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Center of mass for sheet metal

  1. Apr 22, 2005 #1
    I am having the hardest time figuring out the center of mass of this problem.

    A sheet of metal is cut in the shape of a parabola (imagine that you have a parabola shape with the top being flat). The curved edge of the sheet is specified by the equation y=ax^2, and y ranges from 0 to b. Find the center of mass in terms of a and b. (You will need to find the area first.)

    There's the question. The answer is 3b/5. I used center of mass equation

    Ycm=(1/m)(integral of y dm)
  2. jcsd
  3. Apr 22, 2005 #2
    The formula for y_cm looks good. What r u using for "dm"?
  4. Apr 22, 2005 #3

    at first i had (3M)/(ab^3)ydx
    then i found the area a different way and i came up with (Mx^3)ydx/(2ab)

    what is the correct evaulation of this area. this dm is the most important part of the integration.
  5. Apr 22, 2005 #4
    The width of the metal sheet at distance y is 2*Sqrt(y/a). Intergate this from 0 to b to get the area,

    BTW, what is "M"?
    Last edited: Apr 22, 2005
  6. Apr 22, 2005 #5


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    Did you get it?

    I find 3b/4.
  7. Apr 22, 2005 #6
    M comes from this

    Total mass (M) / Total Area (the integration of the function) = dm (tiny portion of mass) / da (tiny portion of area)


    i just solved for dm so i could then substitute in for dm when i find the center of mass
  8. Apr 22, 2005 #7
    That is just saying the density is constant i think. Anyway u dont need the density. Were u able to get the area?

    quasar987: do u get 3b/4 for the final answer or for area or what? hbombs first post says its 3b/5
  9. Apr 22, 2005 #8
    could you show me the steps of how you arrived to this answer
    maybe the answer in the book is wrong.
  10. Apr 22, 2005 #9
    i doubt the book is wrong. First find the area, that is compute the integal

    integral from 0 to b of (2*sqrt(y/a)) dy = Area

    then compute the integral of y "weighted" by mass, or area in this case

    integral from 0 to b of (2*sqrt(y/a)*y)dy

    and divide what u get by the area....
  11. Apr 22, 2005 #10


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    What does "weighted" means?

    According to the dm you found, the integral we want to compute is

    [tex]Y_{CM} = \iint\frac{y}{A}dA[/tex]

    So I found A. HackaB's way it the shortest:

    [tex]A = \int_0^b2\sqrt{\frac{y}{a}}dy = \frac{4b}{3}\sqrt{\frac{b}{a}}[/tex]


    [tex]Y_{CM} = \frac{3}{4b}\sqrt{\frac{a}{b}}\int_{-\sqrt{b/a}}^{\sqrt{b/a}}\int_0^b ydydx = \frac{3}{4}b[/tex]
  12. Apr 22, 2005 #11
    what i meant by weighted is this: when u compute the center of mass of an ojbect, u are finding the average position "weighted" by how much mass is at a given position. In this prob., we know the center o mass is on the y axis by symmetry so we just have to find the "weighted average" of y. The amount of mass at distance y above the x-axis is dm = 2*sqrt(y/a)*dy. Actuallly this is da...area...but we can take density to be 1 cuz it will cancel anyway. So what you need is

    integral of y dm = integral of 2*y*sqrt(y/a)*dy from 0 to b. For this I get (4/5) * b^(5/2) / sqrt(a). Divide this by what quasar987 gave for A to get the answer.
  13. Apr 22, 2005 #12


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    Could you tell me what is wrong with my method? How is the integral I evaluated not the right one? Isn't it the integral of y/A over the surface of the parabola?
  14. Apr 22, 2005 #13

    I get 3b/5

    I turned the parabola on its side so the new function is:

    y = (x/a)^1/2

    then I integrated (Xcm - x)*y*dx from 0 to b, set the integral to zero and solved for Xcm.
  15. Apr 22, 2005 #14

    [tex]Y_{CM} = \frac{3}{4b}\sqrt{\frac{a}{b}}\int_{-\sqrt{b/a}}^{\sqrt{b/a}}\int_0^b ydydx = \frac{3}{4}b[/tex]

    should be

    [tex]Y_{CM} = \frac{3}{4b}\sqrt{\frac{a}{b}}\int_0^b\int_{-\sqrt{y/a}}^{\sqrt{y/a}} ydxdy = \frac{3}{5}b[/tex]

    add: u were integrating over a rectangle
  16. Apr 22, 2005 #15


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    HAHAHA. ooook. :tongue2:

    It could have been

    [tex]Y_{CM} = \frac{3}{4b}\sqrt{\frac{a}{b}}\int_{-\sqrt{b/a}}^{\sqrt{b/a}}\int_{ax^2}^b ydydx = \frac{3}{4}b[/tex]

    too... which is what I tought I was doing.
  17. Apr 23, 2005 #16
    quasar987 could you show me how you arrive at getting the stuff before the integral sign and the logic behind the two integrals, thanks.
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