TI-Nspire™ Activities: Physics
January 29, 2008 through February 5, 2008
Heat and Thermodynamics
Part 1: Collecting temperature data |
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| Step 1: Students will use a Vernier EasyTemp™ or Go!™Temp temperature sensor to collect temperature data. When they open the .tns file, they may get the error message “The expected data collection device was not found.” Instruct them to select OK to continue with the activity. When students reach page 1.3, they should connect the temperature sensor to their handhelds or computers. This should activate the temperature sensor, and a temperature display should appear in the data collection box. Q1. What is the ambient temperature? A. The ambient temperature will vary. Make sure students' values are reasonable. |
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Step 2: After making sure all students are wearing safety goggles, give each student approximately 100 mL of boiling water in an insulated container. Students should place the metal ends of their temperature sensors into the water and wait for the temperature reading to stabilize. Then, they should remove the sensor from the water and wipe it off. |
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Step 4: Next, students will advance to page 1.4 and make a scatter plot of the temperature and time data. Q2. Describe the shape of the graph. A. The data curve downward, indicating that temperature decreased over time at a nonuniform rate. The curve should be approximately exponential, as shown. |
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Part 2: Fitting a curve to the data |
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Step 1: Next, students will attempt to fit an exponential curve of the form y = a + b • cx to their data. Q3. Should c be greater than, less than, or equal to 1? Explain your answer and give your prediction for the value of c. A. The variable c should be less than 1 because temperature decreases as time increases. Students may struggle with this reasoning. You may wish to give them several simple examples (e.g.,  ) to illustrate why a decreasing curve implies a base that is less than 1. Their predictions about the value of c will vary.Q4. What value should a have? (Hint: What will happen to the temperature of the sensor as time approaches infinity?) A. The variable a should be equal to the ambient temperature. You can help students understand why this is so by reminding them that the temperature of the sensor will eventually (i.e., as time approaches infinity) equal the ambient temperature and showing them how the equation requires that y approach a as x approaches infinity. Q5. What value should b have? A. The variable b should equal the initial temperature of the sensor minus the ambient temperature. |
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Step 2: Next, students define the variables a, b, and c using the text box tool, the h button, and the Store Var command. |
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| Step 3: Next, students set the function f1(x) equal to a + b • cx. They change the graph to a Function graph and enter the expression in the function line. They should then vary the value of c to obtain the best possible fit to their data set. Q6. What value of c gave you the best fit to your data? A. The value of c will vary from student to student, but it should be very close to 1. |
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Part 3: Exploring the relationship between cooling rate and temperature difference |
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Step 1: Students will now define the function slope(x) as the derivative of f1(x). Students may have a hard time understanding the concept of the derivative. You may wish to discuss the idea of infinitesimally small intervals with them, or show a few simple examples. Note: Make sure that students use the := notation (not just =) when defining the slope function. The = notation will cause the handheld to calculate and display the actual equation for slope(x). |
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Step 2: Next, students set up a Lists & Spreadsheet application to store slope and temperature-difference data before plotting them. They set column A equal to the time data collected in part 1 and column B equal |
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| Step 4: Next, students use a function to fill column D with temperature-difference data. They name the series tdiff. Note: Make sure students use their ambient measured temperatures in the tdiff formula (i.e., that they do not just copy the formula shown in the worksheet). |
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| Step 5: Next, students make a scatter plot of slope versus temperature difference. Q8. Describe the shape of the graph. A. The data lie on a nearly straight line. Q9. What does the shape of the graph tell you about the relationship between cooling rate and temperature difference? A. The shape of the graph implies that cooling rate decreases linearly as temperature difference decreases. Q10. If the graph were extended, what would its x- and y-intercepts be? What does this tell you about the relationship between cooling rate and temperature difference? A. The graph appears to pass through the origin. This implies that, when the temperature difference is 0, the cooling rate is also 0—exactly what would be expected. If students have a hard time with this concept, remind them that an object will never cool to below the ambient temperature. Q11. Which would you expect to cool more quickly, a 90ºC sensor in a 10ºC room, or a 50ºC sensor in a 20ºC room? Explain your answer. A. A 90ºC sensor in a 10ºC room will cool more quickly than a 50ºC sensor in a 20ºC room because the temperature difference is greater in the first example. |
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