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A wire of length l and cross-sectional a...

A wire of length l and cross-sectional are A is suspended at one of its ends from a ceiling. What will be its strain energy due to its own weight, if the density and Young's modulus of the material of the wire be d and Y?

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Consider an elemental length of the wire of length dx, at a distance x from the lower end. This length is acted upon by the external force equal to the weight of the portion of wire below it = xAdg. In equilibrium, the restoring force F is xAdg. .
Thus, stress =`F/A` = xdg
Now elastic potential energy stored in the elemental length will be
`dU 1/2  stress xx strain xx volume`
`1/2 ((“stress”)/(Y))Stress xx volume`
`=(1)/(2)((xdg))^(2)/(Y).Adx`
`(1)/(2)(Ad^2g^2x^2)/(Y).dx`
Total elastic potential energy=`int_(0)^(L)dU`
`int_(0)^(L)1/2(d^2g^2A)/(Y)x^2dx`
`=(d^2g^2AL^3)/(^Y)`
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