Question

Prove that the centre of mass of two particles divides the line joining the particles in the inverse ratio or their masses?

Answer 1

Let us consider a two particle system of masses m1 and m2at the points A and B respectively.

Consider point C as the centre of mass of system. If  and  are the position vector of masses m1 and m2, the position of the centre of mass will be given as-



If the origin of co-ordinate system is shifted at the centre of mass C of the system, then

.

 Using eq. (1) we get



 

Hence, the centre of mass of two particle system divides the line joining the particles in the inverse ratio of their masses.
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Answer 2

Proof:

Let us consider two particles R₁ and R₂ of masses m₁ and m₂ respectively.

[Refer to the attached image 1 for the diagram]

[CASE 1] (Refer to attached image 2)

Let us assume that R₁ is the origin. Then;-

Position of centre of mass from m₁ will be;

                  R₁ = m₂R / m₁ + m₂  _______(1)

[CASE 2] (Refer to attached image 3)

Let us assume that R₂ is the origin. Then;-

Position of centre of mass from m₂ will be;

                  R₂ = - m₁R / m₁ + m₂  _______(2)

Now, divide equation (1) by (2), we get;-

       R₁ / R₂ = m₂ R/ m₁ R

       R₁ / R₂ = m₂ / m₁

Crossing multiplying LHS and RHS, we get;-

         R₁ m₁ = R₂ m₂

Here, R₁ m₂ is called the moment of mass. We conclude from here that centre of mass is a point about which the moment of mass is equal.

          R₁ m₁ = Constant

              R₁ = Constant/ m₁

               R₁ = 1/ m₁   [Hence, Proved!]

Therefore, it is proved that the centre of mass of two particles divides the line joining the particles in the inverse ratio or their masses forming a rectangular hyperbola.

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