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# 7 - Problem 7.1 If a = 2 what is the x coordinate of the...

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Problem 7.1 If a = 2 , what is the x coordinate of the centroid of the area? Strategy: The x coordinate of the centroid is given by Eq. (7.6). For the element of area dA , use a vertical strip of width dx . (See Example 7.1). x y y = x 2 a Solution: x = x dA dA x = a 0 x ( y dx ) a 0 y dx Substituting y = x 2 , we get x = a 0 x 3 dx a 0 x 2 dx = x 4 4 x 3 3 a 0 = 3 a 4 For a = 2 x = 3 2 x (1, 1) y y = x 2 a y x dA = y dx a y = x 2 Problem 7.2 Determine the y coordinate of the cen- troid of the area shown in Problem 7.1 if a = 3 . Solution: y = y dA dA y = a 0 1 2 y ( y dx ) a 0 ( y dx ) Substituting y = x 2 , we get y = a 0 1 2 x 4 dx a 0 x 2 dx = 1 2 x 5 5 a 0 x 3 3 a 0 = a 5 2 . 5 a 3 3 y = 3 10 a 2 For a = 3 , y = 27 10 x y y = x 2 a (1, 1) dA = y dx Midpoint dx a x y y /2

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Problem 7.3 If the x coordinate of the centroid of the area is x = 2 , what is the value of a ? x y y = x 3 a 0 Solution: x = A x dA A dA x = a 0 x ( y dx ) a 0 ( y dx ) . Substituting y = x 3 , x = a 0 x 4 dx a 0 x 3 dx = x 5 5 x 4 4 a 0 = 4 a 5 If x = 2 , 2 = 4 a 5 and a = 10 4 = 2 . 5 x y y = x 3 a 0 dA = y dx a x y y = x 3
Problem 7.4 The x coordinate of the centroid of the area shown in Problem 7.3 is x = 2 . What is the y coordinate of the centroid? Solution: From Problem 7.4, a = 2 . 5 y = y dA dA = 2 . 5 0 1 2 y ( y dx ) 2 . 5 0 y dx = 2 . 5 0 1 2 x 6 dx 2 . 5 0 x 3 dx y = x 7 14 x 4 4 2 . 5 0 = 4 x 3 14 2 . 5 0 = 4 . 46 y = 4 . 46 y x a y = x 3 y /2 x y y = x 3 a 0 Problem 7.5 Consider the area in Problem 7.3. The “center of the area” is defined to be the point for which there is as much area to the right of the point as to the left of it and as much area above the point as below it. If a = 4 , what are the x coordinate of the center of area and the x coordinate of the centroid? Solution: Center of Area: Let X be the coordinate of the center of the area. The area to the left of X is A L = X 0 x 3 dx = X 4 4 . The area to the right is A R = a X x 3 dx = a 4 4 X 4 4 . Equating the two areas, a 4 4 X 4 4 = X 4 4 , from which a 4 = 2 X 4 or X = a (2) 1 4 . For a = 4 , X = 0 . 8408(4) = 3 . 3636 The centroid: The centroid is x = a 0 x 4 dx a 0 x 3 dx = 4 a 5 . For a = 4 , x = 16 5 = 3 . 200

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Problem 7.6 Determine the x coordinate of the cen- troid of the area and compare your answer to the value given in Appendix B . y x 0 a y = cx n Solution: x = x dA dA x = a 0 x ( y dx ) a 0 y dx = a 0 xC / x n dx a 0 C / x n dx x = x n +2 ( n +2) a 0 x n +1 ( n +1) a 0 = ( n + 1) ( n + 2) a x = a ( n + 1) ( n + 2) Checks with result in Appendix y x 0 a y = cx n y x dA = y dx dx y = cx n a Problem 7.7 Determine the y coordinate of the cen- troid of the area and compare your answer to the value given in Appendix B . Solution: y = y dA dA y = a 0 y 1 2 y dx a 0 y dx = 1 2 a 0 y 2 dx a 0 y dx y = a 0 c 2 x 2 n dx 2 a 0 cx n dx = c 2 c x 2 n +1 (2 n +1) x n +1 ( n +1) a 0 y = c 2 ( n + 1) (2 n + 1) a n Checks with Appendix y x 0 a y = cx n a y /2
Problem 7.8 Suppose that an art student wants to paint a panel of wood as shown, with the horizontal and ver- tical lines passing through the centroid of the painted area, and asks you to determine the coordinates of the centroid. What are they?

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