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

[rahul.mishra]

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 :)

 

[Doc Al]

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.

 

[rahul.mishra]

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?

 

[mathman]

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.

 

[rahul.mishra]

Oh...! i m really sorry... i meant a semi-circular ring...!

 

[Doc Al]

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.