Question

When a metal wire is stretched by a load, the fractional change in its volume ∆V/V is proportional to
(a) $\frac{∆l}{l}$
(b) ${\left(\frac{∆l}{l}\right)}^2$
(c) $\sqrt{∆l/l}$
(d) none of these

Answer 1

Correct option: (a) ∆ll

$$\begin{gathered} C.S.A.\hspace{0.17em}=\mathrm{A} \\ \mathrm{Length}=\mathrm{l} \\ \mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{wire}\mathrm{V}=\mathrm{Al} \\ \mathrm{Assuming}\mathrm{no}\mathrm{lateral}\mathrm{strain}\mathrm{when}\mathrm{longitudinal}\mathrm{strain}\mathrm{occurs}: \\ \mathrm{Increase}\mathrm{in}\mathrm{volume}:∆\mathrm{V}=\mathrm{A}∆\mathrm{l} \\ \Rightarrow \frac{∆\mathrm{V}}{\mathrm{V}}=\frac{\mathrm{A}∆\mathrm{l}}{\mathrm{Al}}=\frac{∆\mathrm{l}}{\mathrm{l}} \\ \mathrm{So},\frac{∆\mathrm{V}}{\mathrm{V}}\mathrm{is}\mathrm{directly}\mathrm{proportional}\mathrm{to}\frac{∆\mathrm{l}}{\mathrm{l}}. \end{gathered}$$