Amy Smith: 2004
Abstract
A force probe and motion detector are used to plot force-distance loops for a transverse standing wave on a long steel spring, and on a rubber bungee.
The work input per cycle is compared with an estimate of the energy loss of the standing wave calculated as the maximum kinetic energy as the string becomes straight.
Introduction
A transverse standing wave is formed on a tensioned spring (etc.) when a transverse traveling wave reflects repeatedly between fixed ends. Energy is supplied each cycle by small lateral movements at one end. Energy is lost from the system by absorption in the supports, by hysteresis and by friction due to air resistance. The energy of the wave is much greater than the work done per cycle to maintain the wave.
Each element of mass of the spring performs simple harmonic motion. The maximum speed as the spring becomes instantaneously straight is given at once by Aw where A is the amplitude and w is the angular frequency. The maximum kinetic energy, DE, of each element of mass is given by ...
... where Dm is an element of mass, and Ai is the amplitude for that element.
Since the maximum velocity as a function of displacement along the spring is a sine function the average value to the maximum kinetic energy is given by ...
... where A is the amplitude to the standing wave.
The maximum kinetic energy, (the energy of the standing wave) is therefore ...
...where m is the mass of the spring.
The energy of the standing wave can be calculated if the mass of the spring, the amplitude of the standing wave and the angular frequency of the wave generator are measured.
The work done per cycle at the 'fixed' end can be estimated by plotting force displacement loops with a force probe and a motion detector. The tongue of the force probe is fixed to the end of the spring and a shield is mounted on the body of the probe. The displacement is measured with a motion detector. The force probe is calibrated in Newtons and the motion detector in meters. The area of the loops on the force displacement graphs are in Joules.
Apparatus
A steel spring of 70 gram mass, and one meter unextended length, was stretched to 3 meters, attached to a clamp at one end and to a force probe at the other. The force probe was fitted with a hemispherical reflector (half a cistern ball) so that its position could be graphed with a motion detector.
A rubber bungee of one meter unextended length and 96 gram mass was made from rubber bands and extended to 3 meters.
Decay curves for standing waves were plotted with the motion detector by driving the amplitude of one half wavelength up to 20+ cm and then suddenly clamping the driving end.
Data:
Amplitude decay
Amplitude as a function of time is plotted in Graph 1. The graph is an overlay diagram prepared in PhotoShop.
The amplitude-time functions of both the steel spring and the rubber bungee are approximately exponential. The energy of the standing wave on the rubber decays more rapidly due, probably, to increased air resistance and hysteresis in the rubber.
Energy decay
The energy of a standing wave on the rubber bungee is shown on Graph 2.
The energy of the standing wave on the rubber bungee at 20 cm amplitude is calculated to be ...
From Graph 2a, the energy loss per cycle at an amplitude of 20 cm is ...
The work done at the fixed end per cycle from the force-displacement graph at an amplitude of 20 cm is ...
Analysis/discussion
The measured work done at the closed end to maintain the standing wave is not the energy supplied per cycle to compensate for energy loss due to friction etc. The small driving force is masked by the much larger reactive forces. The reactive forces can be measured by clamping the force probe to the bench at the 'fixed' end, but because they are two or three orders of magnitude greater that the driving force subtracting the work done at the fixed end from the work done at the driving end gives a value of zero within errors.
Some way must be found to separate the reactive forces perpendicular to the wave motion and the driving force.
Suggestion for future work
If a rubber band were attached near the node at the fixed end and the force probe was used to apply a driving force with regular motion of ±10 cm perpendicular to the wave axis the motion of the cord will not then affect the measured work done by more than ±5%. A longer bungee would allow a standing wave of lower frequency to be established which would make accurate measurements easier to obtain.
Evaluation
It has been shown that the energy loss per second from a standing wave on a bungee made from rubber bands and on a steel spring are both nearly exponential. The energy loss per second on the rubber is greater due probably to hysteresis. The energy loss at an amplitude of ±20 cm on the rubber bungee has been estimated to within ±10%. It has not been possible to compare this value directly with the energy supplied to maintain the standing wave because the small amount of work done by the force probe at the driving end is masked by much larger reaction forces.
The present technique can be used to plot reaction forces at a fixed end as a function of standing wave amplitude but not, as had been hoped, the work done to maintain the standing wave at a particular amplitude. Following the suggestion for further work , may achieve the desired result if accurate data can be obtained . This would be more easily achieved with a longer bungee. If this can be done, it would then be possible to make meaningful comparisons between the energy supplied and the energy lost for both the bungee and the steel spring.