Question

A steel cylindrical rod of length $l$ and radius $r$ is suspended by its end from the ceiling. Define $U$ in terms of tensile strain $\frac{\Delta l}{l}$ of the rod.

• A. $U=\frac{2}{3}\pi r^{2}lE{\left(\frac{\Delta l}{l}\right)}^2$
• B. $U=\frac{4}{3}\pi r^{3}lE{\left(\frac{\Delta l}{l}\right)}^3$
• C. $U=\frac{2}{4}\pi r^{3}lE{\left(\frac{\Delta l}{l}\right)}^2$
• D. None of these
  

Answer 1

The correct option is A $U=\frac{2}{3}\pi r^{2}lE{\left(\frac{\Delta l}{l}\right)}^2$

$U=\frac{1}{2}V\sigma ϵ=\frac{1}{2}\times \frac{V}{E}\times {\sigma }^2⟹because(E=\frac{\sigma }{ϵ})$

$⟹V=\frac{4}{3}\pi r^{2}l$

$⟹ϵ=\frac{\Delta l}{l}$

By putting the values of these terms in the equation you will get:

$⟹U=\frac{2}{3}\pi r^{2}lE{\left(\frac{\Delta l}{l}\right)}^2$