Question

A copper rod of diameter 2 cm and length 10 cm is drawn into a wire of uniform thickness and length 10 m. Find the thickness of the wire.
[CBSE 2012]

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{copper}\mathrm{rod},R=\frac{2}{2}=1\mathrm{cm}, \\ \mathrm{the}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{copper}\mathrm{rod},H=10\mathrm{cm}\mathrm{and} \\ \mathrm{the}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{wire},h=10\mathrm{m}=1000\mathrm{cm} \\ \mathrm{Let}\mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{wire}\mathrm{be}r. \\ \mathrm{As}, \\ \mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{wire}=\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{rod} \\ \Rightarrow \mathrm{\pi }{r}^{2}h=\mathrm{\pi }{R}^{2}H \\ \Rightarrow r^{2}h=R^{2}H \\ \Rightarrow r^2\times 1000=1\times 10 \\ \Rightarrow r^2=\frac{10}{1000} \\ \Rightarrow r^2=\frac{1}{100} \\ \Rightarrow r=\sqrt{\frac{1}{100}} \\ \Rightarrow r=\frac{1}{10} \\ \Rightarrow r=0.1\mathrm{cm} \\ \Rightarrow \mathrm{The}\mathrm{diameter}\mathrm{of}\mathrm{the}\mathrm{wire}=2r=2\times 0.1=0.2\mathrm{cm} \\ \therefore \mathrm{The}\mathrm{thickness}\mathrm{of}\mathrm{the}\mathrm{wire}=0.2\mathrm{cm} \end{gathered}$$

So, the thickness of the wire is 0.2 cm or 2 mm.