Explain the theoretical method for estimation of the centre of mass of a solid body.

Explanation

A solid body is made up of microscopic particle (molecules, ions, atoms) distributed continuously inside the body.
As shown in figure, suppose small mass n made up of small mass element each having mass `Deltam` whose position vector is `vecr=xhati+yhatj+zhatk`.
Whole solid body can be considered to be made up of such small mass elements. let the solid body is made up of small mass elements `Deltam_(1),Deltam_(2),...,Deltam_(n)` having position vectors `vec(r_(1)),vec(r_(2)),...,vec(r_(n))` respectively.
The position vector of centre of mass of the solid body is,
`vecR=(Deltam_(1)vec(r_(1))+Deltam_(2)vec(r_(2))+Deltam_(n)vec(r_(n)))/(Deltam_(1)+Deltam_(2)+...+Deltam_(n))`
`therefore` Coordinate of centre of mass,
`X=(Sigma(Deltam_(i))x_(i))/(SigmaDeltam_(i))`
`Y=(SigmaDeltam_(i)y_(i))/(SigmaDeltam_(i))` and
`Z=(SigmaDeltam_(i)z_(i))/(SigmaDeltam_(i))` [where `i=1,2,3,...,n`]
As no. of particles n is more,`Deltam_(i)` is smaller the summation can be represented as an integration.
`therefore SigmaDeltam_(i)rarrintdm=M`
`SigmaDeltam_(i)x_(i)rarrintxdm`
`SigmaDeltam_(i)y_(i)rarrintydm`
`SigmaDeltam_(i)z_(i)rarrintzdm`
where M is the total mass of the body.
The coordinate of centre of mass of solid body,
`X=(1)/(M)int xdm`
`Y=(1)/(M)intydm`
`Z=(1)/(M)intzdm`
If we choose the centre of mass as the origin then,
`vecR(x,y,z)=0`
means `intvecrdm=0`
or `intxdm=intydm=intzdm=0`