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Three bars each of length L are connected with the help of three springs as shown in the figure. Coefficients of linear expansion of A,B and C are $2 α, α, 4 α$ respectively. The surface on which the arrangement is kept is smooth. If the temperature of rods have been increased by $∆ T$ (assume there is no effect of change in temperature on springs), then mark out the correct statement (s) [Assume that thermal stress to be negligible]

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All of the above

The force exerted by C on the wall is $\frac{56 L α ∆ T k}{11}$

The elongation in spring having spring constant 8k is $\frac{7 L α ∆ T k}{11}$

The compression in all the three spring are different.

answer is D.

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Detailed Solution

Due to rise in temperature elongation in three rods are given by

\vartriangle {L_A} = L \times 2\alpha \times \vartriangle T\

\vartriangle {L_B} = L\alpha \vartriangle T\

\vartriangle {L_C} = 4L\alpha \vartriangle T\

Let the compressions in spring constant k, 8 k and 4 k be x₁,x₂ and x₃and, respectively

Then,{x_1} + {x_2} + {x_3} = \Delta {L_A} + \Delta {L_B} + \Delta {L_C} = 7L\alpha \Delta T\

From\,\,equilibrium\,\,equation\,\,k{x_1} = 8k{x_2} = 4k{x_3} = \beta \

sovling\,\,the\,\,equation,we\,\,get\,\,\beta = \frac{{56L\alpha \Delta T}}{{11}} \times k\,\,\,\,\,\,\,so,\,\,{x_2} = \frac{{7L\alpha \Delta T}}{{11}}\

Force exerted by C on the walls is β

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