The length of a conical bar is l, the diameter of the base is d and the weight per unit volume is w. It is fixed at its upper end and hangs freely. The elongation of the bar under the action of its own weight will be

Concept:

Elongation of the elemental strip is given by,

\(\rm\delta=\frac{P_xdx}{A_xE}\)

Here,

P_x = load at x from bottom

dx = Length of the elemental strip,

A_x = area of the cross-section at x.

E = Modulus of elasticity.

Calculation:

P_x = weight per unit volume × volume below axis on side

\(\rm p_x=W\times\left(\frac{1}{3}\pi r_x^2\times x\right)\)

∴ \(\rm\delta=\frac{P_xdx}{A_xE}\) = \(\rm \frac{W\times\left(\frac{1}{3}\pi r_x^2\times x\right)\times dx}{\pi r_x^2\times E}\) = \(\rm \frac{Wxdx}{3E}\)

Elongation of the entire bar,

= \(\rm\int_0^l\frac{Wxdx}{3E}=\frac{Wl^2}{6E}\)