# Centre of mass is a unique point

1. May 2, 2007

### neelakash

1. The problem statement, all variables and given/known data

Prove that centre of mass is a unique point

2. Relevant equations

definition of CM

3. 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

2. May 2, 2007

### mjsd

there will be a vector going from G to G'

3. May 2, 2007

### neelakash

That is obvious.But how to show that?

4. May 3, 2007

### mjsd

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

5. May 3, 2007

### 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.