Finding Center of Mass for Composite Shapes

AI Thread Summary
To find the center of mass for composite shapes, the integral M(Rcm) = ∫(r dm) is essential, where dm represents mass elements. One participant suggests using a conceptual approach by treating cut-outs as "negative" masses that push the center of mass away from them, simplifying the calculation. This method involves calculating the center of mass of the entire shape and adjusting for the cut-outs as if they were separate particles with negative mass. The discussion emphasizes that sometimes intuitive reasoning can be more effective than complex mathematics in solving such problems. Overall, understanding the relationship between mass distribution and geometry is key to determining the center of mass accurately.
Darkalyan
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Homework Statement


http://docs.google.com/Doc?id=d277r7r_60c2235gfg

Homework Equations


<br /> <br /> M(Rcm)=\int(rdm)<br /> <br />

The Attempt at a Solution



Okay, so I've figured out that I have to integrate the radii by the mass element dm, which in this case would be p, because that's the mass/unit area? I think that's right, but even there I'm not sure and the actual integral itself I have no idea how to do for such a weird shape.
 
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Darkalyan said:

Homework Statement


http://docs.google.com/Doc?id=d277r7r_60c2235gfg


Homework Equations


<br /> <br /> M(Rcm)=\int(rdm)<br /> <br />

The Attempt at a Solution



Okay, so I've figured out that I have to integrate the radii by the mass element dm, which in this case would be p, because that's the mass/unit area? I think that's right, but even there I'm not sure and the actual integral itself I have no idea how to do for such a weird shape.
I wouldn't integrate if I were you. There's a way to know where's the center of mass without much mathematics. Thinking is more powerful than mathematics, sometimes.
 
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Hmm. Could I take the center of mass of the entire circle, assuming there's no cut outs. Then I take the center of mass of the cut-outs, and pretend they act like 'negative' masses, so instead of having the centerr of mass of the composite system get closer to them, they are farther away? So basically pretend this is a 3 particle system and combine the locations of the center of masses, but assume the cutouts have 'negative' masses, so they push away the center of mass instead of bringing it closer?
 
Darkalyan said:
Hmm. Could I take the center of mass of the entire circle, assuming there's no cut outs. Then I take the center of mass of the cut-outs, and pretend they act like 'negative' masses, so instead of having the centerr of mass of the composite system get closer to them, they are farther away? So basically pretend this is a 3 particle system and combine the locations of the center of masses, but assume the cutouts have 'negative' masses, so they push away the center of mass instead of bringing it closer?
Exactly.
 

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