Dr. W. Pezzaglia Physics 8C Lab, Spring 04 Page Las Positas College Lab # Harmonic Motion 04Jan3 Lab 0: Harmonic Motion I. Theory: Three experiments. The first we measure the oscillation of a spring, the second of a rubber band (non-linear). A. Springs: The period of oscillation of a spring is given by: M P () k where M is the mass hung on the spring, and k is the spring constant of Hooke s law: F kx () Note that the oscillation should be independent of the amplitude (i.e. its Isochronous ). The above assumes a massless spring. If instead the spring has mass m then there is a correction to the formula, M am P (3) k Hence, as long as M>>m one can use equation (). However, for our experiment, this will not always be the case. Hence a plot of mass M vs squared period should follow the equation: k M P am (4) 4 For an ideal spring, it can be shown that a=/3. According the an article in the Physics Teacher (April 000) by Nathaniel R. Greene, and Ryan J. Dunn of Bloomsberg University, Pa, a conical spring hung "wide end up" has a theoretical value for a=0.40 vs. a theoretical value of a=0.76 for the "thin end up." B. Rubber Band An approximate non-linear equation for rubber band is: F KT x x (5) L x L0 where L 0 is the relaxed length, L is the stretched length, and x is called the extension ratio. The temperature T is in Kelvin, and the constant K is essentially proportional to the density of elastic fibers in the material times the gas constant. Because of this non-linear behavior, we expect the period to depend upon the amplitude ( Hysteresis is the property where the state of a system depends upon the history. For a rubber band, to reach a certain amount of stretch (deformation) it takes more force if you are loading it as opposed to unloading. The effect is more pronounced if you do the experiment quickly. If you wait too long, the effect will become quite small. In other words, it takes more energy to stretch the band than you get back when it is unscratched. Hence energy was lost. Thus we expect a rubber band to exhibit damping.
Dr. W. Pezzaglia Physics 8C Lab, Spring 04 Page Las Positas College Lab # Harmonic Motion 04Jan3 C. Damped Oscillations: All real systems will not oscillate forever as energy will be dissipated due to friction. Oscillations tend to die out over time exponentially. At the right, the oscillation period is second, with a time constant of =.5 seconds for the decay. x( t) Ae 0 t cos t (6) The constant A is the initial amplitude of the oscillation. The constant is called the decay constant or damping constant (the inverse of the time constant ) with units of inverse time. Note that the presence of damping makes the oscillating frequency to be less than the resonant frequency 0. If the friction in the system is higher increases (system decays faster). If 0 then oscillations will not occur. This might be what you want; for example, if you drive your car fast over a speed bump, you don t want the car to bouncy-bounce. The shocks in your car have springs in them, so to prevent oscillation they also have a BIG mechanical resistance. To determine the decay constant, the peaks are measured and plotted vs time. The data is fit with an exponential showing a decay constant of =0.3934 s-, and hence time constant of =/=.54 sec. The R-squared value of 0.9987 tells us that we have a very good statistical fit. Always include it on your graphs! Exponential Envelope Maximum Displacement. 0.8 0.6 0.4 0. y = 0.9873e -0.3934x R = 0.9987 0 0 4 6 8 0 Time ==================================================================
Dr. W. Pezzaglia Physics 8C Lab, Spring 04 Page 3 Las Positas College Lab # Harmonic Motion 04Jan3 II. Experiments: We shall do an experiment with spring, and then one with rubber band A. Spring Constant Be sure to include in your report whether you had the fat end on top or on bottom. According to our references, the correct way is to have the tapered (small) end up. Measure the mass of the spring m Measure the displacement x as a function of hung weight M (5 different values) Plot Force (Mg) vs displacement Fit with best line. Determine slope. This is your spring constant k. Question : Setup: What is the mass of the spring? Which orientation are you using, fat end up or down? Question : Does your plot verify Hooke s law (i.e. is it a line)? What is the value of your spring constant? (be sure to have correct units!). ================================================================== B. Oscillations of Spring You will measure the period of oscillation for at least 5 different mass values M.. Setup Instructions (a) Connect the AC adapter to the LabPro by inserting the round plug on the 6-volt power supply into the side of the interface. Shortly after plugging the power supply into the outlet, the interface will run through a self-test. You will hear a series of beeps and blinking lights (red, yellow, then green) indicating a successful startup. (b) Attach the LabPro to the computer using the USB cable that is Velcro-ed to the side of the computer box (do not unplug the USB cable from the computer!). The LabPro computer connection is located on the right side of the interface. Slide the door on the computer connection to the right and plug the square end of the USB cable into the LabPro USB connection. (c) Connect a motion detector to a digital port (DIG/SONIC) on the LabPro. The digital ports, which accept British Telecom-style plugs with a left-hand connector, are located on the same side as the computer connections. If you are using an older motion detector, you may need to remove the flat gray cable from the sensor, and replace it with the round black cable labeled Motion Detector Cable, with British Telecom plugs on both ends. Place the motion detector on the floor below the hanging mass, keeping in mind that the detector will not register any motion that occurs closer than 0.4 m from its sensor. (d) Open logger pro on the computer once your motion detector is plugged in. You should see a screen that displays graphs for distance vs. time, velocity vs. time and acceleration vs. time. You can ignore the acceleration vs. time graph for the first part of this experiment. (If you don t see the appropriate graphs, look for the logger Pro Experiments folder. Choose the file hookes_law_ This will start the Logger Pro program, and bring up the appropriate experiment file). 3
Dr. W. Pezzaglia Physics 8C Lab, Spring 04 Page 4 Las Positas College Lab # Harmonic Motion 04Jan3. Taking Data Figure Basic Experimental Set-Up with Motion Detector \ Place a 50 to 00 gram mass on the spring and set it oscillating. Verify that the motion detector is working by hitting Collect. Experiment with the setup until you see a sine curve on the display. Try adjusting the Experiment > Data Collection setting to produce the smoothest curve. Take data for each of the masses you used in Part A. Determine the period of oscillation for each mass by highlighting a region of the graph from one oscillation peak to another oscillation peak. The time interval will appear in the bottom left-hand corner of the graph window as dx:. For best accuracy, determine the time for as many complete oscillations as possible, and divide by the number of oscillations to determine the period. Record your data in tabular format. 3. Analysis Plot mass M vs squared period P. Fit it with best line. [Or, you could plot P vs M and do a power fit, but this would not let you determine a very easily] From equation (4), the slope of the line will be related to the spring constant, and the intercept will be related to the spring s mass (i.e. you can determine a in eqn 3 & 4). Question 3: Summarize Results What was the amplitude used? Does your plot show a line as expected? (What is value of R squared?) Question 4: Spring Constant What is the slope of the graph? (be sure to get correct units). From slope, extract the spring constant. Compare to the spring constant that you measured using Hooke s law. Are they the same? Question 5: Spring Mass What is the intercept of your graph? (be sure to get correct units). From intercept, extract the fudge factor a of equation (3). How does your value of a compare to expected value? 4
Dr. W. Pezzaglia Physics 8C Lab, Spring 04 Page 5 Las Positas College Lab # Harmonic Motion 04Jan3 C. Oscillations of Rubber Band Here you should take more initiative to design the experiment. Below are suggestions.. Setup and Calibration Measure relaxed length L 0 of rubber band. Measure its width Mount rubber band next to meter stick for measurements. The breaking weight of our rubber bands is around 5 kg (?). Try to stay below this!. Elastic Measurement Measure the length L of band as a function of increased applied weight. Go up to about 4.5 kilograms. This is the loading curve. Hysteresis: You could perhaps show Hysteresis by now going backwards, reduce the weight and measure the L. This would be the unloading curve. Raw Data: Plot Force vs. L for loading and unloading cases Question 6: Summarize your graph. Does it show a linear behavior, or non-linear? Do you see Hysteresis? What is the approximate spring constant for the rubber band? 3. Oscillations: As with the spring, measure the oscillation of the rubber band for a given weight. If you let it run long enough the amplitude should diminish faster than for the spring as there will be more damping. Measuring Damping: Experiment with the system until a smooth sine curve for displacement vs. time is visible on the display. In this case, you will need to consider the trade-off between data rate and number of cycles. The higher the data rate, the more accurate your measurements, and smaller number of cycles that can be measured. For this experiment a larger number of cycles is desired. Another recommended approach is to use a stopwatch in conjunction with the computer, i.e. start the program and the stopwatch simultaneously, when the program stops, save the graph and then start the program again, making sure to take down the reading from the stopwatch when you start the program. Do two trials using the same mass, waiting 30 seconds to a minute, and starting the program again, making sure to keep a "running" track of time. Do this over several minutes and you will get several curves that better show the damping effects (i.e. values of x and x and corresponding values of t and t that are separated by a sufficiently long time interval to reveal the effects of damping.) Repeat experiment for different masses (take care not to break it!). Suggest you make some graphs to illustrate the behavior. Question 7: Summarize Results (damped oscillator) Can you show that the rubber band approximately behaves like a (damped) spring? What is the time constant? Is eqn () approximately true? Question 8: Non-linear behavior Do you see evidence that the period changes with amplitude? Can you determine a relationship? 5