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Displacement of COM:Application

There are two...

Question

There are two masses $m_{1}$ and $m_{2}$ placed at a distance l apart. Let the centre of mass of this system is at a point named C. If $m_{1}$ is displaced by $l_{1}$ , towards C and $m_{2}$ is displaced by $l_{2}$ away from C. Find the distance, from C where new centre of mass will be located.

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Solution

From given, we have,

$m_{1} x = m_{2} (L - x)$

$\Rightarrow x = \frac{m_{2} L}{m_{1} + m_{2}}$ ..........(1)

$m_{1}$ displaced by $L_{1}$ and $m_{2}$ displaced by $L_{2}$ , with new centre of mass $C$

we have,

$m_{1} (L_{1} + x - y) = m_{2} (L - x + y + L_{2})$ ........(2)

from (1) and (2), we have,

$m_{1} (L_{1} + (\frac{m_{2} L}{m_{1} + m_{2}}) - y) = m_{2} (L - (\frac{m_{2} L}{m_{1} + m_{2}}) + y + L_{2})$

upon simplification, we get the value of "y" as,

$y = \frac{m_{1} L_{1} - m_{2} L}{m_{1} + m_{2}}$

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