Question

A metal rod of Young’s modulus $Y$ and coefficient of linear expansion $\alpha$ has its temperature raised by $\Delta \theta$. The linear stress to prevent the expansion of rod is ($L$ and $\ell$ are original length of rod and expansion respectively)

• A. $\frac{Y L}{\ell}$
• B. $\frac{Y}{\Delta \theta}$
• C. $Y \propto \Delta \theta$
• D. $Y \propto \left(\frac{\ell}{L}\right)^2$

Hint:

Linear stress and expansion are directly related to the temperature change and the material’s properties.

Answer 1

The Correct Option is C

Step 1: Relation for linear stress.
Linear stress is related to the Young’s modulus and the expansion due to temperature change.
\[ \text{Stress} = Y \times \frac{\Delta \ell}{\ell} \]
Step 2: Expansion due to temperature change.
The change in length $\Delta \ell$ is related to the coefficient of linear expansion $\alpha$ and the temperature change $\Delta \theta$:
\[ \Delta \ell = \alpha L \Delta \theta \]
Step 3: Conclusion.
The stress is proportional to the temperature change $\Delta \theta$ because $\Delta \ell \propto \Delta \theta$. Thus, $Y \propto \Delta \theta$.