Question

A smooth uniform string of natural length l, cross-sectional area A and Young's modulus Y is pulled along its length by a force F on a horizontal surface. Find the elastic potential energy stored in the string.

• A. $U=\frac{F^{2}l}{7AY}$
• B. $U=\frac{F^{2}l}{2AY}$
• C. $U=\frac{F^{2}l}{8AY}$
• D. $U=\frac{F^{2}l}{9AY}$

Answer 1

The correct option is B $U=\frac{F^{2}l}{2AY}$

${\rm Y}=\frac{stress}{strain}=\frac{F}{A}\times \frac{l}{△l}$

$△l=$change in length due to force F

Potential energy$=dW=F.dx$

on small increase in length $dx$

$W=\int F.dx={\int }_0^{△l}{\rm Y}\frac{Ax}{l}.dx$

$={\rm Y}\frac{A}{l}.{\left[\frac{x^2}{2}\right]}_0^{△l}$

$={\rm Y}\frac{A}{l}\frac{{\left(△l\right)}^2}{2}$

$=\frac{{\rm Y}A}{2l}{\left[\frac{Fl}{A{\rm Y}}\right]}^2$

$=\frac{{\rm Y}A}{2l}.\frac{F^2{l}^2}{A^2{{\rm Y}}^2}$

$W=\frac{F^{2}l}{2A{\rm Y}}$