A pendulum clock keeps accurate time at sea level but is transported to the top of a mountain where the local gravitational field is 0.30 % weaker than at sea level. The length of the pendulum is not adjusted.
(a) Derive an expression for the fractional change in the pendulum’s period, TΔT, in terms of the fractional change in the local gravitational field, gΔg, assuming small-angle motion.
(b) Using your expression, determine by how many seconds the clock will gain or lose in one 24-hour day at the mountain location. State clearly whether the clock runs fast or slow.
(c) To restore accurate timekeeping, the clockmaker decides to adjust only the length of the pendulum. Determine the percentage change in length required to compensate for the change in g, showing all reasoning.
(d) Discuss one practical consideration that might limit how precisely the pendulum length can be adjusted in an actual clock mechanism.