Swinging Pendulum

Developer:

Charles Scaife
Department of Chemistry
Union College
Schenectady, NY 12308

Grade Levels:

K through 12

Discipline:

Physical Science/Physics

Goals:

  1. Construct a pendulum.
  2. Determine whether the mass of the pendulum affects the number of swings of that pendulum in a given time.
  3. Determine whether the length of the pendulum affects the number of swings of that pendulum in a given time.
  4. Learn how to graph data by sharing data and plotting the results.


Background:

A pendulum is an object suspended from a fixed point and free to swing back and forth due to the action of gravity and acquired momentum. A pendulum is very easy to construct.

This is a straightforward experiment to perform, and all of the equipment is reusable. At least two authors have described experiments of this nature (1, 2). It is said that Galileo first performed these experiments by observing swinging lamps in a cathedral and timing them with his pulse beat.

Teacher's Notes:

This experiment can be used at a variety of levels, depending on whether students test the dependence on mass of the pendulum or the dependence on length of the pendulum and how much graphing and discussion is required. For K through 2, I do the activity only testing dependence on mass as a demonstration and let the students count swings out loud and record data. The experiment can also be done using swings outside on the playground providing you have a swing set and students of sufficiently differing masses. Fourth through 12th graders study the dependence on both mass and length and do all the work themselves. Only 9th through 12th graders do all of the plotting independently.

Prepare several data sheets of this form shown.

At the end of the experiment after each student has obtained and recorded at least one piece of data, you should get the class to plot the data in various ways to see if there are any obvious visual relationships between the number of swings in a given time and either mass or length. You might plot

  • Number of swings in 30 seconds versus Mass or versus Length;
  • Number of swings in 30 seconds versus Square root of length;
  • Time for one swing versus Length;
  • Time for one swing versus Square root of Length; or
  • any other parameters that students want to try.
I like to have pieces of graph paper prepared ahead of time so that students can plot data as soon as they obtain it. These plots can be done manually on pieces of graph paper, but this is also a good opportunity to show the ease and speed of graphing capabilities of a computer if a computer and graphing program are available.

Liem (2) also describes an intriguing way of plotting visually that is especially effective for younger students. He has each student hang his or her pendulum on a horizontal meter stick at the centimeter mark representing the number of swings obtained. The pendulums form an upside down plot of number of swings in 30 seconds (frequency) versus length.

Materials:

DEPENDENCE ON MASS (for every four students or as a demonstration)

  • 1 Ring stand with iron ring or clamp (or any means of holding the pendulums; doorways with hooks work if you have enough doors that they don't get crowded)
  • 1 Approximately 50-centimeter length thread with Christmas bulb hooks at each end
  • 4 Metal washers of radically differing sizes and masses 1 Meter stick
  • 1 Clock or watch with a second hand
  • 1 Balance for weighing


DEPENDENCE ON LENGTH (for every four students)

  • 1 Ring stand with iron ring or clamp (or any means of holding the pendulums; doorways with hooks work if you have enough doors that they don't get crowded)
  • 4 Various length threads up to about 85 centimeters with small bulb hooks at each end
  • 4 Metal washers having the same mass
  • 1 Meter stick
  • 1 Clock or watch with a second hand


Explanation:

The force of gravity causes a washer to fall and a pendulum to swing when you release it. The speed of the pendulum increases as the pendulum approaches the vertical position and decreases as it moves upward where it stops. The acceleration of gravity is a constant so that the actual force on an object is proportional to it's mass. Therefore, the number of swings is the same for a washer of any mass.

The number of swings, however, is dependent on length and actually increases as the length of the pendulum decreases. See Question 4 for a detailed explanation.

Questions:

  1. What force pulling on the washer caused it to fall when you released the pendulum? What force prevented the washer from falling to the desktop or the floor?

    Gravity pulling downward on the washer caused it to fall. The thread pulling upward on the washer prevented it from falling farther than the length of the pendulum.

  2. Did the speed at which the pendulum moved differ during its swing? At what point was it moving fastest? At what point was it moving slowest?

    The speed at which the pendulum moved did differ during its swing. The pendulum moved fastest when it was at its lowest point in the middle of its swing. The pendulum moved slowest and actually stopped when it was at its highest point at the end of its swing.

  3. Were your numbers of swings in 30 seconds for differing masses the same or different considering that there is some experimental uncertainty in counting the swings? Do you have any idea why this behavior was observed?

    The numbers of swings in 30 seconds for differing masses were the same within the limits of experimental uncertainty. They probably differed by only one swing or at the most two. This behavior was observed because the acceleration of gravity is a constant, and the force (which is mass times acceleration) on the washers was proportionally larger for the larger masses. This is the same reason that objects having different masses, when dropped from a four-story building, will, neglecting wind resistance, hit the ground at the same time. Obviously this would not work for a nickel versus a piece of paper because there would be much more wind resistance on the piece of paper.

  4. Were the number of swings in 30 seconds for differing lengths of pendulums the same or different? Do you have any idea why this behavior was observed?

    The number of swings in 30 seconds for different lengths of pendulums were different. The number of swings in 30 seconds increases as the pendulums get shorter. This occurs because, for small displacements from the center, it can be shown that the restoring force pulling the pendulum toward the center is inversely proportional to the length of the pendulum. Since the period or time for one swing is inversely proportional to the square root of that restoring force, it follows that time for one swing is directly proportional to the square root of the length of the pendulum. But the number of swings in 30 seconds is inversely proportional to the time for one swing and is therefore inversely proportional to the square root of the length of the pendulum.


References:

  1. Janice Pratt VanCleave, Physics for Every Kid: 101 Easy Experiments in Motion, Heat, Light, Machines, and Sound, Wiley, New York, NY, 1991, pp. 222-223; Ray Broekel, Sound Experiments, from A New True Book Series, Childrens Press, Chicago, IL, 1983, pp. 74-77.
  2. Tik L. Liem, Invitations to Science Inquiry, 2nd ed., Science Inquiry Enterprises, Chino Hills, CA, 1987, (13.29) p. 345.


Procedure:

DEPENDENCE ON MASS


Suspend the thread from the stand by the bulb hook at one end. If a balance is available, weigh the smallest washer (numbered 1). Record your name, the number of the washer, and, if you weighed it, the mass of the washer on the data sheet nearby. Then hook the smallest washer onto the bulb hook at the bottom end of the thread. Place the zero-centimeter end of the meter stick directly under the suspended meter stick directly under the suspended washer and extend it out to your right or left as space permits. Ask a student in your group to hold the base of the ring stand firmly as you perform the experiment.

Pull the washer and thread forming your pendulum 30 centimeters (12 inches) to one side. Ask a second student in your group to tell you when the second hand of the watch or clock reaches the minute mark. Release the washer exactly at that time. CAUTION: Do not push the pendulum; simply release it. The remaining student in your group should then count the number of swings back and forth until the second student indicates that exactly 30 seconds have passed. Back and forth count as separate swings. Do not lose count!

Record the number of swings that you observed in 30 seconds. Then repeat the same experiment twice more. Finally, calculate and record an average of your number of swings in 30 seconds.

Rotate duties as you repeat the experiment with three more washers of different size and mass, and record additional results under the first set.



DEPENDENCE ON LENGTH


On the data sheet nearby, write your name neatly, and record all data as you obtain it. Choose a length of thread that has not already been used by another person as noted on the data sheet. Suspend the thread from the stand by the bulb hook at one end. Then hook a metal washer numbered 5 onto the bulb hook at hte bottom end of the thread. Using a meter stick, measure the length of your pendulum in centimeters from the top of the upper hook to the center of the washer, and record the length. Ask a student in your group to hold the base of the ring stand firmly as you perform the experiment.

Pull the washer and thread forming your pendulum out to about a 30-degree angle. Ask a second student in your group to tell you when the second hand of the watch or clock re you when the second hand of the watch or clock reaches the minute mark. Release the washer exactly at that time. CAUTION: Do not push the pendulum; simply release it. The remaining student in your group should then count the number of swings back and forth until the time keeper indicates that exactly 30 seconds have passed. Back and forth count as separate swings. Do not lose countÚ

Record your results, repeat the experiment twice more and record your results, and then calculate and record average values of length and number of swings in 30 seconds. Also calculate the time for one swing by dividing 30 seconds by the number of swings in that time. Record your result. Finally, if your teacher asks yuo to do so, estimate or calculate the square root of the length of your pendulum, and record the result.

Plotting Results:

On the graph paper, plot Number of swings in 30 seconds versus Length of your pendulum by reading up on the vertical axis until you come to your number of swings, reading across the graph horizontally until you come to your length of pendulum, and then placing a small circle on the graph. Your circle will not be on either axis. Likewise, on other sheets of graph paper, plot


    Number of swings in 30 seconds versus Square root of length;
    Time for one swing versus Length; and
    Time for one swing versus Square root of length.
    Do you observe any trends based on data plotted so far? Not that this activitiy requires results from several students or even the whole class. If time and space permit, you or your teacher will also show these plots on a computer.

Data Sheets:

Dependence on Mass Data Sheet
Dependence on Length Data Sheet



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