Finding the Center of Mass in a Divided Square

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Discussion Overview

The discussion revolves around finding the center of mass of a modified square shape, specifically a square divided into four smaller squares with a circular cutout and a corner piece removed. The focus is on the theoretical and mathematical aspects of determining the new center of mass after these modifications.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant suggests cutting the circle out of the remaining three quarters of the square to create a symmetrical shape, proposing that the center of mass would then be at the center of the square.
  • Another participant challenges this approach, pointing out that the cutout corner piece complicates the calculation of the center of mass.
  • A later reply acknowledges the oversight regarding the corner cutout and suggests returning to the calculation of x_bar and y_bar, indicating a potential method to approach the problem despite the complexity.
  • There is a suggestion that removing symmetrical areas about both axes could simplify the problem, although the effectiveness of this method is uncertain.

Areas of Agreement / Disagreement

Participants express differing views on the best approach to find the center of mass, indicating that there is no consensus on the method to be used due to the complexities introduced by the corner cutout.

Contextual Notes

Participants have not resolved the mathematical steps necessary to find the center of mass, and there are dependencies on the definitions of the shapes involved and the assumptions made about symmetry.

cmrgator
Imagine a square with side length "a". Now, divide the square into 4 equal squares with side length "a/2". In the top righthand corner of the large square, a circle with radius "a/4" is cut out, which also removes the top corner piece. What is the new center of mass? (The origin is at the center of the large square). Thanks for your help!
 
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Here is an easy way to think about it.

Cut the circle out of the remaining 3 quarters of the large square, now you have a symmetrical shape about both x and y axis. The center of mass is the center, so we can now ignore this part.

Now what is the center of mass of the 3 circles?
 
Sorry, I don't see how that helps because of the cut out corner piece.
Anyway, even with the corner piece, I wouldn't know what to do.
 
Oops, I missed the part about the corner cut out...


I guess its back to x_bar y_bar in the first place then...

Well, it could still be done the way I was suggesting, don't know if its any easier though...but the point was to take out all area symmetrical about both axis, to simplify the problem.
 
Last edited:

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