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The Question

Between two rigid walls separated by a distance $l_1+l_2$, two rods of equal cross-section area $A$, and length, coefficient of linear expansion and Young's modulus $l_1, \alpha_1, Y_1$ and $l_2,\alpha_2,Y_2$. Suppose the temperature of the entire system is raised by $T$ kelvins. What is the force the two rods exert on each other?
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Concepts Involved

  1. Tendency to expand of the two rods
  2. Stress generated due to the constraint on total length which limits their expansion
  3. Different increases in length of both the rods owing to different young's modulus and original lengths

My efforts

Let at the increased temperature, the length of Rod 1 be $l_1+x$ and that of rod two be $l_2-x$. The "natural length" of both the rods at that temperature is $l_1(1+\alpha_1T)$ and $l_2(1+\alpha_2T)$. Therefore the longitudnal stress developed in each is:-

$\delta l_1=l_1\alpha_1T-x$ and $\delta l_2=l_2\alpha_2T+x$.

For the rods in equilibrium, the forces they exert on each other need to be equal. Therefore, using $F=\delta l/l\times AY$, and equating their forces we get:-

$$\frac{Y_1l_2(l_1\alpha_1T-x)}{Y_2l_1(l_2\alpha_2T+x)}=1$$ This can be solved for $x$ which on substituting in $F=(l_1\alpha_1T-x)/l_1\times AY_1$, we should get the force. But no matter how much I rearrange this equation, it does not match with the given answer. Is there a fault in my arguments/equations or calculations? Sorry if there is trivial mistake.

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  • $\begingroup$ All you need to do is find out how much the rods would have expanded without anything holding them back, then consider this as the total compression, and find the stress. This site is not meant as a place for people to check your homework. Ask a friend or teacher. $\endgroup$ – Pranav Hosangadi Feb 12 '14 at 22:25
  • $\begingroup$ @PranavHosangadi I do not want anyone to check my calculations. All I want is to know if my line of thought was correct and my arguments sound. By "fault in my calculations", I only meant the fault evident to the reader which I might have missed, not something you need to look for. Anyway, tell me if this is too offtopic. $\endgroup$ – Satwik Pasani Feb 13 '14 at 16:26

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