hw22 - Muraj Hamza Homework 22 Due 4:00 am Inst Florin This...

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Muraj, Hamza – Homework 22 – Due: Mar 24 2006, 4:00 am – Inst: Florin 1 This print-out should have 11 questions. Multiple-choice 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. 0 1 2 3 4 5 6 7 8 9 10 0 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 x dm m , where m Z dm , and dm = σ y dx , where σ 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 - 0 0 - 10 = - 2 5 . Rewriting the equation, we have y = s ( x - x 2 ) + y 2 = - 2 5 ( x - 10) + 0 . The x -coordinate of the center of mass is x = σ Z x 2 x 1 x y dx σ Z x 2 x 1 y dx = Z x 2 x 1 x s ( x - x 2 ) dx Z x 2 x 1 s ( x - x 2 ) dx = Z x 2 0 x ( x - x 2 ) dx Z x 2 0 ( 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 0 = 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 x -coordinate of the center of mass of the metal plate. The y -coordinate of the center of mass of the metal plate is y = 1 3 y 3 = 1 3 (4) = 1 . 33333 .
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Muraj, Hamza – Homework 22 – Due: Mar 24 2006, 4:00 am – Inst: Florin 2 Note: This problem has a different triangle for each student. 002 (part 1 of 1) 10 points A(n) 93 kg boat that is 8 m in length is initially 7 . 9 m from the pier. A 38 kg child stands at the end of the boat closest to the pier. The child then notices a turtle on a rock
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