TI-Nspire™ Activities: Physics
February 26, 2008 through March 4, 2008
Forces on Point Charges
| Problem 1 – Graphical vector addition | |
| Step 1: In this part of the activity, students use TI-Nspire features to practice graphical methods of vector addition. First, students should open the file PhysWeek05_PointCharges.tns and read the first two pages. They should then proceed to page 1.3, which contains an empty Graphs & Geometry page. Students should change the page to the Plane Geometry view (Menu > View > Plane Geometry View) and hide the scale (Menu > View > Hide Scale). It is recommended that students change the document settings for the angle measurements to be in degrees for this activity. To change the document settings, students should press /© to enter the page sorter view, press b to select the menu, and then select Document Settings. |
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| Step 2: Next, students use the Vector tool (Menu > Points & Lines > Vector) to draw two coterminal vectors, AB and AC, on page 1.3. |
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| Step 3: Next, students should use the Translation tool (Menu > Transformations > Translation) to translate vector AB to point C. They should label the image of point B as D using the Text tool. |
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| Step 4: Next, students construct vector AD, the sum of AB and AC. They then find the magnitude and direction of vector AD using the Length and Slope measurement tools from the Measurement menu. They should then answer questions 1 and 2. Q1. How does the length of vector AD compare with the lengths of vectors AB and AC when AB and AC are collinear? A. When AB and AC are parallel, AD is equal to the sum of the lengths of AB and AC. Q2. How does the direction (slope) of vector AD compare with the slopes of vectors AB and AC? A. The slope of AD is partway between the slopes of AB and AC. Students can use the Slope measurement tool to observe the direction of the resultant vector, or they can find the angle of the vector to the horizontal. Since the slope of the line is equal to the inverse tangent of the angle, students can use the Text and Calculate tools to find the angle of the resultant vector. The instructions for using this method are given below: • Choose the Text tool (Menu > Actions > Text). • Click on a blank space anywhere on the screen. The text box will open in the edit mode. • Enter the expression tan-1(s) and press •. • Choose the Calculate tool (Menu > Actions > Calculate). • Click on the expression. When Select s? appears on the screen, move the cursor and click on the value of the slope. Press • and the value of the slope should appear. Drag the expression to the desired location and press • again. Remind students that, when the angle is in the second or third quadrant, they need to add 180º to the angle calculated using the inverse tangent. |
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| Step 5: Next, students explore the addition of three coterminal vectors. They first construct vector AE, which should point in a different direction than vector AB, AC, or AD. Then, they use the Translation tool to add vector AE to vector AD. They label the endpoint of this vector F and construct vector AF, which is the sum of vectors AB, AC, and AE. | |
| Problem 2 – Interaction between three similarly charged particles |
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| Next, students should move to page 2.1 and read the text there. Page 2.2 shows three positively charged particles. The vectors attached to each point show the forces on that particle that are produced by the other two particles. Students can drag each point and thus vary the distances between the particles. The numbers on the right-hand side of the screen give the magnitudes of the charges on the particles. Students can click on these labels and vary the magnitudes of the charges. The charges are given in units of coulombs (C), and distances are in meters. | |
| Step 1: Next, students should find the net force on each particle using head-to-tail vector addition. They should then hide the individual force vectors on the particles, so only the net forces are visible, as shown to the right. | 
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| Step 2: Next, students should vary the locations and charges of points Q1, Q2, and Q3 and observe the results. Then, they should answer questions 4–9 on the student worksheet. Q4. Describe the changes you observed in the net force on each particle when you varied magnitudes and positions of the particles. Did you notice any patterns? A. Due to the open-ended nature of the question, many different responses are possible. However, students should note that the net forces on the particles decreased as the particles got farther apart and increased as they got closer together. Students should also note that the net forces on the particles are directly related to the charges on the particles (i.e., the larger the charges, the larger the net forces). Q5. What happens to the net force on a particle when its charge is much larger or smaller than the charges on the other two particles? A. When the charge on one particle is much larger than the charges on the other particles, the net forces on all three particles increase significantly. As the charge on one particle approaches zero, the net force on that particle also approaches zero. Note that students may think that the force on a particle can never go to zero if the other two particles have non-zero charge. Guide them to understand that electric forces affect only charged particles, so a particle with a zero charge will experience no net force. Q6. What happens to the net force when one of the particles moves very close to one of the other particles? A. The forces on the particles become very large. Q7. What happens to the net force on a particle when it is moved far away from the other two particles? A. The forces on the particles become very small. Q8. What happens to the net force on a particle when it is located exactly between two equally charged particles? A. The net force on the central particle is zero because the vectors acting on the central particle have equal magnitudes but opposite directions. Q9. Three particles, each with a charge of +11 μC, are located at the corners of an equilateral triangle with sides of length 1 m. Calculate the magnitude and direction of the net force on each particle. (Hint: Use the template on page 2.2, adjust the scale if needed, and reproduce the conditions of the problem. Then calculate magnitude and angle for each net force.) A. Students can use the template on page 2.2 to solve this problem graphically. They can place the three charges, for example, on the points (–1, 0), (0, 0), and (–  , ). They can then display the coordinates of points Q1, Q2, and Q3 using the Coordinates and Equations tool (Menu > Actions > Coordinates and Equations). They can use the Length and Slope measurement tools to determine the magnitudes and directions of the resulting forces. They should find that the magnitude of the force on each particle is 1.9 N and the directions are as follows:For (0, 0),  = –30 ; for (– ,), = 90º; and for (–1, 0), = 210º. Students will probably need to adjust the scale of the graph (Menu > Window > Window Settings) to see the net forces clearly. Students should also calculate the magnitudes and directions analytically, using Coulomb's law and vector addition rules. Coulomb's law yields the net force on each particle, as shown on the next page (make sure students convert μC to C): Students can use the law of cosines to determine the direction of each vector. |
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| Problem 3 – Interaction between particles of opposite charge | |
| In this part of the activity, students explore the interactions between positively and negatively charged particles. Note that in each template, the sign of the charge cannot be changed. Students should read page 3.1 before moving on to the simulations. (Note: To save time, you may construct the net force vectors for this problem ahead of time in the .tns file, before transferring the file to the students' handhelds or computers.) | |
| Step 1: Students should first move to page 3.2, which shows two positively charged particles and one negatively charged particle. They should use head-to-tail vector addition to find the net force on each particle, and then hide the individual force vectors on each particle so only the net forces are visible. They should change the positions and charges of the particles and observe the results. | 
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| Step 2: Next, students should move to page 3.3, which shows two positively charged particles and two negatively charged particles. They should again find the net force on each particle and hide the component forces. After exploring the effects of position and charge on these forces, students should answer questions 10–14 on the student worksheet. Q10. What happens to the net forces when oppositely charged particles move very close to one another? A. The magnitudes of forces on the oppositely charged particles increase significantly and approach infinity. Their directions become opposite. At the same time, the net force on the third particle approaches zero. Q11. What happens to the net force on a negatively charged particle when it is located exactly between two particles of equal positive charge? A. When a negatively charged particle is located exactly between two equally positively charged particles, the net force on the negatively charged particle is zero because the two force vectors acting on the central particle have equal magnitudes but opposite directions. Q12. Explore other symmetrical arrangements of the particles, and describe any patterns that you observe. A. Encourage students to explore different symmetrical arrangements of particles and discuss their results with the class. Encourage them to predict the net forces resulting from various symmetrical arrangements, and then use the templates on pages 3.2 and 3.3 to test their predictions. Q13. A particle of charge +100 μC is located at (–2, 0), and a particle of charge +200 μC is located at (2, 0). Where should a negatively charged particle with charge –50 μC be placed so that the net force on this particle has a magnitude of 2 N and is directed at –135°? (Hint: You can see the coordinates of a point by choosing Menu > Actions > Coordinates and Equations and then clicking on the point.) A. Two possible coordinates are (7, 5.6) and (–0.385, 0.082). This particular problem is too advanced for analytical solution. It is sufficient if students verify their graphical solution by calculations. In order to do that, they will need to measure the magnitudes and angles of all forces and show that the net force in the system is zero. Note: If students struggle with this problem, you may first ask them to explore a simpler problem, such as determining the magnitude and direction of the net forces on the particles when they are placed at specific locations. Q14. A right isosceles triangle is formed by the charges Q1 = +150 μC, Q2 > 0, and Q3 = –120 μC located at points (0,10), (0, 0), and (–10, 0), respectively. The fourth charge, Q4 = –145 μC, is located at the midpoint of the hypotenuse. The net force on charge Q4 is 10 N in the positive x-direction. What is the charge on Q2? A. The charge on particle Q2 should be approximately +270 μC. Once again, students should verify that the net force in the system is zero after they find their solution. If you wish, you may assign additional problems or explorations to student pairs or groups for homework or independent projects. |
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