THE PENDULUM
Knight Foundation Summer Institute
Elizabeth Chesick, Haverford College
Introduction:
This lesson uses the pendulum which some of the students may never have seen before. Basically a pendulum is just a mass on the end of a string that is free to swing. Before showing the students the setup that they will be using for the experiment introduce them to the idea of a pendulum by tying a string around a tennis ball. Let it swing as the students observe it for a little while. Can they measure its period  designated as "T"? (i.e. the time it takes to complete one full swing). Afterwards show them the actual setup they will be using in the experiment.
The pendulum does not always swing at the same rate. There are several factors that may change the time it takes for the pendulum to make one complete swing. First ask the students for their opinions on what may change the pendulum's period. The following factors are the ones that the students will test in today's experiment:
1. size of the mass hanging on the end of the string mass
2. angle of the swing angle
3. length of pendulum string length
4. location of the pendulum on earth location
* To understand factor #4, it is helpful to understand what is meant by "g". For an explanation, see the lab in this booklet, "Acceleration Due to Gravity".
Objectives:
1. To identify which factors influence the period, T, of the pendulum.
2. To make two graphs of the data and draw conclusions from the graphs.
3. To calculate the value of "g", the acceleration due to gravity.
Vocabulary:
period
pendulum
angle
variable
slope of straight line on graph
Materials:
for each group of 23 students

Procedure:
(The exact details of the experiment will vary since each group of students is designing their own.)
1. Decide on which factor you want to test first. This is your variable. How will you be sure you are measuring this factor alone and not any of the others at the same time? (Note: to investigate the location variable, the experiment would have to be carried out at a different location. Usually, this is not very practical.)
2. How will you time the period? Is it accurate enough to time one swing or should you time more than one swing, then divide by the number of swings you timed to get the Period, T?
3. What observations do you need to record? How many trials should you run for each factor? That is, if you are testing the effect of angle on the period, for how many different angles will you test the period? You may think about using a table such as that below. Change it according to which factor you are testing.
4. How will you decide which factor(s) actually effect(s) the period?
L (length) 
Time (for 10 swings) 
T (Period) 
T^{2} 
** Go on to the next part, after you have decided for yourself which factor actually affects the period the most. The following steps are also intended for more advanced students. If the students are not ready to tackle these next steps, make sure they interpret their data by discussing it with their group and reach a conclusion as to which factor effects the swing of the period the most. Compare the conclusions that each group reaches.
5. Make a graph of T versus L, that is T on the y axis and L on the x axis. Start at 0,0. What is the shape of the graph?
6. Because the "curve" is bent rather than straight, it is not so easy to decide what the
mathematical relationship between length and period would be. In other words, it's hard to determine exactly how period is effected by length. For example, does the period increase as length increases or does it decrease? Does the period change in the same proportion as the length does? (i.e. the length is doubled so the period also doubles?) Or does the period change in a different way? To get the exact mathematical relationship between two factors, physicists usually want to have a straight line graph because it is easier to w rite the equation for a straight line. In this case. it is possible to get a straight line from the data if some "massaging" of the data is allowed. Continue the data table above by adding another column, T~ to the right. Calculate T1 for the values in the table. Make another graph of T~on the y axis and L on the x axis. Make the best strajight line. The graph does make a straight line.
7. Find the slope of the line on the graph of T^{2} vs. L. (Slope is rise over run. or the change in the y variable divided by the change in the x variable.)
The slope is equal to a value, 4p ^{2}/g
Slope = 4p ^{2}/g
G = 4p ^{2}/slope
8. Calculate the value of "g" from the equation right above? How does this compare with the accepted value of "g" (9.8 m/s)? In fact, "g" changes slightly at different places over the earth, so this is how the location variable mentioned in the introduction comes into play. '
9. Calculate the relationship between period and length.
** Again, it's okay if the students cannot do steps 59. It's enough to determine which variable most effects the period of the pendulum.
Assessments:
 Have the students turn in their procedure, data, and conclusions in scientific format.
** Note to teachers: The precise relationship between period and length is:
Extensions:
This booklet contains another interactive and exciting lab on "g" in which the students use various methods to calculate the value of "g" for themselves. This lab is entitled "Acceleration due to Gravity".
Another lab dealing with the effects of forces in this booklet is called "Newton's Second Law  Changes inVelocity with Constant Force". Check it out!
Philadelphia Science Content Standards:
SCIENCE CONTENT STANDARD #1: NATURE OF SCIENCE Since the students develop their own procedure in this lab, Benchmark 1 for grades 58 is satisfied: " design, modify, and conduct an investigation through testing, revising, and occasionally discarding ideas, all of which lead to a better understanding of how things work". ~
SCIENCE CONTENT STANDARD #2: PHYSICAL SETTING This lab satisfies Benchmark 4 for grades 58: "Investigate the relationship between force and motion.
Crossreferences:
This is a great lesson for the students to learn about the scientific method since they design their own procedure and draw conclusions based on their data. It also fits well into a physics unit on forces (specifically gravity).