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Q.
Young’s modulus for a steel wire is $2 \times 10^{11}\, Pa$ and its elastic limit is $2.5 \times 10^8\, Pa$. By how much can a steel wire 3 m long and $2 \,mm$ in diameter be stretched before the elastic limit is exceeded?
The cross-sectional area of a wire of radius $r=1 \,mm =10^{-3} m$ is $A=\pi r^{2}=3.14 \times 10^{-6} m ^{2}$
The maximum force that can be applied without exceeding the elastic limit is therefore $F=\left(\frac{F}{A}\right)_{\max } \times(A) $
$=\left(2.5 \times 10^{8} N m ^{-2}\right)\left(3.14 \times 10-6 m ^{2}\right)=785 \,N$
When this force is applied, the wire will stretch by
$\Delta L=\frac{L}{Y} \frac{F}{A}=\frac{(3\, m )(785 \,N )}{\left(2 \times 10^{11} N / m ^{2}\right)\left(3.14 \times 10^{-6} m ^{2}\right)} $
$=3.75 \times 10^{-3} m =3.75\, mm$