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
B
n
C
n3
D
n2

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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