A uniform heavy rod of weight W, cross-sectional area A and length L is hanging from a fixed support. Young modulus of the material of the rod is Y. Neglect the lateral contraction. Find the elongation of the rod.

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Solution

The correct option is B

Consider a small length dx of the rod at a distance x from the fixed end. The part below this small element has length L - x. The tension T of the rod at the element equals the weight of the rod below it.

$T = (L - x) \frac{W}{L}$

Elongation in the element is given by

Elongation = original length x stress/Y
= $\frac{T d x}{A Y} = \frac{(L - x) W d x}{L A Y},$
The total elongation= $\int_{0}^{L} \frac{(L - x) W d x}{L A Y}$
$\frac{W}{L A Y} {(L_{x} - \frac{x^{2}}{2})}_{x}^{L} = \frac{W L}{2 A Y}$

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