Four rods `A, B, C` and `1`) of the same length and material but of different radii `r, rsqrt(2),rsqrt(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

Options

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 the 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

B, C

Explanation

Thermal force `=Yaalphad theta=Ypir^(2)alphad theta`
`r_(1)=r, r_(2)=rsqrt(2), r_(3)=rsqrt(3), r_(4)=2r`
`F_(1):F_(2):F_(3):F_(4)=1:2:3:4`
Thermal stress `=Yalphad theta`
As `Y` and a are same for all the rods, hence stress developed in each rod will be the same. As strain `=alphad theta`, so strain will also be the same.
`E=` energy stored
`=1/2Y("strain")^(2)xxAxxL`
`:. E_(1):E_(2):E_(3):E_(4)=1:2:4`
so, option b and c are correct.