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# Exploring Pendulums

#### Materials

• Pencils
• Three lengths of string (30 cm, 45 cm, and 60 cm)
• Scissors
• Tape
• Washers of various sizes and weights
• Digital watch or stopwatch
• Balances (to weigh the bobs)
• Rulers (to measure the lengths of string)
• Protractors (to measure the angle) Ben Ostrowsky, Foucault Pendulum, CC-BY-2.0 via Wikimedia Commons

### Purpose

To understand the relationship between gravitational forces and the mass of objects, the changes in speed and direction of objects, and the distance between objects.

### Context

This lesson helps students understand concepts related to how gravitational forces act on objects by exploring the motion of pendulums.

Everything in the universe exerts gravitational forces on everything else, although the effects are readily noticeable only when at least one very large mass is involved (such as a star or planet). Gravity is the force behind the fall of rain, the power of rivers, the pulse of tides; it pulls the matter of planets and stars toward their centers to form spheres, holds planets in orbit, and gathers cosmic dust together to form stars.

Gravitational forces are thought of as involving a gravitational field that affects space around any mass. The strength of the field around an object is proportional to its mass and diminishes with distance from its center. For example, the earth's pull on an individual will depend on whether the person is, say, on the beach or far out in space. The image of an astronaut floating in space illustrates this point.

Students should already know that the earth's gravity pulls any object toward it without touching it. (Benchmarks for Science Literacy, p. 94.) The relationship between force and motion can be developed more fully now and the difficult idea of inertia can be given attention. "Students have no trouble believing that an object at rest stays that way unless acted on by a force. The difficult notion is that an object in motion will continue to move unabated unless acted on by a force." (Benchmarks for Science Literacy, p. 90.) To students, the things around them do appear to slow down of their own accord unless constantly pushed or pulled. The more experiences students can have in seeing the effect of reducing friction, the easier it may be to get them to imagine the friction-equals-zero case.

Galileo Galilei was one scientist who studied gravitational forces. In the late 1500s, Galileo began to study the behavior of falling bodies, using pendulums extensively in his experiments to research the characteristics of motion. At the time, virtually all scholars still followed the belief of Aristotle that the rate of fall was proportional to the weight of the body. Galileo showed that this conclusion was erroneous based on the fact that air resistance slowed the fall of light objects. Galileo was able to combine observation, experiment, and theory to prove his hypotheses.

In easily verifiable experiments or demonstrations it can be shown that the period (swing) of a pendulum is independent of the pendulum's mass. It depends instead on the length of the pendulum. This would suggest that objects fall at a rate independent of mass. The greater the amount of the unbalanced force, the more rapidly a given object's speed or direction of motion changes; the more massive an object is, the less rapidly its speed or direction changes in response to any given force.

In this lesson, students will explore websites with simulations of pendulums, where they'll be able to change the length and angle of the bob and observe its effects. They will then construct and test their own controlled-falling systems, or pendulums, to further observe and verify these theories.

### Motivation

Ask students the following questions in order to get a feel for their current knowledge and perceptions of pendulums. Answers to these questions are provided for you, but don't expect or lead students to these answers yet. At this point, simply gather and keep a good record of students' current ideas; students will have a chance to refine these after the website exploration that follows.

• How would you define a pendulum?
(A pendulum is loosely defined as something hanging from a fixed point which, when pulled back and released, is free to swing down by gravity and then out and up because of its inertia, or tendency to stay in motion.)
•  How does a pendulum work? What are the parts of a pendulum?
(A simple pendulum consists of a mass (called the bob) attached to the end of a thin cord, which is attached to a fixed point. When the mass is drawn upwards and let go, the force of gravity accelerates it back to the original position. The momentum built up by the acceleration of gravity causes the mass to then swing in the opposite direction to a height equal to the original position. This force is known as inertia.)
• What is the period of a pendulum?
(A period is one swing of the pendulum over and back.)
• What is the frequency of a pendulum?
(The frequency is the number of back and forth swings in a certain length of time.)
•  What variables affect the rate of a pendulum's swing?
(Students may come up with a variety of answers, but the four that they will be testing in this lesson are:
• Length of the pendulum-Changing the length of a pendulum while keeping other factors constant changes the length of the period of the pendulum. Longer pendulums swing with a lower frequency than shorter pendulums, and thus have a longer period.
• Starting angle of the pendulum-Changing the starting angle of the pendulum (how far you pull it back to get it started) has only a very slight effect on the frequency.
• Mass of the bob at the end of the pendulum-Changing the mass of the pendulum bob does not affect the frequency of the pendulum.
• Force of gravity-This accelerates the pendulum down. The momentum built up by the acceleration of gravity causes the mass to swing in the opposite direction to a height equal to the original position.)

Many students believe that changing any of the variables (string length, mass, or where we release the pendulum) will change the frequency of the pendulum. Give them a chance to debate and discuss their answers before continuing.

• Where do you see pendulums in everyday life? How are they useful?
(Pendulums can be found in swing sets, grandfather clocks, swinging a baseball bat, and the circus trapeze. Pendulums are useful in timekeeping because varying the length of the pendulum can change the frequency.)

After your discussion, have students explore these websites:

After students have explored these sites, review with them their list of answers to the initial questions about pendulums, revising it with the current information based on the students' exploration of the websites. As you review their answers to the question, "What variables affect the rate of a pendulum's swing?" make sure you include the pendulum's length, mass, angle, and the force of gravity in your discussion.

### Development

Begin this part of the lesson by telling students that they will explore websites to learn more about how pendulums help us learn about gravitational forces. In the second part of the lesson, students will work in groups to construct their own pendulums and test what they have observed on the websites.

Have students run the demonstration called the Pendulum Lab. With this lab, students can play with one or two pendulums and discover how the period of a simple pendulum depends on the length of the string, the mass of the pendulum bob, and the amplitude of the swing.

Make sure they understand how to run the experiment by telling them the following:

With this demonstration, you can observe how one or two pendulums suspended on rigid strings behave. You can click on the bob (the object at the end of the string) and drag the pendulum to its starting position. Also, you can adjust the length and mass of the pendulum by adjusting the the controls in the green box on the right side of the page. Once in motion, you can pause the pendulum by clicking on the "pause/play" button. The pendulum can be brought to its new starting position by clicking on the "Reset" button. You also can measure the period by choosing the "photogate timer" option in the green box.

Point out that the program measures the period, or one swing of the pendulum over and back.

• How does changing the length of the bob affect the period?
(The shorter the length of the bob, the shorter the period will be.)
• How does changing its starting point or angle affect the period?
(The smaller the angle, the shorter the period will be.)
• How can you get the shortest period?
(Decrease the length, and decrease the angle.)
• How can you get the longest period?
(Increase the length, and increase the angle.)
• Explain why the pendulum continues to move without stopping or slowing down once it is set in motion.
(According to the law of inertia, a body in motion will continue in motion, unless acted upon by a force.)

Students can also run the pendulum demonstration called The Undamped and Undriven Pendulum, found on the The Pendulum Lab website.

Explain the features of this demonstration to your students:

In this demonstration, you can vary the length of the pendulum and the acceleration of gravity by entering numerical values or by moving the slide bar. Also, you can click on the bob and drag the pendulum to its starting position. This demonstration allows you to measure the period of oscillation of a pendulum.

To participate in this demonstration, students should follow these steps:

1. Press the "Start" button of the stopwatch just at the moment when the pendulum is going through its deepest point.
2. Count "one" when it goes again through its deepest point (coming from the same side).
3. Repeat counting until "ten." At that moment, they should stop the stopwatch. Dividing the time in the display by ten yields the period of oscillation.

Students can also measure the frequency of a pendulum, or the number of back-and-forth swings it makes in a certain length of time. By counting the number of back-and-forth swings that occur in 30 seconds, students can measure the frequency directly.

• What is meant by the period of oscillation?
(It is a way of measuring the back and forth swing of the pendulum.)
• How does changing the length of the bob affect the period of oscillation?
(The longer the length of the bob, the longer the period of oscillation will be.)
• What is meant by the acceleration of gravity? Is the acceleration of gravity always the same on earth?
(The acceleration of gravity is the force gravity exerts on an object. The force of gravity will always be the same on earth. The force of gravity on other planets will be different from earth's force of gravity.)
• How does changing the acceleration of gravity affect the period of oscillation?
(Increasing the acceleration of gravity increases the period of oscillation.)
• How does changing the starting point or angle affect the period of oscillation?
(Increasing the angle increases the period of oscillation.)
• What happens if you start the pendulum in an upside down position of 180 degrees?
(The pendulum will not move.)

At this point, students should understand that gravitational forces cause the pendulum to move. They should also understand that changing the length of the bob or changing the starting point will affect the distance the pendulum falls; and therefore, affect its period and frequency.

Constructing a Pendulum/Testing Falling
Now that students have an understanding about the variables that affect a pendulum's period and frequency, they can create their own pendulum to test these concepts.

Give each student a copy of the Exploring Pendulums, which includes Predictions, Materials, Procedure, Data Table, and Analysis Questions.

Divide students in cooperative groups of two or three to work together to complete this activity. As outlined, students will first make predictions and then construct and test controlled-falling systems, or pendulums, using the materials listed and following the directions on the worksheet.

This controlled-falling system is a weight (bob) suspended by a string from a fixed point so that it can swing freely under the influence of gravity. If the bob is pushed or pulled sideways, it can't move just horizontally, but has to move on the circle whose radius is the length of the supporting string. It has to move upward from where it started as well as sideways. If the bob is now let go, it falls because gravity is pulling it back down. It can't fall straight down, but has to follow the circular path defined by its support. This is "controlled falling": the path is always the same, it can be reproduced time after time, and variations in the set-up can be used to test their effect on the falling behavior.

Note: Make sure that the groups understand that by changing the value of only one variable at a time (mass, starting angle, or length), they can determine the effect that it has on the rate of the pendulum's swing. Also, students should be sure the measurements with all the variables are reproducible, so they are confident about and convinced by their answer.

After students have completed the experiments, discuss their original predictions on the activity sheet and compare them with their conclusions based on the data and the results of the tests.

Students should have been able to arrive at the following conclusions:

• Heavier and lighter masses fall at the same rate.
• Increasing the angle, or amplitude, increases the distance that the bob falls; and therefore, the frequency, or number of back and forth swings in a set time frame will be less.
• Increasing the length of string to which the bob is attached, increases the radius of the circle on which the bob moves; and therefore, the frequency, or number of back and forth swings in a set time frame, will be less.

Older students should probably learn how the downward force of gravity on the bob is split into a component tangential to the circle on which it moves and a component perpendicular to the tangent (coincident with the line made by the supporting string) and directed away from the support. The tangential force moves the bob along the arc and the perpendicular force is exactly balanced by the taut string.

Now, based on these observations, determine what conclusions students can make about the nature of gravity. (Students should conclude that gravitational force acting upon an object changes its speed or direction of motion, or both. If the force acts toward a single center, the object's path may curve into an orbit around the center.)

### Assessment

Assess the students' understanding by having them explore the Pendulums on the Moon lesson, found on the DiscoverySchool.com website. Students should click the link for "online Moon Pendulum," found under the "Procedure" section of the lesson. This activity simulates the gravitational force on the moon. Students should experiment for approximately 5-10 minutes, changing the mass, length, and angle to observe the effect it has on the pendulum.

Instruct students to change only one variable at a time. Then, ask students these questions:

• How do you get the quickest swing?
(Shorten the length of the string and decrease the angle.)
• How do you get the longest swing?
(Increase the length of the string and increase the angle.)
• In your own words, describe the relationship between mass, length of string, and angle.
(Mass does not affect the pendulum's swing. The longer the length of string, the farther the pendulum falls; and therefore, the longer the period, or back and forth swing of the pendulum. The greater the amplitude, or angle, the farther the pendulum falls; and therefore, the longer the period.)
• How does the force of gravity on the Moon compare with the force of gravity on Earth? What effect do you think the difference in gravitational forces would have on the pendulum?
(The force of gravity is less on the moon than on the Earth. Since the force of gravity is less on the Moon, the pendulum would swing slower at the same length and angle and its frequency would be less.)

### Extensions

Make Coupled Resonant Pendulums
This experiment demonstrates that two pendulums suspended from a common support will swing back and forth in intriguing patterns if the support allows the motion of one pendulum to influence the motion of the other. The directions for this experiment are on the Exploratorium website.

Measuring Falling Time
When Galileo was studying medicine at the University of Pisa, he noticed something interesting about the periods of a pendulum. In church one day, he watched a chandelier swing back and forth in what seemed like a steady pattern of swings. He timed each swing and discovered that each period was the same length (same amount of time). In the previous activity, students measured the periods of their pendulums using either digital watches or stopwatches. Galileo did not have these tools, so he used his pulse. In this activity, students will time the periods of their pendulums using their pulses and compare their results with those obtained with a watch.

Show students how to find their pulse by pressing two fingers on the artery next to their wrist. Make sure that students have been at rest for several minutes before doing this so that they can obtain a steady pulse rate. Working in teams, have one student set the pendulum in motion while another measures the pulse beats that occur during five complete swings and then ten complete swings. Students should reproduce the distances they used in the earlier experiment, Testing Falling, for the amplitude and length of string. Record the number of pulse beats. Repeat this procedure with different students measuring their pulse rates. Then have students measure and record five complete swings and ten complete swings using a stopwatch or digital watch.

Share each group's results with the entire class. How do the pulse beat measurements compare with those timed with a watch? What are the advantages of using a stopwatch or digital watch over counting pulse beats as a method of timing?  