Simple Harmonic Motion: The Swinging Pendulum
A pendulum undergoes a type of motion that is predictable. The consistency of pendulum motion has allowed it to be used to drive the timing mechanism of clocks. In this experiment, you investigate what causes a pendulum to swing faster or slower. At least for a pendulum on the surface of the Earth, only one variable determines the time it takes for a pendulum to swing back and forth one time.
What You Need
- several masses that can be attached to a string (such as 20 g, 50 g, 100 g, 200 g)
- several strings of varying lengths from 0.1 to 1.0 m (strong enough to support the masses)
- support for each pendulum
- Set up a basic pendulum with a measured length and mass free to swing.
- Pull the pendulum back to the side through a small (less than 15 degrees) angle and get the stopwatch ready.
- Release the pendulum and start the stopwatch as the pendulum is released.
- Count ten cycles back and forth. Cycle number one is when the pendulum returns to its original position. Be careful not to count "one" when the pendulum is released.
- The length of the pendulum is the distance from the point where the string is supported to the center of the mass.
- Record the time (in seconds) for the pendulum to complete ten complete cycles.
- Divide the time for ten cycles by ten to get the time for one cycle or the period of the pendulum for the conditions you are testing.
- You can proceed in several ways at this point, with many opportunities to develop your own plan. Here are a few suggestions:
- What variable matters: Mass? Length? Angle? Test the selected variable while holding the others constant. For instance, test light, medium, and heavy mass, and then determine whether the period of the pendulum is dependent on mass. This can be done by measuring the period of a pendulum constructed with each of the three masses. It can also be done qualitatively by setting up three pendula and observing how fast they swing compared to each other.
- Once you determine which variable(s) affects how fast the pendulum swings, you can set up an experiment to measure how the period changes over a range of the variables you selected. The other variables should be kept constant.
The only variable that affects the period of a pendulum is length. The mass does not matter at all. For angles smaller than 15 degrees, angle is insignificant. Insignificant means less than 1 percent.
The longer the string, the longer the period (period is the time to go back and forth one time).
The dependence of period on length is not linear.
A graph of period versus length is shown in Figure 66-2. The model for the graph shows the period is dependent on the square root of the length.
Why It Works
The period of a pendulum is the time it takes for the pendulum to move from one position and return to the same position. The period of a pendulum (in seconds) is given by:
where L is the length of the string (in meters) and g is the gravitational acceleration (9.8 m/s2).
This shows the dependence on the square root of the string length. Because there is no mass in the equation, the period does not depend on mass. The period also depends on the gravitational acceleration of the Earth, which under normal circumstances is not a variable.
Other Things to Try
Try this with a pendulum, consisting of a bowling ball attached to a rope. Make sure the point of attachment and the rope can securely handle the weight of the swinging pendulum.
Try this with a playground swing. Is the natural frequency of oscillation what you predict based on the previous equation? What happens if you push with a rhythm consistent with that natural frequency? What happens if you push with a rhythm very different from the natural frequency?
The period of a pendulum depends on only one variable, which is its length.
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