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Math Help - center of mass of a thin plate with a density function

  1. #1
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    center of mass of a thin plate with a density function

    Looking for somone to validate my answer or maybe tell me what Im doing wrong, I get an answer of x=0,y=98/5......my worksheet claims the answer is x=0,y=14

    find the center of mass of the region bounded by y=49-x^2 and the x-axis with density of 7x^2

    x=0 because its centered on the orgin
    area is over the interval of -7 and 7
    I get a mass of (1372p)/3; (p=density)
    Mx=(268912p)/30
    y=Mx/mass; y=98/5

    From my understanding and how Im doing the problem the density cancels out, which is were Im confused on how it will affect my answer, thats were Im probably messing up but my textbook doesnt shine much light on problems with non-uniform density. Can someone please help me out?
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  2. #2
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    I have to go with the 14, sorry.

    What was your argument when you calculated the mass? It wasn't just 7x^{2}, was it?

    Can you use a double integral? It's about 1000 easier to understand this way.
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  3. #3
    MHF Contributor Calculus26's Avatar
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    -- Not sure what you have in mind --you are treating the density as if it were constant--you are correct about the x coordinate

    but be careful. In general you need symmetry over the region and the density function over this region to make this assumption. Here 7x^2 is symmetric over the region so x = 0 is the x coordinate of the center of mass.

    for Calculation of the y coordinate see attachment

    By the way the density is integrated over the region for the mass and y times the density is integrated
    over the region for the moment.

    The density does not cancel
    Attached Thumbnails Attached Thumbnails center of mass of a thin plate with a density function-center.jpg  
    Last edited by Calculus26; January 5th 2010 at 08:47 PM.
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  4. #4
    MHF Contributor Calculus26's Avatar
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    Correction

    Make sure you see the corrected attachment--I had a mistake in the original
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