Question

Two wires of same material and
length are stretched by the same force.
If the ratio of radii of
the two wires is n: 1 then the ratio of
their elongation is
1) n2:1
2) 1:n2
3) 1:n
4) n: 1​

Answer 1

The elongation in a wire of length $\displaystyle\sf {L,}$ cross sectional area $\displaystyle\sf {A,}$ Young's modulus $\displaystyle\sf {Y}$ and acted upon by a force $\displaystyle\sf {F}$ is,

$\displaystyle\sf{\longrightarrow \Delta L=\dfrac {FL}{AY}}$

Here,

• two wires are of same material, i.e., $\displaystyle\sf {Y}$ is constant.
• they are of same length, i.e., $\displaystyle\sf {L}$ is constant.
• they are acted upon by the same force, i.e., $\displaystyle\sf {F}$ is constant.

Hence,

$\displaystyle\sf{\longrightarrow \Delta L\propto\dfrac {1}{A}\quad\quad\dots (1)}$

But, cross sectional area,

$\displaystyle\sf{\longrightarrow A=\pi\,r^2}$

Since $\displaystyle\sf {\pi}$ is constant,

$\displaystyle\sf{\longrightarrow A\propto r^2}$

Hence (1) becomes,

$\displaystyle\sf{\longrightarrow \Delta L\propto\dfrac {1}{r^2}}$

Therefore,

$\displaystyle\sf{\longrightarrow \dfrac {\Delta L_1}{\Delta L_2}=\left (\dfrac {r_2}{r_1}\right)^2}$

Here ratio of radii of the two wires is,

• $\displaystyle\sf {\dfrac {r_1}{r_2}=\dfrac {n}{1}}$

Hence ratio of elongations is,

$\displaystyle\sf{\longrightarrow \dfrac {\Delta L_1}{\Delta L_2}=\left (\dfrac {r_2}{r_1}\right)^2}$

$\displaystyle\sf{\longrightarrow \dfrac {\Delta L_1}{\Delta L_2}=\left (\dfrac {1}{n}\right)^2}$

$\displaystyle\sf{\longrightarrow\underline {\underline {\dfrac {\Delta L_1}{\Delta L_2}=\dfrac {1}{n^2}}}}$

Hence (2) is the answer.

Answer 2

Dear student,

Given :

• wires are of same material implying γ is same.
• wires are of same length, l₁ = l₂
• same forces are acting
• ratio of radii of wires is r₁ : r₂ = n : 1

To find :

• the ratio of their elongation i.e. Δl

Solution :

By formula,

                  Δl = Fl./ Aγ

Considering the given information, we have;

       Δl ∝ 1/ A

       Δl ∝ 1/ πr²

       Δl ∝ 1/ r²

Using the above relation, we have :

      Δl₁/Δl₂ ∝ r₂²/ r₁²

      Δl₁/Δl₂ ∝ 1 / n²

Hence, option (2) 1/ n² is the correct answer.