Question

The pendulum in a grandfather clock is made of brass and keeps perfect time at 20 ∘C. Do the clock go faster or slower if they is kept at 25 ∘C? (Assume the frequency dependence on length for a simple pendulum applies.) How much time is lost in a year?

Answer 1

Take the linear expansion coefficient for brass as α = 19x10⁻⁶ /°C
Length of the pendulum at 25°C is L = L₀(1 + αΔT)
= 0.248(1+ (19x10⁻⁶ )(25-20))
= 0.24802356m
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From 20°C to 25°C, the length has increased a factor of
0.24802356 / 0.248 = 1.000095
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The period of a pendulm is given by T = 2π√(L/g), so the period is proportional to √L.

Since the length has increased by a factor 1.000095, the period has increased by a factor √1.000095 = 1.0000475

The pendulum will therefore swinging slower than at 20°C and sothe clock will run slow by a factor f = 1.0000475

Over an elapsed time E, the clock will register a time of E/f instead of E. So the discrepancy will be E - E/f = E(1 - 1/f)

The number of seconds in 1 year = 365 x 24 x 60 x 60 = 31536000.
So the lost time is 31536000(1 - 1/1.0000475) = 1498s (about 25 minutes).