581. There are some curious properties of circular motion which
remain to be considered. These we shall investigate by means of the
apparatus of Fig. 80. This consists of a pair of wheels B C, by which a
considerable velocity can be given to a horizontal shaft. This shaft is
connected by a pair of bevelled wheels D with a vertical spindle F. The
machine is worked by a handle A, and the object to be experimented upon
is suspended from the spindle.
582. I first take a disk of wood 18" in diameter; a hole is bored in
the margin of this disk; through this hole a rope is fastened, by means
of which the disk is suspended from the spindle. The disk hangs of
course in a vertical plane.
[Illustration: FIG. 80.]
583. I now begin to turn the handle round gently, and you see the
disk begins to rotate about the vertical diameter; but, as the speed
increases, the motion becomes a little unsteady; and finally, when I
turn the handle very rapidly, the disk springs up into a horizontal
plane, and you see it like the surface of a small table: the rope
swings round and round in a cone, so rapidly that it is hardly seen.
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.
[Illustration: 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.