LAB #7: COUPLED PENDULA AND NORMAL MODESB. The Apparatus and the MeasurementsD. Optional Computer ModelingYou can download the Excel program from the Blackboard and use it to model other systems that involve the sum (or difference) between two simple harmonic oscillations.(a) Look at two frequencies which are exactly a factor of two apart. These are most relevant to such instruments as the flute and the organ, in which the overtones tend to be almost exact multiples of the fundamental note being played.Appendix: Coupled OscillationsBEFORE YOU COME TO LAB:1. Read Knight's Chapter 14 (Oscillations), as assigned for lecture. Notethat the equations describing Simple Harmonic Motion haveapplications in physics and engineering extending far beyond thesimple systems described.2. Read the lab writeup. Focus on the relationship between the physicalphenomena and the equations as you carefully go through theIntroduction and then through Appendix I of the writeup.3. Consider the optional PreLab problem set attached.Princeton University Physics 103/105 LabPhysics DepartmentLAB #7: COUPLED PENDULA ANDNORMAL MODESA. IntroductionIn this week's Lecture and Lab, you will be studying the motion of simple harmonic oscillators.Small departures from equilibrium in almost any system result in a restoring force proportionalto the departure, and consequently the motion is Simple Harmonic Motion (SHM). Electrical,acoustical, and optical systems oscillate with SHM, completely analogous to the mass-plus-spring and pendulum systems you’ll be looking at during this Lab. Many physicists use their“physical intuition” about how systems with springs and masses behave to predict how manydifferent systems will act. “Physical intuition” is not an innate human characteristic; a persondevelops physical intuition with experience. This Lab gives you the opportunity to develop suchintuition about SHM in a “simple” mechanical system that exhibits surprising complex behavior.The main apparatus in this Lab is two “identical” physical pendula (wooden 2�4s withan axle atone end), connected by a weak spring. In the absence of the spring, the two pendula wouldoscillate at (nearly) the same angular frequency 2 fw p=, with angle qto the vertical obeying0cos( ),i i itq q w f= +with independent amplitudes 0iqand phases .ifCoupling the two pendula by the spring produces two characteristic frequencies, which in turnlead to a complex motion, which is not a simple sinusoid. But the motion can be analyzed as asum of two sinusoidal motions, each of which obeys the simple equations of SHM, and oscillatesat its characteristic frequency. (See Appendix A.)47As you pursue your interest in science and engineering, you will that a powerful approach toanalyzing the time dependence of a complicated system is to look for special “characteristic”frequencies, and to determine the patterns of motion associated with simple sinusoidaloscillations at those frequencies. With these solutions in hand, then any general motion of a“linear” system can be described as a sum of these simple motions. This is the concept ofnormal modes, which refers to the set of patterns which each leads to simple sinusoidalvariations in time.In this Lab, you should observe various possible patterns of motion of the two-pendulum system,and record carefully descriptions of the patterns of motion that you see. Using a stopwatch, youcan determine the frequencies associated with the various patterns which occur. Then, using acomputer model which sums and graphs the combined effects of two sinusoids, you can simulatethe patterns that you observe, and other patterns as well (for example, the first overtone in amusical instrument).B. The Apparatus and the MeasurementsThis apparatus consists of two pendula connected at their centers of mass with a spring, assketched in Figure 1. Slide the two pendula (2 4 's�) far enough apart that the spring isstretched when at rest. When you swing the pendula, the spring should never collapsecompletely. If it does, restart the pendula with smaller amplitudes, and/or slide the pendulafarther apart. pivotCML2k1Figure 1: Coupled pendulums; each pendulm is made from a length of 2x4 A spring connects the two pendulums at their centers of mass (CMs).Displace one pendulum while holding the other fixed, and then let both go free at the same time.The motion is complicated! At first one pendulum oscillates, but after a while its oscillationsbecome small and the other pendulum oscillates. Later, the first pendulum oscillates again andthe second does not. And then, this cycle of “transference of oscillation” repeats over and over(until friction damps all oscillations to zero).Next, displace both pendula and let them go. You should find special combinations of the initialdisplacements such that there is no “transference of oscillation”, and both pendula oscillate with48the same frequency. How many different ways can you generate these “normal modes” ofoscillation?Using a stopwatch, measure the frequencies of the pendula without the spring, and then thefrequencies of the normal modes that you have discovered when the spring is attached. After you have analyzed the simple patterns of motion, again displace one pendulum whileholding the other one fixed. Then release them simultaneously. Describe the resulting motion.Using the stopwatch, measure any relevant frequencies, and relate these to the frequencies of thenormal modes they you previously observed. Make a VideoPoint movie of the complex motion, using a capture rate of 5 frames per second.(Since you will be interested mainly in frequencies, there is no need to include a meter stick inthe field of view.) Make sure that the video covers at least two (2) complete cycles of the“transference of oscillation” (and so includes 5 times at which the left pendulum has minimalamplitude of oscillation). Digitize the positions of some identifiable point near the bottom ofeach of the pendula. Make plots of horizontal position of these points versus time. The challenge now is to extract the frequencies and amplitudes of the normal modes from yourplots. You will do this by varying the parameters in a computer model of your experiment untilyou obtain a good match between your