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 $2 r$ 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

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the stress in the rods $A$ , $B$ , $C$ and $D$ are in the ratio $1 : 2 : 3 : 4$

the forces on them exerted by the wall are in the ratio $1 : 2 : 3 : 4$

the energy stored in the rods due to elasticity are in the ratio $1 : 2 : 3 : 4$

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

answer is B, C.

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Detailed Solution

$(b, c)$

$F = (Y α ∆ T) A = (Y α ∆ T) π r^{2} S o F \propto r^{2} F_{1} : F_{2} : F_{3} : F_{4} = 1 : 2 : 3 : 4$

Energy stored $= \frac{1}{2} F x = \frac{1}{2} F \times \frac{F l}{A Y} = \frac{F^{2} l}{2 π r^{2} Y}$

Therefore energy stored $\propto r^{2}$

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