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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

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the stress in the rods are in the ratio $1 : 2 : 3 : 4$

the force on the rod 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

$F = y A \propto Δ T F \propto A \Rightarrow F \propto r^{2} s t r e s s = Y \propto Δ T S t r e s s a r e e q u a l U = \frac{{(s t r e s s)}^{2}}{2 y} \times v o l u m e U = \frac{{(s t r e s s)}^{2}}{2 y} \times π r^{2} l U \propto r^{2}$

Stress = $y \times s t r a i n$
As stress are equal $\Rightarrow$ strain are equal

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