Page:The American Cyclopædia (1879) Volume XI.djvu/337

MECHANICS 325 FIG. 22. inequalities in the times of vibration were ob- viated by Huygens by causing the pendulum to vibrate in a cycloidal arc, which he was the first to demonstrate is the curve of quickest descent from one point to another. To pro- duce this cycloidal vibration, it is only neces- sary to cause the a string by which the pendulum is suspended to wind around a semi- cycloid placed at each side, and to unwind from it when it falls from rest, as shown in fig. 22. The prac- tical difficulties in the use of cycloidal arcs for pendulums are however greater than the advantage gained; therefore the pendulums of astronomical clocks are made to vibrate in small circular arcs. It has been said that in a compound pendulum there is a tendency in the different parts which are at different distances from the point of suspension to vibrate in different times. This will appear from a consideration of fig. 23. Suppose several balls, A, B, 0, D, E, sus- pended by sepa- rate strings of un- equal lengths from a horizontal bar at M. If they are all let fall at the same time from the line M E, the ball A on the shortest string will descend more rap- idly than B, B more rapidly than C, &c., so that af- ter a time they will have the positions A', B', &c. If they are all attached to the same wire and kept in the same line while vibrating, the balls moving in the smaller arcs will tend to accelerate the motion of those further from the centre of motion, and those vibrating in the larger arcs will tend to retard the motion of those nearer the centre of motion. Therefore there is a certain point where there is neither a tendency to retard nor to accelerate ; this point is called the centre of oscillation of the system. The distance between this point and the point of suspension measures the length of a compound pendulum. If a homogeneous cylindrical bar is suspended at one end and made to vibrate, the centre of oscillation is two thirds the distance from the point of suspension. The discovery of the centre of oscillation, as we have seen, also marked an era in the science of mechanics, be- ing one of its most important principles, and FIG. 23. FIG. 24. having a wide application. The centre of os- cillation and the point of suspension of a pen- dulum are convertible points ; that is, if the centre of oscillation is made the point of sus- pension, the time of vibration will not be changed ; a principle which allows of the ex- perimental determination of the centre of os- cillation, and therefore of the length of the pendulum. The centre of oscillation may be entirely beyond the pendulum, as in the me- tronome, an instrument used to measure time in music. (See METRONOME.) Its principle is shown in fig. 24, where a horizontal axis sup- ports a rod, upon which there are two balls whose distance from the centre of motion may be varied at pleasure. If the balls are of equal weight and at equal distances from the centre of motion, they will not oscil- late ; but at unequal distances they will, and slowly in propor- tion as the difference of distance is small. The pendulum affords a correct means of finding the value of <?, and therefore the height through which a body will fall from rest in one second of time. Taking the equation I = ^f and trans- posing, we have g = ^. Therefore, if the length of the seconds pendulum is 39-1 in., the equation becomes in numbers g = 3-14159 2 x 39-1 = 32-16 ft., which is twice the space through which a body will fall in one sec- ond of time. The principal use of the pen- dulum is to measure time. To do this accu- rately, it is necessary to keep the point of sus- pension and the centre of oscillation at the same distance from each other, or in other words, to preserve a constant length. In- crease of temperature causes a pendulum made of one piece to lengthen by expansion. If, however, two ma- terials are so combined that while the expan- sion of one tends to lengthen the system, that of the other tends to raise the centre of os- cillation, and the com- bination is such that the expansion of one shall exactly counteract that of the other, the desired end is attained. Such pendulums are called compensation pen- dulums. Two forms are shown in figs. 25 and 26. The bob of fig. 25 consists of a steel frame holding a hollow glass cylinder contain- ing mercury. It is evident that if this mer- cury by its expansion causes the centre of FIG. 25. FIG. 26.