Phys110-Worksheet2-202209-2 (2)

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University of Victoria *

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PHYS-110

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Statistics

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Feb 20, 2024

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Project 2 Worksheet 1. Include a photo of your raw work, including your TA’s signature. Failure to include this data will result in a grade of zero for the entire lab report.

2. Show all of formulas you derived/used for determining each individual Fx and Fz from your measurements, along with the formulas for uncertainties. These formulas will be the ones you use in your spreadsheet to fill out your table of values.

3. Summarize your results in three tables with columns for labeling the mass system and force, the x component of the force and its uncertainty, and the z component of the force and its uncertainty. (See the description in the Analysis and Writeup section.) Column1 mass (+- 0.1 g) r (+- 0.1 cm) x (+- 0.1 cm) z (+- 0.1 cm) F x (N) δF x (N) F z (N) δF z (N) Case 1 m1 200 7.75 6.75 3.8 1.71 0.03 0.96 0.03 m2 200 8.45 -7 4.2 -1.62 0.03 0.97 0.03 m3 200 8 0 -8 0 0 -1.96 0.03 Case2 m1 310 8.7 8.3 2.7 2.9 0.05 0.94 0.04 m2 300 8.9 -8.4 3 -2.77 0.05 0.99 0.03 m3 200 8.8 0 -8.8 0 0 -1.96 0.03 Case 3 m1 270 8.2 8.15 1.5 2.63 0.05 0.48 0.03 m2 300 6.25 -5.4 3.9 -2.54 0.02 1.83 0.06 m3 200 7 0 -7 0 0 -1.96 0.04 4. Show all of your calculations in summing the forces and determining the uncertainty in the sum of forces for each direction and for each combination of masses. (Do not do this in your spreadsheet.)

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5. Explicitly state the expected value for the sum of forces in the x and z directions. Per- form a statistical comparison of each sum of forces, as compared to the appropriate theoretical/expected value. Theoretical/expected value: Σ Fx=0, Σ Fz=0 6. Respond to the following questions/instructions using complete sentences: (a) Why are three points used instead of two points for each branch of the force diagrams? Three points, rather than two, are utilized for each segment in the force diagrams to enhance the accuracy of the best-fit line representing the string on the pulley. This approach is similar to how repeated measurements yield a more precise average. By plotting a line through a greater number of points, it becomes more reliably aligned with the true orientation of the twine on the pulley, ensuring greater accuracy in depicting its direction. (b) Why would putting all three points close together and making your Y shapes small be a bad idea for this experiment? In this experiment, clustering the three points closely and forming small 'Y' configurations is not advisable, as it leads to a less precise line of best fit. Such an arrangement may fail to accurately reflect the actual contour and length of the twine on the pulley, potentially causing errors in the calculated force values. (c) What does it mean if your test statistic is larger or smaller than 2? Why is a value of 2 used? A test statistic below 2 suggests that your equipment and experimental approach are in alignment with the presuppositions regarding Newton's Laws, specifically the idea that the net forces in both the x and z directions should be zero. Conversely, a test statistic greater than 2 indicates a deviation from these assumptions. This discrepancy could stem from flawed equipment or experimental methods, or from

overlooking additional factors influencing the force measurements. Essentially, a threshold of 2 is employed to determine the congruence between theoretical and actual measurements, where a value less than 2 signifies agreement, and a value exceeding 2 points to a lack of correlation. (d) What is the largest contribution to the uncertainty in determining F x and F z ? The primary source of uncertainty in determining Fx and Fz arises from the method employed for line-drawing in the force diagrams, which are crucial for gathering the measurements necessary for force calculations. The accuracy of pinpointing the points corresponding to the string on the pulleys, done through visual estimation with a mirror, cannot be fully assured. This imprecision in line-drawing leads to uncertainties in measuring the lengths of x, z, and r, ultimately impacting the calculated values of Fx and Fz. Additionally, the precision of the ruler used is limited, as it is graduated only in millimeters. This limitation could further influence the accuracy of the line lengths and, consequently, the force values. (e) How can you independently verify that your lines for determining the x and z components are perpendicular? Use this to test several of your lines and show your work. To confirm that the lines representing the x and z components in our diagrams are at right angles to each other, we can apply the Pythagorean theorem, expressed as a² + b² = c². This theorem outlines the relationship between the sides of a right-angled triangle. Therefore, if the equation accurately applies to all values of x, z, and r components in our force diagrams, it indicates that the x and z components are indeed perpendicular to each other.