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 A, B, C and D is in the ratio $1:2:3:4$
• B. the forces on them exerted by the wall are in ratio $1:2:3:4$
• C. the energy stored in the rods due to elasticity is in the ratio $1:2:3:4$
• D. it is independent of area like surface tension while friction depends

Answer 1

Answer: (B), (C)

the energy stored in the rods due to elasticity is in the ratio $1:2:3:4$
the forces on them exerted by the wall are in ratio $1:2:3:4$

$F=Stress\times Area=Y\times Strain\times Area$. Thus, Force is proportional to area (or square of radii).
Thus, forces will be in ratio 1:2:3:4 (but stresses will be same, because stress is independent of area, depending only on strain and Y)
Energy stored per unit volume= $1/2\times stress\times strain=1/2\times Y\times {\left(Strain\right)}^2$
Now, strain = $\alpha \Delta T$, hence independent of area. Thus, energy per unit volume is independent of area. Thus energy stored will be proportional to $Area\times length$, and length being same, it will be proportional to area. Thus 1:2:3:4