An iron rod measuring 50 cm is connected at one end to an aluminum rod measuring 100 cm, both at a temperature of 20°C. Calculate the total length of…

Question

An iron rod measuring 50 cm is connected at one end to an aluminum rod measuring 100 cm, both at a temperature of 20°C. Calculate the total length of the composite system when heated to 100°C and find its average coefficient of linear expansion. The coefficients of linear expansion for iron and aluminum are 12 × 10⁻⁶ K⁻¹ and 24 × 10⁻⁶ K⁻¹ respectively.

Answer 1

An iron rod measuring 50 cm is connected at one end to an aluminum rod measuring 100 cm, both at a temperature of 20°C. Calculate the total length of the composite system when heated to 100°C and find its average coefficient of linear expansion. The coefficients of linear expansion for iron and aluminum are 12 × 10⁻⁶ K⁻¹ and 24 × 10⁻⁶ K⁻¹ respectively.

150.24 cm, 20 × 10⁻⁵ K⁻¹
150.24 cm, 20 × 10⁻⁶ K⁻¹ ✓
150.32 cm, 18 × 10⁻⁶ K⁻¹
150.32 cm, 18 × 10⁻⁵ K⁻¹

Explanation

The temperature rise is 100 − 20 = 80 K. Use ΔL = L₀αΔθ for each rod.

Iron: ΔL = 50 × 12 × 10⁻⁶ × 80 = 0.048 cm.
Aluminium: ΔL = 100 × 24 × 10⁻⁶ × 80 = 0.192 cm.

Total length = 150 + 0.048 + 0.192 = 150.24 cm.

Average α = total expansion ÷ (original length × Δθ) = 0.24 ÷ (150 × 80) = 2.0 × 10⁻⁵ = 20 × 10⁻⁶ K⁻¹.

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