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Lab Activity: Addition of Vectors Phys 221 Names: _____________ Addition of Vectors Purpose: In this lab, we check a special case of Newton’s laws, namely that the sum of the forces acting on an object in equilibrium is zero. Since forces are vectors, we will practice doing vector additions. First, we measure the forces acting on an object in equilibrium (a ring). Then, we check that adding the forces as vectors gives zero. We do this in two ways: graphically and using vector components. Equipment: protractor, ruler, string, 1 ring, 4 mass hangers, set of slotted masses, force table with 4 pulleys, large sheet of white paper What to turn in next week: 1. These lab instructions filled out correctly and completely. No typing necessary but make it clean and legible. Turn in one copy per group. Every group member should check the work. 2. Attached graphs showing clearly the graphical vector addition that you have done for your experiments. These graphs should be to scale, not sketched! Everything should be labeled, your scale should be given (e.g. 1 inch = 1 N, or whatever is a good scale). The graphs should fill the page reasonably well, don’t make them too small. Lab Activity: Addition of Vectors Phys 221 Names: _____________ Addition of vectors 1. In a short paragraph, describe the experiment. A drawing and a few sentences are enough (skip this question and go back to it once you have done question 2). 2. Experiment Part A: Three forces on a ring in equilibrium a) Place a pulley at the 20.0 mark on the force table and place a total of 100g on the end of the string. Be sure to include the 50g of the mass hanger in the total. Calculate the magnitude of the force produced by the mass and hanger (in Newtons. Use F=mg and take g=9.8m/s2). Assume three significant figures for this and for all other calculations of forces in this lab. Record the value of this force as F1 in Data Table 1 (attached with this handout). b) Place a second pulley at the 90.0-degree mark on the force table and place a total of 200g on the end of the string. Calculate the force produced and record this force as F2 in Data Table 1. Lab Activity: Addition of Vectors Phys 221 Names: _____________ c) Determine by trial and error the amount of mass needed and the angle at which it must be placed in order to place the ring in equilibrium. The ring is in equilibrium when it is centered on the force table and is not touching the peg in the middle. Jiggle the ring slightly to make sure that it is truly in equilibrium. Be sure that all the strings lie along a line, which passes through the center of the ring. d) For this third mass, calculate the amount of force produced and from the table markings determine the angle at which the force acts. Record this force as F3 in Data Table 1. Part B: Four forces on a ring in equilibrium a) Place a pulley at 30.0 degrees with 150g on it, one at 100.0 degrees with 200g on it and one at 145.0 degrees with 100g on it. b) Calculate the force produced by those masses and record them as F4, F5, and F6 respectively in Data Table 2. c) Following the procedure outlined in steps c and d in part A, determine the equilibrium force and record it as F7 in Data Table 2. 3. Calculations Part A: Three forces on a ring in equilibrium a) As a group on the large sheet of paper, find the resultant of the two applied forces F1 and F2 by a scaled, graphical construction using the parallelogram method (place the vectors tail to tail). How is F3 related to that resultant? Draw F3 on the graph as well. Record your results in the “graphical solution” table on the “calculations 1” page. Later, outside of class, each individual should construct an 8 ½” x 11” version of the diagram. You will have to make careful use of your protractor and ruler! Lab Activity: Addition of Vectors Phys 221 Names: _____________ b) Using trigonometry, calculate the components of F1 and F2 and record them in the “analytical solution” table (take for x-axis the line going through the 0 mark and for the y-axis the line going through the 90 mark). Add the components and determine the magnitude of the resultant by the Pythagorean theorem. Determine the angle of the resultant from the arc tangent of the components. Record those results in the table. What are the magnitude, direction, and x and y components of F3 given your result for the resultant F1 + F2 above? Write your results in the table. c) Calculate the percentage error of the magnitude of the experimental value of F3 with the analytical solution for F3. Also calculate the percentage error of the magnitude of the graphical solution for F3 compared to the analytical solution. For each of those comparisons, also calculate the magnitude of the difference in the angle. Record all values on the “calculations 1” page. Part B: Four forces on an object in equilibrium a) Find the resultant of the three applied forces F4, F5, and F6 using a scaled graphical construction. This time use the polygon method (place the vectors head to tail). Do it on a large sheet of paper just as in Part A. Record your results in the “graphical solution” table on the “calculations 2” page. Draw also F7 on your graph and record its direction and magnitude in the table. Each individual should hand in a 8 ½” x 11” reproduction of the above construction with their lab report. b) Using trigonometry, calculate the components of all three forces F4, F5, and F6 , the components of the resultant, and the magnitude and direction of the resultant. Record them all in the “analytical solutions” table on the “calculations 2” page. Write also the results for F7. c) Make the same error calculations for this problem as asked for in step 3 above in the calculations for Part A. Lab Activity: Addition of Vectors Phys 221 Names: _____________ 4. Could all four pulleys be placed in the same quadrant or in two adjacent quadrants and still be in equilibrium? Explain. 5. The forces used in this experiment are the weights of known masses, that is, the forces exerted on these masses by gravity. Bearing this in mind, explain the functions of the pulleys. Lab Activity: Addition of Vectors Phys 221 Names: _____________ Data Table 1 Notes: 1. calc. stands for calculated, for example you know the mass is 0.1 kg, therefore the force associated with that is the weight, or mg, or 0.1kg * 9.8 m/s2=0.98N. 2. exp stands for ‘from experiment’. This value you will get directly from your measurement. Force Mass (kg) F1 0.100 kg F2 measured resultant F3, i.e. experiment 0.200 kg Force (N) calc. Direction 20.0 calc. exp. 90.0 calc. exp. Data Table 2 Force Mass (kg) F4 0.150 kg Force (N) calc. Direction 30.0 calc. 0.200 kg F5 100.0 calc. F6 measured resultant F7, i.e. experiment 0.100 kg exp. 145.0 calc. exp. Lab Activity: Addition of Vectors Phys 221 Names: _____________ Calculations 1 Notes: 1. calc. stands for calculated, for example you know the mass is 0.1 kg, therefore the force associated with that is the weight, or mg, or 0.1kg * 9.8 m/s2=0.98N. 2. graph stands for ‘from your graph’. This value you will get directly from your graph, not some calculation! Once you have your graphical result you can calculate F3, which should just be the opposite of F1+F2. Graphical Solution Force Mass (kg) F1 0.100 kg Force (N) calc. Direction 20.0 calc. F2 resultant force F1 + F2 0.200 kg 90.0 calc. graph graph calc. calc. calc. F3 Analytical Solution Force Mass (kg) F1 0.100 kg Force (N) calc. Direction y-component calc. calc. calc. 20.0 calc. F2 resultant force F1 + F2 x-component calc. 0.200 kg calc. calc. 90.0 calc. calc. calc. calc. calc. calc. calc. calc. F3 Error Calculations Determine your relative error for the magnitude only. To do this take the calculated magnitude as your true value and find the ratios F3,anal F3,exp F3,anal F3,anal F3, graph F3,anal Lab Activity: Addition of Vectors Phys 221 Names: _____________ Calculations 2 Graphical Solution Force Mass (kg) F4 0.150 kg Force (N) calc. Direction 30.0 calc. 0.200 kg F5 100.0 calc. F6 resultant force F4 + F5 + F6 0.100 kg calc. graph 145.0 graph calc. calc. calc. F7 Analytical Solution Force Mass (kg) F4 0.150 kg F5 0.200 kg Force (N) calc. Direction y-component calc. calc. calc. calc. calc. calc. calc. calc. calc. 30.0 calc. 100.0 calc. F6 resultant force F4 + F5 + F6 x-component calc. 0.100 kg calc. calc. 145.0 calc. calc. calc. calc. F7 Error Calculations Determine your relative error for the magnitude only. To do this take the calculated magnitude as your true value and find the ratios F7,anal F7,exp F7, anal F7,anal F7, graph F7,anal Lab Activity: Addition of Vectors Phys 221 Names: _____________