Wilberforce pendulum

Key takeaways

Wilberforce pendulum belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Wilberforce pendulum to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Wilberforce pendulum from memory before moving on to harder problems.

Reference excerpt

A Wilberforce pendulum, invented by British physicist Lionel Robert Wilberforce around 1896, consists of a mass suspended by a long helical spring 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 education. 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. Despite the name, in normal operation it does not swing back and forth as ordinary pendulums 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 to 'tune' the torsional vibration period.

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 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.

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 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.

f T = 1 2 π k m ≈ 1 2 π κ I = f R {\displaystyle f_{T}={\frac {1}{2\pi }}{\sqrt {\frac {k}{m}}}\approx {\frac {1}{2\pi }}{\sqrt {\frac {\kappa }{I}}}=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 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.

Alternation or 'beat' frequency The frequency 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, 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. For example, if the pendulum bobs up and down at a rate of fT = 4 Hz, and rotates back and forth about its axis at a rate of fR = 4.1 Hz, the alternation rate falt will be:

f a l t = f R − f T = 0.1 H z {\displaystyle f_{\rm {alt}}=f_{R}-f_{T}=0.1\;\mathrm {Hz} }

T a l t = 1 / f a l t = 10 s {\displaystyle T_{\rm {alt}}=1/f_{\rm {alt}}=10\;\mathrm {s} }

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 will occur.

References

External links Pitre, John. "Wilberforce Pendulum" (PDF). Physics 182S lab. Univ. of Toronto. Retrieved 2008-05-03. Video of Wilberforce pendulum oscillating, by Berkeley Lecture Demonstrations, YouTube.com, retrieved April 25, 2008 Wilberforce pendulum historical demonstration, by Dr. Dominic Dickson, University of Liverpool, 10 December 2008

Illustrations

Wilberforce pendulum: Wilberforce pendulum, 1908 — Wilberforce pendulum, 1908

Worked examples

Example 1 — a first encounter with Wilberforce pendulum

Start with the simplest possible case. Write down what Wilberforce pendulum claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Wilberforce pendulum before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Wilberforce pendulum ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Wilberforce pendulum

In research: Wilberforce pendulum appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Wilberforce pendulum in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Wilberforce pendulum is common in secondary-school and first-year university syllabi. It links to neighbouring topics Dynamics (mechanics), Pendulums, so understanding it makes those chapters shorter.
In everyday life: Look for Wilberforce pendulum outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Wilberforce pendulum in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Wilberforce pendulum means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Wilberforce pendulum out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Wilberforce pendulum in simple terms?

A Wilberforce pendulum, invented by British physicist Lionel Robert Wilberforce around 1896, consists of a mass suspended by a long helical spring 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 physic…

Why does Wilberforce pendulum matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Wilberforce pendulum?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Wilberforce pendulum.