Physical 6BL Math Lab

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University of California, Santa Barbara *

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6BL

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Physics

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Apr 3, 2024

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Lab 1: Math Lab Physics 6BL 4/8/21 INSTRUCTIONS: - To begin, download a copy of this document to your Google Drive - Go to “File” on the top menu, then click “Make a Copy” from the dropdown menu - Fill in your name, section, TA, and date on the top left corner of this page - You should have 4 pages of questions after this initial page so 7 pages total - You will include this instructions page in your report - Tables, sketches, and questions each have a separate section of the document - Tables and sketches may be copy and pasted or uploaded as an image from the lab specific google spreadsheet - The tables are formatted to fit the information asked for in the lab - If there are any exercises with separate questions, please answer them in the closest question box - To clarify: You can edit this lab in Google Docs - If you need extra space, please consider changing the size of your work so it fits in the boxes. Contact us if you have trouble with this. - Please ask on Piazza or email us if you have any questions about this format. We are happy to help. - In the end, you should save your report as a pdf and turn it in to the submission portal. Gradescope will not ask you to match pages to questions because you submitted in this format. Your Final Report Pages should be as follows: Page 1 - This Instructions Page Page 2 - Exercise 2: (graph) Page 3 - Question 1 and Question 2 Page 4 - Exercise 4 (Table) and Exercise 5 (graph) Page 5 - Question 4 and Question 5 Page 6 - Question 6 Page 7 - Conclusion

Exercise 2: (graph)

Question 1 a) The graph looks exponential. b) The points on the graph are in a curved shape. The orbital period (y-axis) is also increasing rapidly as the orbital semimajor (x-axis) increases. c) A linear trendline is not a good fit for the data set. The points in the data set are below and above the trendline, which implies that the linear fit is not a good fit. Question 2 (Exercise 3) Table 2 Name Variable Variable Dependant Variable y log(T) Independent Variable x log(a) Slope m z Y-Intercept b log(k)

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Exercise 4 (Table) Log-Log Data Table: Take the LOG10 of the data above to generate the data for your Log-Log Plot below: Planet LOG10( a ) LOG10( T ) Mecury -0.4121768287 -0.6182713815 Venus -0.1406815349 -0.2109921446 Earth 0 0 Mars 0.182984967 0.2743426161 Jupiter 0.7163372879 1.074150597 Saturn 0.9814561665 1.469182617 Uranus 1.283301229 1.924334603 Neptune 1.477844476 2.216904499 Pluto 1.596377144 2.394346597 Exercise 5 (graph)

Question 3 a) The graph looks linear. b) The points are in a straight line and the linear trendline fits all the points of the data set. c) It does make sense to perform a linear fit with this data set because the trendline fits all the points in the data set. There are no points above or below the trendline, which means the line goes through all the points of the plot. d) In the previous plot, the points were in an exponential shape, while in this plot, the points are in a linear shape. In the previous plot, the linear trendline did not fit the plot. In this plot, the linear trendline fits the points of the plot exactly. Question 4 a) The equation is in the form y=mx+b. The equation for the line of best fit is 1.5x+2.47 × 10 -4 . b) Using the table from Question 2, z is equivalent to m in the equation y=mx+b. Therefore, the value of z is 1.5 years/a.u. c) Using the table from Question 2, log(k) is equivalent to b in the equation y=mx+b. b is 2.47 10 -4 in the equation for the line of best fit. To find k, we have to take the log of b, × which would be 10 2.47 10^-4 . Therefore, the value of k is 1. × Question 5 a) First, we have to take the log of the equation: T=(K) (1/i) a (j/i) . T=(K) (1/i) a (j/i) log(T) = (1/i)log(K) + (j/i)log(a) log(T) = (log(K)/i) + (jlog(a)/i) Using the table from Question 2, log(k) is equal to b, log(a) is equal to x, and log(T) is equal to y. y = (b/i) + (jx/i) Y = b + (j/i)x Due to the equation being in the form of y=mx+b, we can conclude that (j/i) is equivalent to m. We can use the table from Question 2 to state that m is equivalent to z. Therefore, (j/i) is equivalent to z. Because z = 1.5 and (j/i) is equivalent to z, then (j/i) = 1.5. Therefore, (j/i)=1.5. The simplest integer values for j and i would be j = 3 and i = 2. (j/i)=1.5→(3/2)=1.5 Using these integers, Equation 6 would be T 2 = Ka 3 . b) The equation k = (K) (1/i) can be used to determine K. Since k = 1 and i = 2, then the equation is 1 = (K) (1/2) . Therefore, K = 1.

Question 6 a) To convert years to seconds and a.u to meters, we must use the equation (sec/year) 2 /(m/a.u) 3 . One year to seconds is 3.154 10 7 and 1 a.u. is 1.496 10 11 . Using × × this conversion, the equation is (3.154 10 7 ) 2 /(1.496 10 11 ) 3 . Therefore the conversion × × factor is 2.971 10 -19 sec 2 /m 3 . × b) The conversion factor made the intercept of the log-log graph smaller. The value of the new intercept is -7.24 10 -23 . ×

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Exercise 7: CONCLUSION In this lab, we learned how to use two different methods to change the independent variable to linearize data. The first method showed how to linearize data if the form of the equation is known. The second method is used when the form of the equation is not known. The lab showed us how to use the logarithm method in order to linearize plots. We had a set of data, took the log of each of the x and y points. Then once the points were plotted, the graph showed that the data had a linear fit.

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