A clock with a brass pendulum gains 1 minute per day in winter at a temperature of 20 °C. Another clock with a steel pendulum loses 1 minute per day in summer at a temperature 40 °C. The temperature at which the two clocks show correct time is (αBrass=18×10⁻⁶/^oC; αsteel=12×10⁻⁶/^oC)

Explanation

The time gain or loss per day (Δt) due to temperature change can be approximated by:

Δt = (1/2) α ΔT × 86400 seconds

Brass Pendulum

• Coefficient of Linear Expansion: α_Brass = 18 × 10⁻⁶ °C⁻¹
• Time Gain: 1 minute/day = 60 seconds/day
• Equation: 60 = (1/2) × 18 × 10⁻⁶ × (T − 20) × 86400
• Solving for T: T ≈ 20.077 °C

Steel Pendulum

• Coefficient of Linear Expansion: α_Steel = 12 × 10⁻⁶ °C⁻¹
• Time Loss: 1 minute/day = 60 seconds/day
• Equation: 60 = (1/2) × 12 × 10⁻⁶ × (40 − T) × 86400
• Solving for T: T ≈ 39.885 °C