Question

A load of 15N produces an extension of 0.10mm in A metal wire 10m in length. If young modulus of metal is 1.8 x 10^11Pa Calculate the diameter of the wire

Answer 1

Given data:

* The force acting on the wire is 15 N.

* The length of the wire is 10 m.

* The extension of the wire is 0.1 mm.

* The young modulus of the wire is,

$Y=1.8\times10^{11}Pa^{}$

Solution:

The Young modulus of the metal wire in terms of area of the wire is,

$Y=\frac{F\times l}{A\times dl}$

where F is the force acting, l is the length of wire, dl is the extension of the wire, and A is the area of the wire,

Substituting the known values,

$\begin{gathered} 1.8\times10^{11}=\frac{15\times10}{A\times0.1\times10^{-3}} \\ 1.8\times10^{11}\times A\times0.1\times10^{-3}=150 \\ A\times0.18\times10^8=150 \\ A=\frac{150}{0.18\times10^8} \\ A=833.3\times10^{-8}m^2 \end{gathered}$

As the area of wire is same as the area of the circle.

Thus, the area of the wire in terms of the radius of wire is,

$\begin{gathered} A=\pi r^2 \\ 833.3\times10^{-8}=\pi r^2 \\ r^2=\frac{833.3\times10^{-8}}{\pi} \\ r^2=265.25\times10^{-8} \\ r=16.29\times10^{-4}\text{ m} \\ r=1.63\times10^{-3}\text{ m} \end{gathered}$

The diameter of the wire is,

$\begin{gathered} D=2r \\ D=2\times1.63\times10^{-3^{}}\text{ } \\ D=3.26\times10^{-3}\text{ m} \\ D=3.26\text{ mm} \end{gathered}$

Thus, the diameter of the wire is 3.26 mm.