Two rods made of different materials are placed between massive walls (see fig.). The cross-section of the rods is A and their respective lengths are $l_{1}$ and $l_{2}$ . The rods are heated through $t^{o}$ C. Find the force with which the rods act on each other if their coefficients of expansion are $α_{1}$ and $α_{2}$ and their Young's moduli are $E_{1}$ and $E_{2}$ . Neglect the deformation of the walls

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Solution

The correct option is A

Since there is no shift in the junction, the sum net elongation of both the rods should be zero.

The net elongation in each rod is equal to (elongation due to heating - contraction due to the force)

Therefore, if F is the force acting at the junction, we have

$Δ l_{1} = l_{1} α_{1} t - \frac{F l_{1}}{E_{1} A}$

Similarly

$Δ l_{2} = l_{2} α_{2} t - \frac{F l_{2}}{E_{2} A}$

The two elongations should add up to zero.

$l_{1} α_{1} t - \frac{F l_{1}}{E_{1} A} + l_{2} α_{2} t - \frac{F l_{2}}{E_{2} A} = 0$

$\Rightarrow (l_{1} α_{1} + l_{2} α_{2}) t = \frac{F}{A (\frac{l_{1}}{E_{1}} + \frac{l_{2}}{E_{2}})}$

$\Rightarrow F = A t \frac{l_{1} α_{1} + l_{2} α_{2}}{\frac{l_{1}}{E_{1}} + \frac{l_{2}}{E_{2}}}$

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Two rods made of different materials are placed between massive walls (see fig.). The cross-section of the rods is A and their respective lengths are $l_{1}$ and $l_{2}$ . The rods are heated through $t^{o}$ C. Find the force with which the rods act on each other if their coefficients of expansion are $α_{1}$ and $α_{2}$ and their Young's moduli are $E_{1}$ and $E_{2}$ . Neglect the deformation of the walls