Can anyone try to give a proof for the formula of finding centre of mass which is given mostly in any book as a standard formala as
Try it. If you fail, contact me.I have done it.
proof:
Say the object is composed of N pieces with masses 
. Call the
displacment vectors between these pieces and the axis
distances 
between the pieces and the axis, then
When the axis is displaced by a vector 
, then we want to compute
where 
is the
displacment vectors between these pieces and the new axis.
Let's relate the to 
And since 
(dropping the subscript for convenience)
Now plugging this into 1.57 we have
The last term contains
. Dividing this by M, this
would be the center of mass in the plane perpendicular to the axis. It is
reckoned about the center of mass, so by definition, this must be zero.Hi Shashank, This is the proof i saw on the net and that i have learnt, is the proof similiar to yours?
I did not understand your last step.
Anyways my proof is very very simple.
Assume an equilibrant to be applied at the Centre of Mass(CM).Now the body is in rotational equilbrium so applying
Summation of moments about the point where equilibrant acts (i.e at CM)=0
and substituting the position vectors of each weight=mg w.r.t. origin assumed and finaaly dividing throughout by g we get
Summation miri=0