Question

A smooth uniform string of natural length $l$,cross-sectional area $A$ and Young's modulus $Y$ is pulled aling its length by a force $F$ on a horizontal surface. If the elastic potential energy stored in the string is $U=\frac{F^{2}l}{xAY}$. Find $x$.

Answer 1

If $\Delta l$ be the elongation of string, Young modulus $Y=\frac{F/A}{\Delta l/l}=\frac{Fl}{A\Delta l}\Rightarrow \Delta l=\frac{Fl}{AY}$
Elastic potential energy stored in the string is:

$U=\frac{1}{2}F.\Delta l=\frac{1}{2}F.\left(\frac{Fl}{AY}\right)=\frac{F^{2}l}{2AY}$

$⟹x=2$