We are accustomed to measure the pressure of the air as it changes in varying weather by means of the barometer and hence we call the changes in the swing of the pendulum due to varying air pressure the “barometric error.” The barometric error of pendulums is only considered in the very finest of clocks for astronomical observatories, master clocks for watch factories, etc., but the resistance of the air is closely considered when we come to shape our bob. This is why bobs are either double-convex or cylindrical in shape, as these two forms offer the least resistance to the air and (which is more important) they offer equal resistance on both sides of the center of the bob and thus tend to keep the pendulum swinging in a straight line back and forth.
Fig. 4. A, arc of circle. B, cycloid path of pendulum, exaggerated.
The Circular Error.—As the pendulum swings over a greater arc it will occupy more time in doing it and thus the rate of the clock will be affected, if the barometric changes are very great. This is called the circular error. In ancient times, when it was customary to make pendulums vibrate at least fifteen degrees, this error was of importance and clock makers tried to make the bob take a cycloidal path, as is shown in [Fig. 4], greatly exaggerated. This was accomplished by suspending the pendulum by a cord which swung between cycloidal checks, but it created so much friction that it was abandoned in favor of the spring as used to-day. It has since been proved that the long and short arcs of the pendulum’s vibration are practically isochronous (with a spring of proper length and thickness) up to about six degrees of arc (three degrees each side of zero on the degree plate at the foot of the pendulum) and hence small variations of power in spring operated clocks and also the barometric error are taken care of, except for greatly increased variations of power, or for too great arcs of vibration. Here we see the reasons for and the amount of swing we can properly give to our pendulum.
Temperature Error.—The temperature error is the greatest which we shall have to consider. It is this which makes the compound pendulum necessary for accurate time, and we shall consequently give it a great amount of space, as the methods of overcoming it should be fully understood.
Expansion of Metals.—The materials commonly used in making pendulums are wood (deal, pine and mahogany), steel, cast iron, zinc, brass and mercury. Wood expands .0004 of its length between 32° and 212° F.; lead, .0028; steel, .0011; mercury, .0180; zinc, .0028; cast iron, .0011; brass, .0020. Now the length of a seconds pendulum, by our tables (3600 beats per hour) is 0.9939 meter; if the rod is brass it will lengthen .002 with such a range of temperature. As this is practically two-thousandths of a meter, this is a gain of two millimeters, which would produce a variation of one minute and forty seconds every twenty-four hours; consequently a brass rod would be a very bad one.
If we take two of these materials, with as wide a difference in expansion ratios as possible, and use the least variable for the rod and the other for the bob, supporting it at the bottom, we can make the expansion of the rod counterbalance the expansion of the bob and thus keep the effective length of our pendulum constant, or nearly so. This is the theory of the compensating pendulum.