Question

The pressure that has to be applied to the ends of a steel wire of length $10\text{cm}$ to keep its length constant when its temperature is raised by ${100}^{\circ }\text{C}$ is:-
Given: $Y=2\times {10}^{11}{\text{N/m}}^2$ and $\alpha =1.1\times {10}^{-5}{/}^{\circ }\text{C}$

• A. $2.2\times {10}^9\text{Pa}$
• B. $2.2\times {10}^7\text{Pa}$
• C. $2.2\times {10}^8\text{Pa}$
• D. $2.2\times {10}^6\text{Pa}$

Answer 1

The correct option is C $2.2\times {10}^8\text{Pa}$

When temperature is increased, length of the rod also increases. To keep the rod at constant length, we have to apply pressure at the ends to oppose expansion.
Let the pressure applied be $P$.


We know, Young's modulus $Y=\frac{\text{stress}}{\text{strain}}=\frac{P}{\frac{\Delta L}{L}}=\frac{PL}{\Delta L}$
$[\because \text{stress}=\frac{F}{A}=P]$
$\Rightarrow Y=\frac{PL}{\Delta L}$
$\Rightarrow P=\frac{Y\Delta L}{L}$
[From thermal expansion, the change in length, $\Delta L=L\alpha \Delta T$]
$\Rightarrow P=\frac{Y\left(L\alpha \Delta T\right)}{L}$
$\Rightarrow P=Y\alpha \Delta T$

Substituting the values,
$\left[\begin{array}{c}Y=2\times {10}^{11}{\text{N/m}}^2\\ \alpha =1.1\times {10}^{-5}{/}^{\circ }\text{C}\\ \Delta T={100}^{\circ }\text{C,}\end{array}\right]$
we get, $P=2\times {10}^{11}\times 1.1\times {10}^{-5}\times 100$
$\Rightarrow P=2.2\times {10}^8{\text{N/m}}^2$
$\therefore P=2.2\times {10}^8\text{Pa}$
This is the required pressure that has to be applied to keep the length of the rod constant.