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Thermal Expansion

Two uniform m...

Question

Two uniform metal rods of length $l_{1}$ and $l_{2}$ and the linear coefficients of expansion $α_{1}$ and $α_{2}$ respectively are connected to form a single rod of length $l_{1} + l_{2}$ . When the temperature of the combined rod is raised by $t^{o} C$ , the length of each rod increases by the same amount. Then $\frac{α_{2}}{α_{1} + α_{2}}$ is :

A

\frac{l_{1}}{l_{1} + l_{2}}

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B

\frac{l_{1} + l_{2}}{l_{1}}

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C

\frac{l_{2}}{l_{1} + l_{2}}

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D

\frac{l_{1} + l_{2}}{l_{2}}

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Solution

The correct option is A $\frac{l_{1}}{l_{1} + l_{2}}$
We know, $d l = l α d t$ , where, $d l =$ change in length, $l =$ original length and $d t =$ change in temperature
Since change in temperature is the same,
we have, $l_{1} α_{1} = l_{2} α_{2}$
Or $\frac{α_{1}}{α_{2}} = \frac{l_{2}}{l_{1}}$
Adding 1 to both sides, we get, $\frac{α_{1} + α_{2}}{α_{2}} = \frac{l_{1} + l_{2}}{l_{1}}$
Or, $\frac{α_{2}}{α_{1} + α_{2}} = \frac{l_{1}}{l_{1} + l_{2}}$

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