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Phys 221 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 To Do: Follow the worksheet Your write-up is due on the due date indicated on class calendar on the web site (no late write-up will be accepted). Phys 221 Addition of vectors 1. 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=9.80 m/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. 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. Phys 221 2. 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, construct an 8 ½” x 11” version of the diagram. You will have to make careful use of your protractor and ruler! 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. Phys 221 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. Include a 8 ½” x 11” reproduction of the above construction in your 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. Phys 221 Data Table 1 Mass (kg) Force F1 F2 Force (N) Direction 0.100 kg 20.0 0.200 kg 90.0 measured resultant F3 F1 F2 Data Table 2 Mass (kg) Force F4 F5 F6 measured resultant F7 F4 F5 F6 Force (N) Direction 0.150 kg 30.0 0.200 kg 100.0 0.100 kg 145.0 Phys 221 Calculations 1 Graphical Solution Mass (kg) Force F1 F2 Force (N) Direction 0.100 kg 20.0 0.200 kg 90.0 sum of the forces F F 1 2 F3 F1 F2 Analytical Solution Mass (kg) Force F1 F2 20.0 0.200 kg 90.0 F F 1 2 F3 F1 F2 Direction 0.100 kg sum of the forces Force (N) Error Calculations x-component y-component Phys 221 Calculations 2 Graphical Solution Mass (kg) Force F4 F5 F6 Force (N) Direction 0.150 kg 30.0 0.200 kg 100.0 0.100 kg 145.0 sum of the forces F4 F5 F6 F7 F4 F5 F6 Analytical Solution Mass (kg) Force F4 F5 F6 Direction 0.150 kg 30.0 0.200 kg 100.0 0.100 kg 145.0 sum of the forces F4 F5 F6 F7 F4 F5 F6 Force (N) Error Calculations x-component y-component Phys 221 Write-up 1. In a short paragraph, describe the experiment. A drawing and a few sentences are enough 2. Three forces on a ring. 2.1. Data table: explain what you measured. 2.2. Tail to tail vector diagram and graphical table: explain how you drew the vectors and what you measured on your vector diagram. 2.3. Analytical table: explain how you did your computations. 2.4. Error calculations 3. Four forces on a ring. 3.1. Data table: explain what you measured. 3.2. Head to tail vector diagram and graphical table: explain how you drew the vectors and what you measured on your vector diagram. 3.3. Analytical table: explain how you did your computations. 3.4. Error calculations 4. Answer these two questions 4.1. Could all four pulleys be placed in the same quadrant or in two adjacent quadrants and still be in equilibrium? Explain. 4.2. 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. 5. Turn in one typed report per group on the due date (listed on the class web site).
