LAB #8: COUPLED PENDULUMS AND NORMAL MODESB. The Apparatus and the MeasurementsC. Computer Modeling of Your ResultsD. Further Computer ModelingWhile you have the Excel program running, use it to look at other interesting phenomena, involved in such things as musical instruments and their “tone.” In each case, print out a copy of the computer's result and write down in your notebook a summary of what you find notable in the waveforms. Note that, up to now, you have been looking at two frequencies which are close to each other. (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 I: 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 Taylor’s Section 3.2, on “The Square Root Rule for a CountingExperiment.” While not specifically relevant to this week’s experiment,the argument that counts of things (such as nuclear decays, occurrencesof rare diseases in human populations, etc.) have uncertainties of theorder of the square root of the counts is of crucial importance inmaking both scientific and political decisions.3. 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.4. Consider the optional PreLab problem set attached.Princeton University Physics 103/105 LabPhysics DepartmentLAB #8: COUPLED PENDULUMS ANDNORMAL MODESA. IntroductionIn this week's lecture and lab, you will be studying the motions of simple harmonicoscillators. The set of concepts involved in Simple Harmonic Motion (SHM) is one of ahandful of extremely useful building blocks for many advanced areas in physics.Electrical, acoustical, and optical systems oscillate with SHM, completely analogous tothe mass-plus-spring and pendulum systems you’ll be looking at during these two weeks.Your radio selects the station you want by using electrical resonance tuned to the desiredfrequency. Some theoretical physicists in search of Grand Unified Theories (GUTs) haveproposed describing the smallest fundamental particles as comprised of oscillating“strings.” 1 Many physicists use their physical intuition about how systems with springsand masses are likely to behave in order to predict how many different systems will act.Contrary to its name, “physical intuition” isn’t always an innate characteristic; a persondevelops physical intuition with experience. This lab gives you the chance to developsuch intuition about SHM in simple mechanical systems. 1 In this case, the word “string” is used evocatively rather than literally.65This week's lab is a study in describing seemingly-complex motion with simpleequations. The techniques underlying the work will recur throughout your scientific andtechnical futures. Our focus is first on the equations describing Simple Harmonic Motion(SHM). These follow from any situation in which there is(a) an equilibrium situation (x = 0, say) generated by a restoring term which islinear in the variable:F = - k x , for example, for a spring.(b) an “inertial term” (the mass, say) which prevents the restoring term fromeasily and immediately pushing the system back to its equilibriumsituation. Newton tells us thatF = m a = m d2x/dt2 .Combining the above terms gives us the basic equation of SHM:m d2x/dt2 = - k x .Any system described by an equation of this form will exhibit SHM behavior, includinghaving simple sinusoidal solutions, a characteristic frequency leading to resonancebehavior, etc. To a mathematician, this is an example of a linear differential equation, inwhich the unknown function and its derivatives appear only in the first degree. (Thereare no squares, cubes, square roots, or strange functions of x involved.)Linear differential equations of the above form, which pervade science and engineering,have one wonderful characteristic – their solutions may be superposed. That is, if thereare two functions u(t) and v(t), each of which satisfy the equation, then any linear sum ofthe two functions is also a solution. As you may know, Fourier Analysis shows that complex functions can be described assums of sine and cosine functions. Electrical engineers build their careers on this simplefact. The response of a system described by the SHM model to any (additional) drivingforce f(t) can be described by investigating its response to a family of sinusoidal drivingfunctions which, when summed, equate to the complicated function f(t). As you wouldexpect, a sine function tuned to the system's resonant frequency will produce a hugeresponse, and other frequencies will have less effect. (This is how you select the radiostation you want, starting from the fact that a whole host of stations are broadcastingsimultaneously, with their outputs superposed linearly in the air.)More generally, it is common in physics to find that the equations describing a complexsystem allow not one, but a whole host of resonant frequencies. The overtones producedby a musical instrument are such frequencies, and their relative frequencies andamplitudes are what makes one instrument have a characteristic “sound.”In this week's lab, we will study a fairly simple system – two identical pendulums,connected by a weak spring. In the absence of the spring, the two would swing at thesame sinusoidal frequency, independent of each other. Adding the spring produces twocharacteristic frequencies, which in turn lead to a complex motion, which is not a simple66sinusoid. But the motion can be analyzed as a sum of two sinusoidal motions, each ofwhich obeys the simple equations of SHM, and oscillates at its characteristic frequency.As you pursue your interest in science and engineering, you will find that the approach toanalyzing a complicated system often hinges on an ability to look for