Modeling Prompt

Some students who test the effect of the angle of release might try some extreme angles that won’t work for setting the pendulum in a predictable swinging motion. Encourage them to stick to angles less than \(90^\circ\) from vertical that result in a smooth swinging motion, because it would not be reasonable to try and model more extreme angles.

Consider pausing the class for a discussion after all students have made a decision about the variable that has the greatest effect on the period. It would be the best use of time for students to focus on creating a model for the relationship between length and period.

It is possible that students might choose to place period on the horizontal axis and length on the vertical axis. Their analysis will work out, and which quantities to use as the independent and dependent variable is the choice of the modeler. In that case, if \(x\) represents time in seconds and \(y\) represents length in inches, the best model (for the sample data) would be \(y=11.42(x-0.17)^2+0.42\).

In the early 1600s, Galileo began to study pendulums when he noticed that a chandelier in the Tower of Pisa had a regular swinging motion. He later figured out how to design a pendulum that took exactly two seconds to swing back and forth, 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.

1. What are some variables that might affect the period of a pendulum?
2. Collect some data. Which variable appears to have the biggest effect on the period of the pendulum? Justify your response.
3. Create a mathematical model relating the variable you identified to the period of a pendulum.
4. Think carefully about how you decided how many digits to record in your measurements. Explain these decisions.
5. Use your model to determine the characteristics of a pendulum that would have a period of 2 seconds.
6. 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?
7. Think carefully about how you decided how many digits to include in the lengths of the 2-second, 1-minute, and 1-hour pendulums. Explain these decisions.
    

Note that if you are using this version of the task, the third question sort of gives away the answer to the second question. You might decide to pose the first two questions to students and give them time to work on them before distributing the rest of the questions.

If students have trouble getting started with their model, it may help to give them the general form of each type of equation so they can experiment with different values:

• Linear:  \(y = a \boldcdot x + k\)
• Quadratic: \(y = a \boldcdot (x - h)^2 + k\)
• Square root: \(y = a \boldcdot \sqrt{x - h} + k\)

In the early 1600s, Galileo began to study pendulums when he noticed that a chandelier in the Tower of Pisa had a regular swinging motion. He later figured out how to design a pendulum that took exactly two seconds to swing back and forth, 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.

1. What are some variables that might affect the period of a pendulum?
2. Collect some data. Which variable appears to have the biggest effect on the period of the pendulum? Justify your response.
3. Build a pendulum where you can adjust the length by making the string longer or shorter. Use your stopwatch to time the period of a pendulum with each length in inches: 30, 20, 15, 10, 5.0, 3.0, 1.0. Record the period of each pendulum in an organized way. 
4. Think carefully about how you decided how many digits to record in your measurements. Explain these decisions.
5. Use technology to create a scatter plot of your data, placing length in inches on the horizontal axis and time in seconds on the vertical axis.
6. Which type of model seems like it would be the best fit: linear, quadratic, or square root? Create a model that fits the data.
7. Use your model to determine the characteristics of a pendulum that would have a period of 2 seconds.
8. 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?
9. Think carefully about how you decided how many digits to include in the lengths of the 2-second, 1-minute, and 1-hour pendulums. Explain these decisions.