Length of Pendulum.—A pendulum is a falling body and as such is subject to the laws which govern falling bodies. This statement may not be clear at first, as the pendulum generally moves through such a small arc that it does not appear to be falling. Yet if we take a pendulum and raise the ball by swinging it up until the ball is level with the point of suspension, as in Fig. 1, and then let it go, we shall see it fall rapidly until it reaches its lowest point, and then rise until it exhausts the momentum it acquired in falling, when it will again fall and rise again on the other side; this process will be repeated through constantly smaller arcs until the resistance of the air and that of the pendulum spring shall overcome the other forces which operate to keep it in motion and it finally assumes a position of rest at the lowest point (nearest the earth) which the pendulum rod will allow it to assume. When it stops, it will be in line between the center of the earth (center of gravity) and the fixed point from which it is suspended. True, the pendulum bob, when it falls, falls under control of the pendulum rod and has its actions modified by the rod; but it falls just the same, no matter how small its arc of motion may be, and it is this influence of gravity—that force which makes any free body move toward the earth’s center—which keeps the pendulum constantly returning to its lowest point and which governs very largely the time taken in moving. Hence, in estimating the length of a pendulum, we must consider gravity as being the prime mover of our pendulum.

The next forces to consider are mass and weight, which, when put in motion, tend to continue that motion indefinitely unless brought to rest by other forces opposing it. This is known as momentum. A heavy bob will swing longer than a light one, because the momentum stored up during its fall will be greater in proportion to the resistance which it encounters from the air and the suspension spring.

As the length of the rod governs the distance through which our bob is allowed to fall, and also controls the direction of its motion, we must consider this motion. Referring again to Fig. 1, we see that the bob moves along the circumference of a circle, with the rod acting as the radius of that circle; this opens up another series of facts. The circumference of a circle equals 3.1416 times its diameter, and the radius is half the diameter (the radius in this case being the pendulum rod). The areas of circles are proportional to the squares of their diameters and the circumferences are also proportional to their areas. Hence, the lengths of the paths of bobs moving along these circumferences are in proportion to the squares of the lengths of the pendulum rods. This is why a pendulum of half the length will oscillate four times as fast.

Now we will apply these figures to our pendulum. A body falling in vacuo, in London, moves 32.2 feet in one second. This distance has by common consent among mathematicians been designated as g. The circumference of a circle equals 3.1416 times its diameter. This is represented as p. Now, if we call the time t, we shall have the formula:

t=pv(1/g)

Substituting the time, one second, for t, and doing the same with the others, we shall have:

(32.2 ft.)
1=—————=3.2616feet.
(3.1416)²

Turning this into its equivalent in inches by multiplying by 12, we shall have 39.1393 inches as the length of a one-second pendulum at London.

Now, as the force of gravity varies somewhat with its distance from the center of the earth, we shall find the value of g in the above formula varying slightly, and this will give us slightly different lengths of pendulum at different places. These values have been found to be as follows:

Now, taking another look at our formula, we shall see that we may get the length of any pendulum by multiplying p (which is 3.1416) by the square of the time required: To find the length of a pendulum to beat three seconds:

3² = 9.

39.1393×9=352.2537inches =29.3544feet.

A pendulum beating two-thirds of a second, or 90 beats:

(?)²=4/9

39.1393×4
—————=17.3952inches.
9

A pendulum beating half-seconds or 120 beats:

(½)²=¼

39.1393×1
——————=9.7848inches.
4

Center of Oscillation.—Having now briefly considered the basic facts governing the time of oscillation of the pendulum, let us examine it still further. The pendulum shown in Fig. 1 has all its weight in a mass at its end, but we cannot make a pendulum that way to run a clock, because of physical limitations. We shall have to use a rod stiff enough to transmit power from the clock movement to the pendulum bob and that rod will weigh something. If we use a compensated rod, so as to keep it the same length in varying temperature, it may weigh a good deal in proportion to the bob. How will this affect the pendulum?

If we suspend a rod from its upper end and place along-side of it our ideal pendulum, as in Fig. 2, we shall find that they will not vibrate in equal times if they are of equal lengths. Why not? Because when the rod is swinging (being stiff) a part of its weight rests upon the fixed point of suspension and that part of the rod is consequently not entirely subject to the force of gravity. Now, as the time in which our pendulum will swing depends upon the distance of the effective center of its mass from the point of suspension, and as, owing to the difference in construction, the center of mass of one of our pendulums is at the center of its ball, while that of the other is somewhere along the rod, they will naturally swing in different times.

Our other pendulum (the rod) is of the same size all the way up and the center of its effective mass would be the center of its weight (gravity) if it were not for the fact which we stated a moment ago that part of the weight is upheld and rendered ineffective by the fixed support of the pendulum rod, all the while the pendulum is not in a vertical position. If we support the rod in a horizontal position, as in Fig. 3, by holding up the lower end, the point of suspension, A, will support half the weight of the rod; if we hold it at 45 degrees the point of suspension will hold less than half the weight of the rod and more of the rod will be affected by gravity; and so on down until we reach the vertical or up and down position. Thus we see that the force of gravity pulling on our pendulum varies in its effects according to the position of the rod and consequently the effective center of its mass also varies with its position and we can only calculate what this mean (or average) position is by a long series of calculations and then taking an average of these results.

We shall find it simpler to measure the time of swing of the rod which we will do by shortening our ball and cord until it will swing in the same time as the rod. This will be at about two-thirds of the length of the rod, so that the effective length of our rod is about two-thirds of its real length. This effective length, which governs the time of vibration, is called the theoretical length of the pendulum and the point at which it is located is called its center of oscillation. The distance from the center of oscillation to the point of suspension is called the theoretical length of the pendulum and is always the distance which is given in all tables of lengths of pendulums. This length is the one given for two reasons: First, because, it is the timekeeping length, which is what we are after, and second, because, as we have just seen in Fig. 3, the real length of the pendulum increases as more of the weight of the instrument is put into the rod. This explains why the heavy gridiron compensation pendulum beating seconds so common in regulators and which measures from 56 to 60 inches over all, beats in the same time as the wood rod and lead bob measuring 45 inches over all, while one is apparently a third longer than the other.

Table Showing the Length of a Simple Pendulum

That performs in one hour any given number of oscillations, from 1 to 20,000, and the variation in this length that will occasion a difference of 1 minute in 24 hours.

Calculated by E. Gourdin.

Table of the Length of a Simple Pendulum,
(CONTINUED.)

In the foregoing tables all dimensions are given in meters and millimeters. If it is desirable to express them in feet and inches, the necessary conversion can be at once effected in any given case by employing the following conversion table, which will prove of considerable value to the watchmaker for various purposes:

Conversion Table of Inches, Millimeters and French Lines.

Center of Gravity.—The watchmaker is concerned only with the theoretical or timekeeping lengths of pendulums, as his pendulum comes to him ready for use; but the clock maker who has to build the pendulum to fit not only the movement, but also the case, needs to know more about it, as he must so distribute the weight along its length that it may be given a length of 60 inches or of 44 inches, or anything between them, and still beat seconds, in the case of a regulator. He must also do the same thing in other clocks having pendulums which beat other numbers than 60. Therefore he must know the center of his weights; this is called the center of gravity. This center of gravity is often confused by many with the center of oscillation as its real purpose is not understood. It is simply used as a starting point in building pendulums, because there must be a starting point, and this point is chosen because it is always present in every pendulum and it is convenient to work both ways from the center of weight or gravity. In Fig. 2 we have two pendulums, in one of which (the ball and string) the center of gravity is the center of the ball and the center of oscillation is also at the center (practically) of the ball. Such a pendulum is about as short as it can be constructed for any given number of oscillations. The other (the rod) has its center of gravity manifestly at the center of the rod, as the rod is of the same size throughout; yet we found by comparison with the other that its center of oscillation was at two-thirds the length of the rod, measured from the point of suspension, and the real length of the pendulum was consequently one-half longer than its timekeeping length, which is at the center of oscillation. This is farther apart than the center of gravity and oscillation will ever get in actual practice, the most extreme distance in practice being that of the gridiron pendulum previously mentioned. The center of gravity of a pendulum is found at that point at which the pendulum can be balanced horizontally on a knife edge and is marked to measure from when cutting off the rod.

The center of oscillation of a compound pendulum must always be below its center of gravity an amount depending upon the proportions of weight between the rod and the bob. Where the rod is kept as light as it should be in proportion to the bob this difference should come well within the limits of the adjusting screw. In an ordinary plain seconds pendulum, without compensation, with a bob of eighteen or twenty pounds and a rod of six ounces, the difference in the two points is of no practical account, and adjustments for seconds are within the screw of any ordinary pendulum, if the screw is the right length for safety, and the adjusting nut is placed in the middle of the length of the screw threads when the top of the rod is cut off, to place the suspension spring by measurement from the center of gravity as has been already described; also a zinc and iron compensation is within range of the screw if the compensating rods are not made in undue weight to the bob. The whole weight of the compensating parts of a pendulum can be safely made within one and a half pounds or lighter, and carry a bob of twenty-five pounds or over without buckling the rods, and the two points, the center of gravity and the center of oscillation, will be within the range of the screw.

There are still some other forces to be considered as affecting the performance of our pendulum. These are the resistance to its momentum offered by the air and the resistance of the suspension spring.

Barometric Error.—If we adjust a pendulum in a clock with an air-tight case so that the pendulum swings a certain number of degrees of arc, as noted on the degree plate in the case at the foot of the pendulum, and then start to pump out the air from the case while the clock is running, we shall find the pendulum swinging over longer arcs as the air becomes less until we reach as perfect a vacuum as we can produce. If we note this point and slowly admit air to the case again we shall find that the arcs of the pendulum’s swing will be slowly shortened until the pressure in the case equals that of the surrounding air, when they will be the same as when our experiment was started. If we now pump air into our clock case, the vibrations will become still shorter as the pressure of the air increases, proving conclusively that the resistance of the air has an effect on the swinging of the pendulum.

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.

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.