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# Spring System Graphing

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A mass M is connected to the end of a spring with mass m and force constant k. The mass of the spring is 7.40 g. When the mass/spring system is set into simple harmonic motion, the period of its motion is

T = 2*PI*SQRT((M + (m/3))/K)

Part 1: A two-part experiment is conducted with use of various masses suspended vertically from the spring. First, the masses are hung from the spring and allowed to come to rest. The static extensions are measured for six masses, M. The following is a list of the extensions for each mass:

Mass, M (g) Extension (cm)
20.0 17.0
40.0 29.3
50.0 35.3
60.0 41.3
70.0 47.1
80.0 49.3

Construct a graph of Weight (Mg) versus extension (x). Add a linear trendline to the data. From the slope of your graph, determine a value for k for this spring. (Re-write your equation in terms of weight and extension. Separately, write the physics equation you would use to answer this question by hand. Comparing the two equations, you should see that the slope and k are related.)

Part 2: The mass/spring system is now set into simple harmonic motion. The experiment is repeated with the same various masses and the periods are measured using a stopwatch. For ease of measuring, the total time is measured for 10 oscillations. The following is a list of the periods for 10 oscillations obtained for each mass:

Mass, M (g) Period (s) -- for 10 oscillations
20.0 7.03
40.0 9.62
50.0 10.67
60.0 11.67
70.0 12.52
80.0 13.41

a) If you notice, the measured periods are slightly different than the predicted values from Part 1.b. What could be the cause of the differences (list at least three plausible causes)?

b) Construct a graph of T2 versus mass, M, and add a linear trendline to the data. From the slope of the line, determine a value for k (compare the trendline equation to the equation given at the beginning of the problem squared). Compare this value to the value of k obtained in part 1. What is the percentage difference between the two values?

%Difference = ((K"-K')/K')*100%

c) Using the trendline equation and k value obtained in Part 2.b, determine a value for m. Compare this value with the given value of 7.40 g.

%Difference = ((m"given" - m"measured")/m"given")*100%