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Question:

A pendulum clock loses 12s a day if the temperature is 40oC and gains 4s a day if the temperature is 20oC. The temperature at which the clock will show correct time, and the coefficient of linear expansion (α) of the metal of the pendulum shaft are respectively.

25oC; α=1.85×10⁻⁵/oC

60oC; α=1.85×10⁻⁸/oC

30oC; α=1.85×10⁻⁷/oC

55oC; α=1.85×10⁻⁶/oC

Solution:

The time period of a pendulum is given by, t = 2π√(l/g)
Δt = 2π√(g) * (Δl/2√l)
Δl = αΔθ
∴ Δt = π√(g/l) αΔT
∴ Δt₁/Δt₂ = ΔT₁/ΔT₂
∴ 12/4 = (40 - T)/(T - 20)
∴ 3T - 60 = 40 - T
∴ 4T = 100
∴ T = 25°C