Question

A metal bar of length $1\text{m}$ and area of cross-section $1{\text{mm}}^2$ is clamped between two rigid supports. For the material of rod, its Young's modulus is $Y={10}^{11}{\text{N/m}}^2$ and coefficient of linear expansion is $\alpha =1.8\times {10}^{-5}{}^{\circ }{C}^{-1}$. If the temperature of the rod is decreased by $\Delta T={10}^{\circ }C$, the force exerted by the rod on the supports is

• A. $9\text{N}$
• B. $6\text{N}$
• C. $16\text{N}$
• D. $18\text{N}$

Answer 1

The correct option is D $18\text{N}$

We know,
Thermal stress $=Y\alpha \Delta T$
For cooling,
$\Delta T$ is negative.
$\therefore$ We use $\left|\Delta T\right|$.
$\Rightarrow$ Thermal stress $=Y\alpha \left|\Delta T\right|$
$={10}^{11}\times 1.8\times {10}^{-5}\times 10$
$=1.8\times {10}^7{\text{N/m}}^2$

Also,
Force on the suppport,
$F=YA\alpha \left|\Delta T\right|=A\times$ stress
$=1\times {10}^{-6}\times 1.8\times {10}^7\text{N}$
$\Rightarrow F=18\text{N}$