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Compression waves in collisions between balls and rods
Jacob Tambornino & Constant Chin
University of Minnesota MXP 2018
Introduction
In a classical model of a elastic collision between two objects the assumption made is that everything is perfectly rigid (Figure 1). However that is not the case. Small deformation occurs between the two objects and the aim of this project is to show that through the use of a steel rod and a steel ball bearing. In our case what happens is that the ball bearing collides with the rod and sends a small compression wave up and down the rod. This causes an continuous step motion to the end of the rod opposite the collision while the rod as a whole moves in a linear fashion. The way this will be measured is through the use of a Quadrature Michelson Interferometer which allows for a precise measurement the movement of the end of rod.
Figure 1. Motions before and after collision of ball Mball with initial velocity vi and stationary rod Mrod.
Setup
The set up for our system is fairly simple. We have a ball bearing attached to a pendulum while simultaneously having a rod in suspension as well (Figure 2). This allows for us to have a low amount of friction while also allowing the rod to swing as freely as possible. This is important as it allows for the compression wave to be unhindered as it moves up and down the length of the rod after the collision. In addition a small mirror is epoxied to the end of the rod which will be the measurement arm mirror of interferometer.
Figure 2. Experiment setup including ball, rod, trigger signal circuit and interferometer.
Figure 3 below shows a design of Quadrature Michelson Interferometer using 633nm He-Ne laser. Comparing to a normal Michelson Interferometer a half wave plate is placed before the beam is split by a non-polarized beam splitter and two quarter wave plates are placed on the reference arm and output beam (Ellsworth & James, 1973).
Figure 4. The polarization of laser beam inside the Quadrature Michelson Interferometer.
Then the signal of two photon detectors was connected to two channels of oscilloscope. When put oscilloscope into x-y mode the changing intensities trace out a circle (Figure 5). We took average of all X and Y voltages and find the coordinates of center of the circle in x-y plot. Then we used arc-tangent convert the coordinates of x and y of each data points relative to the center of circle into phase angles 𝜑. The displacement is given by d=𝜑𝜆/4𝜋.
Figure 3. Diagram showing the integral part of the Quadrature Michelson Interferometer. Half wave plate (HWP). Quarter wave plate (QWP). Non-polarized beam splitter (NBS). Polarized beam splitter (PBS).
The input beam first passes through the half wave plate which allows us to select the direction of polarization. The quarter wave plate on reference arm is set 22.5° from the vertical. Because the beam passes twice through this plate the new linear beam will have a polarization 45° from the vertical. The measuring arm made to the polarization of the beam. The new recombined beam then passes through the second wave plate with the fast axis angled 45° different from the polarization angle of input beam. This does two things, the first being that the reference beam will pass through unaffected since it is parallel to the fast axis and the second being that it will make the measurement beam become circularly polarized. The x and y component of the output beam will be split by a polarized beam splitter and detected by two photon detectors. Those whole process was shown in Figure 4.
Figure 5. Graph of the average intensity of the X and Y components detected by the two photon detectors.
Figure 6. Model of non-rigid rod.
After deriving the equations of motion of all the elements written by Newton Second Law, Hooke’s law, and classic Hertz equation (equation 1), which can describe the force between the ball and rod (Graff, 1991). We got the theoretical results of the motion of the end of the rod (Figure 7).
Theory
We came up with a model of the ball and rod system to estimate how the rod reacts after the collision. The model assumes that the rod is made of N identical ‘dots’ with mass m and N-1 identical springs, each with a known spring constant k and initial length ∆l (Figure 6). The length of rod is L=(N-1)∆l, and mass of rod is Mrod=Nm. The value of spring constant will depend on the cross-section area of rod A and Young's modulus Y of the material that the rod made of.
Figure 7. Plot of displacement (a) and velocity (b) of the end of rod versus time.
Results
The over all movement of the end of the rod over one second after the collision was plotted in figure 8. Since the rod was hanged like a pendulum, the overall movement looks like a pendulum.
Figure 8. Displacement (a) and velocity (b) of the end of rod as function of time.
The movement caused by compression wave was much smaller than the overall movement and the velocity plot overlapping together due to the high frequency of compression wave. To study the compression wave, we zoomed in the data in 4 different time stages, which is shown from Figure 9 to 12. Figure 9 shows the situation at 4.8 ms. Displacement looks like the ‘steps’ in the modeling results. In the plot of velocity, between each peak, there are ‘flat bottom’. Figure 12 shows the situation at 900 ms, which is long enough for compression wave to become ‘steady’ stage. Distance versus time no longer like ‘steps’, and the velocity are almost a sinusoidal function of time.
Figure 9. Displacement (a) and velocity (b) as function of time at 4.8 ms.
Figure 10. Displacement (a) and velocity (b) as function of time at 50 ms.
Figure 11. Displacement (a) and velocity (b) as function of time at 150 ms.
Figure 12. Displacement (a) and velocity (b) as function of time at 900 ms.
In the Fourier transform of the velocity (Figure 13), the fundamental frequency is 4.5kHz and there are other harmonic frequencies at about 9kHz and 13kHz etc. From the four Fourier transform figures at different time, the trend can be observed. Amplitudes of all frequencies are decreasing through the time. The fundamental frequency decreases slowest and harmonic frequencies decreases much faster than fundamental frequency. At 900 ms, harmonic frequencies almost vanish and only the fundamental frequency exists.
Figure 13. Fourier transform of velocity at four different time. 4.8 ms(a) 50 ms(b) 150 ms(c) 900 ms(d).
Conclusion
The measured features match well with our theoretical model. It indicates that our ‘dots’ and springs model is a successful model to estimate the movement caused by compression waves at the beginning few milliseconds after collision. From the frequency domain of the compression wave, we observed that different frequencies components of compression wave have different attenuation rates. The fundamental frequency has the lowest attenuation rate and finally, only the fundamental frequency can survive. The attenuation of compression wave should depend on other mechanics characteristics of the material and the environment, for example air resistance. This feature needs further study and will help us design and improve a feasible model based on current one that can also estimate the time evolution of the compression wave.
Acknowledgements
We are very thankful that Kurt Wick, as our project adviser, answered all our questions and helped us solve problem throughout the whole project. We also thank professor Kevin Booth and professor Dan Dahlberg for their assistance and advise during our project.
References
1. Elsworth, Y, and J F James. “An Optical Screw with a Pitch of One Wavelength.” Journal of Physics E: Scientific Instruments, vol. 6, no. 11, 1973, pp. 1134–1136., doi:10.1088/0022-3735/6/11/027.
2. Freschi, A. A., et al. “Compression Waves and Kinetic Energy Losses in Collisions between Balls and Rods of Different Lengths.” American Journal of Physics, vol. 82, no. 4, 2014, pp. 280–286., doi:10.1119/1.4864273.
3. K. F. Graff, Wave Motion in Elastic Solids (Dover Publications, Inc., NY,1991), Chap. 2. Longitudinal waves in thin rods.
4. T. John R, Classical Mechanics (2005), Chap. 1. Newton’s Laws of motion. & Chap. 5. Oscillations
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