Consider a two-particle system with the particles having masses $m_{1}$ and $m_{2}$ . If the first particle is pushed towards the centre of mass through a distance $d$ , by what distance should be the second particle be moved so as to keep the centre of mass at the same position?

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Solution

Consider Fig. Suppose the distance of $m_{1}$ from the centre of mass $C$ is $x_{1}$ and that of $m_{2}$ . Suppose mass $m_{2}$ is moved through a distance $d^{'}$ towards $C$ so as to keep the centre of mass at $C$ .
Then, $m_{1} x_{1} = m_{2} x_{2}$ ...........(i)
and $m_{1} (x_{1} - d_{)} = m_{2} (x_{2} - d^{'})$ ............(ii)
Subtracting Eq. (ii) from Eq. (i), we get
$m_{1} d = m_{2} d^{'}$ or $d^{'} = (m_{1} / m_{2}) d$

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