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According to my Halliday-Resnick, the period of a pendulum clock can be kept constant with temperature by attaching vertical tubes of mercury to the bottom of the pendulum. How does this work?

My guesses:

  • Air resistance somehow changes with the temperature of the air, and this is offset by mercury evaporating (thus shifting the center-mass)
  • Air resistance somehow changes with the temperature of the air, and this is offset by thermal expansion of the tube (thus shifting the center-mass)

Any help is greatly appreciated.

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  • $\begingroup$ Oddly, some woods have a very low thermal expansion along the grain : 3-4ppm/K, an order of magnitude smaller than brass. Austrian wall clocks often have a wood pendulum with a really large brass bob (diameter 20% of the pendulum length) supported only at the bottom. Simple, much cheaper, and surprisingly accurate. $\endgroup$
    – user_1818839
    Jul 30, 2021 at 14:31
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    $\begingroup$ You don't need to use hazardous materials like mercury, or unpredictable natural materials like wood. The "gridiron" pendulum was invented almost 300 years ago by the John Harrison (famous for designing the first clocks which could keep accurate time on board ships, to determine longitude). See en.wikipedia.org/wiki/Gridiron_pendulum $\endgroup$
    – alephzero
    Jul 30, 2021 at 18:32

2 Answers 2

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It's actually an ingenious, but relatively simple bit of physics and engineering.

It works by compensating for the linear thermal expansion of the pendulum rod, utilizing the thermal expansion of the mercury but in the opposite direction and thus preserving the position of the center of gravity.

Note that the period of the (compound) pendulum is given by $$T=2\pi\sqrt{\frac{I}{mgL}}$$ where $L$ is the distance to the center of mass of the rod from its pivot point, and $I$ is its moment of inertia also about the pivot point. This means the period varies if this distance varies.

With the adding of mercury tubes, as the temperature rises for example, this will cause expansion in the pendulum rod downward $^1$. But at the same time, this temperature rise causes the expansion of the mercury in the tube that moves the mercury upward.

The exact opposite effect happens for temperature drops. That is, the pendulum arm decreases its length, so that $L$ decreases, while the mercury in the tube decreases its height in the other direction.

The net result is that the arm keeps a constant location of the center of gravity, and therefore keeps a constant period.

$^1$ The rod is fixed to a pivot at the opposite end, so even though the whole thing may expand, it's the change in the distance $L$ from the pivot that's important.

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  • $\begingroup$ Thanks! That makes a lot of sense. $\endgroup$
    – DarthBaguette
    Jul 30, 2021 at 3:54
  • $\begingroup$ No worries and good luck with your studies. Cheers. $\endgroup$
    – joseph h
    Jul 30, 2021 at 3:56
  • $\begingroup$ You might be explicit that the thermal expansion coefficient of Hg is larger than that of the pendulum rod, so a lesser length can be used to cancel any vertical movement of the total pendulum COG. $\endgroup$
    – Neil_UK
    Jul 31, 2021 at 5:15
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My guess is that the effect for which the mercury compensates is due to the thermal expansion of the pendulum arm (not air resistance). The center of mass of mercury in tall tubes could rise significantly with temperature, keeping the period constant.

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