Solving Center of Mass Doubt: What is 'y' in Integration?

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Discussion Overview

The discussion revolves around the concept of center of mass (COM) in the context of continuous mass distributions, particularly focusing on the interpretation of coordinates used in integration to find the COM. Participants explore specific cases, such as rings and semi-circular discs, and the implications of different coordinate choices.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant expresses confusion about the variable 'y' in the integration formula for calculating the center of mass, particularly in different geometrical contexts.
  • Another participant clarifies that 'x' and 'y' represent the coordinates of the mass element dm and that the center of mass can be viewed as the average location of all mass.
  • There are repeated inquiries about the position of the center of mass for a semi-circular ring, with a focus on whether it lies on the circumference or elsewhere when the origin is at the center of the diameter.
  • A participant asserts that the center of mass of a uniform circle (ring) is at the center of the circle, but this is later corrected to specify a semi-circular ring.
  • One participant suggests looking up centroids for reference but emphasizes the importance of practicing integration independently.

Areas of Agreement / Disagreement

The discussion contains multiple competing views regarding the interpretation of coordinates in the context of different shapes. There is no consensus on the specific application of 'y' in various cases, and participants express differing levels of understanding and approaches to the problem.

Contextual Notes

Participants do not fully resolve the assumptions regarding the definitions of 'x' and 'y' in different geometrical contexts, nor do they clarify the integration steps necessary for finding the center of mass in specific cases.

rahul.mishra
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A silly doubt regarding center of mass...
As we know for bodies having continuous distribution of mass we can know their center of mass by the method of integration...
like, Xcm = 1/M∫x.dm

but what is x here?
in many cases...
like in finding the COM of a ring
Xcm = 0 and Ycm = 2r/∏ (ofcouse when the center is taken on the mid point of diameter)

but on finding the COM of a semi-circular disc or plate...
we can assume it to be formed with different rings...
finally in the formula Ycm = 1/M∫y.dm we set y = 2r/pi ...
so i just can't understand what this 'y' is... and how do we decide it in different cases...
i think you can get what i want to ask actually...
and please help me... i m new to this site...
rahul :)
 
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I'm not sure I get your question, but x and y represent the coordinates of the mass element dm. And the center of mass, (Xcm, Ycm), can be thought of as the 'average' location of all the mass.
 
can you tell me the position of the center of mass of a ring of radius R and mass M?
Does it lie on the circumference or somewhere else when origin is assumed to be at the
center of the diameter joining the two ends of the ring?
 
rahul.mishra said:
can you tell me the position of the center of mass of a ring of radius R and mass M?
Does it lie on the circumference or somewhere else when origin is assumed to be at the
center of the diameter joining the two ends of the ring?

The center of mass (assuming uniform density) of a circle (ring, torus) is at the center of the circle.
 
Oh...! i m really sorry... i meant a semi-circular ring...!
 
rahul.mishra said:
Oh...! i m really sorry... i meant a semi-circular ring...!
You can just look it up. (Like here: List of centroids.) But you'd better practice so you can do the integration on your own.
 

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