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 $$l = \frac{L}{x}$$, the acceleration of the chain is $$\frac{g}{2}$$. The value of $$x$$ is

1. Identify the Forces acting on the Chain

Let the linear mass density of the chain be λ. Since the chain is uniform, it is given by:

At the given point in time:

• The length of the chain hanging on one side is l. The mass of this section is m1 = λl.
• The length of the chain hanging on the other side is L - l. The mass of this section is m2 = λ(L - l).

Assuming the longer side (L - l) moves downward, the net driving force (Fnet) on the chain is the difference between the gravitational forces acting on both sides:

2. Formulate the Acceleration Equation

The total mass being accelerated is the entire mass of the chain, m = λL.

Using Newton's second law (Fnet = m · a):

3. Substitute the Given Values

We are given that when l = L / x, the acceleration of the chain is a = g / 2. Substituting these into our acceleration equation:

Cancel out g from both sides:

Rearranging the terms to solve for x:

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