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Q.
Elongation of a wire under its own weight is independent of
Mechanical Properties of Solids
Solution:
Mass $= M$
Force $=\rho ALg$
Let weight of the rod is $W$.
Consider a small length $dx$ of the rod at distance $x$ from the fixed end. The tension $T$ in element equals the weight of rod below it.
$T =( L - x ) \frac{ W }{ L }$
Elongation $=\frac{ L \times \text { Stress }}{ y }=\frac{( L - x ) W dx }{ LAy }$
Total elongation $=\int_{ O }^{ L } \frac{( L - x ) W dx }{ LAy }=\frac{ W }{ LAy }\left( Lx -\frac{ x ^{2}}{2}\right)_{0}^{ L }=\frac{ WL }{2 Ay }$
Now, $W =\rho ALg$
Putting it in there: $\Delta L =\frac{\rho AL ^{2} g }{2 Ay }=\frac{\rho L ^{2} g }{2 y }$