Centre of mass is a unique point

AI Thread Summary
The discussion centers on proving that the center of mass (CM) is a unique point in a system of particles. The author begins by defining the CM using the equations for two assumed centers of mass, G and G', and notes the absence of a unique relationship between their position vectors. They conclude that since the center of mass is derived from the masses and their positions, it must yield a unique value. Additionally, they argue that if multiple centers of mass existed, it would lead to a contradiction by implying the existence of even more centers. The proof ultimately reinforces the uniqueness of the center of mass in a particle system.
neelakash
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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
 
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there will be a vector going from G to G'
 
That is obvious.But how to show that?
 
I believe now you have a relation between r_i and r'_i for each i.
 
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.
 
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