A uniform metallic wire is elongated by \(0.04\, m\) when subjected to a linear force \(F\). The elongation, if its length and diameter is doubled and subjected to the same force will be ..... \(cm .\)

\(\Delta \ell=\frac{ F }{ Y.A } \cdot \ell\)

\(\Delta \ell=\frac{ F \ell}{ Y \cdot \pi r ^{2}}\)

\(\Delta \ell \propto \frac{\ell}{ r ^{2}}\)

\(\frac{\Delta \ell_{2}}{\Delta \ell_{1}}=\left(\frac{\ell_{2}}{\ell_{1}}\right)\left(\frac{ r _{1}}{ r _{2}}\right)^{2}\)

\(=(2)(\frac{1}{2})^2\)

\(\frac{\Delta \ell_{2}}{\Delta \ell_{1}}=\frac{1}{2}\)

\(\Delta \ell_{2}=\frac{\Delta \ell_{1}}{2}\)

\(=\frac{0.04}{2}\)

\(=0.02\, m\)

\(\Delta \ell_{2}=2\, cm\)