Pendulum Clock
A pendulum clock counts swings. If the pendulum gets a little longer, warms up in summer or moves to a place with different gravity, every swing takes a little longer or shorter — and the clock gains or loses time. How much a day, and how do you fix it?
Why Pendulum Clocks Drift
The gears of a pendulum clock move the hands on by a fixed amount for every swing. The clock is only right if each swing takes exactly as long as when it was set. A longer pendulum swings more slowly, so the clock loses time; stronger gravity makes it swing faster, so it gains.
A metal rod grows a little in summer: steel by about 12 millionths per degree. That is tiny, but a day has 86 400 seconds, so it adds up to seconds a day. Precision clocks therefore used rods of Invar, which hardly expands, and a nut under the bob to adjust the length.
Gain or Loss per Day
s/dayA Seconds Pendulum, 10 °C Warmer
L = 0.994 m| Rod | α (10⁻⁶/K) | Grows by | Loses per day |
|---|---|---|---|
| Aluminium | 23 | 0.23 mm | 9.9 s |
| Copper | 17 | 0.17 mm | 7.3 s |
| Steel | 12 | 0.12 mm | 5.2 s |
| Invar | 1.2 | 0.012 mm | 0.52 s |
Worked Techniques
For small changes the period changes by half the relative change in length, and by minus half the relative change in gravity.
The rod grows in proportion to its length and to the temperature rise. A clock that is right in winter runs slow in summer.
To get the original period back at the new gravity, the pendulum must have this length. The difference is how far to turn the nut under the bob.
Sources & Verification
- D. Halliday, R. Resnick & J. Walker, Fundamentals of Physics, 10th ed., Wiley, 2014 — Ch. 15 (Oscillations) and Ch. 18 (Temperature, Heat, and the First Law of Thermodynamics).