examf_sol - PHY 3221: Mechanics I Fall Term 2010 Final...

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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 right-hand 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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examf_sol - PHY 3221: Mechanics I Fall Term 2010 Final...

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