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  4. The following four wires of length L and radius r are made of the same material. When they are under same tension, the largest extension is produced in

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Q. The following four wires of length $L$ and radius $r$ are made of the same material. When they are under same tension, the largest extension is produced in

Mechanical Properties of Solids

Solution:

(Young's modulus, $Y=\frac{F L}{A \Delta L}=\frac{T L}{\pi r^{2} \Delta L}$
$\Rightarrow \Delta L=\frac{T L}{\pi r^{2} Y}$
As wires are of same material,
so $Y$ is same for all.
Also $T$ is same,
so $\Delta L \propto \frac{L}{r^{2}}$.
Thus, $\frac{L}{r^{2}}$ is maximum for length,
$L=250 \,cm$ and radius, $r=0.1\, mm$,
i.e. $25 \times 10^{7} m$,
so its extension is largest.