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Centrifugal Force Illustrated.

energetic. They are connected with some turning part of the engine, so that if it move too fast the balls fly asunder, and by so doing are made to turn a valve and reduce the quantity of steam admitted to the piston; or, on the other hand, if it move too slow, they collapse and allow this throttle-valve to open.

Water in a vessel caused to spin round is, by centrifugal force, raised up all round against the sides of the vessel, forming a hollow liquid basin.

Wet linen may have its moisture shaken out of it by putting it in a cylindrical case with holes pierced all round, which is caused to whirl with considerable rapidity.

A half-formed cylindrical vessel of soft clay, placed on the centre of the potter's table-which is made to whirl and is called his wheel-opens out or widens merely by the centrifugal force of its sides, and thus assists the worker in giving its form.

A ball of soft clay, with a spindle fixed through its centre, if made to turn quickly, soon ceases to be a perfect ball. It bulges out in the middle, where the centrifugal force is greatest, and is flattened towards the ends.

This change of form is exactly what has happened to the ball of our earth. It has bulged out 13 miles at the equator in consequence of its daily rotation, and is flattened at the poles in a corresponding degree.

A mass that weighs 287 pounds at the north pole would weigh just 286 pounds at the equator, owing conjointly to the centrifugal force and the greater distance from the centre produced by the bulge.

If our earth turned seventeen times faster than it now does the centrifugal force at the equator would be nearly equal to the weight of any mass placed there, and the greater velocity would cause them to fly off altogether, or to rise and form a ring round the earth like that which surrounds Saturn.

160. The effects of centrifugal force may be exhibited as well as

Fig. 13.

measured by a piece of apparatus called a whirling-table (fig. 13), which is a round disc of wood mounted on an upright axle, to which, by means of grooved wheels and bands, we can communicate a rapid rotation. Upon this a ball of lead may be placed.

fixed at the end of a small spring, which is attached by its other end to the centre or axle of the disc.

Laws of Centrifugal Force.

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Thus, by the extent to which the ball flies out when it is whirled, we can estimate the amount of centrifugal force with great accuracy and readiness. With such an apparatus the following laws of centrifugal force may be experimentally verified, though they can be reasoned out also from general or abstract principles.

"Laws of Centrifugal Force."

161. The centrifugal force in any rotating body is proportioned,

Ist. To the mass or weight of the body whirled.

This we might expect, because each particle requires to be drawn towards the centre independently.

2nd. To the size of circle described in a second or unit of time.

For a body moving round a circle of double diameter has to be bent towards the centre twice as far every second.

3rd. To the number of turns made per second (or unit of time),

and so that for double, triple, &c., the number of turns, the centrifugal force is increased four, nine, &c., times.

The reason of this increase of the centrifugal force at the square of the rate of increase of the number of turns, will appear when we consider that with, say, triple the number of revolutions per second there will be triple the number of pulls or impulses towards the centre; and as each pull has to be made in one-third of the time, it must be done with triple energy; so that, altogether, there is nine times the amount of centrifugal exertion in the second.

162. It is found that a body revolving in a circle of, say, four feet diameter, must complete its revolution in one second and a half, that it may have centrifugal force just equal to its weight.

It may give some idea of the relation of the centrifugal force to the weight of a body when we say that thirty pounds of metal at the rim of a fly-wheel six feet in diameter turning once a second or sixty times a minute, will exert an outward strain on the wheel of about one hundredweight, and if its speed were increased to five times this the centrifugal strain would amount to one ton and a quarter. If the tenacity of the wheel is unequal to bear this, such a speed would scatter the wheel in pieces.

163. The centrifugal railway (fig. 14) is a philosophical toy intended to illustrate the power of centrifugal force to overcome

82

Newton's Second Law of Motion.

gravity. A small iron carriage, starting from A, sweeps down

A

Fig. 14.

D

the incline, acquiring such a momentum at B, that it is carried round the loop, C, bottom upwards, and lands safely at its destination, D.

164. An artificial imitation of Saturn's ring also il

lustrates this subject beautifully. We provide ourselves with a square bottle, or make one with panes of common glass, and arrange in it, as

D

Fig. 15.

shown in the figure, a spindle with a small metal disc, D, fixed on it, so that we can turn the spindle and disc by means of a crank handle, c. We next fill the vessel with a mixture of alcohol (or spirits of wine) and water of such a density that olive oil justs floats in it. Then, having

previously smeared the disc, D, with oil, we pour a little oil into the liquid;

it will collect round the disc and adhere in the shape of a flattened globe. On gently turning the handle we see it gradually spread out, till the centrifugal force at last ruptures it away from the disc; and it forms, while the motion continues, a miniature facsimile of the wonderful ring of Saturn. .

"The Second Law of Motion."

165. Any change in the amount or the direction of a body's motion must be due to the action of some force impressed on the body in the direction of that change, and is a measure of that impressed force.

This, it will be seen, follows at once from the first law, and is in fact but an expansion of it. Let us, however, consider exactly what it implies.

When we project a ball in a horizontal direction, the force of projection would impel it straight in that line; but when we see it gradually move down towards the earth, we infer the action of another and a vertical force. We also measure this force by the change of momentum it produces in that vertical direction in the unit of time.

The Second Law of Motion.

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It is to be carefully observed that the law applies equally to bodies at rest and to bodies possessing any kind of motion; for the same amount of change will always be produced by the same force, whatever forces or motions may be already in operation on the body, and whether the change conspire with or oppose any motion already possessed.

In other words, whatever number of forces be acting on a body, cach force may be regarded as producing independently its own change of motion; so that the resulting force must be in the direction and in the proportion of the resulting change of motion, and may thus be found from a consideration of the motions solely. This we have already anticipated in explaining how the rule of the parallelogram of forces reduces itself practically to that of the parallelogram of motions.

166. Further, this second law is the ground of our measurement of force, as well as of mass.

By the definition of force, different forces acting on the same body for the same length of time will produce in them velocities proportional to those forces; and so we have the means of comparing forces by the velocities they generate per second in the same

mass.

Thus, two boys compare their force of muscle by trying how far each will roll the same stone or ball along the ground, the distances being proportioned to the velocities in this instance.

In this way we compare the force of gravity at different parts of the earth's surface, by its accelerating power on the same body carried from place to place.

Again, the same force acting on different masses will produce velocities which diminish in proportion as the masses increase.

With the same exertion a soldier will roll a ten-pound cannon ball twice as far as a twenty-pound ball; and the same charge of gunpowder will send the latter ball only half as far in a horizontal line as the former.

Lastly, if the force be changed and the mass it moves be changed, and yet the velocity remain unchanged, it is manifest that the force and the mass go hand in hand in their changes, or the force is proportioned to the mass. This, we have seen, is illustrated by the case of gravity acting on bodies falling freely; all bodies require the same velocity in the same time under the attractive power of the earth, and hence this force must be proportioned to the

mass.

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Newton's Third Law of Motion.

167. There is no action or motion in the universe but at the expense of an unequal and opposite concomitant action: or, as it is usually quoted, "Action and reaction are equal and opposite.

If a man in one boat pull at a rope attached to another boat, the two will approach. If they be equal in size and load, they will both move at the same rate in whichever the man may be; if unequal, the lesser mass will move the faster. With the rope attached to a large ship, the man's boat alone would seem to move; yet he really moves the ship a little; only if the ship be a thousand times as heavy as the boat, its motion will be but a thousandth part of that of the boat, and it would require a thousand men in a thousand boats pulling all together to make the ship meet them half way.

A magnet and a piece of iron attract each other equally, whatever disproportion there is between the masses. If either be balanced in a scale, and the other be then brought within a certain distance beneath it, the very same counterpoise will be required to prevent their approach, whichever be in the scale. If the two were hanging near each other as pendulums, they would approach and perhaps meet; but the smaller mass would perform the greater portion of the journey.

So a pound of lead and the earth attract each other with equal force; but that force makes the lead approach sixteen feet in a second towards the earth, while the contrary motion is as much less than this, as the earth is heavier than one pound, and is, therefore, utterly inappreciable.

The attraction of the earth and moon is mutual: but, as the earth is much the larger body, the attraction alters the motion of the moon far more than it does that of the earth; still, just as a boy, holding on by a man's coat, draws the coat away from the man to a certain extent by his reaction, though he may be compelled to move with the man, so the reacting force of the moon draws towards it the waters of the ocean—the loose jacket of the earth as it were—and so exhibits in the phenomenon of the tides the mutual character of the attraction.

168. A cannon, when fired, recoils with even greater force than the momentum given to the ball, for it suffers the reaction of the expelled gunpowder as well as of the ball; but the large mass of the cannon diffuses the reacting momentum and gives a small