Question

A fixed volume of iron is drawn into a wire of length l. The extension produced in this wire by a constant force F is proportional to then:
$A)\dfrac{1}{{{l^2}}} \\
  B)\dfrac{1}{l} \\
  C){{\text{l}}^2} \\
  D)l \\ $

Answer 1

Hint- Here we will proceed by applying the concept of Young’s modulus related to elasticity where tensile stress will be the ratio of force to area and tensile strain will be the ratio of change in dimension to original dimension. Then we will substitute the values of area, force and dimension to get the required answer.

Complete step-by-step solution -
Formula used-
1. $E = \dfrac{{tensile{\text{ stress}}}}{{tensile{\text{ strain}}}}$
2. $Tensile\;stress = \dfrac{F}{A}$
3. $Tensile{\text{ strain = }}\dfrac{{\Delta l}}{l}$

Let us assume the area of the cross section of iron wire be A and volume of iron be V And we are given that length of wire as l and constant force as F. Here we will apply the concept of Young’s modulus related to elasticity as the question is about extension produced in the wire,
The formula of young’s modulus is $E = \dfrac{{tensile{\text{ stress}}}}{{tensile{\text{ strain}}}}$
Where tensile stress will be ratio of force (F) to area (A)
And tensile strain will be ratio of change in dimension (l) to original dimension (l)
Now using this formula of young’s modulus,
We get-
$E = \dfrac{{\dfrac{F}{A}}}{{\dfrac{{\Delta l}}{l}}}$
$ \Rightarrow \Delta l = \dfrac{{Fl}}{{AE}}$
Multiplying numerator and denominator by l,
We get-
$ = \dfrac{{F{l^2}}}{{\left( {Al} \right)E}}$
But we know that area multiplied by length is volume,
So we get-
$\Delta l = \dfrac{{F{l^2}}}{{VE}}$
And if volume is fixed (given)
then,
$\Delta l\alpha {l^2}$
Which means the extension produced in this wire by a constant force F is proportional to ${l^2}$.
Hence, option C is right.

Note- While solving this question, we must know that the E used for young’s modulus formula can be replaced with abbreviation Y. Also we know that area multiplied by length is volume. Hence we will get our desired result.