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Unformatted text preview: PHY 3221: Mechanics I Fall Term 2010 Final Exam, December 14, 2010 This is a closed book exam lasting 90 minutes. Since calculators are not allowed on this test, if the problem asks for a numerical answer, answering 2 + 2 is as good as 4, and 2 is as good as 1 . 4142 ... . There are six equally weighted problems worth a total of 30 points. The problems appear on the second and third page of this test. Begin each problem on a fresh sheet of paper. Use only one side of the paper. Avoid microscopic handwriting. Put your name, the problem number and the page number in the upper righthand corner of each sheet. To receive partial credit you must explain what you are doing. Carefully labelled figures are important! Randomly scrawled equations are not helpful. Draw a box around important results (or at least results which you think might be im portant). Good luck! 1 Problem 1. [5 pts] You use a rope to pull a crate across a level floor. The maximum tension that the rope can have without breaking is T max , and the coefficient of kinetic friction is k . Suppose that you tug on the rope so that it makes an angle with the horizontal (refer to Fig. 1). Find the maximum possible mass M max of the crate that you can pull at constant speed without breaking the rope, and the optimal angle opt that will allow you to do that. Hint: make sure to draw a force diagram and clearly label all forces. M vectorg k x y Figure 1: An illustration for the crate pulling problem. Solution. The fources which act on the crate are: gravity G = Mg , tension T , normal reaction N and friction F = k N . When we pull the crate at constant speed, the net force is zero. In components: x : F + T x = k N + T cos = 0 , y : G + T y + N = Mg + T sin + N = 0 . Eliminate N N = T cos k Mg + T sin + T cos k = 0 and solve for M : M = T k g (cos + k sin ) Obviously, the mass will be maximum when T = T max . To find the corresponding optimum angle opt , find the extremum of M ( ): dM d = 0 = sin opt + k cos opt = 0 = tan opt = k = opt = tan 1 k Explicitly: sin opt = k radicalBig 1 + 2 k , cos opt = 1 radicalBig 1 + 2 k 2 and M max = T max k g (cos opt + k sin opt ) = T max k g radicalBig 1 + 2 k Problem 2. [5 pts.] Obtain the Fourier expansion of the periodic function F ( t ) = +1 , if...
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 Spring '08
 CHAN
 mechanics

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