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Homework Help: Mass of a cardioid

  1. Nov 13, 2011 #1
    1. The problem statement, all variables and given/known data

    Find the mass and center of mass of the region bounded by the cardioid r =
    1 + cos(θ), assuming the density function is given by ρ = r.

    3. The attempt at a solution

    The first part is simple enough, I set up the integral

    [itex]\int_{0}^{2\pi}[/itex][itex]\int_{0}^{1+cos(\theta)}r dr d(\theta)[/itex]

    I find the mass is 3pi/2

    Now for the centre of, how do I set up the integrals? I know it would be the double integral of xρ drdθ, and then yρ drdθ, but do I keep the same limits? x would be rcos(θ) and y is rsin(θ), but this gives me some really weird functions to integrate.
    Last edited: Nov 13, 2011
  2. jcsd
  3. Nov 13, 2011 #2


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    I'm not sure what's so weird about the functions to integrate, but you are making a mistake already. In polar coordinates to find area you integrate the volume element dxdy=r*dr*d(theta). So I think you just computed the area. If the density is also r, that should give you an extra factor of r.
  4. Nov 13, 2011 #3
    Oh so in that case I get 5/6 pi. But still, what about the centre of mass? I need My and Mx, do I keep the same limits of integration?
  5. Nov 13, 2011 #4


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    Well, I don't get 5*pi/6. I'd check that. But, yes, keep the same limits of integration. Why not? Just put a factor of x or y into the integrand and work them out. Then you divide by total mass, right?
  6. Nov 13, 2011 #5
    sorry 5pi/3. Thank you!
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