Lab 3, Vector Addition Justin Fan 2

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Indiana University, Bloomington *

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P221

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Mechanical Engineering

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Jan 9, 2024

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Lab 3: Vector Addition Introduction Vectors are quantities that possess both magnitude and direction. They differ from scalars, which are defined by magnitude alone. In this lab vectors are measured in the form of Forces that are at equilibrium. The equilibrium of forces is established through altering the angle and weights on a force measuring instrument called the force table. We will use data collected of mass at fixed and differ angles to compare theoretical value to actual value. As well as also trying to determine an unknown force vector. Experimental Methods Part A 1. Level the Force Table: • Check the built-in bubble level. Ensure the air bubble centers within the circle. • Adjust the table legs as needed until level. 2. Setting Up Vector ? : • Hang a total of 120 g* (including a 20 g* hanger) at an angle between 15° and 75° (avoid 50°). • Document the selected angle. 3. Equilibrium for F x and F y : • Place pulleys on the –x and –y axes. • Adjust masses until the center ring aligns perfectly. • Log weights, adding the 20 g* hanger weight. • Conduct 5 trials using the consistent θ from step 2. • Reset weights and pulleys each time. Part B 1. Setting Up the Weights:  Hang the unknown mass, Fun , on the -x-axis.  For F 1 : Use 200 g* in Quadrant I (0° to 90°).  For F 2 : Use 150 g* in Quadrant IV (270° to 0°). 2. Balancing F 1 and F 2 :

 Adjust the pulleys for F 1 and F 2 within their respective quadrants to achieve balance.  Document the final angles in Table 2. 3. Trials:  Conduct 5 trials with consistent weights, adjusting angles to balance forces each time.  Consider string alignment for accurate readings. 4. Determine Unknown Mass:  Measure and record the actual weight of F un using a digital scale. 5. Analysis of x-components:  Calculate the x-component F 1x from F 1 ’s magnitude and direction for all trials.  Repeat the calculation for F 2x using F 2 . 6. Identifying the Unknown Force:  Calculate Σ Fx by summing x-components for each trial.  Compute the mean, standard deviation, and uncertainty for Σ Fx .  Clearly highlight the best value and uncertainty of the unknown force in your report. 7. Analysis of y-components:  Reiterate steps 5 and 6 for y-components.  Update your data table accordingly. Data and Analysis

Part A Table 1 Trial Fx g* Fy g* Note 1 102 60 F=120g* 2 102 60 Angle=30(degree ) 3 104 60 4 102.8 60 5 103.6 62 Mean 102.88 60.4 St.Dev 0.912 0.894 Uncertainty 0.408 0.4 Best Value Fx= 102.9g* uncertainty 0.408 Fy= 60.40g* uncertainty 0.4 Theoretical Fx=103.92g* Fy=60g*

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Part B Table 2/3 Trial Θ 1 (°) Θ 2 (°) F 1x g* F 2x g* ΣFx F 1y g* F 2y g* ΣF y 1 40.2 302.5 152.8 80.5 9 233.3 9 129. 1 - 126. 5 2.6 2 41.0 302.0 150.9 79.4 9 230.3 9 131. 2 - 127. 2 4.0 3 40.9 301.0 151.2 77.2 6 228.4 6 130. 9 - 128. 6 2.3 4 40.1 301.1 153.0 77.4 8 230.4 8 128. 8 - 128. 4 0.4 5 40.4 301.5 152.3 78.3 7 230.6 7 129. 6 - 127. 9 1.7 Mea n 230.7 2.83 St, Dev 1.76 0.99 Unc 0.79 0.44

Notes F1=200 g F2=150 g Fun=208. 3 Best value Fx= 230.7g* Uncertainty 0.79 Fy= 2.83g* Uncertainty 0.44 Theoretical Value: 208.3g* Analysis Question: Do the experimental and theoretical values agree within their uncertainties? If not, speculate what may be responsible for the disagreement. The y component of the experimental value and theoretical value agrees. But the X component does not agree. The reason of such error could be caused by misreading during the experiment by the reader. This error could also be contributed by the instrument itself as we had issues aligning the instrument to equilibrium during the experiment as the ring kept shaking back and forth. Question: At equilibrium, explain how the sum of the forces in the x- direction is related to the magnitude of the unknown force. At equilibrium the sum of the forces in the X direction is equal to the magnitude of the unknown force. The reason is because the Y component cancels out and the result vector of the X component has the equivalence magnitude to the unknown force. Question: Does your calculation of the unknown force agree with the value measured from the scale within its uncertainty? If not, speculate what may be responsible for the disagreement. Our calculation of the unknown force does not agree with the measured results of the unknown force. I believe the reason is because that the instrument contributed the errors since we could not establish complete equilibrium. As a result of such we had to interpret our results to the best of our ability. Therefore, we have a disagreement with the experimental value.

Question: Based of your knowledge of force components, what would you expect the components to sum to? Does your final result agree with this within its uncertainty? Based on my knowledge of force components. I expected the sum of the y component to be zero. However, the experimental results do not agree with me. I believe the reason of this error is contributed by the instrument itself. The reason is because the ring that is used to make sure that the forces are at equilibrium kept on shaking to the point that our group could not align it, so we had to give our best interpretation of the results. Conclusion In our experiment our goal was to measure vectors of forces, weight, and their angles. Then use the data we collected to determine unknown forces. Our results of the experiment did not align with our theoretical predictions. I would like to point out that our experiment experienced various limitations. Which are the contribution factors to our incorrect results. The most noticeable limitation was that our instrument was not accurate. As a result of limitation errors, we had to give our best interpretation of the results we have gained. Appendix Calculation done in class.

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