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.

Transcription

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 represent three displacement vectors as shown in Figure 4. 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). ^ j y ^ i Figure 3: Unit Vectors x a) Use a ruler to measure 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. Note: your measurements should not necessarily be the same as your lab partners : printers produce diagrams with slightly different magnifications. Vector A: B C A Vector B: Figure 4: Sample Vectors 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: 2 a) Vector C can be also represented using Cartesian components. The Cartesian components are scalars 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, sketch the sum of C + B + A. Should the sums for A + B + C and C + B + A be the same? 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 (a rock). 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. B c b a C A 1) In the apparatus, note the labeling of the cords (A, B and C) and of the angles between the cords (a, b and c). Note that cord B can be adjusted at its point of connection to the vertical support. Write an equation to solve for angle c in terms of angles a and b. 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 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 and record the force measured by the spring scale. For this trial only, be sure that each lab group member measures the angles and reads the scale independently to establish uncertainty in the measurements. You can use these uncertainties for all three trials. Record all data below, being sure to include units. Be sure to include the number of the rock you used!! 6

7 b) Repeat the previous step, adjusting cord B by moving the clamp on 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. 4) You will be analyzing the Cartesian components of the force vectors (representing tensions in the cords) to determine the unknown mass. In theory, we could pick any perpendicular axes to analyze the situation, but we re going to pick vertical (y axis) and horizontal (x axis) axes here. Why does this make sense? 5a) For the case when the clamp was in its upward position, sketch vectors representing the tension in each of the three cords in the apparatus from the point where the cords converge. 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 as shown and label components with their magnitudes in terms of A, B, α and β and appropriate trigonometry functions. B β c α A C 7

8 5b) Write an equation to solve for i) α in terms of a and ii) β in terms of b 6) Using data for the horizontal position of cord B and 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) Break the tension vector for cord A into Cartesian components: first, sketch the vector with its components. Then, label angle α in the sketch with its value in degrees and label the vector and vector components with A and with appropriate trig functions (including angle in degrees). 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 equations you can use to solve for the unknown mass in which variables will represent measured tensions and angles. With these, 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) Sketch a vector diagram of the trial where the clamp holding cord B was in the downward position. Rather than including actual measured values, though, use A, B, and C, to label the tensions in cords A, B, and C respectively. Use a, b and c to represent angles, according to the figure on page 6. Finally, label the components of A and B in terms of tensions A, B, and/or C, angles α and β (as defined in question 5 of the in-lab procedure) and appropriate trigonometry functions. 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 for when the clamp on cord B is in the downward position. Use the equation to solve for the tension in C. 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, alter the equation you developed in question 1b above to allow solving for the tension in cord C when in each of the other positions. Then use the equations to solve for the tension. a) clamp on cord B is horizontal: b) clamp on B is in the upward position: 3 a) Write the number of the rock you used: b) Solve for the mass of the rock for each of the three positions of cord B. 10

11 c) Find the worst-case uncertainty in your value for the unknown mass using the values from the three positions of cord B you have calculated. d) For the case where cord B was horizontal, perform a worst case uncertainty analysis for the mass by adding or subtracting uncertainty in each measured quantity as necessary and solving your equation from question 2a in the homework. 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: vertical and horizontal axes often make sense in situations where gravity is a consideration. When analyzing vectors for a system in which an object moves in 15 kg a straight line, often the appropriate choice for the coordinate system is to pick the direction of motion as one coordinate direction and for the second coordinate, 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, you might 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 we are using for this system. Label the coordinate axis in the direction of motion x and the other axis y (where +y is upwards, away from the incline). Remember that they must be perpendicular. b) To the right, draw a free body diagram showing the actual physical forces acting on the box. 11

12 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 25. d) Redraw your free body diagram from part b, replacing any physical force vectors with their components so that your free body diagram contains 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