Two bodies with masses $m_{1}$ and $m_{2} (m_{1} > m_{2})$ are joined by a string passing over fixed pulley. Assume masses of the pulley and thread are negligible. Then the acceleration of the centre of mass of the system $(m_{1} + m_{2})$ is

(\frac{m_{1} - m_{2}}{m_{1} + m_{2}}) g

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{(\frac{m_{1} - m_{2}}{m_{1} + m_{2}})}^{2} g

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\frac{m_{1}}{g} m_{1} + m_{2}

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\frac{m_{2} g}{m_{1} + m_{2}}

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Solution

The correct option is B ${(\frac{m_{1} - m_{2}}{m_{1} + m_{2}})}^{2} g$
If we consider both $m_{1}$ and $m_{2}$ as the system( neglect the mass of the pulley and string),
$a =$ Acceleration of the system $=$ Net force/total mass;
Since $m_{1}$ > $m_{2}$ , $m_{1}$ moves up and $m_{2}$ moves down.

$a = \frac{m_{1} g - m_{2} g}{m_{1} + m_{2}}$
$a_{c m} = \frac{m_{1} a_{1} + m_{2} a_{2}}{m_{1} + m_{2}}$ ...................................................................(1)

Putting the values for $a_{1} = a$ and $a_{2} = - a$ and $a = \frac{m_{1} g - m_{2} g}{m_{1} + m_{2}}$ into equation (1)
$a_{c m} = \frac{m_{1} \times a + m_{2} \times (- a)}{m_{1} + m_{2}} = a \times \frac{m_{1} - m_{2}}{m_{1} + m_{2}} = g {(\frac{m_{1} - m_{2}}{m_{1} + m_{2}})}^{2}$ ;

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