Wilberforce pendulum

The Wilberforce pendulum alternates between two different modes of oscillation.

The Wilberforce pendulum is a combination of spring and torsion pendulum . It consists of a mass that is suspended from a long coil spring , so that the coil spring is twisted slightly when the mass is rotated around its vertical axis. It is a coupled pendulum that is used in physics didactics as an example experiment. It was designed by the British physicist Lionel Robert Wilberforce around 1896 .

The mass on the spring can swing up and down and, like a torsion pendulum, rotate back and forth around its vertical axis. With a certain setting, an interesting movement appears, in which an exclusive rotational oscillation and an exclusive "up and down" oscillation constantly replace one another. The energy of the pendulum is alternately transferred from the translational oscillation to a rotational oscillation and back until the movement slowly decays due to the damping .

Despite its name, the pendulum does not perform a horizontal pendulum motion during normal operation, as does normal pendulums . Usually horizontally opposite "arms" are attached to the mass, to which small weights can be screwed in order to adjust the moment of inertia for the torsional vibration.

Explanation

A Wilberforce pendulum from 1908

The initially surprising property results from a slight coupling of the two movements due to the geometry of the spring. As the mass moves up and down, any downward movement causes the spring to unwind slightly, causing the mass to experience a small amount of torque . An upward movement of the mass, on the other hand, causes the number of turns of the spring to increase slightly, i.e. it winds it up and thus gives the mass a small torque in the other direction. Every "up and down" oscillation thus leads to a torque that is then shown in a slight right and left turn. In the case of a pendulum that only swings up and down, energy is transferred from the translational oscillation to the rotational oscillation with each oscillation, so that the amplitude of the rotational oscillation increases and that of the translational oscillation decreases at the same time, until after a while the mass only performs a rotational movement.

The situation is similar if the mass initially only rotates back and forth. Then rotation in one direction causes the spring to be wound up, rotation in the other direction causes it to unwind. Each development reduces the tensile force of the spring so that the mass can sink further; a winding makes the tensile force increase, pulls the mass than further upwards. Each rotation ensures that the mass swings up and down more strongly until the energy is transferred back from the rotational movement back into the translational movement and the pendulum only moves up and down.

Frequency of change of movement

The pendulum can be viewed as two coupled harmonic oscillators . The movement can be described as a harmonic oscillation with a variable amplitude, with the amplitudes oscillating out of phase with a sinusoidal function.

The frequency with which the pendulum alternates between the two modes of oscillation is the difference between the natural frequencies of the individual oscillations. The smaller the distance between these two natural frequencies, the slower the change between the two forms of oscillation. This behavior, which can be seen with all coupled pendulums , can be compared with the acoustic phenomenon of a beat in musical instruments. There two sine tones are combined and thus produce a tone with a frequency that corresponds to the difference between the individual frequencies.

For example, in the case of a Wilberforce pendulum, in which the mass swings up and down at a frequency and rotates back and forth with a frequency , the frequency and the corresponding period with which both forms of oscillation alternate are included ${\ displaystyle f _ {\ mathrm {T}} = 4 \, \ mathrm {Hz}}$ ${\ displaystyle f _ {\ mathrm {R}} = 4 {,} 1 \, \ mathrm {Hz}}$ ${\ displaystyle f _ {\ mathrm {alt}}}$

{\ displaystyle f _ {\ mathrm {alt}} = f _ {\ text {R}} - f _ {\ mathrm {T}} = 0 {,} 1 \; \ mathrm {Hz},}

{\ displaystyle T _ {\ mathrm {alt}} = {\ frac {1} {f _ {\ mathrm {alt}}}} = 10 \; \ mathrm {s}.}

The movement therefore changes from translation to rotation within five seconds and back to translation in the following five seconds.

Usually the moment of inertia of the mass is adjusted until the rotation frequency is very close to the translation frequency so that the transition between the two oscillation forms can be clearly seen. This is usually possible by unscrewing and unscrewing the weights on the arms.

Web links

Video of a Wilberforce pendulum swinging , by Berkeley Lecture Demonstrations, YouTube.com, accessed January 19, 2013
John Pitre: Wilberforce Pendulum (PDF; 290 kB) In: Physics 182S lab . Univ. of Toronto. Retrieved January 19, 2013.

Individual evidence

^ Lionel Robert Wilberforce: On the vibrations of a loaded spiral spring . In: Philosophical Magazine . 38, 1896, pp. 386-392. Retrieved January 9, 2008.
^ Arnold Sommerfeld : Mechanics of deformable media (= lectures on theoretical physics. Volume II). 6th edition. Akademische Verlagsgesellschaft, Leipzig 1970, § 42 Torsion and bending in helical springs, pp. 286–291.
^ Richard E. Berg, Marshall, Todd S .: Wilberforce pendulum oscillations and normal modes . In: American Journal of Physics . 59, No. 1, May 4, 1990, pp. 32-37. doi : 10.1119 / 1.16702 . Retrieved May 3, 2008.

[1] Lionel Robert Wilberforce: On the vibrations of a loaded spiral spring . In: Philosophical Magazine . 38, 1896, pp. 386-392. Retrieved January 9, 2008.

[2] Arnold Sommerfeld : Mechanics of deformable media (= lectures on theoretical physics. Volume II). 6th edition. Akademische Verlagsgesellschaft, Leipzig 1970, § 42 Torsion and bending in helical springs, pp. 286–291.

[3] Richard E. Berg, Marshall, Todd S .: Wilberforce pendulum oscillations and normal modes . In: American Journal of Physics . 59, No. 1, May 4, 1990, pp. 32-37. doi : 10.1119 / 1.16702 . Retrieved May 3, 2008.