Question

A copper rod of radius 1cm and length 2cm is drawn into a wire of length 18m of uniform thickness. Find the thickness of the wire.

Answer 1

Hint: Since there is no loss in material, the volume of the copper rod is the same as the wire. So we can equate their volumes to determine the thickness of the wire.

Complete step-by-step answer:
Given,
Radius of the copper rod, ${r_1} = 1cm$
Length of copper rod, \[{l_1} = 2cm\]
Length of wire, ${l_2} = 18m = 1800cm$
Let the thickness of wire be ${t_2}$
Volume of a cylinder of radius r and length l is given by : $V = \pi {r^2}l$
Volume of copper rod is equal to the volume of wire as density and mass of the material does not change.
$ \Rightarrow {V_1} = {V_2}$
$ \Rightarrow \pi r_1^2{l_1} = \pi r_2^2{l_2}$
$ \Rightarrow {r_2} = \sqrt {\dfrac{{r_1^2{l_1}}}{{{l_2}}}} = \sqrt {\dfrac{{{1^2} \times 2}}{{1800}}} = \sqrt {\dfrac{1}{{900}}} = \dfrac{1}{{30}}cm$
Thickness of the wire is equal to the diameter which is twice the radius
$ \Rightarrow {t_2} = 2{r_2} = \dfrac{2}{{30}} = \dfrac{1}{{15}}cm$
So, this is your desired answer.

Note: In these types of questions, there will be a condition given directly or indirectly using which we have to determine the value of the unknown. In this situation since the mass was constant (as there was no material loss) & density remains constant as the material is same so therefore the volume was constant using which we found the thickness of the wire.