Work, Energy and Power

Question

A uniform chain of length L and mass m is lying on a smooth table and one-nth part of its length is hanging vertically down over the edge of the table. Find the work required to pull the hanging part on to the table.

Solution

Let $space straight lambda$ be linear mass density of chain.

To pull the chain we have to do work against the weight of hanging part of chain.
Let at any instant length of hanging part be x.
Therefore the weight of hanging part of chain is,
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The work done in pulling the chain by small distance dx is,
$d W equals negative lambda x g d x$
Total work done to pull the whole of hanging part of chain is,
$W equals integral d W space space space space equals space integral subscript L divided by n end subscript superscript 0 minus lambda g x space d x$
$equals right enclose negative lambda g x squared over 2 end enclose subscript bevelled L over n end subscript superscript 0$
$space space equals fraction numerator lambda g L squared over denominator 2 n squared end fraction space equals fraction numerator m g L over denominator 2 n squared end fraction$

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Work, Energy And Power

A uniform chain of length L and mass m is lying on a smooth table and one-nth part of its length is hanging vertically down over the edge of the table. Find the work required to pull the hanging part on to the table.

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