Q&A: Pendulums and Grandfather Clocks

Question: We recently moved from Denver to Sacramento. After a couple months we noticed our grandfather clock was running fast. It used to keep excellent time, even compared to electronic devices. A friend told me it had to do with the “difference in gravity” between my old and new home, but he couldn’t explain why. I’m hoping you can help me out. The clock has been in my family for three generations and it came with no owners manual, so I may need to call a repairman. Thanks! — CG, Sacramento, CA

Answer: Your friend is correct. Pendulum clocks are powered by gravity, and gravity gets weaker the higher your altitude. Gravity is about 0.05% less in Denver than Sacramento. That means your clock will run faster by 0.05% in your new home. Doesn’t sound like much, but it works out to around 43 seconds/day.

The best pendulum clocks can be accurate to within one second per day, or around 0.0007%, so you’ll definitely notice an error of 0.05%.

Grandfather clocks use what’s called a seconds pendulum, which means it takes one second to swing from left to right, and one second to swing back again. That’s the way I set it up in the animation. That means T = 2s in the pendulum formula (T = 2π√(L/g), which if solved for L gives a pendulum length of 0.496m. Perhaps surprisingly, the mass of the bob is irrelevant as all masses are accelerated equally by gravity.

The animation shows how a difference in the length of the pendulum affects its swing period. The long pendulum has the usual 2-second period of a grandfather clock, and “tocks” once each 1.0 seconds. The shorter pendulum is half as long and “ticks” once each 0.707 seconds. So the length of the pendulum, as well as gravity, affects how fast it oscillates.

[Note: The animation ends after (approximately) 9.5 cycles of the longer pendulum. It looks like the two pendulums have re-synchronized at that point, but it’s only approximate. The longer pendulum has a period √2 times that of the shorter, and since √2 is an irrational number they can never exactly sync. At the end of the animation the number of long pendulum cycles is actually 9.458, and that’s pretty close visually. The shorter pendulum is at 13.585 cycles, so they appear to be synced.]

The length of the pendulum is measured from the pivot point to its center of mass (COM). In an “ideal” pendulum the COM is essentially the center of the bob. In a real grandfather clock the COM is higher because of the mass of the arm:

So here’s what  you need to do, CG. Somewhere in the arm/bob assembly there’s a set screw or other type of adjustment hardware. It allows you to shorten or lengthen the pendulum to compensate for changes in gravity. In your case, you’ll need to lengthen the pendulum slightly to get it back in sync with the time of day.

For a first approximation, set the pendulum length to 0.05% longer than its original setting. That should get you pretty close. Observe the accuracy over a period of a few days to see what error remains, then make a smaller additional adjustment depending on whether it’s running fast or slow. If it’s running fast, you need to lengthen the pendulum and vice versa. Repeat as needed until you have it synced to within a few seconds a day. Then you’ll be good until next time you move.

Pendulum clocks were invented in 1656 by Dutch scientist and inventor Christiaan Huygens, and patented the following year. Since that time, the machining of accurate gears has improved and the clocks available today are sufficiently accurate for everyday use. They’re becoming less common because they do need “winding” and occasional lubrication, and they do take up some space, but many people still enjoy them for their artistic craftsmanship and historical significance.

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