Question

A student plots a graph from the readings obtained on performing the experiment for determination of Young's modulus of a metal wire but he forgets to put the labels on the graph. The quantities on $x$ and $y-$ axis (respectively) shown in the graph may be:

• A. Weight hung and length increased
• B. Stress applied and length increased.
• C. Stress applied and strain developed.
• D. Length increased and weight hung.

Answer 1

Answer: (A), (B), (C), (D)

Length increased and weight hung.

Weight hung with the metal wire will apply force and stress and strain both will develop in wire.

We know, Young's modulus is,
$Y=\frac{\text{stress}}{\text{strain}}$
$Y=\frac{\sigma }{\epsilon }$
$\Rightarrow \epsilon =\frac{1}{Y}\sigma ...\left(1\right)$
On comparing Eq.$\left(1\right)$ with equation of a straight line $y=mx+c$, we can see $m=\frac{1}{Y}$ and $\left(1\right)$ represents equation of a straight line passing through origin.

We can also write,
$Y=\frac{FL}{A\Delta L}$
$\Rightarrow \Delta L=\left(\frac{L}{AY}\right)F...\left(2\right)$
where, $F$ is weight, $\Delta L$ is increase in length.

On comparing Eq.$\left(2\right)$ with equation of straight line $y=mx+c$, we can see $m=\frac{L}{AY},c=0$ and $\left(2\right)$ is an equation of a straight line passing through origin.

Again $Y=\frac{FL}{A\Delta L}=\frac{\sigma L}{\Delta L}$
$\because [\sigma =\frac{F}{A}]$
$\Rightarrow \Delta L=\frac{L}{Y}\sigma ...\left(3\right)$
$[\sigma =\text{stress and}\Delta L=\text{Length increased}]$
on comparing Eq.$\left(3\right)$ with equation of straight line $y=mx+c$
we can see $m=\frac{L}{Y}$ and $\left(3\right)$ is also an equation of straight line passing through origin.

Again,
$Y=\frac{FL}{A\Delta L}$
$\Rightarrow F=\frac{YA}{L}\Delta L...\left(4\right)$
Here, $F=$ weight and $\Delta L=$ Length increased
On comparing Eq.$\left(4\right)$ with equation of staright line $y=mx+c$, we can see $m=\frac{YA}{L}$ and Eq. $\left(4\right)$ also represents equation of straight line passing through origin.
$\therefore$ Hence, all options $\left(a\right),\left(b\right),\left(c\right),\left(d\right)$ are correct.