Centre of mass is a unique point

[neelakash]

Homework Statement

Prove that centre of mass is a unique point

Homework Equations

definition of CM

The Attempt at a Solution

I started with O as origin.G as CM and G' as assumed 2nd CM in the same sysyem of particles.

OG=R=(1/M) sum(i) [m_i*r_i]

OG'=R'=(1/M) sum(k) [m_k*r'_k]

The problem is theere is no unique relation between r_i and r'_k

 

[mjsd]

there will be a vector going from G to G'

 

[neelakash]

That is obvious.But how to show that?

 

[mjsd]

I believe now you have a relation between $$r_i$$ and $$r'_i$$ for each i.

 

[neelakash]

OK,I found it.R_CM is a function of (m_i,r_i).So, it has a unique value.

Another way to look at it:Suppose,there are exactly x(>1) CMs.Then you can show there are atleast x+1 CMs.So,it is a contradiction.