Finding the center of mass of a homogeneous object

AI Thread Summary
The discussion revolves around calculating the center of mass for a homogeneous object divided into three rectangles. The original poster calculated the center of mass coordinates as (1.82, 4.72) but found discrepancies with the provided solutions of (1.77, 4.23). Participants identified a potential error in the area calculation for the second rectangle, m2, which was incorrectly assumed to be 6σ. After reviewing the calculations, it was confirmed that the mistake stemmed from miscalculating the area, leading to the correct coordinates being closer to the provided solutions. The conversation highlights the importance of careful arithmetic and verification in solving physics problems.
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Homework Statement
Find the coordinates of the center of mass of the object represented in the figure.
Relevant Equations
Equation of the center of mass.
The object is:
242762

My attempt at a solution:
I divided the object into 3 different rectangles and found the coordinates for the center of mass of each one, considering the origin at point "O".
242763

Then I found the mass of each rectangle, assuming the object has an area density of σ.
m1 = 15σ; m2= 6σ; m3 = 8σ.
After that I applied the center of mass position equation for x and y coordinates.
rx = (r1x*m1+r2x*m2+r3x*m3)/(m1+m2+m3) = (0.75*15σ+2.25*6σ+3.5*8σ)/(29σ) = 1.82
ry = (5*15σ + 9*6σ + 1*8σ)/(29σ) = 4.72
So the coordinates of the center of mass would be (1.82, 4.72).
However the solutions given are: (1.77, 4.23). Please help me figure out what I did wrong.
 
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Check your area for m2. Looks like that may be the issue.
 
ChemAir said:
Check your area for m2. Looks like that may be the issue.
So you are suggesting that 2x4 is not equal to 8?
 
Near as I can tell your mistake is thinking that the book has the right answer and you don't.
 
phinds said:
So you are suggesting that 2x4 is not equal to 8?

No. I see BC(2)x CD(1.5 ) = 3. OP says 6.

And the arithmetic worked out afterward, rx=1.769... I didn't check ry.

But I was eating lunch, and I could have missed something.
 
ChemAir said:
No. I see BC(2)x CD(1.5 ) = 3. OP says 6.
Ah. My bad. I was looking at M3 even though you clearly said M2.

Interestingly (and embarassingly) enough, I made exactly the same mistake which is why I thought he had the right answer.o:)
 
ChemAir said:
Check your area for m2. Looks like that may be the issue.
Yeah that was it. Looks like I got distracted while I was solving the problem. Thanks for spotting my mistake!
 
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