A copper wire of length ' $l$ ' is suspended from the ceiling by one of its ends, it is made up of a material of density P, Young’s modulus y and radius r, the elastic potential energy stored in the wire due its own weight is $\frac{π r^{2} P^{2} l^{3} g^{2}}{k y}$ where $k$ = _____

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Detailed Solution

$U = \frac{1}{2} s t r e s s \times s t r a i n \times v o l u m e$

$U = \frac{1}{2} \frac{{(s t r e s s)}^{2}}{y} v o l u m e$

Stress in small part of length dx at a distance of x from the bottom is given by

$S t r e s s = \frac{P \times A \times x \times g}{A} = P x g$

$d U = \frac{1}{2} (P x g)^{2} \frac{1}{y} \times π r^{2} \times d x$

$\Rightarrow U = \int_{0}^{l} \frac{1}{2} (P x g)^{2} \times \frac{1}{y} \times π r^{2} \times d x = \frac{π r^{2} P^{2} l^{3} g^{2}}{6 y}$

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