Question

The length of a wire is increased by 1 mm on the application of a given load. In a wire of the same material but of length and radius twice that of the first, an application of the same load. Extension in length is
(A) 0.25 mm
(B) 0.5 mm
(C) 2 mm
(D) 4 mm

Answer 1

In this problem we have to use the concept of extension of the length of the material when a force acts on it. The given question can easily be solved by using the Hooke’s law of elasticity. On making two equations according to the question and then by equating their young’s modulus because they are of the same material, we can easily determine the extension in the length in the second case. Young's modulus is a measure of the ability of a material to withstand changes in length when under lengthwise tension or compression.

Complete step by step answer:
${{F}_{1}}={{F}_{2}}$
${{l}_{1}}=l$
${{l}_{2}}=2l$
${{r}_{1}}=r$
${{r}_{2}}=2r$
$\Delta {{l}_{1}}=1\text{ mm=1}{{\text{0}}^{-3}}\text{ m}$
We know that by Hooke’s law of elasticity, young’s modulus of a wire is given by
$y=\dfrac{Fl}{A\Delta l}$
$y=\dfrac{Fl}{A\Delta l}$
Where,$y$= Young’s modulus of wire, F = load, l= Original length of the wire, A = Area of cross section of the wire and
$\Delta l$ = Extension in length

Given that two wires of the same material are used in both the cases therefore their young’s modulus will also be equal. Hence by equating their young’s modulus we get
${{y}_{1}}={{y}_{2}}$
$\dfrac{{{F}_{1}}{{l}_{1}}}{{{A}_{1}}\Delta {{l}_{1}}}=\dfrac{{{F}_{2}}{{l}_{2}}}{{{A}_{2}}\Delta {{l}_{2}}}$
Since, load are same and equal
${{F}_{1}}={{F}_{2}}$
On putting the corresponding values, we get
$\dfrac{l}{\pi {{r}^{2}}\times {{10}^{-3}}\text{m}}=\dfrac{2l}{\pi {{\left( 2r \right)}^{2}}\Delta {{l}_{2}}}$
$\implies \dfrac{1}{{{10}^{-3}}\text{m}}=\dfrac{1}{2\Delta {{l}_{2}}}$
On further solving we get
$\Delta {{l}_{2}}=\dfrac{{{10}^{-3}}\text{m}}{2}\Leftrightarrow \Delta {{l}_{2}}=0.5\text{ mm}$

Hence, the option (B) is correct.

Note:
For the wires of the same material young’s modulus is the same as young’s modulus of elasticity depending upon the nature of material of the substance. If the material of the wires is different than their values of young’s modulus will also be different. Young’s modulus is the property of the material which measures the elasticity of the material.