Which of the following statements is correct regarding the COM of a two mass system?

If masses are real, centre of mass always lies at the midpoint joining the two masses.

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Centre of mass divides internally the line joining the two masses in the ratio of their masses.

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Centre of mass divides externally the line joining the two masses in the ratio of their masses.

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Centre of mass divides internally the line joining the two masses in the inverse ratio of their masses.

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Solution

The correct option is D Centre of mass divides internally the line joining the two masses in the inverse ratio of their masses.
Let $O$ and $P$ be the positions of the two masses seperated by distance $^{'} d^{'}$ and $C$ be the centre of mass of system, on the line joining the two masses.

From figure,
$O C + C P = d$
$O C = \frac{m_{2} d}{m_{1} + m_{2}} & C P = \frac{m_{1} d}{m_{1} + m_{2}}$
$\Rightarrow \frac{O C}{C P} = \frac{m_{2}}{m_{1}}$

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