The Gravity Pendulum and its Horological Quirks
by Walt Arnstein
On one occasion or another, most of us have watched a grandfather clock or other timepiece based on a simple pendulum and marveled at this device as it swung back and forth majestically, often seeming hardly to move, yet maintaining remarkable accuracy day after day. The elegance and simplicity of this oscillator suggests that there is little to go wrong with it. After all, there is no hairspring to magnetize, rust, or tangle with regulator pins, no balance wheel to expand or contract with temperature or unbalance from one orientation to another. The oscillatory energy and the forces that shuttle it between its potential and kinetic forms are provided by gravity. On some of the clocks, even the energy that moves the main gear train originates with gravity, by means of weights and pulleys. What can be simpler?
In actuality, a gravity pendulum is a fairly complex mechanism, depending on a number of variables for which the designer and user must always be ready to adjust. In simplest terms, the pendulum's period of oscillation is a function of:
- Its effective length
- The acceleration of gravity
- The swing amplitude.
So what is the problem? To begin with, the effective length varies with temperature. In simplest terms, the effective length of a pendulum is the distance from its pivot to its center of oscillation, also known as center of percussion. This magical spot is instinctively known to any baseball player as the 'sweet spot on the bat that can send a ball into the stands without so much as stinging the batter's wrists. On a pendulum, it is the spot at which a point mass suspended from the pivot by a weightless string would have the same period as the pendulum. The overall mass doesn't matter, but the location of the spot does.
Well, temperature plays havoc with a typical pendulum particularly one several feet long. To offset this effect, large tubes of mercury are often located in the pendulum's weight assembly, with the surface of the mercury free to move upward as it expands with temperature, thus shortening the effective length of the pendulum, hopefully just enough to compensate for the expansion of the pendulum's shaft.
Then, there is gravity. The simplified (linearized) equation for the period T of a simple pendulum of effective length L is
T = 2Pi Sqrt(L/g)
where g is the local acceleration of gravity. Most people intuitively treat gravity as a constant, about 32.2 ft/sec2 or 9.82 m/sec2 , which is the value of gravity at Earth's sea level. In actuality, the value of gravity above the surface of the Earth varies inversely as the square of the location's distance from the center of the Earth. Below Earth's surface -- say, deep beneath the ocean, the value of gravity varies directly in proportion to the location's distance from the center of the Earth. In other words, gravity on the surface is kind of a maximum. So, believe it or not, a pendulum clock will slow down whether you strap it to your back and tote it to the top of Mount Everest or dive with it in a bathyscaph to the bottom of the Philippine Trench.
How much? A pendulum clock that runs perfectly in your beach-front living room on Cape Cod, say, will lose almost exactly 2 minutes per day on top of Mount Everest due to changes in the acceleration of gravity. In the Philippine Trench, it will lose about 76 seconds per day. But we don't need to go to these extremes to become aware of the pendulum's sensitivity to altitude. Suppose, for example that a work transfer takes you to Santa Fe, NM, altitude 8000 feet. Assuming your grandfather clock, which ran perfectly on Cape Cod, survives Bekins' gentle ministrations, you will probably be shocked to find it inexplicably losing about 33 seconds per day in Santa Fe! And to think of all the money you paid that Hyannis clock shop to set that Ridgeway up properly.
Obviously, a pendulum clock would not be a very useful timekeeper aboard an airliner. Maybe Lufthansa and TWA should continue to use quartz or spring-powered balance wheel clocks. On the other hand, if we can shrink its size, it could make a pretty good altimeter!
Then, there is the matter of swing amplitude. The pendulum is not a linear device. That is, there is not a direct linear proportionality between a pendulum's angular displacement from the vertical and the gravity's restoring torque -- the relation is trigonometric. So, the oscillatory frequency is not entirely independent of the amplitude and the effect is more pronounced than it is with a balance wheel and hairspring (which is at least theoretically independent of amplitude). In the weight-driven clock, the amplitude is at least constant regardless of the location of the weights, i.e., power reserve. (Even so, the amplitude of fine weight-driven pendulum clocks is kept very small to maintain approximate linearity). But in a spring-driven pendulum clock, like some wall-mounted regulators, the state of wind of the spring affects the amplitude of the pendulum and hence, to a varying degree, the clock's rate.
What about transporting a clock into outer space? Alas, its period would vary |with the individual planet, due to changes in the acceleration of gravity. This will depend on the mass of the planet in question and your distance from its center of mass. On the surface of Jupiter, for example, local g is about 6 of our g's. (Also, since the surface is liquid only and is constantly astir with colossal storms, the feasibility of setting up a pendulum clock might present a few added logistical problems). We won't even think about the likelihood of the weight cables or pulleys snapping under the five-fold increase in the tug of the weights.
In view of the above considerations, I am sure you will appreciate the simplicity of the wonderful balance wheel mechanism on your wrist. Just think: It's relatively independent of gravity and it does pretty well in the presence of temperature changes. Those 'simple' pendulums, on the other hand, are not all that simple!