584. We may repeat the experiment in a different manner. I take a piece of iron chain about 2' long, g; I pass the rope through the two last links of its extremities, and suspend the rope from the spindle. When I commence to turn the handle, you see the chain gradually opens out into a loop h; and as the speed increases, the loop becomes a complete ring. Still increasing the speed, I find the ring becomes unsteady, till finally it rises into a horizontal plane. The ring of chain in the horizontal plane is shown at i. When the motion is further increased, the ring swings about violently, and so I cease turning the handle.
Fig. 81.
585. The principles already enunciated will explain these remarkable results; we shall only describe that of the chain, as the same explanation will include that of the disk of wood. We shall begin with the chain hanging vertically from the spindle: the moment rotation commences, the chain begins to spin about a vertical axis; the parts of the chain fly outwards from this axis just as the ball flies outwards in [Fig. 74]; this is the cause of the looped form h which the chain assumes. As the speed is increased the loop gradually opens more and more, just as the diameter of the circle [Fig. 74] increases with the velocity. But we have also to inquire into the cause of the remarkable change of position which the ring undergoes; instead of continuing to rotate about a vertical diameter, it comes into a horizontal plane. This will be easily understood with the help of [Fig. 81]. Let o p represent the rope attached to the ring, and o c be the vertical axis. Suppose the ring to be spinning about the axis o c, when o c was a diameter; if then, from any cause, the ring be slightly displaced, we can show that the circular motion will tend to drive the ring further from the vertical plane, and force it into the horizontal plane. Let the ring be in the position represented in the figure; then, since it revolves about the vertical line o c, the tendency of p p and q q is to move outwards in the directions of the arrows, thus evidently tending to bring the ring into the horizontal plane.
586. In [Art. 103] we have explained what is meant by stable and unstable equilibrium; we have here found a precisely analogous phenomenon in motion. The rotation of the ring about its diameter is unstable, for the minutest deviation of the ring from this position is fatal; circumstances immediately combine to augment the deviation more and more, until finally the ring is raised into the horizontal plane. Once in the horizontal plane, the motion there is stable, for if the ring be displaced the tendency is to restore it to the horizontal.
587. The ring, when in a horizontal plane, rotates permanently about the vertical axis through its centre; this axis is called permanent, to distinguish it from all other directions, as being the only axis about which the motion is stable.
588. We may show another experiment with the chain: if instead of passing the rope through the links at its ends, I pass the rope through the centre of the chain, and allow the ends of the chain to hang downwards. I now turn the handle; instantly the parts of the chain fly outwards in a curved form; and by increasing the velocity, the parts of the chain at length come to lie almost in a straight line.
LECTURE XVIII.
THE SIMPLE PENDULUM.
Introduction.—The Circular Pendulum.—Law connecting the Time of Vibration with the Length.—The Force of Gravity determined by the Pendulum.—The Cycloid.