Question

A rod is acted by two equal forces as shown in the figure. The coefficient of thermal expansion of the rod is $\alpha$ and area of cross section is $A$. When the temperature the rod increased by$∆T$. The length of the rod does not change. The young’s modulus$Y$ will be.

• A. $\frac{F}{2A\alpha ∆T}$
• B. $\frac{F}{A\alpha ∆T}$
• C. $\frac{2F}{A\alpha ∆T}$
• D. $\frac{F}{3A\alpha ∆T}$

Answer 1

The correct option is B

$\frac{F}{A\alpha ∆T}$

Step 1: Given data

$\alpha =$Coefficient of thermal expansion

$A=$Area of cross-section

$∆T=$Temperature increase

$Y=$Young's modulus

$L=$Length

$∆L=$ Change in length

Here when temperature increases there will be expansion and due to force applied there will be compression, as the length remains the same.

Step 2 : To find Young's modulus

Increase in length (due to rise in temperature) $=$decrease in length

we know

$$\begin{gathered} Y=\frac{FL}{A∆L} \\ ∆L=\frac{FL}{YA} \end{gathered}$$

$\left(∆L\right)$ due to force

$$\begin{gathered} K\alpha ∆T=\frac{FL}{YA} \\ Y=\frac{FK}{K\alpha ∆TA} \end{gathered}$$

$\left(K\right)$ is a constant, therefore $Y=\frac{F}{\alpha ∆TA}$

Hence the Young's modulus $Y=\frac{F}{A\alpha ∆T}$

Therefore Option B is the correct answer.