TI-Nspire™ Activities: Physics
March 25, 2008 through April 1, 2008
The Period of a Pendulum
| Part 1 – Exploring the length-period relationship | |
| Step 1: Students should open the file PhysWeek09_pendulum.tns and read the first four pages. Page 1.5 contains an animation of a pendulum. In the simulation, students will change the length (l) of the pendulum, and the program will calculate the period (time) for them. Students may use the animation to observe the changing periods of the pendulum. Note: The animated pendulum has a period longer than that displayed (e.g., when time = 1.554 sec, the animated pendulum will take longer than 1.554 sec to complete one oscillation). |
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| Step 2: Students should record the value of time for each of the values of l in the Lists & Spreadsheet application on page 1.6. Students should observe that increasing the length increases the period. They should enter the time data from page 1.5 into the spreadsheet on page 1.6. |
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| Step 3: Next, the students should create a scatter plot (/b > Scatter Plot) of the data on page 1.7. They should use length for the x-values and period for the y-values. Discuss the shape of the graph with the students. Some students will probably assume that this is a linear relationship. Ask them how they could determine whether the relationship is linear. They should reason that a linear regression can provide information about the relationship. You may need to guide them to this conclusion. |
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| Step 4: Next, students should move back to page 1.6 and use a linear regression (Menu > Statistics > Stat Calculations > Linear Regression (mx + b)) to find an equation that fits the data points. Students should use length for the X List and period for the Y List. They should store the values in Column C of the Lists & Spreadsheet application on page 1.6. |
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| Step 5: Next, students should move to page 1.7 and plot their best-fit lines on the scatter plot of period vs. length. To do this, they should change the plot type to Function and then select the function in which the best-fit line from the linear regression is stored. Students should pay particular attention to the y-intercept of the best-fit line. Students should then answer questions 1–3 on pages 1.8 and 1.9. The questions, and their answers in italics, are given below. Discuss the calculated value of the intercept (zero length) with students. They should conclude that the relationship is not, in fact, a linear relationship. Q1. What does your calculated linear regression model predict for the period of a pendulum with zero length? A. 0.848 sec Q2. What should the period of a pendulum of zero length be? A. 0 sec Q3. Based on your linear regression model, is the relationship between period and pendulum length linear? Explain your answer. A. The linear regression does not yield the correct period for a zero-length pendulum. Therefore, the relationship between period and pendulum length is probably not linear. |
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| Part 2 – Building a correct model for the relationship | |
Step 1: Next, students will attempt to find the best-fit equation of the form  to model the data. Students should read page 1.10 and then move on to the Lists & Spreadsheet application on page 1.11. This spreadsheet has the square root of the length in Column A and the period in Column B. Students should perform a linear regression with rootl for the X List and period for the Y List. Students should store the values in Column C of the Lists & Spreadsheet application on page 1.11. |
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| Step 2: Next, students should create a scatter plot of period vs. rootl. They should also plot the best-fit line for the data. For this plot, students should not resize the window. They should examine the y-intercept of the best-fit line and use their observations to help them answer questions 4–6 on pages 1.13 and 1.14. Q4. What is the slope of the best-fit line relating period and rootl? A. 0.2004 sec/cm1/2 Q5. Does this best-fit equation correctly predict the period of a pendulum with zero length? A. Yes. Q6. The standard equation for period is as follows:  where L is the pendulum length and g = 9.807 m/s2. Do your results confirm this relationship? Show your work. A. Setting the best-fit expression for period equal to the ideal expression for period yields the following:  The two models agree well. Note that some students will forget to use the correct units; remind them that the best-fit equation uses centimeters for length, but the value of g given in the problem uses meters. |
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