Kinetic Pendulum: What's the Relationship Between Distance and Swing Time?

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Author: Bob Bonnet and Dan Keen

Purpose or Problem

The purpose is to understand one of the principles of pendular motion.


A pendulum is a weight hung by a tether (a rope, string, or rod) from a fixed point, and made to swing. When the pendulum is pulled away from its motionless hanging state (perpendicular to the Earth), the weight gains potential or stored energy. When released, the potential energy is turned into kinetic or working energy.

Once released, the pendulum is pulled down toward the Earth by gravity, but it does not stop when it returns to the Earth's perpendicular plane (called plumb). At that point, the moving pendulum has momentum (mass multiplied by velocity), which causes it to continue to swing past the plumb point, until the force of gravity slows it to a stop. The pendulum then swings back through the plumb and up to the point where it was first released. This swing out and back is called one oscillation period. Then, once again, gravity continues its effect, and the pendulum continues to swing back and forth.

If it were not for the friction with air against the pendulum and the friction at the point where it is secured to a fixed point, the swinging would continue indefinitely.

Many early scientists, including Lord Kelvin (1824–1907), Jean Foucault (1819–1868), and Galileo (1564–1642), devoted time to the study of the natural laws of pendular motion. Galileo was reported to note, while sitting in church, that a chandelier swung with the same time period, regardless of whether it was swinging in a small arc or a large arc (the sermon must not have been very interesting that day!). This project will attempt to prove this natural law of pendular motion discovered by Galileo.


Hypothesize that the swing period of a pendulum with a fixed rope length is the same, regardless of the arc distance traveled. (Because of air resistance and other factors, we will state this hypothesis is true for the first five oscillation periods of our constructed apparatus.)

Materials' List

  • Two bowling balls of the same weight
  • Two plastic bags with handles (used at grocery and retail stores)
  • Rope
  • String
  • Child's outdoor swing set
  • Yard stick or tape measure
  • Large, heavy metal washer
  • A day with negligible or no wind
  • A friend to assist
  • Possible adult supervision needed


The mass of the bowling balls, the tape measure, the length of the ropes, and the environment (temperature, humidity, wind) are held constant. The distance each ball is pulled back from the plumb line is varied.

Find two bowling balls of equal weight and set each one in a plastic bag, the kind used at grocery and retail stores to carry products home. Although these bags are very thin, they are strong and have convenient handles. Bowling balls can be hazardous if they fall on your feet. Pay extra attention and take safety precautions when you work with the bowling balls. Place them on the ground, never on a table where they could unexpectedly roll off.

Tie a long piece of rope through the two handles on one of the bags. Tie another long piece of rope through the handles on the other bag.

Tie a long piece of string onto a heavy metal washer. From a child's backyard swing set, tie the other end of the string to the top pipe, letting the washer hang about one or two inches from the ground. Be sure the washer hangs freely and does not touch any of the swings.

Similarly, tie the two bowling balls in their bags from the top pipe. Be sure they hang freely and do not touch any of the swings or each other. Using a tape measure, make the distance from the top pipe to the top of each bowling ball exactly the same length.

The washer on a string acts as a plumb line, also called a plumb bob, which is a weighted line that is perpendicular to the ground.

Pull one of the bowling balls back about four feet from the plumb line. Have your friend pull the other ball back about one foot. On the count of three, both of you should let go of the balls at the same time. It is important for both of you to let go simultaneously.

Notice that even though your ball has farther to travel, it will cross over the plumb-line point at the same time as the ball your friend let go.

Watch the balls swing through five periods, and note they are still hitting the plumb line at the same time, proving the hypothesis correct.

Because of other variables, including friction with the air (one ball moves through more air than the other and, thus, experiences more friction), the balls may eventually stop meeting at the plumb point.

You may want to measure the distance the bowling balls travel by measuring the length of the arcs. When the ball is pulled back one foot from plumb, how many degrees is the angle from plumb? How many degrees is the angle when the ball is pulled back four feet?


Write down the results of your experiment. Document all observations and data collected.


Come to a conclusion as to whether or not your hypothesis was correct.

Something More
  1. A common natural law of gravity and astronomy (celestial mechanics) that also applies to pendulums is the inverse-square law, which states the following: if one pendulum is twice as long as another, the longer one will have a period that is "one over the square of two," or one fourth, as fast:

Prove this expression by experimentation.

  1. Pendular mechanisms have been used throughout history to keep time. Construct a pendulum that completes one period in one second (clue: the length of the string should be about 39.1 inches).
  2. Research the work of the English scientist Lord Kelvin and his discoveries with bifilar pendulums (having two strings instead of one).
  3. Research the work of the French scientist Foucault, who used a large iron ball on a wire to show that the Earth rotates.
  4. Could you use pendulums or plumb lines to detect earthquakes or other vibrations in the Earth?
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