lionel Robert Wilberforce around 1896,[1] consists of a mass suspended by a long helical spring and free to turn on its vertical axis, twisting the spring.

The mass can both bob up and down on the spring, and rotate back and forth about its vertical axis with torsional vibrations.

When correctly adjusted and set in motion, it exhibits a curious motion in which periods of purely rotational oscillation gradually alternate with periods of purely up and down oscillation.

The energy stored in the device shifts slowly back and forth between the translational 'up and down' oscillation mode and the torsional 'clockwise and counterclockwise' oscillation mode, until the motion eventually dies away.

The mass usually has opposing pairs of radial 'arms' sticking out horizontally, threaded with small weights that can be screwed in or out to adjust the moment of inertia to 'tune' the torsional vibration period.

The device's intriguing behavior is caused by a slight coupling between the two motions or degrees of freedom, due to the geometry of the spring.

So each oscillation of the weight back and forth causes it to bob up and down more, until all the energy is transferred back from the rotational mode into the translational mode and it is just bobbing up and down, not rotating.

A Wilberforce pendulum can be designed by approximately equating the frequency of harmonic oscillations of the spring-mass oscillator fT, which is dependent on the spring constant k of the spring and the mass m of the system, and the frequency of the rotating oscillator fR, which is dependent on the moment of inertia I and the torsional coefficient κ of the system.

The pendulum is usually adjusted by moving the moment of inertia adjustment weights towards or away from the centre of the mass by equal amounts on each side in order to modify fR, until the rotational frequency is close to the translational frequency, so the alternation period will be slow enough to allow the change between the two modes to be clearly seen.

This behavior, common to all coupled oscillators, is analogous to the phenomenon of beats in musical instruments, in which two tones combine to produce a 'beat' tone at the difference between their frequencies.