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Exploration of the Ball-Rod Physical System using a Michelson Interferometer
Hosni Kaissi and Long Zhao
University of Minnesota
Abstract
We use a Michelson interferometer with a 633 nm HeNe laser to investigate the motion of two steel rods of lengths 39.8 cm and 55.1 cm after being struck by a steel ball. The collision launches an elastic wave through the rod with observable displacements of the end mirror by interferometry. The speed of sound in thin, steel rods was found to be consistent with literature at 4830 ± 10 m/s for the 39.8 cm rod and 4870 ± 10 m/s for the 55.1 cm rod by observing rod lengths traveled within 43 round trip times of the induced compression wave. The evolution of the rod’s motion with time is also analyzed with position and velocity vs. time graphs for the two rods. The rods initially experience start-stop motion due to compression wave dominated movement with no gross rod motion between bursts and evolve to include gross motion as the compression wave transfers energy to the rod. Rod friction is neglected for short timescales less than 1 ms but play a significant factor in slowing the rod subsequently.
Introduction and Theory
Elastic collision theory is a widely studied subject in physics and other physical sciences. Colliding objects undergo deformations due to the compressibility of matter. If the forces from the collision are not relatively high for the material, only elastic deformations occur which are temporary changes in shape . When an object’s shape is elastically deformed by a collision, a compression wave is launched from the point of contact and then subsequently reflect within the object until the wave dies out and the original shape resumes. Post-collision parameters such as speed of sound in the material are found by studying the compression wave.
In this experiment, a steel rod is struck by a steel ball of constant initial energy by means of a pendulum. A compression wave is formed that travels through the rod reflecting at each end successively as the energy in the wave is transformed into energy in the gross motion of the rod consistent with momentum and energy conservation. Conservation of momentum and energy equations for this system yield a final velocity for the rod independent of the ball's ricochet velocity and is given by:
(1)
where m1 and m2 are the masses of the ball and rod, respectively, and u10 is the velocity of the ball at the moment of impact. This velocity will apply to the rod's average velocity since the instantaneous velocity is undefined for the initial wave motion. The average velocity must be computed within 1 ms of impact since frictional forces play a non-negligible factor in energy loss afterwards.
In a Michelson interferometer, the path difference between two beams corresponds to a phase difference that determines whether the waves constructively or destructively interfere. When the phase difference is an integral or half integral number of 2π, the waves interfere constructively or destructively, respectively. The total distance traveled by the rod’s back end may be found by counting the number of cycles that have elapsed on the interference graph within the compression wave's round-trip time. The displacement of the rod's end is given by the following equation:
(2)
where the wavelength of the laser is 633 nm and n is number of complete cycles of interference. This equation limits our resolution and we cannot discern any motion less than half the laser wavelength (316.5 nm).
Experimental Apparatus
The experimental setup is composed of a mechanical apparatus containing the ball-rod system and an optical apparatus containing the Michelson interferometer and measurement tools. The mechanical apparatus is displayed in the figure below.
A steel ball is held by a copper wire in bifilar suspension so that it moves along one plane. The wire is attached to two insulated supports and the triggering circuit keeps the ball at +3 V using a power supply. The ball is pulled back and released from a drop-height h which remains constant for the experiment at 0.6 cm. The trigger voltage, VT , connects to an oscilloscope and data collection begins at t=0 once the ball strikes the rod and shorts the trigger circuit. Laser from the interferometer reflects off a mirror attached to the rod. The interferometer and measurement tools are shown below.
Light from a 633 nm HeNe laser reflects off two mirrors that align the beam before reaching a beamsplitter (BS). The beamsplitter splits the laser into a reference beam and a signal beam reflecting and transmitting about 50% of the incoming light in perpendicular directions within each arm of the interferometer. The signal beam reflects off the mirror on the rod’s back end then recombines with the reference beam which reflected off the reference mirror. Initially, the length of the interferometer’s arms are set to be equal to within a few millimeters which is much smaller than the laser’s coherence length. The changing path difference between the signal and reference beams due to the rod’s motion corresponds to a changing phase difference upon recombination. This yields an interference pattern at the photodetector which produces voltage readings proportional to the detected intensity as a function of time. The photodetector voltage vs. time signals for each tested rod are exported to a USB memory stick and subsequently to a computer for analysis in MATLAB.
Results
The photodetector voltage vs. time signals consist of multiple bursts of interference produced by the rod's motion. The rod does not move between the segments of interference suggesting start-stop movement due to compression wave dominated motion. This is similar to the way a slinky moves after being struck. Below is an example of unfiltered photodetector voltage graph for a 1 ms time interval as well as a zoom-in. The red line is used as a crossings counter discussed further below.
Equation 2 is used to find the position vs. time graph of the rod’s mirror end within some time interval by counting the number of interference cycles within this interval. This is equivalent to half the number of times the voltage signal crosses an adjustable median line through the data for that interval. The original data is bandpass filtered to remove the low frequency background that the signal sits on and to remove the high frequency noise to avoid extra counts. A discreteness exists in the position graph because our resolution is half the laser’s wavelength (316.5 nm). The position graph must be smoothed and then the velocity vs. time graph for that time interval may be obtained by differentiating. Below are photodetector voltage, position, and velocity vs. time graphs for t=0 (left column) and t=15 ms (right column). All graphs are within 1 ms time intervals. The the first and second set of graphs correspond to the 39.8 cm rod and the 55.1 cm rod, respectively.
Before comparing the different length rods, note that each rod behaves in a start-stop motion for t=0 graphs (left column) evident from the position and velocity graphs. However, at t=15 ms (right column) there is interference between the bursts. The position graphs approach linearity and there exist smaller velocity peaks between the larger ones. This is because the compression wave has transferred some of its momentum to the rod and thus the rod has acquired a velocity in between the compression wave bursts. The 39.8 cm rod has fewer interference bursts within the 1 ms interval compared to the 55.1 cm rod. This makes sense since the compression wave's round-trip time is lower for a shorter rod. Also, note the higher average velocity of the shorter rod within the t=0, 1 ms time interval due to its lower mass. This average velocity is in agreement with equation 1, but breaks down for higher times due to energy loss from friction as well as energy loss from the reflections of the compression wave. The speed of sound in steel in thin rods was found for each rod by averaging over 43 cycles of bursts or round trip times. The compression wave travels 2 times the length of the rod within 1 round trip time. Dividing the total rod lengths traveled by the total time gives a speed of sound of 4830 ± 10 m/s for the 39.8 cm rod and 4870 ± 10 m/s for the 55.1 cm rod. These values are consistent for the expected velocity of sound in thin, alloy 316 stainless steel rods.
Conclusion
Our method of analysis is dependent on the assumption that the rod’s mirror end only moves forward. This breaks down for times above t=50 ms where a great number of reversals in the photodetector voltage graphs reveal possible backwards motion of the rod end. These reversals are sudden changes in the direction of the voltage data and suggest a switch in direction of the rod end. The rod comes to a stop before the compression wave dies out due to a high level of friction and the compression wave then vibrates the rod in place until it loses its energy. Observing larger time scales is possible by reducing friction significantly with a lubricant or a ball-bearing mechanism. Nevertheless, despite the limitations for larger timescales, the rod end's motion was observed to behave in start-stop movement and changed to include gross motion of the rod as the compression wave transferred its energy to the rod. By analyzing the resulting graphs, the speed of sound in thin, steel rods was found to be consistent with literature at 4830 ± 10 m/s for the 39.8 cm rod and 4870 ± 10 m/s for the 55.1 cm rod.
References
[1] Mile Young, “Elastic vs Plastic Deformation”, August 28, 2010. http://elitetrack.com/blogs-details-5337/
[2] A.A Freschi, R. Hessel, M. Yoshida, D.L. Chinaglia, “Compression waves and kinetic energy losses in collisions between balls and rods of different lengths”, Am. J. Phys. 82, 280 (2014).
[3] Churchill, Ruel V. and Brown, James W. Fourier Series and Boundary Value Problems. 8th Edition. New York: McGraw-Hill, 2012.
[4] “Velocity of Sound in Various Media”, http://www.rfcafe.com/references/general/velocity-sound-media.htm
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