Interactive Dynamics The Equation of Motion for a Simple Pendulum and the Experimental Determination of the Acceleration Due to Gravity

Introduction
As a first activity, we will tackle the problem of a swinging pendulum in different ways and will use what we learn to determine the local acceleration due to gravity in this classroom. This value for the acceleration due to gravity can, if you like, be used for all additional experimentation and calculation this semester. This will also provide your first opportunity to present technical information in a very short report, something you may find to be difficult in the beginning. In addition, you will begin to learn how to compute the propagation of error.
Experimental Determining Local Gravity (Done the first day.)
We have all been told that acceleration due to gravity, g = 9.8 m/s^{2} or g = 9.81 m/s^{2}, however, due to a myriad of factors, g in this room may be slightly different than the g that you have always been told to use. An extreme example of how g would vary measurably is to measure it while flying in an airplane at 35,000 ft versus measuring it on the surface of the earth. One of the easiest ways to measure g is to take advantage of the principle on which all grandfather clocks are based: the simple pendulum.
The time it takes a pendulum to make a complete swing from one extreme to another (to complete one cycle) is called the period of oscillation. This swinging back and forth repeats itself over equal time periods and therefore this motion is called periodic motion. It turns out that we can describe this periodic motion in terms of circular functions, sine and cosines, and therefore it is called harmonic motion. (All harmonic motion is periodic, but not all periodic motion is harmonic.) It turns out that the period of oscillation of a pendulum depends on the following three factors:
 the length of the pendulum;
 the amplitude of the swing;
 the acceleration due to gravity.
One question we might ask at this point is what factors are we ignoring which might affect the period of the pendulum? Do you think that they have a significant impact on the period? We will show that if the amplitude of the swing (i.e., the swing angle) is small enough, the effect of the amplitude can be ignored and we are thus left with the knowledge that the period of oscillation is related to only two quantities: the length of the pendulum and the value of g.
The circular frequency of the pendulum, w, is the frequency of a periodic quantity expressed in radians per second and the period of the pendulum's motion is related to the circular frequency through the relation:
period = p = 2p/w.
So, where does this leave us? Well, we can measure the length of the pendulum and we can measure the period of oscillation, but how are all of these things related to the acceleration due to gravity? This is something we will show you in class.
Experimental Proposal
Before you go out to Kunkle Lounge to begin the experiment, we need a hand or typewritten experimental proposal detailing each step in your experimental procedure and the role of each team member at each step of the experimental procedure.
Experimental  Part 1
Begin by building a simple pendulum and measuring its period of oscillation in Kunkle Lounge. You will design the entire experiment and consider that you want your measurements to be as accurate as possible. Also be sure and take into account the theory and assumptions we just covered in class. Be sure and note what factors you are ignoring and what approximations you are making as you do the experiment.
After making your measurements, use the theory presented in class to compute g with an estimate of the accuracy (i.e., give us an estimate, with supporting arguments, of how many decimal places really mean something).
Experimental  Part 2
For the second part of your experimental work, we would like you to experimentally determine the dependence of the period of the pendulum on the amplitude of the swing. To do this, carefully measure the period for as many amplitudes as you reasonably can (do at least 3) and go to as large an amplitude as you can.
Numerical Investigation (Done the second day.)
For the second part of the activity, you will use Mathematica to solve the equation of motion for the nonlinear pendulum presented in class.
RampUp Problems: Euler's Method
Implement Euler's method as presented in class solve the nonlinear pendulum equation of motion for an initial angle of 45 degrees and a zero initial rotation rate. We would like you to integrate for one second starting at zero seconds. To give each team member a task, break the one second interval into 3, 4, and 5 steps. Then have each team member integrate/solve the differential equation for one of the three step sizes (i.e., 0.333, 0.25, or 0.2 seconds). Do this using pencil, paper, and your calculator. Compare your answers. Note: if you only have two team members, then do it for 4 and 5 steps.
Euler's method is very simple, but even the most sophisticated ordinary differential equation solvers work very much like Euler's method. Sophisticated solvers usually just have better ways of estimating derivatives.
The Nonlinear System — Mathematica
Use Mathematica to solve the nonlinear pendulum equation for a range of starting amplitudes. Plot the period versus the amplitude. To find the period of the solution in Mathematica, you will find that the FindRoot command will be very useful. On the same plot, plot the points you obtained in the experimental part of the activity for period vs. amplitude (this plot, by itself and without the error bars, should be turned in Sept. 14 for TR sections and Sept. 15 for the MW section so that we know you are on the right track). Be sure to include error bars on your experimental points.
Activity Report
You will turn in one report per team and the reports will be due on September 21 for the TR sections and September 22 for the MW section.
This activity will not require a full activity report. We simply want you to turn in the following:

A brief description of the experimental setup.

Tables of measurements made during your experimental tests.

Your estimated value of g, along with your error estimates and analysis.

Comments on the origins of differences between your measured value of g and 9.81.

A plot of your Mathematica results showing that for the nonlinear pendulum, the period does depend on the amplitude. In addition this plot should also contain your experimental results. You will find that the ListPlot command is nice for plotting data points.

Do not turn in your Mathematica notebook. Put it in your ftp directory and name it: sectxyourteamnameact1.nb, where you fill in the section number and your team name. That way we can quickly execute it to see if it works properly.
Handwritten work is fine as long as it is very neat.
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people have accessed this page since July 27, 1998.
Prepared by Gary L. Gray, Francesco Costanzo, Ben Conaway, Chris Watterson, and Molly Riley.
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© Copyright 1998—1999 by Ben Conaway, Molly Riley, Chris Watterson, Gary L. Gray, and Francesco Costanzo. All rights reserved.