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

Question

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 = L x , the acceleration of the chain is g 2 . The value of x is _____.

TestHub · Accepted Answer

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,
$a = \frac{(m_{2} - m_{1})}{(m_{2} + m_{1})} g$
Since $a = \frac{g}{2}$
$\frac{g}{2} = \frac{(m_{2} - m_{1})}{(m_{2} + m_{1})} g \Rightarrow \frac{g}{2} = \frac{(L - l) - l}{L} g \Rightarrow l = \frac{L}{4} = \frac{L}{x}$
$\Rightarrow x = 4$

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