Consider a two particle system with particles having masses m₁ and m₂. If the first particle is pushed towards the centre of mass through a distance d, by what distance should the second particle be moved, to keep the centre of mass at the same position? 

If x1 and x2 are the positions of masses m1 and m2, the position of the center of mass is given by

\(\rm X_{cm}=\frac{m_1x_1+m_2x_2}{m_1+m_2}\)

If x1 changes by Δ x1 and x2 changes by Δ x2, the change in xcm will be

\(\rm Δ x_{cm}=\frac{m_1Δ x_1+m_2Δ x_2}{m_1+m_2}\)      .......(1)

Given Δ xcm = 0 and Δ x1 = d.

Using these values in eq. (1) we get m1d + m2 Δ x2 = 0

or \(\rm \Delta x_2=-\frac{m_1d}{m_2}\)

∴ Distance moved by \(\rm m_2=\frac{m_1d}{m_2}\)