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

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

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

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

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

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Solution

The correct option is A
$F = A t \frac{l_{1} α_{1} + l_{2} α_{2}}{\frac{l_{1}}{E_{1}} + \frac{l_{2}}{E_{2}}}$

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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Similar questions

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

Q. Two rods of different materials having coefficients of linear expansion

α_{1}

and

α_{2}

and Young's moduli,

Y_{1}

and

Y_{2}

, respectively, are fixed between two rigid massive walls. The rods are heated such that they undergo the same increase in temperature. There is no bending of rods. If

α_{1} / α_{2} = 2 / 3

, then the thermal stresses developed in the two rods are equal, provided

Y_{1} / Y_{2}

is equal to

Two rods of different materials, having coefficients of linear expansion $α_{1}$ and $α_{2}$ and Young's moduli $Y_{1}$ and $Y_{2}$ respectively, are fixed between two rigid walls. The rods are heated to the same temperature. There is no bending of the rods. If $α_{1} : α_{2} = 2 : 3$ , the thermal stress developed in the two rods will be equal provided $Y_{1} : Y_{2}$ is equal to