Question

Four rods $A,B,C$ and $D$ of the 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 through the same range. If the rods do not bend, then

• A. The stress in the rods are in the ratio $1:2:3:4$
• B. The force on the rod exerted by the wall are in the ratio $1:2:3:4$
• C. The energy stored in the rods due to elasticity are in the ratio $1:2:3:4$
• D. The strains produced in the rods are in the ratio $1:2:3:4$

Answer 1

Answer: (B), (C)

The correct options are
B The force on the rod exerted by the wall are in the ratio $1:2:3:4$
C The energy stored in the rods due to elasticity are in the ratio $1:2:3:4$

Thermal force $=YA\alpha d\theta =Y\pi r^3\alpha d\theta$
$r_1=r,r_2=r\sqrt{2},r_3=r\sqrt{3},r_4=2r$
$F_1:F_2:F_3:F_4=1:2:3:4$
Thermal stress $=Y\alpha d\theta$
As $Y$ and $\alpha$ are same for all the rods, hence stress developed in each rod will be the same. As strain $=\alpha d\theta$, so strain will also be the same.
$E=Energystored$
$=\frac{1}{2}Y(strain{)}^2\times A\times L$
$\therefore E_1:E_2:E_3:E_4=1:2:4$
So, options (b) and (c) are correct.