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Addition of Vectors Purpose: In this lab we check a special case of Newton's second law, 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 . 4 pulleys
What to turn in next week: 1. These lab instructions filled out correctly and completely. No typing necessary but make it is clear 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 carried out. These graphs need to be to scale, bot sketched! Everything should be labeled and your scale given, c. inch -1 N, or whatever is a good scale to fit your paper. The graphs should fill the page reasonably well don't make them too small! Addition of Vectors 1. Introduction In this lab we will explore physically, geometrically, and algebraically the addition of vectors 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 100 grams on the end of the string. Be sure to include the 50.0 grams 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 g-9.80m/s). Work to three significant figures for this and for all other calculations of forces in this lab. Record the value of this force as F in Data Table 1 (attached with this handout). b) Place a second pulley at the 90.0° 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 F 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 bring the ring to 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 Data Table 1 Direction Force (N) Force Mass (kg) 20.0° calc F: 0.100 b.98 calc, 90.00 0.200 ܕ . ܕ܂ exp. cale. 249 exp F3 0.250 12:45 rt A: Calculations a) As a group, find the resultant of the two applied forces and Fz by a scaled, to tail). Use the graph paper attached to this lab. graphical construction using the parallelogram method (place the vectors tail Question 1: How is F3 related to that resultant? Draw Fs on the graph as well. Record your results in the graphical solution" table below. Graphical Solution Direction Force Mass (kg) Force (N) calc. 20.0° F1 0.100 kg calc. 90.0° F2 graph graph F1+F2 0.200 kg cale. 2.300 calc. 0.252 calc. calc. F3
b) Calculate the percentage error between the graphical calculation and the analytical calculation for the magnitude of F). c) Using trigonometry, calculate the components of Ft and Fz and record them i the "analytical solution" table below. Define the x-axis as the line going through the 0° mark and the y-axis the line going through the 90° mark. Add the components and determine the resultant F1+F2, both magnitude and direction Fill out the remainder of the "analytical solution" table. Analytical Solution 0.98 Force Mass (kg) Force (N) Direction X-comp (N) Y-comp calc. cale. calc. F: 0.100 kg 20.0° calc. calc. calc. F2 0.200 kg 1,96 90.0° calc. calc. calc. calc. cale. F1+F2 300 calc calc. calc. calc. calc. F3 332 13, 253 0.330 d) Calculate the percentage error between the experimental value and the analytical calculation for the magnitude of F3. 2.94 - 1267
Part B: Four forces on a ring in equilibrium a) Place a pulley at 30.0 degrees with 150. grams on it, one at 100 degrees with 200 grams on it and one at 145 degrees with 100. grams on it. b) Calculate the force produced by those masses and record them as F. Fs, and c) As in part A, by trial and error add the appropriate mass configuration to give a net force of zero. Record the added force as Fin Data Table 2 Forespectively in Data Table 2. Data Table 2 Direction Force Mass (kg) Force (N) calc. 30.00 F4 0.150 kg calc. 100.0° Fs 0.200 kg calc. 145.00 FG 0.100 kg exp calc. exp. F Part B: Calculations a) As a group, find the resultant of the two applied forces F and Fs and Fby a scaled, graphical construction using the parallelogram method (place the vectors head to tail to visualize the multistep addition). Use the graph paper attached to this lab. Draw F7 on the graph as well. Record your results in the "graphical solution" table below.
Graphical Solution 2 Force Direction Mass (kg) Force N calc F4 30.00 0.150 kg calc 100.00 FS 0.200 kg calc F 0.100 kg calc. 145.0° graph Graph F +Fs + F calc. cale calc F b) Calculate the percentage error between the experimental value and the graphical calculation for the magnitude of F. Question 2: Could all four pulleys be placed in the same quadrant or in two adjacent quadrants and still be in equilibrium? Explain using a sketch of vectors. Question 3: 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 function of the pulleys.
>> Using trigonometry, calculate the components of F, Fs, and Fs and record them in the analytical solution table below. Define the x-axis as the line going through the 0° mark and the y-axis the line going through the 90mark Add the components and determine the resultant F+Fs + F. both magnitude and direction Fill out the remainder of the "analytical solution table. Analytical Solution Y-comp- Force Mass (kg) Direction Force (N) calc. X-comp (N) cale. calc. F4 0.150 kg 30.0° calc cale. calc, FS 100.0° cale. 0.200 kg calc. calc. 145.0° calc. FG calc. 0.100 kg calc. calc. calc. calc calc F4 + F + F calc. calc calc. F d) Calculate the percentage error between the experimental value and the analytical calculation for the magnitude of F1. Notes: 1. "calc." stands for calculated. For example, if the mass is 0.1 kg, the force is the weight = mg. or 0.1kg * 9.8m/s 0.98N. 2. "exp" stands for experimental. These values you get directly from measurements,