Unit 2: Vectors Student Papers

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1 INQUIRY PHYSICS A Modified Learning Cycle Curriculum by Granger Meador Unit 2: Vectors Student Papers inquiryphysics.org 2010 these SAMPLE NOTES, the STUDENT PAPERS, and any PRESENTATIONS for each unit have a creative commons attribution non-commercial share-alike license; you may freely duplicate, modify, and distribute them for non-commercial purposes if you give attribution to Granger Meador and reference however, please note that the TEACHER S GUIDES are copyrighted and all rights are reserved so you may NOT distribute them or modified versions of them to others

2 2 Vectors Name Lab: Force Table Version In this laboratory, you will explore the properties and relationships of vector quantities. Such quantities have both magnitude and direction. These quantities can be manipulated mathematically using special rules of vector math. We will be studying force relationships because forces are vector quantities; they have both magnitude and direction. Your group will be using a force table. It is a circular platform with four rotating arms attached at a central point. You will be using three of the four arms in this lab. On the three arms you will place strings attached to a central ring. Weights will be hung from weight holders on the ends of the strings. The arms (and strings and weights) will then be moved until the central ring is centered. A. To begin, check that three of the four arms on your table have strings threaded over the pulleys. Check that each string has a weight holder attached to it. B. Place weights on each weight holder so that the total weight on each arm is different from the others. (The fourth arm will remain unused throughout the lab.) C. Once you have placed the weights, rotate the three arms until you can center the ring that the strings are attached to. Make certain that the ring is perfectly centered before proceeding. D. Decide which arm is arm 1, which is arm 2, and which is arm 3. E. Record in the table under "Trial 1" the total force on each arm. Add to each reading the weight of the weight holders themselves. F. Use the markings on the force table to measure the angle between each pair of arms. Record the angles in the table. See the diagram for help in deciding how the angles are labeled. G. Finally, repeat the entire procedure using different angles and weights. Record the new data under "Trial 2" in the table. Trial Force Angles between arms 1 arm 1 1,2 arm 2 2,3 arm 3 3,1 2 arm 1 1,2 arm 2 2,3 arm 3 3,1 Unit 2: Vectors, Lab Page 1 of by G. Meador

3 2 Vectors Name Lab: Composition of Forces Apparatus/Force Board Version In this laboratory, you will explore the properties and relationships of vector quantities. Such quantities have both magnitude and direction. These quantities can be manipulated mathematically using special rules of vector math. We will be studying force relationships because forces are vector quantities; they have both magnitude and direction. In this experiment, your lab group may be using a Composition of Forces Apparatus (CFA) or a force board. The basic instructions for using both devices are similar. There are three arms on a CFA which can be adjusted to different angles. A spring scale attached to a central ring is placed on each of the arms (see Figure 1). The force board substitutes a large circular board for the three-armed CFA. Like the CFA, the force board has three spring scales on it which are attached to a central ring (see Figure 2). Figure 1: Comp. of Forces Apparatus After checking that all three scales are on your apparatus and are attached to the central ring, pull each scale tight enough so that it produces a reading. The CFA scales are stretched by pulling on a piece of string that attaches them to the arms of the CFA. The force board scales are stretched by pulling on small chains and locking these chains in the notches of the force board. The reading on the stretched scales is a measurement of force. Change the angles between the springs until the ring in the middle of the apparatus is centered. You change the angle on the CFA by moving the arms around. You change the angle on the force board by moving the scale chains to different notches on the board. Figure 2: Force Board Place your apparatus on top of a large piece of paper, keeping the center ring centered. CAREFULLY mark on the piece of paper the positions of your scales, noting which scale corresponds to arm 1, arm 2, and arm 3. (There are slots in the CFA arms where you can make a mark.) Record the scale reading on the large piece of paper next to each marked scale position. Also record the data under Trial 1 in the table. When you remove your group's piece of paper, you should be able to reconstruct the positions of the arms by drawing lines along the marks you've made to represent each arm. The lines will of course all meet at a center point. Then measure the three angles between the arms and record your measurements under Trial 1 in Table I. Refer back to figure 1 or 2 if you're uncertain about how to designate the various angles. Finally, repeat the entire procedure using different angles on the back of your sheet of paper and record the data under Trial 2 in the table. Trial Force Angles between arms 1 arm 1 1,2 arm 2 2,3 arm 3 3,1 2 arm 1 1,2 arm 2 2,3 arm 3 3,1 Unit 2: Vectors, Lab Page 1 of by G. Meador

4 The Idea 1. Consider arm 2. The force on arm 2 is pulling in a different direction than the forces on arms 1 and 3. The ring remains centered, however. Explain how the size and direction of the force on arm 2 compares to the size and direction of the combination of forces 1 and 3. The class will need to discuss the above question before we continue. The class will work together to form a class hypothesis regarding the lab. Write it in the space below. Class Hypothesis (expressed as a written statement) The relationships between force vectors can be written as vector equations. For example, if we wanted to say that force Q is equal to half of the sum of forces D and Z, we would write: The class hypothesis expresses in words a relationship between,, and. 2. In the space below, show that relationship as a vector equation: Testing the Hypothesis We can easily test our class hypothesis by graphically adding two of the vectors and finding their resultant. You are going to draw a scale drawing of the Trial 1 data on a sheet of graph paper. The scale drawing will be a picture of the three force vectors you experimented with. In that picture, the length of an arrow represents a specific amount of force. Pick either a scale of 1 cm = 20 g OR a scale of 1 cm = 10 g; use whichever makes the diagram large without going off the edge of the paper. For example, a scale of 1 cm = 20 g means that if a force is 114 grams, an arrow 114 / 20 or 5.7 centimeters long will represent that force. Unit 2: Vectors, Lab Page 2 of by G. Meador

5 Your scale diagram will have three arrows coming from a central point. Each arrow represents one of the three force vectors. The direction of each arrow is determined using the angle data from the table. (See Figure 1.) A. Draw the vector for force 1 first. It will be an arrow pointing straight upward (north) from the center of the graph paper. Make the arrow the appropriate length to match the size of force 1. Label your arrow. B. Now draw the vector for force 2. It will also be an arrow pointing outward from the center of the graph paper. You must use the data you recorded in the table to determine how many degrees away from the force 1 vector (from north) you should draw the force 2 vector. Make the force 2 vector the appropriate length to match the size of force 2. Label your arrow. Figure 2 Figure 1 C. Finally draw the vector for force 3. Check that the angles between it and vectors 1 and 2 match your data. Make the force 3 vector the appropriate length to match the size of force 3. Label your arrow. D. CAREFULLY check that each vector is the proper length and that you have properly measured the angles between the various vectors. E. Add vector F1 to F3 by drawing a copy of F1 at the head of F3 as shown in Figure 2. Figure 3 F. Since the two vectors we are adding are now drawn head to tail, we can find their resultant. Draw an arrow from the center of the paper (where F1, F2, and F3 originate) to the head of the last added vector. Label that new arrow vector R. Your drawing should now look something like Figure 3. TRIAL 1 DIAGRAM 3. Vector R is the resultant of the two vectors F1 and F3. Measure the length of your resultant R. What is its corresponding force value in grams? 4. What theoretical value (in grams) does the class hypothesis predict for R? 5. We will determine how well your data fits your prediction by comparing your measured value from question 3 to the theoretical value from question 4. We will compute the percent error between those two values with the formula at right. Note that the absolute value is taken of the numerator, so percent error is always positive. What was the percent error between your actual and predicted values for R? Vectors have direction as well as magnitude. You have measured the magnitude or size of R and compared it to its hypothetical or theoretical value. Now you will examine how well your data supports the portion of your hypothesis regarding direction. 6. Look at the class hypothesis on page 2. It can be interpreted so that it predicts the theoretical angle between F2 and R. Give that theoretical value (in degrees): 7. Now use your protractor to measure the actual angle between vectors R and F2. Give that measured value (in degrees): 8. Use answers 6 and 7 to compute the percent error for the direction of the resultant. Unit 2: Vectors, Lab Page 3 of by G. Meador

6 TRIAL 2 DIAGRAM Draw another scale diagram using the Trial 2 data from the table. Add F1 and F3 and measure the magnitude of the new resultant R, and answer the questions below. 9. New R for Trial 2 = grams; Theoretical R for Trial 2 = grams 10. Calculate the percent error for the Trial 2 resultant's length. Now show the measured and theoretical angles between R and F2 for the second trial. 11. New Angle Between R and F2 for Trial 2 = ; Theoretical Angle = 12. Calculate the percent error for direction of the new resultant. Conclusions 13. What do the results tell you about force vectors? Conclusion Guidelines: W rite in complete sentences and do not use the first person (I, we, our, etc.). Include the following: A. Restate the hypothesis. B. State the range (lowest and highest values) of percentage error. C. Discuss whether that range of error is explainable by systematic errors, and discuss the likely sources of such errors (equipm ent problem s/lim itations and other significant issues that could not be corrected). Be specific and reasonable, avoiding vague term s like human error and ignoring trifles like the difficulty of using a pencil or protractor. D. Given above discussion, indicate if the experim ent supported or refuted the hypothesis. Unit 2: Vectors, Lab Page 4 of by G. Meador

7 2 Vectors Name Worksheet A: Guided Practice To add vectors, you first draw each vector one after another, HEAD-TO-TAIL. Then you draw the resultant from the tail of the first vector straight to the head of the last vector (from START TO FINISH). Example: Add 5.00 g north to 6.00 g west to 4.00 g at 40.0 S of W. First draw each vector Then draw the resultant from ead-to-tail as shown below: the start to the finish: So the answer is 9.38 g at 15.0 N of W. Notice that the resultant's angle is measured from an imaginary compass set at its TAIL, not its head. PRACTICE PROBLEMS (answers given below) 1. Add these three displacements: 3.00 m south, 5.00 m west, and 2.00 m at 40.0 north of west. Draw a graphical solution on the 1-cm grid below. Show your final results in the spaces to the right of the diagram. Scale: 1.00 cm equals m Resultant magnitude (in m): Resultant direction: (give degrees & compass headings) 2. A plane moving at 300 m/s east encounters a 100 m/s wind blowing north. a. Find the size and direction of the plane's resultant velocity by drawing a scale diagram on the grid below. Scale: 1.00 cm equals m/s Resultant speed in m/s: Resultant direction: b. How long will it take for the plane to travel 3750 m in its resultant direction? c. How far east will the plane have travelled in that amount of time? Unit 2: Vectors, Worksheet A: Guided Practice 2010 by G. Meador

8 2 Vectors Nam e W orksheet B: Concepts and Calculations Write the letter corresponding to the best answer in the blank at the left of each question. 1. A cat runs 40.0 meters due east and then turns around and runs 30.0 meters due west. The cat ran a total scalar distance of: a meters b meters c meters d meters 2. The cat in the previous question had an actual vector displacement of: a meters b meters west of east c meters east d meters west Which of the following quantities is a scalar? a. speed b. acceleration c. displacement d. force 5. Your experiment with the force table showed that when three vectors are balanced... a. one vector must be larger than the other two. b. each vector must be equal to the scalar sum of the other vectors' magnitudes. c. the resultant of any two vectors must be smaller than the remaining vector. d. each vector is an equilibrant for the other two. 6. It is technically incorrect to say that "5.00 m/s" is a velocity vector because velocity... a. must include a direction. b. is a scalar quantity. c. would have different units. d. is the same thing as speed When you add two vectors having magnitudes of 2 and 8, it is impossible for the resultant to have a magnitude of: a. 6 b. 7 c. 10 d. 12 Unit 2: Vectors, Worksheet B: Concepts and Calculations 2010 by G. Meador

9 9. Three children were fighting over a cat. One child pulled on the cat with a force of 300 grams south; another applied 200 grams east; the last applied 400 grams at 30.0 north of east. Find the magnitude and direction of the resultant force on the cat. Draw a graphical solution on the lines below. Show your final results in the spaces to the right of the diagram. (The lines are spaced 1.00 cm apart.) Scale: 1.00 cm equals g Resultant magnitude: (in grams) Resultant direction: (specify degrees and compass headings) 10. An ocean liner was heading south at 7.00 m/s when it encountered a current which headed west at 4.00 m/s. a. Find the magnitude and direction of the liner's resultant velocity by drawing a scale diagram on the lines below. Show your final results in the spaces to the right of the diagram. Scale: 1.00 cm equals m/s Resultant speed in m/s: Resultant direction: (specify degrees and compass headings) b. Use your results to calculate how long it would take the liner to travel 1500 meters along its resultant path. c. Calculate how far the liner would have travelled directly south in that time. Unit 2: Vectors, Worksheet B: Concepts and Calculations 2010 by G. Meador

Vector Addition INTRODUCTION THEORY

Vector Addition INTRODUCTION THEORY Vector Addition INTRODUCTION All measurable quantities may be classified either as vector quantities or as scalar quantities. Scalar quantities are described completely by a single number (with appropriate