Question

Work done in stretching a wire through $1\text{mm}$ is $2\text{J}$. Find the amount of work done $\left(\text{in J}\right)$ in elongating another wire (of same material with half the length and double the radius of cross section ) through the same length .

Answer 1

Work done is stretching wire is
$W=\frac{1}{2}\times \left(\frac{YA}{L}\right)\times (\Delta L{)}^2...(i)$
Since work done is $2\text{J}$
$2=\frac{1}{2}\times \left(\frac{YA}{L}\right)\times (\Delta L{)}^2...(ii)$
If length is halved and radius of cross section is doubled.
$L^{\prime }=\frac{L}{2}$
$A^{\prime }=4A[\text{Area}=\pi r^2]$
Now, Work done
$W^{\prime }=\frac{1}{2}\times \left(\frac{Y{A}^{\prime }}{L^{\prime }}\right)\times (\Delta L{)}^2$
Material is same and stretched thorugh same length, hence $Y$ and $\Delta L$ will remain same.
$W^{\prime }=\frac{1}{2}\times \left(\frac{Y\times 4A\times 2}{L}\right)\times \Delta L^2$
$\Rightarrow W^{\prime }=8\times \frac{1}{2}\times \left(\frac{YA}{L}\right)\times \Delta L^2$
Substituting from Eq.$\left(ii\right)$, we get
$W^{\prime }=8\times W=8\times 2$
$\therefore W^{\prime }=16\text{J}$