Purpose To understand more about vectors, by using three completely different methods of vector addition. We will use the data collected by a group of students to complete this work. The instruction below was given to the student. Read carefully to follow how the student collected the data Theory Several of the practice problems about vectors refer to forces. Even though we will address forces in a later chapter, it is still possible to use them as examples of vectors. Each group was given a set of three vectors (A, B, and C) expressed in terms of a magnitude (in arbitrary units called "arbs"), and an angle in degrees for the direction. You will find the sum of these three vectors, R = A + B + C, in three different ways. You will be given this resultant vector Table Top - Central Pin String Bubble Level Hooked Weight Pulley Double Scale Tripod Base Levering Screw Support Rod determined by the students. In the first part of the lab the students used a force table, as shown above. The ring in the center of the force table is a single object which feels the forces (a "pull" which can be represented as a vector) from each of the attached strings. Those forces are directly proportional to the masses hanging from the other ends of the strings, so we will say that "1 arb is equivalent to 1 gram.if The students hanged masses equal to the magnitudes of vectors A, B, and C from three of the strings The strings were passed over pulleys placed at the appropriate angles for each vector. The net effect of these three pulls (the "net force") is the same as that of a single string pulling with a strength equal to the magnitude of the resultant, in the same direction as the resultant. It is difficult to determine exactly what size and direction this resultant force has. However, it is easy to find when the net force is (nearly) zero: the ring will stay put, without needing to rest against the center post. By adding a fourth force (which we will call the "Equilibrant," E), we get a net force equal to zero. That is, A + B + C + E - 0. But since R = A+ B + C, that means R + E - 0, or R - -E. So, by experimentally determining E, we can say that R has the same magnitude as E, but is in exactly the opposite direction (that is, 180 away)-Preparation 1. The students used a force table, four hanging mass sets, and a ruler and a protractor to perform the experiment. 2. Get a set of three vectors and their resultant from the instructor. These are the vectors you will use in this lab. Procedure Part A: Force Table In this part of the lab, the students used the force table to "physically" add the three vectors following the instruction below. 1. Hang masses corresponding to vectors A, B, and C at their appropriate angles, as described above. Gently clamp the pulleys in place - no need to over-tighten! 2. Tug on the remaining string to get a rough estimate of the strength and direction of the force needed to cancel out the resultant of those three forces. 3. Place a pulley at the rough location needed to balance the forces and gently tighten the clamp so it cannot move. Pass the free string over that pulley, and hang an amount of mass from it roughly equal to what you need to balance the forces. 4. Experiment with adding or removing masses until you find the range of masses for which the ring remains hovering without touching the pin. Record the minimum and maximum masses in your template, and then use them to determine the best value and uncertainty for the magnitude of the equilibrant E. 5. Next, set the mass of the equilibrant to the best value you just recorded. Loosen the pulley clamp and slide the pulley from side to side to determine the minimum and maximum angles for which the ring remains hovering without touching the pin. Record these angles in your template and use them to determine the best value and uncertainty for the direction of the equilibrant E 6. On the Rough Sketch, draw and label the locations of the strings for vectors A, B, C, and E. 7. The resultant found from this step has a magnitude equal to that of the equilibrant, and a direction exactly opposite to that of the equilibrant. Record the magnitude and direction of your resultant vector, with uncertainty, on the template. Part B: Graphical Vector Addition In this part of the lab, you will find the vector sum of the same three vectors using the graphical method, which is described in further detail in your textbook Note: like usual, each partner should make his or her own diagram and measurements in the template However, you should discuss with each other how to perform this part correctly, and compare answers it's a good way to catch errors).1. Take a sheet of paper (you may use the given page on the template), a ruler, and a protractor. Draw an X-y coordinate system on the paper, being sure to make the two axes perpendicular. 2. Choose a convenient scale for your drawing. For example, a map may have a legend that reads "1 in. = 1.5 mi." meaning that one inch on the map means 1.5 miles of the actual ground that the map refers to. In your case, you should have something like #1 cm - X arbs," where X is chosen so that the graphical method will work well. That is, if X is too small, then the sum of vectors will not all fit on your page. If X is too large, then the vectors will be so small that they will be very difficult to measure accurately. Be sure to write your scale somewhere on your paper. You should all use the same scale for your drawings. 3. Using a ruler and protractor, draw vector A starting from the origin, of the appropriate size and angle. 4. Lightly sketch a new x-y axis at the end (tip) of vector A. The new x-y axis must be parallel to the original x-y axis. Then, again using ruler and protractor, draw vector B starting from the end of vector A, of the appropriate size and direction. 5. Lightly sketch a new x-y axis at the end (tip) of vector B. Then, again using ruler and protractor, draw vector C starting from the end of vector B, of the appropriate size and direction. 6. Draw the resultant R, starting at the origin of the original coordinate system, and ending at the end of vector C. Measure its size and direction NOTE: Do not try to draw your resultant R from part A! The point is to employ three different ways to find the vector R. 7. Since it is not possible to draw lines of exactly the right length, at exactly the right angle, starting from exactly the right point, it shouldn't be surprising if your result is not perfectly accurate. Based on how well you were able to use pencil and paper, try to get a reasonable estimate of how far off the correct values may be from your measured values. This rough judgment call on your part will represent your uncertainty in this result. Record the magnitude and direction of your resultant vector, with uncertainty, on the template. Convert the magnitude from em to arbs. Part C: Component Method In this part of the lab, you will calculate the vector sum of the same three vectors (A, B, and C). This step is purely mathematical: there are no measurements. NOTE: For this part, you should assume all of the given vectors are exact. When you report your final answers for R, round them to the nearest 0.1 arb or nearest 0.1 degree. 1. Draw a rough triangle showing each vector with its vector components. Then, using the methods described in class, calculate each vector's x- and y- components (Ax, Ay, By. By, etc.). Note that these components may be positive or negative, depending on the direction2. Add the three x-components to find the x-component of the resultant vector, R. 3. Add the three y-components to find the y-component of the resultant vector, Ry 4. Draw a rough triangle showing the components Ry and Ry, and the resultant vector R. Use this triangle to help you calculate the magnitude and direction of the resultant vector R. Data Analysis 1. Calculate the percent error in the magnitude of the result for Part A, assuming that the result of Part C is the correct value. 2. Calculate the percent error in the magnitude of the result for Part B, assuming that the result of Part C is the correct value. [measured - actuall % Error x 100% actual 3. Summarize your results by filling out the table with your results from each of the three parts including uncertainty for Parts A and B. along with the percent errors you just calculated. Conclusions 1. Do the calculated values of magnitude and angle from Part C fall within the experimental range you found in Part A? Based on your answer, do you think you overestimated your uncertainty, underestimated, your uncertainty, or chose about the right uncertainty in Part A? 2. Do the calculated values of magnitude and angle from Part C fall within the experimental range you found in Part B? Based on your answer, do you think you overestimated your uncertainty, underestimated, your uncertainty, or chose about the right uncertainty in Part B? 3. Explain why it is true to say R - A + B + C in this lab, but it is false to say that R = A + B + C. What to submit for your lab report: 1. Your lab template with your calculations clearly shownWhich set of vectors did you use? (1-8): Vector Magnitude Direction OWL A B C Draw a rough sketch of A, B, and C on the figure at right: (3 pts.) 20 350 Part A: Force Table Record your minimum and maximum mass measurements for the equilibrant E. Use your data to calculate the best 280 0 290 value and uncertainty of the magnitude of E. (4 pts.) Minimum Maximum Magnitude of E Mass Record your minimum and maximum angle measurements (using the best value of the mass) for the equilibrant E. Use your data to calculate the best value and uncertainty of the direction of E. (4 pts.) Minimum Maximum Direction of E Angle Draw a rough sketch of E on the figure above. This figure should now look exactly like the setup. (1 pts.) Record the magnitude (in arbs) and direction of the resultant R, with uncertainty: (2 pts.) Magnitude of R Direction of RFor the resultant, record the following values, with uncertainty: (2 pts.) Units Magnitude Direction cm I arbs 2 PHYS 211 Lab: Summer 2022 Lab II: Vectors-Force Table Penn State Altoona Part C: Component Method 1. Draw individual triangles roughly showing each vector with its vector components. Then, using the methods described in class, calculate the x- and y- components of each vector (Ax, A,, etc.) (9 pts.) Vector Triangle X-component calculation y-component calculation A2. Add the three x-components to find the x-component of the resultant, Rx. (2 pts.) 3. Add the three y-components to find the y-component of the resultant, Ry. (2 pts.) Rx Ry 4. Draw a triangle showing the components Rx and Ry, and the resultant vector R. Use this triangle to calculate the magnitude and direction of the resultant vector R, to the nearest 0.1 arb or degree. Magnitude: (3 pts.) Direction: (3 pts.)1. Calculate the percent error in the magnitude of the resultant for Part A, assuming that the result from Part C is the correct value. (2 pts.) 2. Calculate the percent error in the magnitude of the resultant for Part B, assuming that the result from Part C is the correct value. (2 pts.) PHYS 211 Lab: Summer 2022 Lab II: Vectors-Force Table Penn State Altoona 3. Summarize your results by filling out the table with your results from each of the three parts, including uncertainty for Parts A and B, along with the percent errors you just calculated (1 pts.) Resultant Magnitude Direction % Error Part A + Part B Part CConclusions 1. Do the calculated values of magnitude and angle from Part C fall within the experimental range you found in Part A? Based on your answer, do you think you overestimated your uncertainty, underestimated, your uncertainty, or chose about the right uncertainty in Part A? (2 pts.) Magnitude: Direction: 2. Do the calculated values of magnitude and angle from Part C fall within the experimental range you found in Part B? Based on your answer, do you think you overestimated your uncertainty, underestimated, your uncertainty, or chose about the right uncertainty in Part B? (2 pts.) Magnitude: Direction: 3. Explain why it is true to say R - A + B + C in this lab, but it is false to say that R = A + B + C (2 pts.) BEFORE YOU LEAVE LAB: Please put the slotted mass sets back together the way you found them! Each set should have one 50-g hanger, nine 20-g, one 10-g, and two 5-g masses. 5