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Difficulty: MediumThermal Expansion of Solids (Linear, Area, and Volume Expansivity)

An iron ring has an internal cross-sectional area of 0.50 m20.50\text{ m}^2 at 30∘C30^\circ\text{C}. If the linear expansivity of iron is 1.2×10−5 K−11.2 \times 10^{-5}\text{ K}^{-1}, what is the increase in its internal cross-sectional area when heated to 130∘C130^\circ\text{C}?

  1. 1.2×10−3 m21.2 \times 10^{-3}\text{ m}^2Answer
  2. B
    6.0×10−4 m26.0 \times 10^{-4}\text{ m}^2
  3. C
    1.8×10−3 m21.8 \times 10^{-3}\text{ m}^2
  4. D
    2.4×10−3 m22.4 \times 10^{-3}\text{ m}^2

Answer

The increase in the internal cross-sectional area of the ring is 1.2×10−3 m21.2 \times 10^{-3}\text{ m}^2.
For two-dimensional (area) expansion of solids, the area expansivity β\beta is equal to twice the linear expansivity (2α2\alpha). Given α=1.2×10−5 K−1\alpha = 1.2 \times 10^{-5}\text{ K}^{-1}, β=2.4×10−5 K−1\beta = 2.4 \times 10^{-5}\text{ K}^{-1}. Multiplying by the initial area (0.50 m20.50\text{ m}^2) and temperature rise (100 K100\text{ K}) yields an area increase of 1.2×10−3 m21.2 \times 10^{-3}\text{ m}^2.

Step-by-Step Solution

1
Determine the temperature change (ΔT\Delta T) and the superficial expansivity (β\beta).
ΔT=130∘C−30∘C=100 K\Delta T = 130^\circ\text{C} - 30^\circ\text{C} = 100\text{ K}, and β=2α=2×(1.2×10−5 K−1)=2.4×10−5 K−1\beta = 2\alpha = 2 \times (1.2 \times 10^{-5}\text{ K}^{-1}) = 2.4 \times 10^{-5}\text{ K}^{-1}.
Area expansion depends on superficial expansivity, which is twice the linear expansivity for an isotropic solid.
2
Calculate the increase in area (ΔA\Delta A) using the area expansion formula.
ΔA=A0βΔT=0.50 m2×(2.4×10−5 K−1)×100 K=1.2×10−3 m2\Delta A = A_0 \beta \Delta T = 0.50\text{ m}^2 \times (2.4 \times 10^{-5}\text{ K}^{-1}) \times 100\text{ K} = 1.2 \times 10^{-3}\text{ m}^2.
The fractional change in area is directly proportional to initial area, superficial expansivity, and temperature change.

Key Concept

Relationship between linear expansivity (α\alpha) and superficial expansivity (β=2α\beta = 2\alpha) in thermal expansion of area.
Estimated Time:1m 30s
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Topics

Thermal Expansion of Solids (Linear, Area, and Volume Expansivity)

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