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Find the Center of Mass of a Lamina

  1. Dec 4, 2014 #1
    1. The problem statement, all variables and given/known data

    Find the x-coordinate of the center of mass of the lamina that occupies the region cut from the first quadrant by the lines x = 6 and y = 1 if the density function is p(x,y) = x + y + 1

    2. Relevant equations

    Mass = ∫∫spdσ
    X Coordinate of center of Mass = Myz/M
    Myz = ∫∫sxpdσ

    3. The attempt at a solution

    I am not sure how to set this one up, as I have not seen one like this without a bounding equation (ie x2 + y2 + z2 = a2 or similar.)

    I attempted simply setting up the integral as being from 0≤x≤6 and 0≤y≤1 so M=∫∫x+y+1dydx = 27 and Myz = ∫∫x2 +xy +x dydx = 126

    However, dividing 126 by 27 results in 14/3, not 11/3 as the answer theoretically should be.

    Any help would be greatly appreciated!
     
  2. jcsd
  3. Dec 5, 2014 #2

    SteamKing

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    Staff Emeritus
    Science Advisor
    Homework Helper

    The problem appears to be in your calculation of Myz. If you can't find the problem, please post your work here so it can be reviewed in detail.
     
  4. Dec 5, 2014 #3

    Mark44

    Staff: Mentor

    Your two formulas above are a bit confusing. You have a two-dimensional object, so the two moments are with respect to the x axis or the y axis.

    The notation Myz is the moment about the y-z plane, which isn't applicable here because the object is two-dimensional.

    The formula to use for the x-coordinate is My/M
    Here the boundaries are a lot simpler: the region is a rectangle, with x ranging between 0 and 6 and y ranging between 0 and 1.
     
  5. Dec 5, 2014 #4

    Mark44

    Staff: Mentor

    Yes, I agree. BTW, I get 11/3 for the x-coord. at CM.

    Please show us your calculations for My.
     
  6. Dec 5, 2014 #5
    I did end up finding my error, of course it was just a simple issue that I had missed. Thank you all!
     
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