TI-Nspire™ Activities: Physics
April 15, 2008 through April 22, 2008
Damped and Driven Harmonic Motion
| Problem 1 – Damped simple harmonic motion | |
| Step 1: Students should open the file PhysWeek12_DampedSHM.tns and read the first three pages. Page 1.4 shows a graph of displacement vs. time for a damped simple harmonic oscillator. The variable m represents the damping coefficient for the oscillator. Students should vary m and observe the effects on the motion of the oscillator. Then, they should answer questions 1 and 2. Q1. How does the value of m affect the shape of the curve? A. The larger m becomes, the more quickly the motion of the oscillator is damped. Q2. Imagine a line connecting the peaks of the curve. What form would that line have? A. Students may recognize that the peaks of the curve decrease along approximately an exponential curve. If they do not, you may wish to guide them to this observation. |
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| Step 2: Next, students should use the Graph Trace tool (Menu > Trace > Graph Trace) to identify the coordinates of successive peaks of the curve. After selecting the Graph Trace tool, students should move the cursor to the leftmost peak of the curve. The TI-Nspire will display a capital M when the cursor is on the peak of the curve. Students should record the x- and y-coordinates of this peak. They should then move the cursor to the next peak to the right and record its coordinates. They should continue in this way until they have recorded the coordinates of all the peaks that are visible on screen. |
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| Step 3: Next, students should move to the Lists & Spreadsheet application on page 1.6. They should enter the x- and y-coordinates they recorded for the curve peaks into the spreadsheet. They should enter the x-values into the time column and the y-values into the amp column. Note that students will have different x- and y-values for their maxima, depending on the value of m that they have set on page 1.4. |
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| Step 4: Next, students should make a scatter plot of amp vs. time on page 1.7. Students should discuss the shape of the curve and attempt to identify its functional form. Encourage students to change the plot to a Function plot and plot various functions of x in an attempt to fit the data points. Note: When students graph ex, make sure that they use the u key or type exp(–m•x), and not e–m•x where e is the E key, into the function bar. Then, students should answer questions 3–5. Q3. What function best fits the graph of amplitude vs. time? A. Students should realize that the curve is exponential with a negative exponent (i.e., of the form y = e–x). You may need to provide guidance to the students to help them realize this. Q4. Was your prediction in question 2 correct? If not, explain any errors in your reasoning. A. Student answers will vary. Encourage discussion of why the students predicted the shapes they did. Q5. Predict the form of the equation that describes damped simple harmonic motion. Explain your answer. A. The equation will have the form  . Remind students that undamped simple harmonic motion is described by functions of the form y(x) = A sin x, where A is the amplitude of the motion. In this case, the amplitude of the curve decreases according to an exponential curve (e–x). |
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| Problem 2 – Driven, damped simple harmonic motion | |
| Step 1: Next, students should read page 2.1. The graph on page 2.2 shows displacement vs. time for a spring that is driven by a motor. The spring's fundamental frequency is represented by the variable fund, and the driving frequency of the motor is represented by the variable w. The amplitude of the spring's displacement is represented by the variable A. Before altering the graph on page 2.2, students should answer question 6. Q6. Predict the value of w that will produce the largest amplitude (A) for the spring, assuming that fund is fixed at 7. A. Student answers will vary. |
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| Step 2: Next, students should increase the value of w incrementally (increments of 0.1 will yield the best results). They should record the amplitude that each value of w yields. Students should increase the value of w from 6 to approximately 7.5. |
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| Step 3: Next, students should move to the Lists & Spreadsheet application on page 2.4. In column A (variable w1), they should enter the incremental values of w that they recorded in step 2. In column B (variable diff), they should use a formula to calculate the difference between fund and the values of w they entered (i.e., diff = fund – w). To do this, students should type =7-a[] into the formula bar of column B. In column C (variable amp), they should enter the amplitude of the motion associated with each frequency difference. |
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| Step 4: Next, students should create a scatter plot of amp vs. diff on page 2.5. They should resize the graph (Menu > Window > Zoom - Data). | 
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| Step 5: Next, students should move back to page 2.2 and change the value of m, the damping coefficient, to 0.38. They should then reset w to 6 and repeat step 2, using the same values of w that they used in step 2. Then, they should move to page 2.4 and enter the amplitudes they recorded for each value of w into column D (variable amp2). |
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| Step 6: Next, students should move to page 2.5 and plot amp2 vs. diff on the same scatter plot with the graph of amp vs. diff. They should compare the two plots and then answer questions 7–9. Q7. What value of diff produced the maximum value of A in each case? A. For m = 0.335, the maximum amplitude occurs when w is approximately 6.8. For m = 0.38, the maximum amplitude occurs when w is between 6.9 and 7. Q8. Was the prediction you made in question 6 correct? If not, explain any errors in your reasoning. A. Student answers will vary. Most students will probably have predicted that the maximum amplitude would occur at fund = w. The graph will show that the maximum amplitude actually occurs when w is slightly less than fund. Encourage students to discuss why they made the predictions they made. Q9. What do you think causes the maximum amplitude to occur when fund and w are not equal? A. Student answers will vary. The maximum amplitude occurs when fund and w are not equal because of the effects of damping on the oscillator. If the oscillator were not damped, the amplitude would approach infinity when fund = w. |
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