Finding the center of mass of a homogeneous object

[Lone Wolf]

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:

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".

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.

 

[ChemAir]

Check your area for m2. Looks like that may be the issue.

 

[phinds]

Check your area for m2. Looks like that may be the issue.

So you are suggesting that 2x4 is not equal to 8?

 

[phinds]

Near as I can tell your mistake is thinking that the book has the right answer and you don't.

 

[ChemAir]

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.

 

[phinds]

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.

 

[Lone Wolf]

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!