Question

A metal bar of length L and area of cross-section A is rigidly clamped between two walls. The Young's modulus of its material is Y and the coefficient of linear expansion is $\alpha$. The bar is heated so that its temperature increases by ${\theta }^{\circ }C$. The force exerted at the ends of the bar is given by

• A. $YL\alpha \theta$
• B. $YL\alpha \frac{\theta }{A}$
• C. $YA\alpha \theta$
• D. $Y\theta \alpha \frac{L}{A}$

Answer 1

The correct option is C

$YA\alpha \theta$

The coefficient of linear expansion is defined as
$\alpha =\frac{Increaseinlength}{Originallength\times Temperaturerise}$ $=\frac{l}{L\theta }$
$\therefore$ Increase in length, $l=\alpha L\theta$.
Now, we know that, $Y=\frac{FL}{Al}$
$\Rightarrow F=\frac{YAl}{L}=\frac{YA\alpha L\theta }{L}=YA\alpha \theta$
Hence, the correct choice is (c).