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Mass of a cardioid

  • Thread starter Locoism
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  • #1
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Homework Statement



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.

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:

Answers and Replies

  • #2
Dick
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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.
 
  • #3
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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?
 
  • #4
Dick
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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?
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?
 
  • #5
81
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sorry 5pi/3. Thank you!
 

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