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Thermal Expansion

Two rods havi...

Question

Two rods having length $l_{1}$ and $l_{2}$ made of materials with the linear expansion coefficient $a_{1}$ and $a_{2}$ were soldered together. The equivalent coefficients of linear expansion for the obtained rod:

A

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

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B

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

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C

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

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D

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

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Solution

The correct option is B $\frac{l_{1} a_{1} + l_{2} a_{2}}{l_{1} + l_{2}}$
STEP 1: The coefficient of the linear expansion of first rod of length $L_{1}$ is:
$α 1 = \frac{Δ I 1}{I_{1} \times Δ t}$
$Δ I 1 = α 1 I_{1} Δ t . . . . . (1)$
STEP 2: The coefficient of the linear expansion of second rod of length $I_{2}$ is:
$α 2 = \frac{Δ I 2}{I_{2} \times Δ t}$
$Δ I 2 = α 2 I_{2} Δ t . . . . . . . (2)$
STEP 3: Now, if a single rod of linear expansion $α$ is taken instead of two rods.
$α = \frac{Δ I 1 + Δ I 2}{(I 1 + I 2) \times Δ t} . . . . . . (3)$
STEP 4: Putting the values of $Δ I 1 a n d Δ I 2$ in equation 3, we get;
$α = \frac{α 1 I 1 Δ t + α 2 I 2 Δ t}{(I 1 + I 2) Δ t}$
$α = \frac{I 1 α 1 + I 2 α 2}{I 1 + I 2}$
Hence, the correct option is (c) $\frac{I 1 α 1 + I 2 α 2}{I 1 + I 2}$

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