Modeling Prompt 5
Swing Time
Algebra 2
In-Class Launch
Use after Unit 6, Lesson 9
Before class, prepare a demonstration pendulum, using a washer and a piece of string.
Arrange students in groups of 2–4. Distribute to each group:
- 4 washers (or other appropriate objects) of different weights
- string (at least 3 feet)
- scissors
- a stopwatch
A pendulum is a weight suspended from a pivot point so that it may freely swing. Demonstrate, using your prepared sample pendulum. Simply hold the top of the string between your fingers, or tie the string around a pencil to allow the pendulum to swing more freely.
Ask, “Where have you seen pendulums before?” (clocks, swings, wrecking ball)
Tell students that the period of a pendulum is the time it takes for one full swing, back and forth. Demonstrate by pulling the weight back, letting it swing forward and back, and then catching it.
Ask students what they think you could change about your pendulum to change its period. Some possibilities are: the amount of weight at the end, the length of the string, or how far you pull the weight back before letting go. Record the different potential variables that students mention, for all to see. Tell students that they will run experiments to figure out which variables affect the period.
Teacher Instructions
Students, who test the effect of the angle of release, might try extreme angles that won’t set the pendulum in a predictable swinging motion. Encourage them to use angles less than \(90^\circ\) from the vertical, which result in a smooth swinging motion, because modeling more extreme angles would not be reasonable.
Consider pausing the class for a discussion after all students have decided which variable has the greatest effect on the period. Students’ best use of time is to focus on creating a model for the relationship between the length and the period.
It is possible that students choose to place the period on the horizontal axis and the length on the vertical axis. Their analysis will work, and the choice of quantities to use as the independent and dependent variables belongs to the modeler. In such a case, if \(x\) represents time, in seconds, and \(y\) represents length, in inches, then best model (for the sample data) is \(y=11.42(x-0.17)^2+0.42\).
In the early 1600s, Galileo began to study pendulums after noticing that a chandelier in the Tower of Pisa had a regular swinging motion. He later figured out how to design a pendulum that swung back and forth in exactly two seconds, which allowed people to build clocks that used pendulums to keep time. The time it takes a pendulum to complete one back-and-forth swing is called its period.
- What variables might affect the period of a pendulum?
- Collect data. Which variable appears to have the greatest effect on the period of the pendulum? Justify your response.
- Create a mathematical model, relating the variable you identified to the period of a pendulum.
- Think carefully about how you decided on the number of digits to record in your measurements. Explain these decisions.
- Use your model to determine the characteristics of a pendulum with a period of 2 seconds.
- Would it be possible to create a pendulum with a period of 1 minute? 1 hour? If so, what would you need to create these pendulums? If not, why not?
- Think carefully about how you decided on the number of digits to include in the lengths of the 2-second, 1-minute, and 1-hour pendulums. Explain these decisions.