Interactive DynamicsThe 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.

# Determining Local Gravity

We have all been told that acceleration due to gravity, g = 9.8 m/s2 or g = 9.81 m/s2, 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:

1. the length of the pendulum;
2. the amplitude of the swing;
3. 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 - Part 1

You will begin the activity 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 5) and go to as large an amplitude as you can.

# Numerical Investigation

For the second part of the activity, you will use Microsoft Excel and Mathematica to solve different versions of the equation(s) of motion describing the behavior of the pendulum.

## The Linear System — Excel

Implement Euler's method (presented in class) in Microsoft Excel to solve the linearized pendulum equation(s) for the same starting amplitudes you used experimentally. Show that for the linear equation(s), the period does not depend on the starting amplitude. 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(s) 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 for period vs. amplitude.

# Activity Report

This activity will not require a full activity report. We simply want you to turn in the following:
1. A brief description of the experimental setup.

3. Your estimated value of g, along with a justification of the number of significant figures used.

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

5. A plot of your Excel results showing that the period of the linearized pendulum does not depend on the amplitude of oscillation.

6. 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.

7. You should also turn in your Mathematica notebook but do not turn in your Excel spreadsheet as it is likely to be very large.
Handwritten work is fine as long as it is very neat.

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