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The Coupled Pendulum - Lab Instruction

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					                        The Coupled Pendulum Experiment
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   In this lab you will briefly study the motion of a simple pendulum, after which you will
couple two pendulums and study the properties of this system.
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   1. Introduction to the Software

           The experimental apparatus consists of two, nearly identical, simple pendulums, as
       well as springs that can be used to provide a coupling force between the two. Each of the
       pendulums has a potentiometer, which is used as a position transducer, at their fulcrum
       and both of them are connected to a PC which is equipped to measure their position as a
       function of time. The software used to perform this data acquisition is located on the
       desktop of the lab PC titled “The Coupled Pendulum Experiment”. Double-click this
       icon, and allow time for the software to initialize. Once the software has opened, you’ll
       see two tabs located at the top of the front panel; “Introduction” and “Data Acquisition”.
       The introduction tab holds information regarding the process of data acquisition used for
       this experiment, as well as “Calibrate Data Acquisition” button; it is important that you
       press this button with the pendulums at rest, at the bottom of their swing, before
       continuing on to the next tab. After having finished with the introduction tab, one will
       find five main controls on the “Data Acquisition” tab. The first of which is a numeric
       field labelled “Time to Collect (s)” this is where you’ll entered the amount of time you
       would like data to be acquired for when the “Start Data Collection” button is pressed. If
       for any reason would like to stop the data collection during a run, you may press the
       “Stop Data Collection” button; this will allow the data that has already been recorded to
       be saved. Once data has finished being acquired you may press the “Save Pendulum
       Position Data” button or the “Save Power Spectrum FFT Data” button to save the
       required data into a Microsoft Excel spreadsheet.

   2. The Power Spectrum Fast Fourier Transform (FFT)

         As data is acquired in the lab, the software used in the lab continuously performs a
      Power Spectrum FFT on the data that has already been acquired. This “transforms” the
      wave form acquired by the data acquisition card into the frequency domain. Peaks on the
      plot of this transformed function correspond to the oscillatory modes of the pendulum’s
      motion. This operation is only performed on the data from the left pendulum, and is most
      important when the pendulums are coupled together; when the left pendulum swings free
      the FFT will provide one peak, which corresponds to the frequency at which it is
      swinging. The outcome of this operation when the pendulums coupled is explained
      further on in this instruction.

   3. The Simple Pendulum

          During this part of the experiment you will determine the damping constant associated
       with each pendulum’s oscillatory motion, and the distance between their centers of
       gravity and their fulcrums; the pendulum’s effective length. One may write the following
       equation of motion for a simple pendulum in the absence of friction:

                                               1
  Where θ0 is the angle from which the pendulum is released, g is the acceleration due to gravity, l is the
                             effective length of the pendulum, and t is time.
    This equation neglects any sort of damping, but in the real world we know that there
is air resistance, as well as friction in the bearings located at the top of each pendulum.
Please note that air resistance and bearing friction cause very different forms of damping.
Air resistance is a form of “viscous” damping, which means that the damping force that
acts on the pendulum is proportional to its velocity; this is the cause most often discussed
in physics texts on “damped harmonic oscillators”. The damping caused by the bearings
is known as “dry friction”; the friction in the bearings is constant, much like that
observed on a surface with a normal force and a coefficient of kinetic friction. For further
reference, please see “Viscous-vs-Dry.pdf” located on the lab PC’s desktop.
    It is important to note that although each of these two sources of damping cause the
pendulums oscillatory amplitude to decrease over time, they do so in different ways;
viscous damping causes exponential decay, while dry friction causes amplitude to decay
linearly. For example, one may write the following equation of motion for the case of
viscous damping:


                               Where B is the coefficient of viscous friction
   In the case of viscous friction, one may make observe the following plot:




                       Figure 1.0 – Plot of exponential decay due to viscous friction

   In the case of dry friction, one may write the following equation of motion, and plot as
shown below:




                        Where a is the constant coefficient of linear decay



                                                2
                           Figure 2.0 – Plot of linear decay due to dry friction
   Procedure:
   1. Perform trial runs to estimate how long it takes for the pendulum to decay from a
      certain initial position θ0. Record this time in your lab notebook.
   2. Start with θ0 from step one, and record data for as long as necessary; be sure to save
      this data to an Excel spreadsheet.
   3. Scroll through the data that you’ve received in Excel, copy and paste every point that
      corresponds to a local peak into a new column. Plot these points versus their
      respective times; they should fit nicely to the bounding function describing the
      damping in the system.
   4. Determine the dominant type of damping in the system, and its appropriate damping
      parameters; record these in your lab notebook.
   5. Plot t (time) versus 2πn, where n is an integer corresponding to the t-values from step
      three. To clarify, the time of the first peak would correspond to n=0, the second to
      n=1 and so on. The slope of this will be equal to        . Also use this data to calculate
      and record the period of your pendulums motion.
   6. Extract a value for l, then measure the pendulum and compare the values. Determine
      why the two values are different as well as the proper way to calculating the true
      value of l.
   7. Repeat steps 1-6 for the other pendulum; record all results.

4. The Coupled Pendulum
      Now that you’ve studied each pendulum individually, and determined that they are
   similar, it’s almost time to couple them. First you must determine the spring constants of
   the springs that you’ll be using. You have been given a set of springs, some weights, a
   measuring device, and the knowledge of Hooke’s Law (                 ); find k.
      Once you’ve determine the spring constants for the springs that you’ll be using you
   should familiarize yourself with the lab apparatus. It is important that the pendulums
   swing in the same plane; therefore they’ve been precision machined to help ensure this.
   When connecting the coupling spring, remember that it important to couple the
   pendulums as close to the position of their centers of gravity as possible. The following is
   a diagram of the coupled pendulum system:

                                               3
                      Figure 3.0 – Diagram of the coupled pendulum system
    In coupling the pendulums you’re creating a mechanical system in which there are
two natural modes of oscillation, each with a characteristic frequency. Any un-driven
motion in the system will be a linear combination of these two natural modes, with the
following frequencies:



   Now you’re ready to study the system’s behaviour. The best way to do this is to
examine various initial conditions of θ1 and θ2 at time t=0, and observe the systems
behaviour when the pendulums are released. There are three cases of initial conditions
that we’ll use to observe our systems behaviour, they are as follows:




   The beating case shows a phenomenon that occurs in many areas of physics, known as
beating. In optics for example, beating occurs when two light waves exist at the same
point in space and interfere with each other. The mathematics behind this can be derived
using a few simple trig identities, the general idea being:
                                                $" # "b ' $"a + "b '
               Acos(" a t) + Acos(" b t) = 2Acos& a     t ) cos&   t)
                                                % 2       ( % 2     (
with the first term being the low-frequency “beat envelope”, and the second the higher
frequency “carrier” oscillations occurring within the beat envelope. In our case the
parallel and anti-parallel are interfering with each other.
      !
   When data is acquired in the lab, the Power Spectrum FFT plot should reflect peaks at
these natural modes, the lower of which will be ωa. By noting the frequencies at which
these peaks occur, one may calculate the frequencies involved in the beating case.
                                          4
Procedure:
          Note: All measured and calculated values must include uncertainties!
1. Using the above equations, l determined earlier, and the spring constants found
   before coupling the pendulums, calculate ωa and ωb.
2. Set-up and run the experiment for the parallel case. Record and save the data for this
   trial. Use this data to determine the frequency of oscillation for both pendulums.
   Write an explanation towards why ωb is not observed in this case.
3. Run the experiment again with the anti-parallel case.
4. Compare the frequencies observed in the parallel and anti-parallel cases with those
   calculated in step one.
5. Run the experiment again using the beating case. Note the trends observed on the plot
   in your lab notebook. Save your data, including the Power Spectrum FFT data, and
   use it to determine the beat frequency as well the pendulums individual oscillatory
   frequencies; record all calculations and plots in lab notebook. This may be done by
   performing the following calculations:
                                      " # "b                          " + "b
                  Beat frequency = a           Carrier frequency = a
                                          2                              2
6. Compare the experimentally observed frequencies from step five with those
   calculated from the above equation.
      !




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