Center of Mass (Subtraction over addition)

In summary, the conversation is about finding the coordinates for the center of mass of a uniform plate. The equations used are Ms/Ms + M1/Ms + M2/Ms - M1X1/Ms - M2X2/Ms = Xs and Ms/Ms + M1/Ms + M2/Ms - M1X1/Ms - M2X2/Ms = Ys. The asker initially struggles to find the correct answer using these equations, but eventually realizes that the given answer is incorrect and their initial calculation was correct.
  • #1
Ignitia

Homework Statement


b06OECX.png

Find the coordinates for the center of mass in the picture above. Plate is uniform.

Homework Equations


Since it's uniform, the lengths can be converted to the mass. So:

coord x =(M1X1 +M2X2 + M3X3 + ... +MiXi)/(M1 +M2 + M3 + ... +Mi)
coord y =(M1Y1 +M2Y2 + M3Y3 + ... +MiXi)/(M1 +M2 + M3 + ... +Mi)

I saw it as 3 pieces: the original square, 6x7, and two missing pieces, 2x2 and 4x4.

I'll only put the X equation since Y is similar.

s = plate (shown)
1 = 4x4 missing piece
2 = 2x2 missing piece.
I simply need to find Xs = coords of the plate.

coord x (total, 6x7 square) = 1 = (MsXs +M1X1 + M2X2) / (Ms +M1 + M2)

(Ms +M1 + M2) * 1 = (MsXs +M1X1 + M2X2)
Ms +M1 + M2 - M1X1 - M2X2= MsXs
Ms/Ms +M1/Ms + M2/Ms - M1X1/Ms - M2X2/Ms = Xs

Plugged in the values, it didn't come out correctly. Same for the Y coordinate.

Edit:
Ms/Ms +M1/Ms + M2/Ms - M1X1/Ms - M2X2/Ms = Xs
22/22 + 16/22 + 4/22 - (16/22 (2)) - (4/22 (3)) = -0.090
However, answer given is -0.90

The Attempt at a Solution


I was shown the solution of cutting up the plate into 4 pieces, and finding the CM that way. That came out correct. But I want to know what went wrong when I went 'take whole plate, remove two pieces, find CM' There's probably something really obvious that I'm missing.
 
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  • #2
Ignitia said:
coord x (total, 6x7 square) = 1 = (MsXs +M1X1 + M2X2) / (Ms +M1 + M2)

(Ms +M1 + M2) * 1 = (MsXs +M1X1 + M2X2)
Ms +M1 + M2 - M1X1 - M2X2= MsXs
Ms/Ms +M1/Ms + M2/Ms - M1X1/Ms - M2X2/Ms = Xs
I do not understand your notation here. Or the equations that you assert. But if M1 is missing then M1 should be regarded as negative.
 
  • #3
jbriggs444 said:
M1 should be regarded as negative.
Or is it 0? It doesn't take much to prove me wrong sometimes, but I see this as 1's and 0's... is it 1 and -1 or am I totally missing the concept...
 
  • #4
Sorry, guess I didn't explain it clearly.

mr9Em2z.png


The original plate is grey + red + green, where its CM is x=1. So what I wanted to do was find the X coord of they grey plate only, but I could deduce the red and green's x coords, and their area as well.

So, X (grey+red+green aka total plate) = 1 = (MgreyXgrey + MredXred + MgreenXgreen) / (Mgrey + Mred + Mgreen)
 
  • #5
Ignitia said:
Ms/Ms +M1/Ms + M2/Ms - M1X1/Ms - M2X2/Ms = Xs
Looks fine to there. Please post subsequent working.
 
  • #6
Ms/Ms +M1/Ms + M2/Ms - M1X1/Ms - M2X2/Ms = Xs
22/22 + 16/22 + 4/22 - (16/22 (2)) - (4/22 (3)) = -0.090
However, answer given is -0.90
 
  • #7
Ignitia said:
answer given is -0.90
It could not possibly be that close to the left hand edge of the shape. -0.09 is correct.
 
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  • #8
haruspex said:
It could not possibly be that close to the left hand edge of the shape. -0.09 is correct.

But the given answer -

...

You know what, I'll just accept the fact I spent an hour unable to figure out why my correct answer wasn't the given answer. Thank you.
 

1. What is the concept of "Center of Mass (Subtraction over addition)"?

The center of mass is a point in a system where the mass of the object is concentrated. This concept uses the principle of subtraction over addition, where the center of mass is calculated by finding the position of the center of mass of a smaller system and then subtracting the mass and position of the smaller system from the original system. This results in a more simplified and accurate calculation of the center of mass.

2. How is the center of mass calculated using the principle of subtraction over addition?

The center of mass is calculated by finding the position of the center of mass of a smaller system and then subtracting the mass and position of the smaller system from the original system. Mathematically, this can be represented as: Xcm = (m1x1 + m2x2 + ... + mnxn) / (m1 + m2 + ... + mn), where Xcm is the position of the center of mass, m is the mass, and x is the position of each individual component.

3. Why is the principle of subtraction over addition used in calculating the center of mass?

The principle of subtraction over addition is used in calculating the center of mass because it simplifies the calculation and makes it more accurate. By breaking down a system into smaller components, the calculation becomes more manageable and the accuracy of the center of mass is improved.

4. How is the center of mass used in physics and engineering?

The center of mass is an important concept in physics and engineering as it helps in understanding the stability, motion, and balance of a system. It is used in designing structures, vehicles, and machines to ensure they are balanced and will not tip over. It is also used in physics to analyze the motion of objects and to calculate forces acting on them.

5. Can the center of mass be located outside of the physical object?

Yes, the center of mass can be located outside of the physical object. This can occur when the mass distribution is not uniform and there are higher concentrations of mass in certain areas. In this case, the center of mass will be closer to the area with higher mass concentration. However, the center of mass will always be within the boundaries of the object.

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