Question

The pressure that has to be applied to the ends of a steel wire of length $10cm$ to keep its length constant when its temperature is raised by $100^\circ C$ is:
(For steel Young’s modulus is $2 \times {10^{11}}N{m^{ - 2}}$ and coefficient of thermal expansion is $1.1 \times {10^{ - 5}}{K^{ - 1}}$ )
(A) $2.2 \times {10^7}Pa$
(B) $2.2 \times {10^6}Pa$
(C) $2.2 \times {10^8}Pa$
(D) $2.2 \times {10^9}Pa$

Answer 1

Hint We need to find the change in length in the steel wire due to the change in the temperature. Then by using the formula for Young’s modulus, we need to find the pressure by using the given values.

Formula Used: In the solution we will be using the following formula,
$\Rightarrow \Delta L = \alpha L\Delta T$ where $\Delta L$ is the change in length,
$\alpha$ is the coefficient of thermal expansion
$L$ is the length and $\Delta T$ is the change in temperature.
$\Rightarrow Y = \dfrac{{F/A}}{{\Delta L/L}}$
where $Y$ is the Young’s modulus,
$F$ is the force and $A$ is the area of cross section

Complete step by step answer
In this question we are told that the temperature of the steel wire is to be raised by the amount of $100^\circ C$ . Now due to the increase in the temperature, the length of the steel wire will also increase. Now the value of this increase in length will be given by the formula,
$\Rightarrow \Delta L = \alpha L\Delta T$
In the problem we are given, $L = 10cm$ . So in SI units it will be, $L = 0.1m$ . Now the thermal coefficient is given as, $\alpha = 1.1 \times {10^{ - 5}}{K^{ - 1}}$ and the change in temperature is given as, $\Delta T = 100^\circ C$ , which is equal to $\Delta T = 100K$ . Now substituting the values we get,
$\Rightarrow \Delta L = 1.1 \times {10^{ - 5}} \times 0.1 \times 100$
On calculating we have,
$\Rightarrow \Delta L = 1.1 \times {10^{ - 4}}m$
So pressure has to be applied to prevent this change in length.
Now from the formula for the Young’s modulus we have,
$\Rightarrow Y = \dfrac{{F/A}}{{\Delta L/L}}$
Here the term $\dfrac{F}{A}$ is the force per unit area, which is the pressure. Therefore we can write this formula for the pressure as,
$\Rightarrow P = \dfrac{F}{A} = \dfrac{Y}{L}\Delta L$
Now we are given $Y = 2 \times {10^{11}}N{m^{ - 2}}$
So substituting we get the value,
$\Rightarrow P = \dfrac{{2 \times {{10}^{11}}}}{{0.1}} \times 1.1 \times {10^{ - 4}}$
Hence on calculating we get,
$\Rightarrow P = 2.2 \times {10^8}Pa$
So the correct answer is option C.

Note
The Young’s modulus is the modulus of elasticity of a substance. It is the mechanical property that measures the tensile stiffness of a solid. It can be calculated by the ratio of the tensile stress to the axial strain of the body.