Lab: Vectors. You are required to finish this section before coming to the lab. It will be checked by one of the lab instructors when the lab begins.

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1 Lab: Vectors Lab Section (circle): Day: Monday Tuesday Time: 8:00 9:30 1:10 2:40 Name Partners Pre-Lab You are required to finish this section before coming to the lab. It will be checked by one of the lab instructors when the lab begins. Review your text, sections and 2.9. Then, respond to each of the following questions: 1 a) Vectors A, B, and C are shown to the right. Using the tipto-tail method, sketch the sum of A + B + C in the space below. Show the vector sum clearly on your sketch. Note: do the sum in the order indicated; for example, when adding any arbitrary vectors X + Y, X should be drawn first, and the tail of Y should be placed at the tip of X. B A C b) Does the sum B + A + C yield the same vector sum as A + B + C? Support your response by sketching the tip-to-tail sum in the space below, clearly indicating the vector sum. 1

2 c) In the space provided, sketch the following vector operations using the tip-to-tail method. Indicate the vector sum clearly. i) 3A iii) C ii) 2A + B iv) 2A - B - end of pre-lab - I. Addition of Vectors. In the pre-lab, you sketched the results of simple vector operations. While these sketches provide a good qualitative example of how vector operations work, such sketches cannot provide more exact quantitative results. Quantitative results can be approximated by creating a scale drawing of the vectors using rulers and protractors or other angle measuring devices. Typically, the most exact results are possible by combining vectors mathematically using geometry and/or trigonometry; however, to combine vectors mathematically you must know or be able to obtain the numerical values of the individual vectors magnitudes and directions. Since a vector quantity contains both magnitude and direction, both values must be specified to describe the vector completely. The magnitude is simply specified by a value, like 14 meters/sec, or 60 Newtons. The direction is often expressed as an angle; in the Cartesian coordinate system (see Figure 1) it is most often expressed as angles related to the axes. As an example, the direction of the vector shown in the two-dimensional coordinate system in Figure 2 could be described as forty degrees counterclockwise from the x-direction. For the sake of simplicity and given that we are limited to two dimensions when drawing vectors on paper, in this lab we will limit ourselves to vectors in two dimensions, x and y. The x and y (and z when used) directions are often represented with unit vectors. The Cartesian unit vectors are dimensionless vectors having magnitudes of 1 in each of the Cartesian directions x and y (and z). The unit vector y x z (out of page) Figure 1: Cartesian Coordinates y 40 x Figure 2: Vector Direction 2

3 in the x-direction (see Figure 3) is called i-hat and is represented as an i with a caret above it. The unit vectors in the y- and z- directions are called j-hat and k-hat. If you walk 15 meters in the x-direction the vector representing your displacement is 15 m ^ i. 1) Let vectors A, B and C (right) represent three displacement vectors as shown in Figure 4. The vectors are scaled so that 1.0 cm = 3.0 km of actual displacement. Note that A and B point purely in the x- and y-directions, while C points at an angle relative to the x- direction (shown as the dotted line). a) Use a ruler to find the magnitudes of vector A and B to the nearest 0.05 cm. Then, express vectors A and B (magnitude and direction) using unit vectors. ^ j y ^ i Figure 3: Unit Vectors x A Vector A: Vector B: B C b) Use a ruler and protractor to find values for the magnitude and direction of vector C. Express the magnitude in cm to the nearest 0.05 cm and direction in degrees to the nearest 0.5. Write your values below, expressing your response relative to the x-axis (not using unit vectors). Vector C: Figure 4: Sample Vectors 2 a) Vector C can be also represented using Cartesian components. The Cartesian components are vectors that point in the x and/or y (and/or z) directions that, when added, yield the original vector. Vector C has two Cartesian components, one in the x- direction and one in the y- direction (parallel and perpendicular to the dotted line shown to the right). On the diagram, carefully draw the components of C. Then, measure the components to the nearest 0.05 cm and write them below. C x-component: y-component: 3

4 b) Solve for the magnitude and direction of vector C by combining the x- and y- components you measured in 2a using geometry and trigonometry. Compare this with your result for vector C from question 1b. Show all work. How well do they compare? Check with an instructor before you continue. 3 a) On the grid below, use your values for vectors A, B, and C (from question 1) to show the vector sum A + B + C. Use ruler and protractor carefully to create as accurate a sum as possible. b) What are the magnitude and direction of the sum A + B + C (using ruler and protractor)? c) On the same grid, add C + B + A. Are the sums for A + B + C and C + B + A exactly the same? Why might they be slightly different? 4 a) In question 3b, you found the vector sum graphically. When you know all of the Cartesian components of the vectors to be added, you can also find the vector sum mathematically: first, combine the vector components in each Cartesian direction (combine all of the vectors in the x-direction, then the y-direction...). Then, you can combine the resulting vectors, which are perpendicular to each other. For vectors A, B and C, sketch the addition of the x-direction vectors, then y-direction vectors, using 4

5 components of vector C from question 2b. It doesn t need to be to exact scale because you are now solving mathematically. Then, calculate the sum for each direction. Show your results below. x-direction: y-direction: b) Using your results from question 4a, find the vector sum of A, B and C, by adding the x- and y- component vectors (tip-to-tail, of course!). Sketch this below (it doesn t need to be to exact scale because you are now solving mathematically). Then, find the vector sum using geometry and trigonometry to find its magnitude and direction. Show all work below. c) Compare your result to the graphical sum you found in question 3b. How do they compare? Which method (graphical or mathematical) do you think is more accurate? Explain why. Check with an instructor before you continue. 5

6 II. Components of Forces. You will use the apparatus provided to solve for the mass of an unknown object. For simplicity, we will consider that the unknown object is the only object in the apparatus that has significant mass (we will assume that the masses of cords, etc. are all zero). One of the legs of the apparatus contains a spring scale, which is used to measure force exerted on it in Newtons. The spring scale will be stretched by the tension in the cord connected to it, so the reading on the spring scale is the measure of the tension in the cord to which it is connected. 1) Sketch the apparatus below, labeling the cord attached to the spring scale as A, the cord supporting the unknown mass C, and the third cord B. Note that cord B can be adjusted at its point of connection to the vertical support; be sure to indicate this on your sketch. 2) The cords are all attached to a ring at the point where the cords converge. The total force on this ring is the sum of the forces acting on it by each of the cords attached to it. What is the total force acting on the central ring? How do you know? 3) You will be analyzing the Cartesian components of the force vectors (representing tensions in the cords) to determine the unknown mass. You can pick any perpendicular axes to use as your x- and y-directions, but in cases that include gravitational force such as this, a specific choice for the x- and y-directions makes a lot of sense. What do you think it is? Why? Check with an instructor before you continue. 6

7 4) Using your definition for x- and y-directions as chosen in question 3, sketch vectors representing the tension in each of the three cords in the apparatus. Then, clearly show how you can break the vectors into their x- and y-components. Use α and β to represent angles between the vectors and the horizontal components. 5 a) Now you will take data that will be used to determine the unknown mass. Adjust cord B so that it is approximately horizontal, measure the angles between the cords (with the vertices at the center of the central ring) and record the force measured by the spring scale. Be sure that each lab group member measures the angles and reads the scale independently to establish uncertainty in the measurement. Record all data below, being sure to include units. b) Repeat the previous step, adjusting cord B by moving the clamp point at which it is attached to the vertical support downward as far as possible. Record all data below. Then, adjust cord B by moving the clamp upward (above the original horizontal position) as far as possible. Record all data below. 7

8 6) Using data for the first position of cord B, using average values for angles and force (not considering uncertainty), estimate the value of the unknown mass by doing the following (show all work!): a) Using your response from question 4, break the tension vectors in cords A, B, and C into Cartesian components: sketch each vector, showing its components. Label all angles in the sketches with their values in degrees and label the vectors and vector components with A, B, or C and with appropriate trig functions (including angle in degrees) for the components. b) How can you find the magnitude of the tension in cord B? Explain what physical principles you can use to do so. Then apply this to solve for the magnitude of the tension in cord B. c) Solve for the tension in cord C. d) What is the value in grams of the mass supported by cord C? Check with an instructor before you leave. Make sure that your lab station is left as you found it. 8

9 III. Homework. Show all calculations to receive full credit. Label all vector diagrams thoroughly: each vector should be clearly labeled. 1) In the sample calculations, you used measured values for force and angles to determine the unknown mass. Now you are going to derive an equation you can use to solve for the unknown mass in which variables will represent measured tensions and angles. With this, you should later be able to simply enter your measured values to calculate the unknown mass. Suggestion: If you enter the equation into Excel or any other spreadsheet or calculating routine, it will speed up the efficiency of repeated calculations. If you do so, you may print the results of your calculations in spreadsheet format and attach it to the lab, but make sure you clearly show all equations you used at the appropriate point in this handout. a) First, to make sure your derived equation is clear and consistent, sketch a vector diagram of the trial in the lab for which you found the unknown mass. Rather than including actual measurements, though, use A, B, and C, to represent the tensions in cords A, B, and C respectively. Use a, b and c to represent angles, according to the figure (right). Finally, show the components of the vectors you will need to use to solve for the unknown mass in terms of tensions A, B, and/or C, angles a, b and/or c and appropriate trigonometry functions. B b c a C A b) Using your diagram above, derive an equation to solve for the tension in cord C in terms of the variables you measured in lab. Hint: you should easily be able to check to see if your equation seems correct by checking it with the result of your sample calculations. If you do the calculations in Excel, be sure to show this sample calculation on the spreadsheet to prove that you entered the correct equation into the spreadsheet. 9

10 2) Now, calculate the unknown mass based on your data for the remaining two positions of cord B using the equation you derived in question 1b. a) 2 nd position: d) 3 rd position: 3 a) Find the uncertainty in your value for the unknown mass using the values from the three positions of cord B you have calculated. b) Now perform a worst-case uncertainty analysis on one of your positions of cord B. You will probably find the equation you derived in question 1b will prove most helpful here! If you are unsure of how to perform worst case uncertainty analysis, be sure to refer to the uncertainty supplement links on the Lab Handouts webpage and/or the uncertainty tutorial link on the Physics 107 webpage. 10

11 4) Which uncertainty from question 3 is greater? Why do you think this is the case? Explain thoroughly. 5) In the lab you performed, since the entire system was stationary, you chose the directions for the coordinate system using a little common sense thinking: the vertical cord C and the vertical gravitational force made vertical and horizontal axes a sensible choice. Vertical and horizontal axes often make sense in situations where gravity is a consideration. 15 kg When analyzing vectors for a system in which an object moves in a straight line, often the appropriate choice for the coordinate system is to pick the direction of motion as one coordinate direction and the second coordinate is a direction perpendicular 25 to the direction of motion. As an example, consider a 15 kg box sliding down a frictionless incline (see right). Since the box slides down the incline, one coordinate direction will point in the direction along the incline. Therefore, the other coordinate direction is the direction perpendicular to the surface of the incline (or in the direction normal to the incline). Since the box slides down the incline because of gravity, it is reasonable to think, as in the lab you performed, that the coordinates should be vertical and horizontal. In fact, it really doesn t matter what axes you pick and you could certainly analyze this situation using vertical and horizontal axes. In the following problems, however, you are going to use the parallel and normal axes, partly to use a coordinate system other than vertical/horizontal and partly because many aspects of the situation are easier to analyze using this system. a) On the figure (above right) sketch the coordinate axes that make sense for this system. Label the coordinate axis in the direction of motion x and the other axis y (where +y is upwards or away from the incline). Remember that they must be perpendicular. b) To the right, draw a free body diagram showing the physical forces acting on the box. c) Now, draw a vector diagram that shows the weight of the box and the x- and y-components that add to yield the weight (tip-to-tail of course). Be sure to label the angle that, through geometric reasoning, must be

12 d) Redraw your free body diagram from part b, replacing any physical force vectors that don t point purely in the x- or y-directions with components of the vectors. Your free body diagram, then, should contain only vectors that point along the axes. e) Is there a nonzero total force acting on the box in the y-direction? Your diagram in part d should be helpful here. Explain how you arrived at your answer. f) Is there a nonzero total force acting on the box in the x-direction? Explain how you arrived at your answer. g) The total force in the direction of the box s motion is responsible for the acceleration of the box down the incline. What is the acceleration of the box down the incline? h) What minimum force (including direction) would I need to apply to the box to cause it to slide with an acceleration of 0.50 m/s 2 up the incline? 12

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