TI-Nspire™ Activities: Physics
May 13, 2008 through May 20, 2008
Concave Mirrors
| Problem 1 – Image formation in a concave parabolic mirror | |
| Step 1: Students should open the file PhysWeek16_concavemirrors.tns and read the first two pages. Page 1.3 contains an object (Object) and a simulated concave parabolic mirror (Mirror). The height and location of the object can be altered. Students should explore the simulation by changing the size and location of the object and observing how those changes affect the object's image (Image). Note that the image seems to disappear when the object is moved to within a certain radius of the focal point (fp). This occurs because the image of an object so close to the focal point is too large to be shown on the screen. (The image size and distance from a mirror approach infinity as an object approaches the focal point of a mirror.) |
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| Step 2: Next, students should answer questions 1–7 on pages 1.4–1.7. Encourage students to refer back to the simulation as they answer the questions. Q1. Describe the image formed when the object is farther from the mirror than the focal point (fp). A. The image is on the same side of the mirror as the object. It always appears farther from the mirror than the focal point. The image is real and inverted. Q2. Describe the image formed when the object is between the focal point and the mirror. A. The image appears on the other side of the mirror from the object. It is as large as the object or larger. The image is virtual and upright. Q3. Under what conditions is the image a real image? A. When an object is farther from the mirror than the focal point, a real image forms. Students may struggle with the difference between real and virtual images. One way they can remember the difference is to remember that a real image can be projected on a screen. A virtual image, in contrast, cannot be projected on a screen. Q4. Under what conditions is the Image a virtual image? A. When an object is between a mirror and the focal point, a virtual image forms. Q5. Under what conditions is the image smaller than the object ? A. When the distance between an object and a mirror is greater than approximately twice the focal length, the image appears smaller than the object. The farther away the object moves, the smaller the image gets, and the more the image moves toward the focal point. You may wish to explore this idea with students further to help them understand how these observations can lead to the conclusion that a parabolic mirror focuses light rays coming from a great distance toward the focal point. Note that students have not taken any formal measurements at this point, so words such as approximately are appropriate in this context. These conjectures can be verified later as quantitative relationships are determined. Q6. Under what conditions is the image the same size as the object? A. When the distance between an object and a mirror is twice the focal length of the mirror, the image and object appear to be the same height. Q7. Under what conditions is the image larger than the object? A. There are two conditions under which an image is larger than an object. When the object is between one and two focal lengths from the mirror, the image is larger than the object, but it is inverted. When the object is between the focal point and the mirror, the image appears larger than the object. However, in this case, the image is upright. |
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| Problem 2 – Establishing quantitative relationships between object and image | |
| Step 1: Students should now move to pages 2.1 and 2.2 and read the description of the next simulation. Page 2.3 contains a diagram similar to that on page 1.3. However, page 2.3 includes additional information on the dimensions of the simulation, as follows: f = focal length Ho = Object height So = distance between Object and fp Hi = Image height Si = distance between Image and fp |
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| Step 2: Page 2.4 is a Lists & Spreadsheet application that contains columns to store data on the variables listed on page 2.3. The variables f, Ho, Hi, So, and Si have all been set to manual data capture in this application. Students should drag the object on page 2.3 through a series of locations, all farther from the mirror than the focal point. As students move the object, they should press /^ to capture the values of f, Ho, So, Hi, and Si at various points. Students should capture at least 20 data points from a wide range of object locations. Note: Students must not capture data when So is negative, zero, or undefined, or the upcoming calculations will not work correctly. In the Lists & Spreadsheet application on page 2.4, values of f are stored in variable focal (Column A), values of Ho are stored in variable objh (Column B), values of Hi are stored in variable imgh (Column C), values of So are stored in variable objd (Column D), and values of Si are stored in variable imgd (Column E). |
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| Step 3: Next, students should graph the distance between the image and the focal point (imgd) vs. the distance between the object and the focal point (objd) for the data points they captured. They should use the Data & Statistics application on page 2.5 to make the graph. |
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| Step 4: Next, students should use the Regression tool (Menu > Actions > Regression) to attempt to fit different curves to the data. They should then answer questions 8 and 9 on page 2.6. Q8. Describe the relationship between objd and imgd (So and Si). A. The relationship is an inverse nonlinear one. Encourage students to suggest possible mathematical relationships, such as hyperbolic relationships, that may match the data. After using the Regression tool to try different curves, students should conclude that a Power curve best fits the data. Q9. Write the equation for the curve that best fits your data. A. For the initial focal length, the equation is Si = 25.6•So–1. |
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| Step 5: Next, students should change the focal length of the mirror by moving point fp on page 2.3. Then, they should delete the data they recorded on page 2.4 and repeat the data-collection procedure from step 2. Finally, they should answer questions 10 and 11 on page 2.8. Note: When deleting data, students should make sure to select only the numbered rows, not the whole spreadsheet. If they delete the formulas or variable names in the formula and title boxes of the spreadsheet, the data capture will not work properly. Q10. Write the equation that fits your new data set. Record the new focal length (f), as well. A. The equation and focal length will vary from student to student. If you wish, you may assign different students in the class to collect data for different focal lengths, and then have the class compare their data. Q11. Based on the two equations you have recorded, what is the general relationship between So, Si, and f? A. After comparing their data, students should realize that the general relationship is Si = f2•So–1. |
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| Step 6: Next, students explore the magnification provided by the mirror. They should read pages 2.9 and 2.10. Then, they should return to the spreadsheet on page 2.4 and use Column F to calculate the magnification of each data point collected. Students should assign this column to the variable mag. Note: Make sure students enter the magnification equation in the formula bar (gray box) of Column F and the variable name in the label bar (white box) of Column F. |
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| Step 7: Next, students should graph mag vs. objd for their data on page 2.11. They should use the Regression tool to find the equation of the best-fit curve for the data. Then, they should answer question 12 on page 2.12. Q12. What is the relationship between magnification and object distance? A. The relationship is mag = f•So–1. Students should recognize the focal length in the regression equation. Encourage students to compute the result from the previous question as a verification that this relationship is valid. You can also encourage students to establish other relationships using algebraic manipulation. They can then test these relationships using the data capture method. |
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