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

Explanation

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.

Elongation in the element is given by
`elongation =origiN/Al length xxstress/Y`
`(Tdx)/(AY)=((L-x)Wdx)/(LAY)
The total elongation `=int_0^L((L-x)W dx)/(LAY)`
`=W/(LAY)(Lx-x^2/2)_0^L=(WL)/(2AY)`