Question

A wire of length $L$ and cross-sectional area $A$ is made of a material of Young's modulus $Y$. If the wire is stretched by an amount $x$, the work done during the stretching is

• A. $\frac{YA{x}^2}{2L}$
• B. $\frac{YA{L}^2}{2x}$
• C. $\frac{YAx}{2}$
• D. $\frac{YAL}{2}$

Answer 1

The correct option is A $\frac{YA{x}^2}{2L}$

Let the extension of the wire at an instant of time be $x$ and let it be further stretched by $dx$ at that instant.


The force acting on the wire at that instant is given by,
$F=AY\frac{\Delta L}{L}=AY\frac{x}{L}$
$\Rightarrow F=\frac{YxA}{L}$
The small amount of work done by the applied force during the stretching of wire through $dx$,
$dW=Fdx=\left(\frac{YxA}{L}\right)dx...\left(i\right)$
Total work done in stretching the wire through length $x$ can be found by integrating Eq.$\left(i\right)$ with limits from $0$ to $x$.
$\Rightarrow W={\int }_0^x\frac{YxAdx}{L}=\frac{YA}{L}{\int }_0^{x}xdx$
$\therefore W=\frac{YA{x}^2}{2L}$