Question 22 of 210

A uniform metal chain of mass $m$ and length $L$ passes over a massless and frictionless pulley. It is released from rest with a part of its length $l$ is hanging on one side and rest of its length $L-l$ is hanging on the other side of the pulley. At a certain point of time, when $l=\dfrac{L}{x}$, the acceleration of the chain is $\dfrac{g}{2}$. The value of $x$ is _____.

A$6$B$2$C$1.5$D$4$

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Answer: (D) $4$

According to the information given in the question, the part of the chain that envelopes the pulley is very small. Therefore, the acceleration of the chain can be written as,
$a=\dfrac{\left({m}_{2}-{m}_{1}\right)}{\left({m}_{2}+{m}_{1}\right)}g$
Since $a=\dfrac{g}{2}$
$\dfrac{g}{2}=\dfrac{\left({m}_{2}-{m}_{1}\right)}{\left({m}_{2}+{m}_{1}\right)}g \Rightarrow \dfrac{g}{2}=\dfrac{\left(L-l\right)-l}{L}g \Rightarrow l=\dfrac{L}{4}=\dfrac{L}{x}$
$\Rightarrow x=4$