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Center of Oscillation

That point at which, if the whole matter of a suspended body were collected, the time of oscillation would be the same. In a long cone suspended from its apex, the center of oscillation is at four-fifths of its length from the apex, and in a bar suspended from one end is at two-thirds of its length. A pendulum being irregular in form it is difficult to calculate its center of oscillation, but it always is situated below its center of gravity. The following explanation may aid the student in locating the center of oscillation:

All know that a simple theoretical pendulum is one where the whole weight is centered in one point, suspended from, and oscillating about, a fixed point, or center of suspension. A sphere of platinum, suspended by a fibre of silk, would probably be the nearest approximation to a perfectly simple pendulum. A compound pendulum is one where the weight is not centered in or about one point, but is extended for some distance up and down the rod. Suppose there are fixed upon the fibre, at equal distances, three platinum balls. From the well known fact that a short pendulum vibrates quicker than a long one, the upper or short pendulum will endeavor to make its vibrations in the short time due to its length as a pendulum. The middle ball will endeavor to make its oscillations in the time its length of support demands, and the lower and longest will attempt the slow and regular vibrations of the long pendulum. Suppose that these three balls, representing three pendulums of three different lengths, be drawn aside from the perpendicular 5 deg. and suddenly released, the consequence will be that the upper one will have made its full excursion by the time the middle one has descended to the perpendicular, and before the lower one has arrived there; the momentum of the three balls bending the fibre of silk into such a curve as will accommodate the tendencies of the three balls.

If the silk fibre be replaced by an inflexible rod, and the now rigid compound pendulum be drawn aside as before, the upper ball will endeavor to hasten forward the middle one to its own speed, and the middle and upper one will both combine to hasten the lower one. So also, the middle one will retard somewhat the rapidity of the upper one, and the slow-moving lower one will do its best to restrain the haste of both those above it, and the consequence of all these tendencies will be that the lower one will be somewhat accelerated, and the upper one proportionally retarded; the whole assuming a vibration which is the mean (middle ball) of the two extremes, provided the three masses are equal, thus compelling the whole to oscillate as a pendulum whose length is that of the middle ball. But if the lower ball be the largest, its control over those parts above it will be in proportion to its mass and the time of its vibrations will nearly coincide with those made by its center of gravity. Suppose, again, the largest amount of matter to be in the upper ball. then will its influence be more potent toward forcing the lower and longer pendulums to accommodate their rate to that of the upper one, and their vibrations will be thereby increased to a degree which will approximate the normal vibrations of that short pendulum. Thus you see the difficulty of exactly fixing upon the exact length of any compound pendulum by simple computation. Every particle of matter from the top of the rod to the lower extremity, which differs in its distance from the point of suspension, has its own time for making an oscillation about that point; and the greater the number of particles that have an equal distance from that point, the greater influence they possess in determining the time of vibration; in this case, as in republics, the mass rules. To obviate these counteracting influences that are constantly at work in the oscillations of the compound pendulum, it becomes necessary to concentrate, as far as possible, all the matter of the pendulum at such a distance from the point of suspension as will produce the number of vibrations desired, and this center of oscillation will always fall in a line produced through the center of gravity and the point of suspension, and will always be below the center of gravity.

The center of oscillation and suspension are convertible points; that is, a pendulum inverted and suspended from the center of oscillation will vibrate in the same time. Huygens, the Dutch scientist, discovered this remarkable fact, and it affords a ready means of determining experimentally the length of a compound pendulum, which may be measured by means of a platinum or lead ball, suspended by a fibre of silk from the same point, and in front of the pendulum to be measured, and of such a length that the vibrations will perfectly coincide in time. The distance from the point of suspension to the center of the ball (which is also the center of oscillation) is nearly the length of that compound pendulum.

It should be remembered that the center of oscillation is the point to be affected in all compensations for temperature. The difficulty in producing a perfect compensation pendulum is to harmonize and bring into coincidence the antagonistic tendencies of the center of gravity, center of oscillation and moment of inertia, all of which are properties and peculiarities of compound pendulums, and must be taken into consideration by those who are experimenting upon them with the expectation of producing any arrangement in advance of those in use at present.