Table of the Length of a Simple Pendulum,
(CONTINUED.)
| Number of Oscillations per Hour. | Length in Meters. | To Produce in 24 Hours 1 Minute. | |
|---|---|---|---|
| Loss, Lengthen by Millimeters. | Gain, Shorten by Millimeters. | ||
| 3,600 | 0.9939 | 1.38 | 1.32 |
| 3,550 | 1.0221 | 1.42 | 1.36 |
| 3,500 | 1.0515 | 1.46 | 1.40 |
| 3,450 | 1.0822 | 1.50 | 1.44 |
| 3,400 | 1.1143 | 1.55 | 1.48 |
| 3,350 | 1.1477 | 1.60 | 1.53 |
| 3,300 | 1.1828 | 1.64 | 1.57 |
| 3,250 | 1.2194 | 1.69 | 1.62 |
| 3,200 | 1.2578 | 1.75 | 1.67 |
| 3,150 | 1.2981 | 1.80 | 1.73 |
| 3,100 | 1.3403 | 1.86 | 1.78 |
| 3,050 | 1.3846 | 1.93 | 1.84 |
| 3,000 | 1.4312 | 1.99 | 1.90 |
| 2,900 | 1.5316 | 2.13 | 2.04 |
| 2,800 | 1.6429 | 2.28 | 2.18 |
| 2,700 | 1.7669 | 2.46 | 2.35 |
| 2,600 | 1.9054 | 2.65 | 2.53 |
| 2,500 | 2.0609 | 2.87 | 2.74 |
| 2,400 | 2.2362 | 3.11 | 2.97 |
| 2,300 | 2.4349 | 3.38 | 3.24 |
| 2,200 | 2.6612 | 3.70 | 3.54 |
| 2,100 | 2.9207 | 4.06 | 3.88 |
| 2,000 | 3.2201 | 4.48 | 4.28 |
| by Meters. | by Meters. | ||
| 1,900 | 3.568 | 0.0050 | 0.0048 |
| 1,800 | 3.975 | 0.0055 | 0.0053 |
| 1,700 | 4.457 | 0.0062 | 0.0059 |
| 1,600 | 5.031 | 0.0070 | 0.0067 |
| 1,500 | 5.725 | 0.0080 | 0.0076 |
| 1,400 | 6.572 | 0.0091 | 0.0087 |
| 1,300 | 7.622 | 0.0106 | 0.0101 |
| 1,200 | 8.945 | 0.0124 | 0.0119 |
| 1,100 | 10.645 | 0.0148 | 0.0142 |
| 1,000 | 12.880 | 0.0179 | 0.0171 |
| 900 | 15.902 | 0.0221 | 0.0211 |
| 800 | 20.126 | 0.0280 | 0.0268 |
| 700 | 26.287 | 0.0365 | 0.0350 |
| 600 | 35.779 | 0.0497 | 0.0476 |
| 500 | 51.521 | 0.0716 | 0.0685 |
| 400 | 80.502 | 0.1119 | 0.1071 |
| 300 | 143.115 | 0.1989 | 0.1903 |
| 200 | 322.008 | 0.4476 | 0.4282 |
| 100 | 1,288.034 | 1.7904 | 1.7131 |
| 60 | 3,577.871 | 4.9732 | 4.7586 |
| 50 | 5,152.135 | 7.1613 | 6.8521 |
| 1 | 12,880,337.930 | 17,903.6700 | 17,130.8500 |
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.
| Inches expressed in Millimeters and French Lines. | ||
|---|---|---|
| Inches. | Equal to | |
| Millimeters. | French Lines. | |
| 1 | 25.39954 | 11.25951 |
| 2 | 50.79908 | 22.51903 |
| 3 | 76.19862 | 33.77854 |
| 4 | 101.59816 | 45.03806 |
| 5 | 126.99771 | 56.29757 |
| 6 | 152.39725 | 67.55709 |
| 7 | 177.79679 | 78.81660 |
| 8 | 203.19633 | 90.07612 |
| 9 | 228.59587 | 101.33563 |
| 10 | 253.99541 | 112.59516 |
| Millimeters expressed in Inches and French Lines. | ||
| Millimeters. | Equal to | |
| Inches. | French Lines. | |
| 1 | 0.0393708 | 0.44329 |
| 2 | 0.0787416 | 0.88659 |
| 3 | 0.1181124 | 1.32989 |
| 4 | 0.1574832 | 1.77318 |
| 5 | 0.1968539 | 2.21648 |
| 6 | 0.2362247 | 2.65978 |
| 7 | 0.2755955 | 3.10307 |
| 8 | 0.3149664 | 3.54637 |
| 9 | 0.3543371 | 3.98966 |
| 10 | 0.3937079 | 4.43296 |
| French Lines expressed in Inches and Millimeters. | ||
| French Lines. | Equal to | |
| Inches. | Millimeters. | |
| 1 | 0.088414 | 2.25583 |
| 2 | 0.177628 | 4.51166 |
| 3 | 0.266441 | 6.76749 |
| 4 | 0.355255 | 9.02332 |
| 5 | 0.444069 | 11.27915 |
| 6 | 0.532883 | 13.53497 |
| 7 | 0.621697 | 15.79080 |
| 8 | 0.710510 | 18.04663 |
| 9 | 0.799324 | 20.30246 |
| 10 | 0.888138 | 22.55829 |
| 11 | 0.976952 | 24.81412 |
| 12 | 1.065766 | 27.06995 |
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.