What a chain! 4

A chain of uniform mass density \(\lambda\) and mass \(M\) and length \(L\) is hanging from ceiling through a string. The other end of the chain is held at rest at the position shown in the figure. At time \(t=0\) the held part of chain is left to fall under gravity. Find the tension in the string as a function of time \((t)\) (neglecting friction). Your answer can be represented as

\[T(t) = \frac{Mg}{a} + \frac{b Mg^2t^2}{c L}\]

where \(a\), \(b\), and \(c\) are positive coprime integers. Enter your answer as \(a + b + c\).

Note: Figures are not to scale.

This is inspired by Depanshu's Chain please don't fall! and is a part of my set Aniket's Mechanics Challenges.


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