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Find the center of mass of a plate

  • #51
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For part A it was not necessary to do a double integral because the density was constant. You could just write down that the mass of the vertical strip on the right was σ(5-(x2+1))dx. That was instead of the integral (∫y=x2+1y=5σ.dy).dx
So I am correct on part A, I just need to correct part B, right?
 
  • #53
402
13
Your part A is ok.
Ok, thanks. And for part B I just need to correct ##Y_{cm}##? Or ##X_{cm}## too?
 
  • #54
haruspex
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Ok, thanks. And for part B I just need to correct ##Y_{cm}##? Or ##X_{cm}## too?
You need to use double integrals everywhere in part B.
By always doing the y integral first you may avoid the square roots.
 
  • #55
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You need to use double integrals everywhere in part B.
By always doing the y integral first you may avoid the square roots.
What do you mean by doing the y integral first? Why would that make me avoid square roots?
 
  • #56
haruspex
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What do you mean by doing the y integral first? Why would that make me avoid square roots?
You have a double integral in y and x. You can do either first.
If you do x first it will have a bound that depends on y and involves a square root, so the result of the integral may involve a square root, which then becomes part of the integrand for the second integral.
If you do the y integral first it has bounds that involve x but no square root. The result will be a function of x, which you then need to integrate, but it will not have a square root in the integrand.
If you are feeling energetic you can try it both ways and compare.
 
  • #57
402
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You have a double integral in y and x. You can do either first.
If you do x first it will have a bound that depends on y and involves a square root, so the result of the integral may involve a square root, which then becomes part of the integrand for the second integral.
If you do the y integral first it has bounds that involve x but no square root. The result will be a function of x, which you then need to integrate, but it will not have a square root in the integrand.
If you are feeling energetic you can try it both ways and compare.
I didn't know I was able to iintegrate double integrals in any order. It's like partial derivatives, right?
 
  • #58
haruspex
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I didn't know I was able to iintegrate double integrals in any order. It's like partial derivatives, right?
Yes.
 

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