Centre of mass is a unique point

In summary, the centre of mass is a unique point in a system of particles, as proven by the definition of CM and the fact that there is a unique value for R_CM, which is a function of (m_i, r_i). This is also supported by the fact that if there were multiple CMs, there would be at least one more CM, leading to a contradiction.
  • #1
neelakash
511
1

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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  • #2
there will be a vector going from G to G'
 
  • #3
That is obvious.But how to show that?
 
  • #4
I believe now you have a relation between [tex]r_i[/tex] and [tex]r'_i[/tex] for each i.
 
  • #5
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.
 

Related to Centre of mass is a unique point

1. What is the centre of mass?

The centre of mass is a point in an object or system where its entire mass can be considered to be concentrated. It is also known as the center of gravity.

2. Why is the centre of mass important?

The centre of mass is important because it helps us understand the overall motion and stability of an object or system. It is also used in various calculations and equations in physics and engineering.

3. How is the centre of mass calculated?

The centre of mass can be calculated by finding the weighted average of the positions of all the particles in an object or system. This can be done using the formula: xcm = (m1x1 + m2x2 + ... + mnxn) / (m1 + m2 + ... + mn)

4. Is the centre of mass always located within the object or system?

No, the centre of mass can be located both inside or outside of an object or system. For example, in a ring or a hollow sphere, the centre of mass is located at the geometric centre of the object.

5. How does the distribution of mass affect the location of the centre of mass?

The distribution of mass within an object or system affects the location of the centre of mass. Objects with more mass concentrated towards one side will have their centre of mass located closer to that side. Objects with symmetrical mass distribution will have their centre of mass located at the geometric centre.

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