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Question

A chain of length $$L$$ and mass $$M$$ is held on a frictionless table with $${\left(\dfrac{1}{n}\right)}^{th}$$ part hanging over the edge. Work done in pulling the chain is directly proportional to:


A
n
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B
n
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C
n3
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D
n2
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Solution

The correct option is D $${ n }^{ -2 }$$

The length of the chain is L and mass is M. Its $$\dfrac{1}{n}th $$ part is hanging.

So, mass of $$\dfrac{L}{n}$$ length chain will be , $$\dfrac{M}{n}$$

This mass will be at mid-point of the length.

So, effective height from the point will be, $$h =  \dfrac{L}{2n}.$$

Here work done will be equal to its potential energy.

Thus $$P.E. = m \times g \times h $$

i.e. $$P.E.= \dfrac{M}{n} \times g \times \dfrac{L}{2n}\\$$

Thus work done will be $$= \dfrac{mgl}{2n^2}$$

So, work done is directly proportional to $$n^{-2}$$, as other terms are constant.

Option D is correct.


Physics

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