Fig. 3.
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.
| Number of Oscillations per Hour. | Length in Millimeters. | Variation in Length for One Minute in 24 Hours in Millimeters. |
|---|---|---|
| 20,000 | 32.2 | 0.04 |
| 19,000 | 35.7 | 0.05 |
| 18,000 | 39.8 | 0.05 |
| 17,900 | 40.2 | 0.06 |
| 17,800 | 40.7 | 0.06 |
| 17,700 | 41.1 | 0.06 |
| 17,600 | 41.6 | 0.06 |
| 17,500 | 42.1 | 0.06 |
| 17,400 | 42.4 | 0.06 |
| 17,300 | 43.0 | 0.06 |
| 17,200 | 43.5 | 0.06 |
| 17,100 | 44.0 | 0.06 |
| 17,000 | 44.6 | 0.06 |
| 16,900 | 45.1 | 0.06 |
| 16,800 | 45.7 | 0.06 |
| 16,700 | 46.3 | 0.06 |
| 16,600 | 46.7 | 0.07 |
| 16,500 | 47.3 | 0.07 |
| 16,400 | 47.9 | 0.07 |
| 16,300 | 48.5 | 0.07 |
| 16,200 | 49.1 | 0.07 |
| 16,100 | 49.7 | 0.07 |
| 16,000 | 50.0 | 0.07 |
| 15,900 | 51.0 | 0.07 |
| 15,800 | 51.6 | 0.07 |
| 15,700 | 52.3 | 0.07 |
| 15,600 | 52.9 | 0.07 |
| 15,500 | 53.6 | 0.07 |
| 15,400 | 54.3 | 0.08 |
| 15,300 | 55.0 | 0.08 |
| 15,200 | 55.7 | 0.08 |
| 15,100 | 56.5 | 0.08 |
| 15,000 | 57.3 | 0.08 |
| 14,900 | 58.0 | 0.08 |
| 14,800 | 58.8 | 0.08 |
| 14,700 | 59.6 | 0.08 |
| 14,600 | 60.4 | 0.08 |
| 14,500 | 61.3 | 0.08 |
| 14,400 | 62.1 | 0.09 |
| 14,300 | 63.0 | 0.09 |
| 14,200 | 63.9 | 0.09 |
| 14,100 | 64.8 | 0.09 |
| 14,000 | 65.7 | 0.09 |
| 13,900 | 66.7 | 0.09 |
| 13,800 | 67.6 | 0.09 |
| 13,700 | 68.6 | 0.09 |
| 13,600 | 69.6 | 0.09 |
| 13,500 | 70.7 | 0.09 |
| 13,400 | 71.7 | 0.10 |
| 13,300 | 72.8 | 0.10 |
| 13,200 | 73.9 | 0.10 |
| 13,100 | 75.1 | 0.10 |
| 13,000 | 76.2 | 0.10 |
| 12,900 | 77.4 | 0.11 |
| 12,800 | 78.6 | 0.11 |
| 12,700 | 79.9 | 0.11 |
| 12,600 | 81.1 | 0.11 |
| 12,500 | 82.4 | 0.11 |
| 12,400 | 83.8 | 0.11 |
| 12,300 | 85.1 | 0.12 |
| 12,200 | 86.5 | 0.12 |
| 12,100 | 88.0 | 0.12 |
| 12,000 | 89.5 | 0.12 |
| 11,900 | 91.0 | 0.12 |
| 11,800 | 92.5 | 0.13 |
| 11,700 | 94.1 | 0.13 |
| 11,600 | 95.7 | 0.13 |
| 11,500 | 97.4 | 0.13 |
| 11,400 | 99.1 | 0.13 |
| 11,300 | 100.9 | 0.14 |
| 11,200 | 102.7 | 0.14 |
| 11,100 | 104.5 | 0.14 |
| 11,000 | 106.5 | 0.14 |
| 10,900 | 108.4 | 0.15 |
| 10,800 | 110.5 | 0.15 |
| 10,700 | 112.5 | 0.15 |
| 10,600 | 114.6 | 0.16 |
| 10,500 | 116.8 | 0.16 |
| 10,400 | 119.1 | 0.16 |
| 10,300 | 121.4 | 0.17 |
| 10,200 | 123.8 | 0.17 |
| 10,100 | 126.3 | 0.17 |
| 10,000 | 128.8 | 0.18 |
| 9,900 | 131.4 | 0.18 |
| 9,800 | 134.1 | 0.18 |
| 9,700 | 136.9 | 0.19 |
| 9,600 | 139.8 | 0.19 |
| 9,500 | 142.7 | 0.19 |
| 9,400 | 145.8 | 0.20 |
| 9,300 | 148.9 | 0.20 |
| 9,200 | 152.2 | 0.21 |
| 9,100 | 155.5 | 0.21 |
| 9,000 | 159.0 | 0.22 |
| 8,900 | 162.6 | 0.22 |
| 8,800 | 166.3 | 0.23 |
| 8,700 | 170.2 | 0.23 |
| 8,600 | 173.7 | 0.24 |
| 8,500 | 178.3 | 0.24 |
| 8,400 | 182.5 | 0.25 |
| 8,300 | 187.0 | 0.25 |
| 8,200 | 191.5 | 0.26 |
| 8,100 | 196.3 | 0.27 |
| 8,000 | 201.3 | 0.27 |
| 7,900 | 206.4 | 0.28 |
| 7,800 | 211.7 | 0.29 |
| 7,700 | 217.2 | 0.30 |
| 7,600 | 223.0 | 0.30 |
| 7,500 | 229.0 | 0.31 |
| 7,400 | 235.2 | 0.32 |
| 7,300 | 241.7 | 0.33 |
| 7,200 | 248.5 | 0.34 |
| 7,100 | 255.7 | 0.35 |
| 7,000 | 262.9 | 0.36 |
| 6,900 | 270.5 | 0.37 |
| 6,800 | 278.6 | 0.38 |
| 6,700 | 286.9 | 0.39 |
| 6,600 | 295.7 | 0.40 |
| 6,500 | 304.9 | 0.41 |
| 6,400 | 314.5 | 0.43 |
| 6,300 | 324.5 | 0.44 |
| 6,200 | 335.1 | 0.46 |
| 6,100 | 346.2 | 0.47 |
| 6,000 | 357.8 | 0.48 |
| 5,900 | 370.0 | 0.50 |
| 5,800 | 382.9 | 0.52 |
| 5,700 | 396.4 | 0.54 |
| 5,600 | 410.7 | 0.56 |
| 5,500 | 425.8 | 0.58 |
| 5,400 | 440.1 | 0.60 |
| 5,300 | 458.5 | 0.62 |
| 5,200 | 476.3 | 0.65 |
| 5,100 | 495.2 | 0.67 |
| 5,000 | 515.2 | 0.70 |
| 4,900 | 536.5 | 0.73 |
| 4,800 | 559.1 | 0.76 |
| 4,700 | 583.1 | 0.79 |
| 4,600 | 608.7 | 0.83 |
| 4,500 | 636.1 | 0.86 |
| 4,400 | 665.3 | 0.90 |
| 4,300 | 696.7 | 0.95 |
| 4,200 | 730.2 | 0.99 |
| 4,100 | 766.3 | 1.04 |
| 4,000 | 805.0 | 1.09 |
| 3,950 | 825.5 | 1.12 |
| 3,900 | 846.8 | 1.15 |
| 3,850 | 869.0 | 1.16 |
| 3,800 | 892.0 | 1.21 |
| 3,750 | 915.9 | 1.25 |
| 3,700 | 940.1 | 1.28 |
| 3,650 | 966.8 | 1.31 |