|Position-Time Graph of a Pendulum
by Greg Jacobs
Woodberry Forest School
After completing this experiment, students will be able to:
Why Use This Lab in the AP Physics Course?
- Explain how and why the position-time graph of a pendulum is sinusoidal
- Produce a position-time graph for an object in one-dimensional motion
- Explain the relationship between the period and length of a simple pendulum
To show mathematically that the position-time graph of an object in simple harmonic motion (SHM) is sinusoidal requires a complicated derivation using differential equations. Such a derivation, especially at the Physics B level, tends to cause great confusion -- for example, students tend to think that anything associated with SHM involves a sine function, including the restoring force or even the path of the object. Kinesthetic experience with SHM is often essential for understanding this topic.
Furthermore, the production of a position-time graph is mysterious to the inexperienced physics student. Laboratory work with the sonic motion detector, or with machines that make 60 dots per second on a strip of paper, can help students visualize how position-time graphs work. This experiment offers a low-tech, creative opportunity for the investigation of position-time graphs.
Correlation to the Topic Outline in the Course Description
I.A.1. Motion in one dimension
I.B.2. Dynamics of a single particle
I.F.1. Simple harmonic motion
I.F.3. Pendulum and other oscillations
Simple harmonic motion, and its application to a pendulum, is discussed in all AP-level textbooks. See, for example, Physics, 5th ed., by Douglas C. Giancoli, sections 11-3 and 11-4.
For a less mathematical discussion of simple harmonic motion with specific reference to the AP Exam, see McGraw-Hill's 5 Steps to a 5: AP Physics, by Greg Jacobs and Josh Schulman.
For a detailed version of the derivation in the introduction of this lab, consult a Physics C-level text. See, for example, Physics for Scientists and Engineers, 4th ed., by Paul A. Tipler, section 14-1.
A Java applet showing, among other things, the position-time graph of a pendulum can be found at http://www.walter-fendt.de/ph11e/pendulum.htm.
Consider a mass m in simple harmonic motion, which experiences a restoring force F of the form F = -kx, where x is the displacement from the equilibrium position. By setting F equal to
, the second-order differential equation can be solved to obtain x = A cos
For the special case of the simple pendulum, the proportionality constant k is equal to
, where L is the length of the pendulum. Thus, the period T of the harmonic motion becomes
While this derivation is only of interest to bright Physics C students, the result should be understood by all: the position-time graph of an object in simple harmonic motion looks like a cosine function that repeats over a time interval given by the expression
. In this lab, students experimentally verify both the form and the period of the position-time function.
This experiment should be done in small groups, preferably of no more than two people.
The total time from original assignment to final assessment should be about two weeks. However, the data-collection process itself (including setup and cleanup) can easily be done in a single 80-minute lab period or less. The amount of class time allotted to this experiment can vary; the more you assign as homework, the less class time you will use.
Here is one possible "calendar of events" related to this experiment:
Monday, week 1: Pass out the student copy of the lab. Solicit questions; show and discuss available supplies. Announce Thursday's quiz, and state expectations about preparation for the experiment. Allow some time to explore or brainstorm about supplies and equipment (about 20-30 minutes in class).
Thursday, week 1: Give a brief quiz or interview to check each group's progress:
Question 1: What are you going to use as your pendulum?
Then use some class time to answer questions or fiddle with equipment (about 20 minutes in class).
Question 2: How are you going to collect position-time data?
Monday, week 2: Do the experiment. Students are expected to set up, collect reasonable data, and clean up within the lab period (full 60-80 minute lab period).
Tuesday, week 2: Allow students to use some class time to draw the scaled version of their position-time graph in their lab book and to begin their calculations. (The necessary time will vary. It is useful, though, to allow students to work on the analysis in class to foster group-to-group comparisons and collaboration.)
Thursday, week 2: The final report is due from each student.
Preparation and Prep Time
It is important to the objectives of the experiment that the students do all of the setup. Teacher preparation consists of obtaining a variety of craft-type materials, as described below.
Materials and Equipment
Standard laboratory hardware should be available, such as ring stands, clamps, rods, masses, and so on. Graph paper is necessary for the analysis.
It is very useful for students to have access to craft supplies, as might be used in an elementary school art class: scissors, glue, balloons, baggies, string, and such have been used by many of my students in this exercise.
Long (at least 1 m) strips of paper are necessary. These can be salvaged: everywhere I go I seem to find people with stacks of decade-old dot matrix printer paper. Ask around. Another source of useful paper could be rolls in the art department or in the shipping room. In the worst case, students can staple or tape sheets together.
If you have access to a sonic motion detector, one group will likely request to use it; however, electronic equipment is not necessary.
All necessary materials can be obtained at retail convenience or craft stores, if they can't be salvaged.
Safety and Disposal
Assuming common-sense lab protocols (i.e., no poking each other with ring stands!), there should be no safety issues with this experiment.
- The most common method of obtaining the position-time graphs involves pendulum bobs that drip or leak. The large paper can be dragged underneath the pendulum at a constant speed. Or, the pendulum can be attached to a ring stand, and the ring stand itself can be carried over the paper at an approximately constant speed. The example procedure given in the students' handout is more involved than is strictly necessary.
- If you have a sonic motion detector, it is likely that students will want to use it. On one hand, such high-tech equipment partly defeats the purpose of the experiment; students should develop some kinesthetic experience with position-time graphs. On the other hand, a graph from the motion detector does fulfill all the requirements of the assigned task. I do not explicitly offer use of the detector. But I usually allow only the first group who comes up with the motion detector idea to use it. (Getting a good position-time graph from the motion detector is not necessarily easy. Students will need to use a bob at least the size of a baseball and make small-amplitude oscillations.)
- For groups not using the motion detector, the position-time graph does not have to look like a beautiful cosine function. A reasonable period can be determined even if the graph looks pointy, or if the amplitude is rapidly decreasing, or if small parts of the graph are missing.
- Don't let the students waste time trying to get a "perfect" graph. As long as five or so peak-to-peak or trough-to-trough period measurements can be made, the graph is sufficient. Students' time is better spent on the analysis once they have obtained a reasonable graph.
- Students using strips of paper may need guidance to figure out how to determine a time scale. The easiest method is to use a stopwatch to measure the total time it takes to produce the graph, then to scale the time axis accordingly. If one group is shown this method, they can disseminate the information throughout the class.
- One way to ensure that the paper moves at a constant speed is to attach it to one of those bulldozer-style constant speed vehicles. This technique is not necessary to obtain good data, but it will satisfy the perfectionists in the class.
- Others with an eye for detail may point out that the leaking or dripping bobs are changing mass throughout the experiment -- won't this change the period? The short answer is no, the mass of a pendulum does not affect its period. The more involved answer is yes: Imagine a pendulum with a water-filled baggie for a bob, which traces its position-time graph onto paper pulled beneath it. As the water drips out, the bob's center of mass moves farther from the pendulum's base; thus, the length is increasing, which increases the period. But then the precision of the measurement must be considered. Students can calculate how much the period should change based on the extended length; they will find that the uncertainty on their period measurement will be much greater.
1 and 2. A sketch of what some of my students' data looks like is shown below.
- The x-t graph is not a beautiful sine curve. There are points, strange humps -- that's okay.
- The motion of the pendulum was only captured for a little more than two full cycles. That's enough. The experiment requires five period measurements, but these do not all have to be peak-to-peak or trough-to-trough; look at
T4 and T5, for example.
3. In this student's graph, 1 cm along the horizontal axis represents 1 second of time. The labeled periods work out to an average of 1.2 ± 0.1 s.
4. To convert the period to a length, we solve the equation for the period of a pendulum for length:
. The length of this student's pendulum is calculated to be 36 ± 6 cm.
In both steps 3 and 4, students will likely just eyeball an uncertainty. If they have a larger number of period measurements, they can use the standard deviation to find an uncertainty, but that is not strictly necessary for the goals of this experiment. It is enough that students get used to including a reasonable uncertainty with each measurement.
Sample Discussion Question Answers
1. Based on your observation, when in its cycle was the pendulum moving the fastest? When was it moving slowest? How are these observations consistent with the position-time graph that you made?
We noticed that the pendulum didn't move much when it was near the edges of its motion, while it moved very quickly past the center of the paper.
The slope of a position-time graph indicates velocity. So, the pendulum moves fastest where the graph we made is steepest. Sure enough, our graph is steepest near the x = 0 point.
2. Where does the pendulum bob spend more time: near the center of its motion, or near the endpoints? Justify your answer with reference to your graph and/or your observations.
Since the pendulum moves faster near the center of its motion (as discussed in question 1), it spends less time there. It spends more of its time near the endpoints of its motion, where it's moving slower.
We used a dripping bag as our pendulum bob. We tended to see more drips near the endpoints of the motion than in the middle.
3. Compare the length calculated from the period measurements with the length of the pendulum that you measured in lab.
The length calculated from the period measurements was 36 ± 6 cm. We measured a length of 35 cm to the top of the dripping bag, and 40 cm to the bottom of the bag. The calculated and measured lengths match.
Merely by doing this experiment, and by taking the data-collection process seriously, students gain important kinesthetic experience, both with position-time graphs and with the properties of harmonic motion. Therefore, in my assessment, I give substantial weight to just the production of the graph.
I print out the rubric below, tape it onto the students' notebooks that they turn in, and fill it out.
Experiment: 10 points. Criteria:
Analysis: 10 points. Criteria:
- You designed an experiment that worked.
- You made a reasonable position-time graph in the time allotted.
- You exercised care and attention to detail in the data-collection process.
Experiment points: _______
- You made a useful scale version of your original position-time graph.
- You measured the pendulum's period appropriately.
- You correctly calculated the pendulum's length from the period.
- You indicated a reasonable uncertainty in the period and length measurements.
- You answered the analysis questions carefully and thoroughly.
Analysis points: _______
Total score (out of 20): _________
Greg Jacobs teaches AP Physics B and C at Woodberry Forest School in central Virginia. He is a graduate of Haverford College, and has a master's degree in engineering from Northwestern University. When he is not teaching, Greg broadcasts Woodberry Forest varsity baseball games over the Internet; he is a reporter for STATS, Inc., covering baseball, basketball, and football; and he is a Reader and consultant for the College Board's AP Physics program. Greg lives on campus at Woodberry with his wife Shari and their son Milo Cebu.