{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 2
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
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 4
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?
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern