Coefficient of linear expansion of brass and steel rods are and . Lengths of brass and steel rods are and respectively. If is maintained same at all temperatures, which one of the following relations holds good? [2016]

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Solution

Q.
Coefficient of linear expansion of brass and steel rods are $α_{1}$ and $α_{2}$ . Lengths of brass and steel rods are $l_{1}$ and $l_{2}$ respectively. If $(l_{2} - l_{1})$ is maintained same at all temperatures, which one of the following relations holds good [2016, 1999]

1 $α_{1}^{2} l_{2} = α_{2}^{2} l_{1}$

2 $α_{1} l_{1} = α_{2} l_{2}$

3 $α_{1} l_{2} = α_{2} l_{1}$

4 $α_{1} l_{2}^{2} = α_{2} l_{1}^{2}$

Ans.
(2)

Linear expansion of brass = $α_{1}$

Linear expansion of steel = $α_{2}$

Length of brass rod = $l_{1},$ Length of steel rod = $l_{2}$

On increasing the temperature of the rods by $Δ T$ , new lengths would be

$l_{1}' = l_{1} (1 + α_{1} Δ T)$ ...(i)

$l_{2}' = l_{2} (1 + α_{2} Δ T)$ ...(ii)

Subtracting eqn. (i) from eqn. (ii), we get

$l_{2}' - l_{1}' = (l_{2} - l_{1}) + (l_{2} α_{2} - l_{1} α_{1}) Δ T$

According to the question,

$l_{2}' - l_{1}' = l_{2} - l_{1}$ (for all temperatures)

$∴$ $l_{2} α_{2} - l_{1} α_{1} = 0 or l_{1} α_{1} = l_{2} α_{2}$

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