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Unformatted text preview: Muraj, Hamza Homework 22 Due: Mar 24 2006, 4:00 am Inst: Florin 1 This printout should have 11 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points A uniform flat plate of metal is situated in the reference frame shown in the figure below. 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 x y Calculate the x coordinate of the center of mass of the metal plate. Correct answer: 3 . 33333 . Explanation: Basic Concept: The center of mass coor dinate is x Z xdm m , where m Z dm, and dm = y dx, where is the areal density mass area of the plate. Solution: Let ( x 1 ,y 1 ) = (0 , 0) ( x 2 ,y 2 ) = (10 , 0) ( x 3 ,y 3 ) = (0 , 4) . The equation for the hypotenuse is y y 2 x x 2 = y 3 y 2 x 3 x 2 . The slope of the hypotenuse is s = y 3 y 2 x 3 x 2 = 4 10 = 2 5 . Rewriting the equation, we have y = s ( x x 2 ) + y 2 =  2 5 ( x 10) + 0 . The xcoordinate of the center of mass is x = Z x 2 x 1 xy dx Z x 2 x 1 y dx = Z x 2 x 1 xs ( x x 2 ) dx Z x 2 x 1 s ( x x 2 ) dx = Z x 2 x ( x x 2 ) dx Z x 2 ( x x 2 ) dx = 1 3 x 3 1 2 ( x 2 ) x 2 1 2 x 2 ( x 2 ) x fl fl fl fl fl fl fl x 2 = 1 3 x 3 2 1 2 ( x 2 ) x 2 2 1 2 x 2 2 ( x 2 ) x 2 = x 3 2 3 x 2 2 = 1 3 x 2 (1) = 1 3 (10) = 3 . 33333 . Alternate solution: The center of mass of a right triangle is 1 3 of the height or base of the triangle measured from its right angle. Therefore Eq. 1 is the xcoordinate of the center of mass of the metal plate. The ycoordinate of the center of mass of the metal plate is y = 1 3 y 3 = 1 3 (4) = 1 . 33333 . Muraj, Hamza Homework 22 Due: Mar 24 2006, 4:00 am Inst: Florin 2 Note: This problem has a different triangle for each student....
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 Spring '09
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