A Wilberforce pendulum, invented by British physicist

Lionel Robert Wilberforce Lionel Robert Wilberforce (18 April 1861 – 1 April 1944) was a British physicist. He is best known for the invention of the Wilberforce pendulum, which exhibits a curious motion in which periods of purely rotational oscillation gradually alter ...

around 1896, consists of a

mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...

suspended by a long

helical spring A selection of conical coil springs The most common type of spring is the coil spring, which is made out of a long piece of metal that is wound around itself. Coil springs were in use in Roman times, evidence of this can be found in bronze Fib ...

and free to turn on its vertical axis, twisting the spring. It is an example of a coupled mechanical oscillator, often used as a demonstration in

physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

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

torsional In the field of solid mechanics, torsion is the twisting of an object due to an applied torque. Torsion is expressed in either the pascal (Pa), an SI unit for newtons per square metre, or in pounds per square inch (psi) while torque is expressed ...

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. Despite the name, in normal operation it does not swing back and forth as ordinary

pendulum A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the ...

s do. 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 The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...

to 'tune' the torsional vibration

period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in musical composition * Periodic sentence (or rhetorical period), a concept ...

Explanation

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. When the weight is moving up and down, each downward excursion of the spring causes it to unwind slightly, giving the weight a slight twist. When the weight moves up, it causes the spring to wind slightly tighter, giving the weight a slight twist in the other direction. So when the weight is moving up and down, each oscillation gives a slight alternating rotational

torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...

to the weight. In other words, during each oscillation some of the energy in the translational mode leaks into the rotational mode. Slowly the up and down movement gets less, and the rotational movement gets greater, until the weight is just rotating and not bobbing. The Wilberforce Pendulum

Similarly, when the weight is rotating back and forth, each twist of the weight in the direction that unwinds the spring also reduces the spring tension slightly, causing the weight to sag a little lower. Conversely, each twist of the weight in the direction of winding the spring tighter causes the tension to increase, pulling the weight up slightly. 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 ''f_T'', which is dependent on the

spring constant In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that ...

''k'' of the spring and the mass ''m'' of the system, and the frequency of the rotating oscillator ''f''_R, which is dependent on the

''I'' and the torsional coefficient ''κ'' of the system.

f_T=\sqrt\frac \approx \sqrt\frac=f_R

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 ''f''_R, 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.

Alternation or 'beat' frequency

The

frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...

at which the two modes alternate is equal to the difference between the oscillation frequencies of the modes. The closer in frequency the two motions are, the slower will be the alternation between them. This behavior, common to all

coupled oscillators Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...

, is analogous to the phenomenon of

beat Beat, beats or beating may refer to: Common uses * Patrol, or beat, a group of personnel assigned to monitor a specific area ** Beat (police), the territory that a police officer patrols ** Gay beat, an area frequented by gay men * Battery (c ...

s in musical instruments, in which two tones combine to produce a 'beat' tone at the difference between their frequencies. For example, if the pendulum bobs up and down at a rate of ''f''_T = 4 Hz, and rotates back and forth about its axis at a rate of ''f''_R = 4.1 Hz, the alternation rate ''f''_alt will be:

f_ = f_R - f_T = 0.1\; \mathrm

T_ = 1 / f_ = 10\; \mathrm

So the motion will change from rotational to translational in 5 seconds and then back to rotational in the next 5 seconds. If the two frequencies are exactly equal, the beat frequency will be zero, and

resonance Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillatin ...

will occur.

References

External links

*{{cite web, last=Pitre, first=John, title=Wilberforce Pendulum, work=Physics 182S lab, publisher=Univ. of Toronto, url=http://faraday.physics.utoronto.ca/PHY182S/WilberforcePendulum.pdf, accessdate=2008-05-03
Video of Wilberforce pendulum oscillating
by Berkeley Lecture Demonstrations, YouTube.com, retrieved April 25, 2008 Pendulums Dynamics (mechanics)