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:

• 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 stains produced in the rods are in the ratio $1:2:3:4$

Answer 1

The correct option is A the stress in the rods are in the ratio $1:2:3:4$

let $\alpha$ be the temperature coefficient of all the rod

L be the length of all the rod.

change in the length change dL due to temperature change dT is $l\alpha dT$.

$dL=L\alpha dT$ eq(1)..

similarly, stress force stop this expansion

$dL=\frac{FL}{A\gamma }$ eq(2)..

where, F=stress , A = area of surface of rod,$\gamma$is a young modulus

due to eq(1) and eq(2)

$\frac{FL}{A\gamma }=L\alpha dT$

$F=\alpha A\gamma dT$

F is directly proportional to area

let k is proportional constant

$F=k{r}^2$

$F_A=k{r}^2$

$F_B=k(\sqrt{2}r{)}^2=2r^2$

$F_C=k(\sqrt{3}r{)}^2=3r^2$

$F_D=k(2r{)}^2=4r^2$

therefore

$F_A:F_B:F_C:F_D=1:2:3:4$

Hence the correct option is A.