Question

Four rods $A,B,C,D$ of same length and material but of different radii $r,r\sqrt{2},r\sqrt{3}$ and $2r$ respectively are held between two rigid walls. The temperature of all rods is increased by same amount. If the rods do not bend, then
$\left(A\right)$ The stress in the rods are in the ratio of $1:2:3:4.$
$\left(B\right)$ The force on the rod exerted by the wall are in the ratio of $1:2:3:4.$
$\left(C\right)$ The energy stored in the rods due to elasticity are in the ratio $1:2:3:4.$
$\left(D\right)$ The strains produced in the rods are in the ratio of $1:2:3:4.$

• A. all $A,B,C$ and $D$ are correct
• B. only $A$ and $C$ are correct
• C. only $B$ and $C$ are correct
• D. only $A$ and $D$ are correct

Answer 1

The correct option is C only $B$ and $C$ are correct

Thermal stress is, $\frac{T}{A}=Y\alpha △T...\left(i\right)$
Force exerted is, $T=YA\alpha △T\Rightarrow T\propto A..\left(ii\right)$
Energy stored is, $E=\frac{1}{2}\times (stress{)}^2\times \frac{Al}{Y}...(iii)$
Strain is, $ϵ=\frac{stress}{Y}...\left(iv\right)$

From $\left(i\right)$ and $\left(iv\right)$, it is clear that stress and strain depend only on the material and $△T$ which are same for all the rods. Thus their ratios are $1:1:1:1.$

From $\left(ii\right)$ and $\left(iii\right)$ force and energy stored are proportional to area of cross-section $A=\pi r^2$. Thus, their ratios are
$\pi r^2:\pi (\sqrt{2}r{)}^2:\pi (\sqrt{3}r{)}^2:\pi (2r{)}^2=1:2:3:4.$