Question

Moment of inertia of particles passing through centre of mass is zero. Why?

Answer 1

Well to be straight forward that is the definition. That is how we calculate the centre of mass as the sum of moment of masses about a point divided by the total mass.

Also you can take this the other way round i.e.

let M is the total mass and rcmrcm as the position of centre of mass.

So,

Mrcm=m1r1+m2r2+.....+mnrnrcm=m1r1+m2r2+.....+mnrn

Now we can write M = m1+m2+...+mnm1+m2+...+mn

so the equation comes out to be

m1r1/cm+m2r2/cm+.....+mnrn/cm=0m1r1/cm+m2r2/cm+.....+mnrn/cm=0

where, ri/cm=ri−rcmri/cm=ri−rcm

which means the respective distance of the point ‘i’ with the centre of mass.

This proves what you asked but still remember it is not this but the other way around. We have defined that the moment of masses about centre of mass is zero and then we find the position of centre