S16_BallRodInterferometer

The Effects of Compression Waves on the Motion of a Steel Rod in a Collision

Andrew Ziegler and Nardin Vahidi-Azar

University of Minnesota MXP 2016

Abstract

The goals of our experiment were two-fold. The first was to obtain a measurement of the speed of sound in a steel rod by observing the period of a compression wave introduced into the rod by a collision with a steel ball. The second was to observe precisely how the compression wave motion introduced into the rod by the impulsive forces of the collision is consistent with constant velocity, rigid body motion predicted by the conservation of momentum for objects in a collision. Specifically, the question we seek to answer is: how does the rod's motion evolve over time from one mode to ?

Introduction and Theory

The physical system that we are considering is shown in Figure 1.

Figure 1. A representation of the collision between the ball and the rod showing the velocities before and after the collision. This representation treats the rod as a rigid body.

Solving for the velocity of the rod by calculating the momentum's before and after the collision and setting them equal we obtain an expression for the rod's velocity after the collision.

According to the conservation of momentum the rod should be moving with a constant speed; however, one assumption that we have made is that the rod is a perfectly rigid body. Of course this is not true so the motion of a real rod subject to a collision will be more complicated. Never-the-less the conservation of momentum is still correct in an average sense as momentum must always be conserved. Thus we know that whatever complicate motion that the rod takes on its average speed must be that predicted by the conservation of momentum. Because the motion of the compression wave in the rod are on the scale of microns (10^-6 m) the conservation of momentum is a prediction of the macroscopic behavior of the rod.

To understand the instantaneous motion of the rod as opposed to its average motion then one must consider the effects of the rod's compressible nature. The compression waves in a rod with a small diameter relative to its overall length are described by the one dimensional wave equation.

This equation is derived by considering an infinitesimal mass element of the rod and applying Hooke's Law for continuous bodies to that small mass element. The variable y represents small displacements from equilibrium of this mass element and the parameter c represents the speed of sound in the material. In this experiment we perform a measurement of c for our rod. If you know anything about solving differential equations then you will know that this equation is impossible to solve without the proper boundary conditions. The boundary conditions for our rod are shown in Figure 2.

Figure 2. The boundary conditions for the one dimensional wave equation that describe a stationary rod subject to a force on one end.

We take our rod to have both zero initial velocity and displacement. One end of the rod is free so that boundary condition is that the first spatial derivative is equal to zero. The other end is subject to an impulsive force that causes the rod to begin to vibrate with a compression wave. The forcing function we use to model the collision is a step function that is a high value for a short time and then zero for all of time after.

Comparing these two solutions for the rod's motion leads to some inconsistencies. The inviolable conservation of momentum predicts that all points on the rod will move with a constant velocity whereas the wave equation predicts that each point on the rod will oscillate sinusoidally while having an average velocity due to the collision. The effect of this is that the end of the rod will appear to start and stop as the compression wave moves up and down the rod. The important question is: how are these two modes of motion consistent with each other. The answer is that while over a short time scale the two predictions will only agree in an average sense, more precisely the average velocity of the rod is the same in models. However, over a longer timescale dissipation factors such as internal friction will cause the compression wave to die off eventually leading to a rod that is moving as a rigid body at a constant speed if external friction is negligible. The question that we seek to answer in this investigation is how the motion of the rod evolves from the compression wave motion to the constant velocity motion.

Experimental Setup

Our experimental setup consisted of two main parts the physical apparatus comprising the rod and pendulum and the optical apparatus that includes the interferometer and detector. A schematic for the physical apparatus shown in Figure 3 and the optical apparatus is displayed in Figure 4.

Figure 3. The physical apparatus used for the experiment. This figure is meant to represent an idealized setup as the actual setup used does not make a very good visual.

The physical part of the experiment consisted of a steel ball held in bifilar suspension by copper wires from a support that was insulated from both the optical table. The ball was held at +5 V by the triggering circuit shown in Figure 3 and when the ball completes the circuit by contacting the grounded rod data acquisition began. The rod used was made of steel and supported by ball bearing rollers to both align the rod with the interferometer and keep friction to a minimum. A mirror was also attached to the end of the rod to reflect the beam back into the interferometer. This setup creates the perfect conditions to study the motion of a rod subject to a collision because after the ball has collided with the rod the rod is then decoupled from all forces assuming friction is negligible.

Figure 4. The optical apparatus consisted of a Michelson interferometer with the rod serving as one of the arms. As the rod moves after being hit by the pendulum it changes the path length of the laser causing an interference pattern to appear at the detector.

A Michelson interferometer was used to monitor the position of the rod's end after the collision to a high degree of accuracy. The laser used allowed us to distinguish distances as small as 316.5 nm i.e. half the wavelength of the laser used. The detector consisted of a silicon photodiode with a built-in amplifier. We added a low pass filter between the detector and the oscilloscope to remove some of the high frequency noise that was introduced to the signal from the amplification. An oscilloscope was used to record the signal because the high frequency of fringes necessitates a very high sampling rate, our oscilloscope was collecting data at 25 MS/s. The signals were then exported to a USB drive which we could then transfer to a computer for further analysis.

An image of our setup as it looked when we had finished building it is shown in Figure 5. This gives a better idea of how the setup looked when we performed our experiment. At the bottom of the image you can see the laser used for the experiment, a 17 mW HeNe laser. The laser was directed by two mirrors into the non-polarizing beam splitter that sent one beam towards our rod and another towards our stationary reference mirror. Clearly is the rod and ball-bearing supports in the center of the image and to the left of this you can see our pendulum. The rod was about 40 cm and length with a diameter of about 1cm. The mirror was attached with epoxy glue. We built our pendulum by threading copper wire through a steel ball in wrapping the wire around nylon screws inserted into 8" optical posts. This gave us an insulated support to which we could connect our triggering circuit. Not shown in the image are the triggering circuit, photo detector and amplifier, low-pass filter, and oscilloscope.

Figure 5. A picture of our setup as it looked when finished. The path of the laser is shown in red. By raising the ball highlighted in green and letting it hit the rod, a compression wave was introduced into the rod that causes it to move. By moving it changes the path length of the laser which we can see as alternating bright and dark fringes at the detector.

Data Analysis Scheme

The data collected consisted of voltage vs time signals that indicate the changing intensity of the laser on the photo-detector as bright and dark fringes pass across it. A typical signal is shown in Figure 6 where the fringes are grouped into distinct packages indicating the start and stop motion spoken about in the introduction.

Figure 6. A typical signal collected by our detector. The fringes are the areas of higher amplitude oscillation and decrease in amplitude as frequency increases due to our low-pass filter.

By counting fringes we can recreate the position of the rod as a function of time because we know that each fringe indicates a displacement of 316.5 nm for the rod's end. However, it is easier to implement an indirect method of fringe counting that measures the frequency of the fringes. Because the frequency of fringes corresponds to the velocity it is a simple matter to then get the position of the rod's end by numerically integrating the velocity. The tool we have from getting from the time domain to the frequency domain is the Fourier Transform which we use to perform a tonal analysis of part of the signal. The tonal analysis extracts the frequency with the highest amplitude from the power spectrum and then we multiply this frequency by 316.5 nm to get the velocity of the rod's end at this point. This is performed point by point for the entire signal. Figure 7 provides a visual of this analysis method.

Figure 7. A representation of the data analysis algorithm. The signal segment currently being analyzed is highlighted in red. This segment is Fourier transformed and the frequency with the highest amplitude is extracted and after being scaled by the half-wavelength scaling factor plotted as the velocity of the rod's end. The program then increment the signal segment forward one data point and repeats.

Results

The figure below shows the velocity of the end of the rod as a function of time, which moves the start-and-stop fashion described in the introduction. This is apparent due to the narrow velocity peaks followed by areas of almost zero velocity. We can calculate the velocity of this wave with taking the average value of period over many velocity peaks, and dividing the total distance traveled by the compression wave (twice the length of the rod) by this average period.

Figure 8. A plot of the velocity of the rod's end vs time. This plot was obtained from the analysis scheme described above and represents the velocity of the rod's end just after the collision.

Averaging over 285 periods we calculated the speed of sound in our steel rod to be 4923 +/- 1 ms. To calculate the displacement of the rod we numerically integrate the velocity graph using the trapezoid method. We can see in the graph below that the displacement of the rod looks like a staircase, the flat areas representing the end of the rod sitting stationary and the regions with higher slopes indicating a velocity peak. Notice that the average speed of the rod's end is constant value which is consistent with the conservation of momentum for our rod.

Figure 9. A plot of the rod's displacement vs time. This plot is obtained by numerically integrating the plot shown in Figure 8. The stop and start motion of the rod is indicated by the staircase shape of the plot.

The motion of the rod changes due to the internal friction of the rod and other dissipative factors. The longitudinal wave dies out after a while and the rod starts acting like a rigid bod, moving with a constant velocity. Figures below show the evolution of the velocity of the rod versus time.

Figure 10. This series of plots is meant to show the evolution of the rod's end velocity over time. We see that the motion near the beginning is highly periodic; however, dissipative factors such as friction and acoustical losses cause the compression wave to die off and make the velocity more uniform.

The series of plots above display the evolution of the velocity of the rod's end over time. While the motion just after the collision is dominated by the compression wave, characterized by the highly periodic stop and start motion, we see that over time some dissipative factors kill off the compression wave and the velocity becomes that of a rigid body. We can characterize the evolution of the velocity by noticing that two main factors seem to be dominant: widening of the velocity peaks with decreasing amplitude and the appearance of a secondary velocity peak in between the main velocity peaks that appeared at the start of the motion. The broadening of the velocity peaks can be explained as dissipation of the compression wave due to internal friction within the rod. The secondary velocity peaks are more difficult to explain due to the ambiguities inherent to the experiment, namely our inability to distinguish between forward and backward fringes; however, numerical simulations provide a tentative explanation for this behavior. We can model the rod as a weighted spring with many mass elements connected by identical springs. However, because the rod is moving in a medium, some of the energy of the compression wave is transferred into the air as a sound wave. Thus the springs at the end of weighted spring are less efficient than the springs in the interior which implies they have a smaller spring constant. Modeling the spring in this manner we see that the motion of the end of the spring evolves in much the same way as described above. The motion starts out periodic due to the compression wave but eventual interference between the two modes of oscillation cause the end of the rod to have multiple velocity peaks and evolve towards a more constant state.

Conclusions

In the experiment performed here we have performed a measurement of the speed of sound in a steel rod. The velocity measured was 4923 m/s which is consistent with the values for the velocity of steel as reported in the literature. More interestingly we have also observed the evolution of the the velocity of a rod from a compression wave dominated to a constant velocity. While the investigation that we performed was not able to provide a precise explanation for this behavior we have succeeded in describing it phenomenologically hopefully laying the foundations for future investigation of this surprisingly interesting physical system.

References

Some useful resources for this experiment

1.Love, A. E. H. A Treatise on the Mathematical Theory of Elasticity. New York: Dover Publications, 1944.

2.Taylor, John R. Classical Mechanics. Sausalito, CA: U Science, 2005.

3.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)

4.Churchill, Ruel V. and Brown, James W. Fourier Series and Boundary Value Problems. 8^th Edition. New York: McGraw-Hill, 2012.

5.Shackelford, James F. CRC Materials Science and Engineering Handbook. Boca Raton: CRC, Taylor & Francis Group, 2016