557. To compel a body to swerve from motion in a straight line force
must be exercised. In this chapter we shall study the comparatively
simple case of a body revolving in a circle.
558. When a body moves round uniformly in a circle force must be
continuously applied, and the first question for us to examine is, as
to the _direction_ of that force. We have to demonstrate the important
fact, that it constantly tends towards the centre.
559. The direction of the force can be exhibited by actual experiment,
and its magnitude will be at the same time clearly indicated by the
extent to which a spring is stretched. The apparatus we use is shown in
Fig. 74.
The essential parts of the machine consist of two balls A, B, each 2"
in diameter: these are thin hollow spheres of silvered brass. The balls
are supported on arms P A, Q B, which are attached to a piece of wood,
P Q, capable of turning on a socket at C. The arm A P is rigidly fixed
to P Q; the other arm, B Q, is capable of turning round a pin at Q. An
india-rubber door-spring is shown at F; one end of this is secured to P
Q, the other end to the movable arm, Q B. If the arm Q B be turned so
as to move B away from C, the spring F must be stretched.
[Illustration: FIG. 74.]
A small toothed wheel is mounted on the same socket as C; this is
behind P Q, and is therefore not seen in the figure: the whole is made
to revolve rapidly by the large wheel E, which is turned by the handle
D.
560. The room being darkened, a beam from the lime-light is allowed to
fall on the apparatus: the reflections of the light are seen in the
two silvered balls as two bright points. When D is turned, the balls
move round rapidly, and you see the points of light reflected from them
describe circles. The ball B when at rest is 4" from C, while A is 8"
from C; hence the circle described by B is smaller than that described
by A. The appearance presented is that of two concentric luminous
circles. As the speed increases, the inner circle enlarges till the two
circles blend into one. By increasing the speed still more, you see the
circle whose diameter is enlarging actually exceeding the fixed circle,
and its size continues to increase until the highest velocity which it
is safe to employ has been communicated to the machine.
561. What is the explanation of this? The arm A is fixed and the
distance A C cannot alter, hence A describes the fixed circle. B, on
the other hand, is not fixed; it can recede from C, and we find that
the quicker the speed the further it recedes. The larger the circle
described by B the more is the spring stretched, and the greater is the
force with which B is attracted towards the centre. This experiment
proves that the force necessary to retain a body in a circular path
must be increased when the speed is increased.
562. Thus we see that uniform motion of a body in a circle can only be
produced by an uniform force directed to the centre.
If the motion, even though circular, have variable speed the law of the
force is not so simple.
563. We can measure the magnitude of this force by the same apparatus.
The ball B weighs 0·1 lb. I find that I must pull it with a force of 3
lbs. in order to draw it to a distance of 8" from C; that is, to the
same distance as A is from C. Hence, when the diameters of the circles
in which the balls move are equal, the central force must be 3 lbs.;
that is, it must be nearly thirty times as great as gravity.
564. The necessity for the central force is thus shown: Let us conceive
a weight attached to a string to be swung round in a circle, a portion
of which is shown in Fig. 75.
[Illustration: FIG. 75.]
Suppose the weight be at S and moving towards P, and let a tangent to
the circle be drawn at P. Take two points on the circle, A and B, very
near P; the small arc A B does not differ perceptibly from the part A
B on the tangent line; hence, when the particle arrives at A, it is
a matter of indifference whether it travels in the arc A B, or along
the line A B. Let us suppose it to move along the line. By the first
law of motion, a particle moving in the line A B would continue to do
so; hence, if the particle be allowed, it will move on to Q: but the
particle is not allowed to move to Q; it is found at R. Hence it must
have been withdrawn by some force.
565. This force is supplied by the string to which the weight is
attached. The incessant change from the rectilinear motion of the
weight requires a constantly applied force, and this is always directed
to the centre. Should the string be released, the body flies off in
the direction of the tangent to the circle at the point which the body
occupied at the instant of release.
566. The central force increases in proportion to the square of the
velocity. If I double the speed with which the weight is whirled round
in the circle, I quadruple the force which the string must exert on the
body. If the speed be trebled, the force is increased ninefold, and so
on. When the speeds with which two equal masses are revolving in two
circles are equal, the central force in the smaller circle is greater
than that of the larger circle, in the proportion of the radius of the
larger circle to that of the smaller.