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examf - PHY 3221 Mechanics I Fall Term 2010 Final Exam This...

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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
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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 μ k . Suppose that you tug on the rope so that it makes an angle θ 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 θ opt that will allow you to do that. Hint: make sure to draw a force diagram and clearly label all forces. M θ vectorg μ k Figure 1: An illustration for the crate pulling problem. Problem 2. [5 pts.] Obtain the Fourier expansion of the periodic function F ( t ) = +1 , if π/ω < t < 0 , 1 , if 0 < ω < + π/ω.
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