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Thermal Expansion

A rod of leng...

Question

A rod of length $20 cm$ is made of metal. It expands by $0.075 cm$ when its temperature is raised from $0^{\circ} C$ to $100^{\circ} C$ . Another rod of a different metal B having the same length expands by $0.045 cm$ for the same change in temperature. A third rod of the same length is composed of two parts one of metal A and the other of metal B. Now, the rod expands by $0.06 cm$ for the same change in temperature. The portion made of metal A has the length (in $cm$ )

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Solution

For rod 1 :-
$L_{1} = 20 cm$
$Δ L_{1} = 0.075 cm$
$Δ L_{1} = L_{1} α_{1} Δ T$
$α_{1} = [\frac{0.075}{20 \times 100}]$
$α_{1} = 37.5 \times 10^{- 6} /^{\circ} C$
For rod 2 :-
$Δ L_{2} = L_{2} α_{2} Δ T$
$0.045 = (20) α_{2} [100]$
$α_{2} = [\frac{0.045}{20 \times 100}]$
$α_{2} = 22.5 \times 10^{- 6} /^{\circ} C$
Now for third rod :-

$Δ L = Δ L_{1} + Δ L_{2}$
$0.06 = L [37.5 \times 10^{- 6}] \times 100 + (20 - L) \times 22.5 \times 10^{- 6} \times 100$
$6 = 0.375 L + 4.5 - 0.225 L$
$1.5 = 0.15 L$
$L = 10 cm$

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