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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 donot bend, then
a) The stress in the rods A, B, C and D are 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 are due to elasticity are in the ratio 1:2:3:4
d) The strains produced in the rods are in the ratio 1:2:3:4

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Both b and d

only d

only a

Both b and c

answer is C.

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Detailed Solution

Thermal stress developed when rods of same materials and length are fixed between two rigid support is $Yα ∆ t$ which is independent of area but force developed is stress $\times$ area( $π r^{2}$ )

so force is proportional to r².

energy stored is

$\frac{{(s t r e s s)}^{2}}{2 Y} v o l u m e$

As lengths are same volume is proportional to area hence energy is also proportional to area ( $π r^{2}$ )

Hence the correct answers are both b and c.

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