revolve about their shorter axes, whatever may be the inclination of these axes to the planes of their orbits. Were the earth, by the action of any extraneous force, constrained to revolve about some other axis than that about which it is found to revolve, it would, as soon as the force ceased to act, return to its present axis of rotation.

Experimental Illustrations.

139. The principles relating to the centrifugal forces admit of experimental illustration. The instrument represented in the figure, may be employed to show the value of the centrifugal force. A repre

forked branch BC, sustaining a stretched wire. D and E are two balls which are pierced by the wire, and are free to move along it. Between B and E is a spiral spring, whose axis coincides with the wire.

Immediately below the spring, on the horizontal part of the fork, is a scale for determining the distance of the ball E, from the axis, and for measuring the degree of compres sion of the spring. Before using the instrument, the force required to produce any degree of compression of the spring is determined experimentally, and marked on the scale.

If now a motion of rotation be communicated to the axis, the ball D will at once recede to C, but the ball E will be restrained by the spiral spring. As the velocity of rotation. is increased, the spring will be compressed more and more, and the ball E, will approach B. By a suitable arrangement of the wheelwork, the angular velocity of the axis corresponding to any degree of compression may be ascer

tained. We have thus all the data necessary to a verification of the law of the centrifugal force.

If a vessel of water be made to revolve about a vertical axis, the interior particles will recede from the axis on account of the centrifugal force, and will be heaped up about the sides of the vessel, imparting a concave form to the upper surface. The concavity will become greater as the angular velocity is increased.

If a circular hoop of flexible metal be fastened so that one of its diameters shall coincide with the axis of a whirling machine, its lower point being fastened to the horizontal beam, and a motion of rotation be imparted, the portions of the hoop farthest from the axis will be most affected by the centrifugal force, and the hoop will be observed to assume an elliptical form.

If a sponge, filled with water, be attached to one of the arms of a whirling machine, and a motion of rotation be imparted, the water will be thrown from the sponge. This principle has been made use of in a machine for drying clothes. An annular trough of copper is mounted upon an axis by means of radial arms, the axis being connected with a train of wheelwork, by means of which it may be put in motion. The outer wall is pierced with holes for the escape of the water, and a lid serves to confine the articles to be dried. To use this instrument, the linen, after being washed, is placed in the annular space, and a rapid motion of rotation imparted to the machine. The linen is thrown, by the centrifugal force, against the outer wall of the instrument, and the water, being partially squeezed out, and partially thrown off by the centrifugal force, escapes through the holes made for the purpose. Sometimes as many as 1,500 revolutions per minute are given to the drying machine, in which case, the drying process is very rapid and very perfect.

If a body be whirled about an axis with sufficient velocity, it may happen that the centrifugal force generated will be greater than the force of cohesion which binds the

particles together, in which case, the body will be torn asunder. It is a common occurrence that large grindstones, when put into a state of rapid rotation, burst, the fragments being thrown with great velocity away from the axis, and often producing much destruction.

When a wagon, or carriage, is driven rapidly around a corner, or is forced to turn about a circular track, the centrifugal force generated is often sufficient to throw out the loose articles from the vehicle, and even to overthrow the vehicle itself. When a car upon a railroad track is forced to turn around a sharp curve, the centrifugal force generated, tends to throw the weight of the cars against the rail, producing a great amount of friction, and contributing to wear out both the track and the car. To obviate this difficulty in a measure, it is customary to raise the outer rail, so that the resultant of the centrifugal force, and the force of gravity, shall be sensibly perpendicular to the plane of the two rails.

Elevation of the outer rail of a curved track.

140. To find the inclination of the track, that is, the elevation of the outer rail, so that the resultant of the weight and centrifugal force

E

BYC

F

Fig. 123.

may be perpendicular to the line joining the two rails. Let G be the centre of gravity of the car, and let the figure represent a vertical section through the centre of gravity and the centre of the curved track. Let GA, parallel to the horizon, represent the acceleration due to the centrifugal force, and GB, perpendicular to the horizon, the acceleration due to the weight of the car. Construct the resultant GC, of these forces, then must the line DE be perpendicular to GC. Denote the velocity of the car, by v, and the radius of the curved track, by r. The acceleration due to the weight will be

equal to g, the force of gravity, and the acceleration due to

the centrifugal force will be equal to

CB
GB

292

The tangent of the

angle CGB will be equal to ; or, denoting the angle

But the angle DEF is equal to the angle CGB. Denoting the distance between the rails, by d, and the elevation of the outer rail above the inner one, by h, we shall have,

Equating the two values of tana, we have,

Hence, the elevation of the outer rail varies as the square of the velocity directly, and as the radius of the curve inversely.

It is obvious that this connection would require to be different for different velocities, which, from the nature of the case, would be manifestly impossible. The correction is, therefore, made for some assumed velocity, and then such a form is given to the tire of the wheels as will complete the correction for different velocities.

The Conical Pendulum.

141. The conical pendulum consists of a solid ball attached to one end of a rod, the other end of which is connected, by means of a hinge-joint, with a vertical axle. When the axle is put in motion, the centrifugal force generated in the ball causes it to recede from the axis, until an equilibrium is established between the weight of the ball, the centrifugal force, and the tension of the connecting rod.

When the velocity is constant, the centrifugal force will be constant, and the centre of the ball will describe a horizontal circle, whose radius will depend upon the velocity. Let it be required to determine the time of revolution.

Let BD be the vertical axis, A the ball, B the hingejoint, and AB the connecting rod, whose

mass is so small, that it may be neglected, B in comparison with that of the ball.

Denote the required time of revolution, by t, the length of the arm, by 7, the acceleration due to the centrifugal force, by f, and the angle ABC, by 9. Draw AC perpendicular to BD, and denote AC, by r, and BC, by h.

D

r

Ꮐ

Fig. 124.

From the triangle ABC, we have, r = lsing; and since is the radius of the circle described by A, we have the distance passed over by 4, in the time t, equal to 2r2lsing. Denoting the velocity of A, by v, we have, from Equation (55),

But the centrifugal force is equal to the square of the velocity, divided by the radius; hence,

The forces which act upon A, are the centrifugal force in the direction AF, the force of gravity in the direction AG, and the tension of the connecting rod in the direction AB. In order that the ball may remain at an invariable distance from the axis, these three forces must be in equilibrium. Hence (Art. 35),

gf sin BAF: sinBAG;

but,

sin BAF sin(90°) cosp;

=