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Question

Two particles having mass ratio n : 1 are interconnected by a light in extensible string that passes over a smooth pulley. If the system is released, then the acceleration of the centre of mass of the system is


A
(n1)2g
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B
(n+1n1)2g
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C
(n1n+1)2g
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D
(n+1n1)9
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Solution

The correct option is C $$\left( \frac { n - 1 } { n + 1 } \right) ^ { 2 } g$$
Given$$,$$
$$\frac{{{m_1}}}{{{m_2}}} = \frac{n}{1} = n$$
Each mass have the acceleration $$a = \frac{{\left( {{m_1} - {m_2}} \right)}}{{{m_1} + {m_2}}}$$
however $${{m_1}}$$ which is heavier will have the will have acceleration $${{a_1}}$$ vertically down while the lighter mass $${{m_2}}$$ will have acceleration $${{a_2}}$$ vertically up $$ \to {a_2} =  - {a_1}$$
The acceleration or the centre of mass of the system$$,$$ $${a_{cm}} = \frac{{{m_1}{a_1} + {m_2}{a_2}}}{{{m_1} + {m_2}}}$$
given that $${a_2} =  - {a_1} \to {a_{cm}} = \frac{{\left( {{m_1} - {m_2}} \right){a_1}}}{{{m_1} + {m_2}}} = \frac{{{m_1} - {m_2}}}{{{m_1} + {m_2}}} \times \frac{{\left( {{m_1} - {m_2}} \right)g}}{{{m_1} + {m_2}}} = \frac{{{{\left( {{m_1} - {m_2}} \right)}^2}g}}{{{m_1} + {m_2}}}$$
Since $$\frac{{{m_1}}}{{{m_2}}} = n$$ diving by $${{m_2}}$$ and simplifying 
$$ \Rightarrow {a_{cm}} = {\left( {\frac{{n - 1}}{{n + 1}}} \right)^2}g$$
Hence,
option $$(C)$$ is correct answer.

Physics

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