What should be lengths of steel and copper rod, so that the length of steel rod is $5 c m$ longer than the copper rod at all temperatures? ( $α$ of copper $- 1.7 \times 10^{- 5}^{o} C^{- 1}$ and $α$ for steel $= 1.1 \times 10^{- 5}^{o} C^{- 1}$ ).

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Solution

Let $α_{1}$ and $α_{2}$ be the coefficients of linear expansion of steel and copper respectively. Suppose that the steel rod is of length $l_{1}$ and cooper rod of length $l_{2}$ so that all temperatures, the steel rod is 5 cm loner than the copper rod i.e.
$l_{1} - l_{2} = 5 c m . . . . . (1)$
It will happen so, if on increase of temperature say by $θ^{o} C$ , the increase in length of the two rods is same i.e.
$Δ l_{1} = Δ l_{2}$
$l_{1} α_{1} θ = l_{2} α_{2} θ$
$l_{2} = \frac{α_{1}}{α_{2}} l_{1}$
Here, $α_{1} = 1.1 \times 10^{- 5} o C^{- 1}$ ,
$α_{2} = 1.7 \times 10^{- 5} C^{- 1}$
$∴ l_{2} = \frac{1.1 \times 10^{- 5}}{1.7 \times 10^{- 5}} l_{1} = \frac{11}{17} l_{1}$
In the equation, (1) substituting for $l_{2}$ , we have
$l_{1} - \frac{11}{17} l_{1} = 5$
$l_{1} = \frac{85}{6} = 14.17 c m$
Also, $l_{2} = l_{1} - 5 = 14.17 - 5 = 9.17 c m$ .

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Similar questions

Q. What should be lengths of steel and copper rod, so that the length of steel rod is

5 c m

longer than the copper rod at all temperature?

(α

for copper

= 1.7 \times 10^{- 5}^{\circ} C^{- 1}

and

α

for steel

= 1.1 \times 10^{- 5}^{\circ} C^{- 1})

The lengths of steel and copper rods are so that the length of the steel rod is 5 cm longer than that of the copper rod at all temperatures, then length of each rod, are ( $α$ for copper = $1.7 \times 10^{- 5} /^{\circ} C$ and $α$ for steel = $1.1 \times 10^{- 5} /^{\circ} C$ ) -