Lesson: Swinging on a StringContributed by: Integrated Teaching and Learning Program, College of Engineering, University of Colorado at Boulder
Educational Standards :
Pre-Req Knowledge (Return to Contents)
Basic understanding of forces such as lift, weight, thrust and drag, and rotational motion and angular momentum.
Learning Objectives (Return to Contents)
After this lesson, students should be able to:
Introduction/Motivation (Return to Contents)
Have you ever played on a swing? As you swing, you smoothly ride from the top of one arc, through the bottom, to the top on the other side of the swing, and back again. When you are on a swing, you move like a pendulum. A pendulum is a string hanging from a fixed spot with a weight (called a bob) at one end that can swing back and forth.
One day in the late 1500s, a man named Galileo Galilei was sitting in church when he noticed the lamps hanging from the ceiling were swinging back and forth. Some of the lamps were making great big swings, and others were only making little swings back and forth, but they all went back and forth pretty regularly. Galileo was a curious man, and so he decided to use his heartbeat to measure how long it took the pendulums to swing back and forth. He was very surprised by what he learned. Today, you will repeat Galileo's experiment to learn about pendulums.
Many people consider Galileo to be the "father of experimental science." Before Galileo, most people tried to understand the world around them just by thinking about the things they saw. Galileo certainly thought a lot, but he did something that not many other people did — he designed experiments to test his ideas. This is how people do science and engineering today!
From his experiments, Galileo was able to describe the motion of a pendulum with the mathematical equation included in this lesson. Eventually, he came up with the idea of using a pendulum as a way to keep track of time. He used his discoveries as a tool for other experiments, in which he made many other discoveries.
Engineers also use inventions and discoveries to build new things. Today, engineers use pendulums in clocks, but they also use them for detecting earthquakes and helping buildings resist shaking. Engineers use pendulums to measure how fast a bullet is flying, and to help robots balance. Maybe you can think of some new ways to use a pendulum, too!
It turns out that understanding a pendulum's motion is really useful. Many other objects move back and forth regularly like pendulums, such as a weight bouncing up and down on a spring, a wheel spinning around — even radio waves go back and forth! The physics of understanding how pendulums behave is an important step towards understanding all kinds of motion.
During his life, Galileo made many scientific discoveries, including descriptions of gravity and the motion of falling objects, moons of Jupiter, new kinds of thermometers and many other things. He was a pioneer of the scientific method of investigating the world around us. Today, we will follow in Galileo's footsteps to learn about how pendulums behave.
Lesson Background & Concepts for Teachers (Return to Contents)
Galileo's interest in pendulums is generally believed to have started when he was sitting in the Cathedral at Pisa, Italy. After he noticed the lamps swinging back and forth regularly, he began experimenting with pendulums to learn about their motion. Pendulums are pretty simple devices, and the factors that could affect their motion are the length of the string, the weight of the bob, and the size of the swing. Galileo experimented to determine which of these variables determined how often a pendulum swings.
In this lesson, students observe that the size of the swing does not affect the time it takes for a pendulum to swing back and forth. Just like Galileo, students find that even when a pendulum swings through a small angle, the time of each swing (the period) remains the same as if it swung through a large angle! Like Galileo, students also find that it does not matter what mass the object at the end of the string is — the time for each oscillation (the period) is still the same. Since Galileo was in medical school when he did his experiments, he decided the pendulum would be useful to measure the pulse of patients. Perhaps the students will think of some new uses, too!
Thanks to Galileo, we now know that the period of a pendulum can be described mathematically by the equation:
P = period; i.e., the time for one pendulum swing [sec]
l = length from the fixed point at the top of the pendulum to the center of mass of the bob [m]
g = gravitational constant (9.8 m/sec2)
π ≈ 3.14 (dimensionless constant)
Note that this equation does not include terms for the mass of the pendulum or the angle it swings through. The only factor that significantly affects the swing of a pendulum on Earth is the length of its string.
Students might wonder why the length of the string is the only thing that affects a pendulum's period. This can be explained by examining possible effects of each of the three variables: the length of the string, the mass of the bob, and the angle displaced. The length of the string affects the pendulum's period because of conservation of angular momentum, which the students learned about in Mechanics Unit, Lesson 8. When a pendulum's string is shorter, the pendulum must travel with a higher velocity to conserve angular momentum. A pendulum on a short string swings faster than a pendulum on a long string for the same reason that an ice skater spins faster when the skater pulls his/her arms in close to his/her body.
The mass of the bob does not affect the period of a pendulum because (as Galileo discovered and Newton explained), the mass of the bob is being accelerated toward the ground at a constant rate — the gravitational constant, g. Just as objects with different masses but similar shapes fall at the same rate (for example, a ping-pong ball and a golf ball, or a grape and a large ball bearing), the pendulum is pulled downward at the same rate no matter how much the bob weighs.
Finally, the angle that the pendulum swings through (a big swing or a small swing) does not affect the period of the pendulum because pendulums swinging through a larger angle accelerate more than pendulums swinging through a small angle. This is because of the way objects fall; when something is falling, it keeps accelerating. As long as an object is not going as fast as it can, it is speeding up. Therefore, something that has been falling longer will be going faster than something that has just been released. A pendulum swinging through a large angle is being pulled down by gravity for a longer part of its swing than a pendulum swinging through a small angle, so it speeds up more, covering the larger distance of its big swing in the same amount of time as the pendulum swinging through a small angle covers its shorter distance traveled.
Vocabulary/Definitions (Return to Contents)
Associated Activities (Return to Contents)
Lesson Closure (Return to Contents)
Ask the students to explain which factors might affect the period of a pendulum. (Answer: Pendulum length, bob weight, angle pendulum swings through.) Which factor(s) really do affect the pendulum's period? (Answer: The length of the pendulum.) Why does the weight not make a difference? (Answer: Because the pendulum, just like falling objects, is not dependent on weight.) Why do pendulums with shorter strings have a shorter period? (Answer: Conservation of angular momentum. For a given mass and a given angular momentum, velocity must increase if the mass is at a smaller distance from the axis of rotation.) Why does the angle the pendulum starts at not affect the period? (Answer: Because pendulums that start at a bigger angle have longer to speed up, so they travel faster than pendulums that start at a small angle.)
Assessment (Return to Contents)
Discussion Questions: Ask the students and discuss as a class.
Voting: Ask a true/false question and have students vote by holding thumbs up for true and thumbs down for false. Count the votes and write the totals on the board. Give the right answer.
Lesson Summary Assessment
Human Matching: On ten pieces of paper, write either the term or the definition of the five vocabulary words. Ask for ten volunteers from the class to come up to the front of the room, and give each person one of the pieces of paper. One at a time, have each volunteer read what is written on his/her paper. Have the remainder of the class match term to definition by voting. Have student "terms" stand by their "definitions." At the end, give a brief explanation of the concepts.
Lesson Extension Activities (Return to Contents)
As a library research project, have the students research Galileo Galilei. What other scientific findings did he make during his lifetime? Have the students' research the ways that engineers use pendulums today. Some suggestions: seismographs, inertial dampeners, in sky-scrapers.
References (Return to Contents)
Galileo's Battle for the Heavens, NOVA programming on air and online, February 2004: http://www.pbs.org/wgbh/nova/galileo/.
Galileo's Pendulum Experiments, The Experiment Group, February 2004: http://galileo.rice.edu/sci/instruments/pendulum.html.
Gamow, George. The Great Physicists from Galileo to Einstein. New York, NY: Harper and Brothers, 1961.
Gittewitt, Paul. Conceptual Physics. Menlo Park, CA: Addison-Wesley, 1992.
Inclined Plane, Tel-Aviv University, Virtual Museum of Science, Technology and Culture, February 2004: http://muse.tau.ac.il/museum/galileo/inclined_plane.html.
Wolfson, Richard and Jay M. Pasachoff. Physics: For Scientists and Engineers. Reading, MA: Addison-Wesley Longman Inc., 1999.
ContributorsSabre Duren, Ben Heavner, Malinda Schaefer Zarske, Denise Carlson
Copyright© 2004 by Regents of the University of Colorado.
The contents of this digital library curriculum were developed under a grant from the Fund for the Improvement of Postsecondary Education (FIPSE), U.S. Department of Education and National Science Foundation GK-12 grant no. 0226322. However, these contents do not necessarily represent the policies of the Department of Education or National Science Foundation, and you should not assume endorsement by the federal government.
Supporting Program (Return to Contents)Integrated Teaching and Learning Program, College of Engineering, University of Colorado at Boulder
Last Modified: August 23, 2010