First slide

Thermal expansion

Question

A thin brass sheet at 10°C and a thin steel sheet at 20°C have the same surface area. The common temperature at which both would have the same area is (coefficient of linear expansion for brass and steel are 19 x 10^-6/°C and 11 x 10^-6/°C respectively)

Moderate

A

-3.75°C
B

-2.75°C
C

2.75°C
D

3.75°C

Solution

The area of brass sheet at 10°C = a₁ b₁
Area of steel sheet at 20°C = a₂b₂
Given that $a_{1} b_{1} = a_{2} b_{2} . . . . (1)$
Let the common temperature be T. Then at temperature T, the lengths of the two sheets are respectively
$a_{1} (1 + α_{B} T) and a_{2} (1 + α_{S} T)$
Now, $a_{1} (1 + α_{B} T) \times b_{1} (1 + α_{B} T)$
$= a_{2} (1 + α_{S} T) \times b_{2} (1 + α_{S} T)$
or ${(1 + α_{B} T)}^{2} = {(1 + α_{S} T)}^{2} ∵ (a_{1} b_{1} = a_{2} b_{2})$
Substituting the values of $α_{B} and α_{S}$ and solving we get
$T = - 3 \cdot 75^{\circ} C$

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