Fig. 175. To prepare ourselves for the mechanics of the problem, we must resort once more to our pendulum; for it also can be caused to oscillate in one direction twice as rapidly as in another. By a complicated mechanical arrangement this might be done in a very perfect manner, but at present simplicity is preferable to completeness. The wire of our pendulum is therefore permitted to descend from its point of suspension, A, Fig. 175, midway between two horizontal glass rods, a b, a′ b′, supported firmly at their ends, and about an inch asunder. The rods cross the wire at a height of 7 feet above the bob of the pendulum. The whole length of the pendulum being 28 feet, the glass rods intercept one-fourth of this length. On drawing the pendulum aside in the direction of the rods, a b, a′ b′, and letting it go, it oscillates freely between them. I bring it to rest and draw it aside in a direction perpendicular to the last; a length of 7 feet only can now oscillate, and by the laws of oscillation a pendulum 7 feet long vibrates with twice the rapidity of a pendulum 28 feet long.
I wish to show you the figure described by the combination of these two rates of vibration. Attached to the copper ball, p, is a camel’s-hair pencil, intended to rub lightly upon a glass plate placed on black paper and over which is strewed white sand. Allowing the pendulum to oscillate as a whole, the sand is rubbed away along a straight line which represents the amplitude of the vibration. Let a b, Fig. 176, represent this line, which, as before, we will assume to be described in one second. When the pendulum is at the limit, b, of its swing, let a rectangular impulse be imparted to it sufficient to carry it to c in one-fourth of a second. If this were the only impulse acting on the pendulum, the bob would reach c and return to b in half a second. But under the actual circumstances it is also urged toward d, which point, through the vibration of the whole pendulum, it ought also to reach in half a second. Both vibrations, therefore, require that the bob shall reach d at the same moment; and to do this it will have to describe the curve b c′ d. Again, in the time required by the long pendulum to pass from d to a, the short pendulum will pass to and fro over the half of its excursion; both vibrations must therefore reach a at the same moment, and to accomplish this the pendulum describes the lower curve between d and a. It is manifest that these two curves will repeat themselves at the opposite sides of a b, the combination of both vibrations producing finally a figure of 8, which you now see fairly drawn upon the sand before you.
The same figure is obtained if the rectangular impulse be imparted when the pendulum is passing its position of rest, d.
| Fig. 176. | Fig. 177. | Fig. 178. |
I have here supposed the time occupied by the pendulum in describing the line a b to be one second. Let us suppose three-fourths of the second exhausted, and the pendulum at d′, Fig. 177, in its excursion toward b; let the rectangular impulse then be imparted to it, sufficient to carry it to c in one-fourth of a second. Now the long pendulum requires that it should move from d′ to b in one-fourth of a second; both impulses are therefore satisfied by the pendulum taking up the position c′ at the end of a quarter of a second. To reach this position it must describe the curve d′ c′. It will manifestly return along the same curve, and at the end of another quarter of a second find itself again at d′. From d′ to d the long pendulum requires a quarter of a second. But at the end of this time the short pendulum must be at the lower limit of its swing: both requirements are satisfied by the pendulum being at e. We thus obtain one arm, c′ e, of a curve, which repeats itself to the left of e; so that the entire curve, due to the combination of the two vibrations, is that represented in [Fig. 165]. This figure is a parabola, whereas the figure of 8 before obtained is a lemniscata.
We have here supposed that, at the moment when the rectangular impulse was applied, the motion of the pendulum was toward b: if it were toward a we should obtain the inverted parabola, as shown in [Fig. 178].
Supposing, finally, the impulse to be applied, not when the pendulum is passing through its position of equilibrium, nor when it is passing a point corresponding to three-fourths or one-fourth of the time of its excursion, but at some other point in the line, a b, between its end and centre. Under these circumstances we should have neither the parabola nor the perfectly symmetrical figure of 8, but a distorted 8.
And now we are prepared to witness with profit the combined vibration of our two tuning-forks, one of which sounds the octave of the other. Permitting the vertical fork, T, [Fig. 172], to remain undisturbed in front of the lamp, we can oppose to it a horizontal fork, which vibrates with twice the rapidity. The first passage of the bow across the two forks reveals the exact similarity of this combination, and that of our pendulum. A very perfect figure of 8 is described upon the screen. Before the lecture the vibrations of these two forks were fixed as nearly as possible to the ratio of 1:2, and the steadiness of the figure indicates the perfection of the tuning. Stopping both forks, and again agitating them, we have the distorted 8 upon the screen. A few trials enable me to bring out the parabola. In all these cases the figure remains fixed upon the screen. But if a morsel of wax be attached to one of the forks, the figure is steady no longer, but passes from the perfect 8 into the distorted one, thence into the parabola, from which it afterward opens out to an 8 once more. By augmenting the discord, we can render those changes as rapid as we please.
When the 8 is steady on the screen, a rotation of the mirror of the fork, T, produces the scroll shown in Fig. 179.