When the temperature of a metal wire is increased from \(0^{\circ} \,C\) to \(10^{\circ}\, C\), its length increases by \(0.02 \% .\) The percentage change in its mass density will be closest to:

\(\therefore \Delta L = L \alpha \Delta T \Rightarrow \frac{\Delta L }{ L }=\alpha \Delta T =0.02 \%\)

\(\therefore \beta=2 \alpha\) (Areal coefficient of expansion) \(\Rightarrow \beta \Delta T =2 \alpha \Delta T =0.04 \%\)

Volume \(=\) Area \(\times\) Length

Density \((\rho)=\frac{\text { Mass }}{\text { Volume }}=\frac{\text { Mass }}{\text { Area } \times \text { Length }}=\frac{ M }{ AL }\)

\(\Rightarrow \frac{\Delta \rho}{\rho}=\frac{\Delta M}{M}+\frac{\Delta A }{ A }+\frac{\Delta L }{ L }\) (Mass remains constant)

\(\Rightarrow\left(\frac{\Delta \rho}{\rho}\right)=\frac{\Delta A }{ A }+\frac{\Delta L }{ L }=\beta \Delta T +\alpha \Delta T\)

\(=0.04 \%+0.02 \%\)

\(=0.06 \%\)