The pressure on the inclined plane is to the weight of the body as the base, BC, of the inclined plane, is to its length, A B.

The accelerated motion of a descending body is to that which it would have had if it fell freely, as the height, A C, of the plane is to its length, A B. The final velocity which the descending body acquires is equal to that

E

B

Fig. 98.

which it would have had if it had fallen freely through

a distance equal to the height of the plane; and, therefore, the velocities acquired on planes of equal heights, but unequal inclinations, are equal.

[It matters little what may be the inclination of the plane along which a body moves; the force must be calculated in the same manner; i. by the parallelogram of forces. For example, suppose that the plane was very oblique, as E F, in Fig. 98, then we should say that C the line, DC, was almost equal to the line D B; and hence that the force of the ball, D, moving on the inclined plane, E F, would differ very little from the F force of the same body falling through air, and unobstructed otherwise than by the resistance of the atmosphere.

The space passed through by a body falling freely is to that gone over an inclined plane, in equal times, as the length of the plane is to its height. [Galileo made several experiments with the inclined plane for the purpose of determining if the laws

of falling bodies were correct. It is easy to see for ourselves, and therefore we will try an experiment with Galileo's inclined plane. Here is a narrow channel of wood, a b, which is made for the purpose of trying this experiment, having its inside made perfectly smooth;

a.

Fig. 99.

it is twelve feet long, and divided into feet and inches on the inside, and marked so that any person can readily distinguish the figures. Let us place one end, b, upon some stones, e d, so as to raise it above the other end, We will now place a ball in the channel, at the upper part, b, and you will see that when it is liberated, it will roll rapidly down the channel, and also that the spaces traversed in 1, 2, 3, and its seconds, are as the squares of the time necessary to traverse the spaces. By this experiment we learn what are the true laws of gravitation, and discover that gravity is an uniform accelerating force.]

If a series of inclined planes be represented, in position and length, by the chords of a circle terminating at the extremity of the vertical diameter, the times of descent down each will be equal, and also equal to the time of descent through that vertical diameter. Thus, let A D, A G, DB, GB, be chords of a circle terminating at the extremities, A B, of the vertical diameter, and regarding these as inclined planes, a body will descend from A to D, or A to G, or D to B, or G to B in the same time that it would fall from A to B.

Fig. 100.

F

If a body descend down a system of several planes, A C, Fig. 101,

B

with

different inclinations, it would acquire the same velocity as it would have had in descending through the same vertical height, A B, though the times of descent are unequal.

If a body which has descended an inclined plane meets at the foot of it a second plane of equal altitude, it will ascend this plane with the velocity acquired in coming down the first, until it has reached the same altitude from which it descended. Its velocity being now expended, it will re-descend, and ascend the first plane as before, oscillating down one plane, up the other, and then back again. The same thing will take place if, instead of being over an inclined plane, the motion be made over a curve, as in Fig. 102. In practice, however, the resistance of the air and friction soon bring these motions to an end.

Fig. 101.

Fig. 102.

In the motions of projectiles two forces are involved the continuous action of gravity, and the momentary force which gave rise to the impulse-such as muscular exertion, the explosion of gunpowder, the action of a spring, &c.

The resulting effects of the combination of these forces will differ with the circumstances under which they act. If a body be projected downward in a vertical line, it follows its ordinary course of descent, its accelerated motion arising from gravity being conjoined to the original projectile force.

e d

Fig. 103.

But if it be thrown vertically upward, the action of gravity is to produce an uniform retardation. Its velocity becomes less and less, until finally it wholly ceases. The body then descends by the action of the earth, the time of its descent being equal to that of its ascent, its final velocity being equal to its initial velocity.

[If a body be thrown in any other than a vertical direction, it will describe a curved line, the form of which may be easily deduced from the laws of falling. Let us assume the simplest case for instance, that the body be urged by any force in a horizontal direction. If there were no such force as gravity, the body would continually move in a horizontal direction, and with an uniform velocity. By reason of the first impelling force, it would traverse the space, a b, in one second, the equally large space, bc, in two seconds, and so on, and must, consequently, at the end of the first, second, third, &c., seconds, have reached the points, bed, &c. But it has sunk from the force of gravity: in the first second it fell 15 feet; consequently, at the end of that time, instead of being at b, it will be 15 feet below it. At the end of the next second, it is 60 feet below c; at the end of the third, 135 feet below it, &c. The curved line, ag, described by the body in this manner, is a parabola.-Professor Müller's Lectures, Lecture X.]

But if the projectile force forms any angle with the direction of gravity, the path of the body is in a parabolic curve, as seen in Fig. 104. If the direction of the projection be horizontal, the path described will be half a parabola.

This, which passes under the title of the parabolic theory of projectiles, is found to be entirely departed from in practice. The curve described by shot thrown from guns is not a parabola, but another curve, the Ballistic. In vertical projections, instead of the times of ascent and descent being equal, the former is less. The final velocity is not the same as the initial, but less. Nor is the descending motion uniformly accelerated; but, after a certain point, it is constant. Analogous differences are discovered in angular projections.

Fig. 104.

The distance through which a projectile could go upon the parabolic theory, with an initial velocity of 2,000 feet per second, is about 24 miles ; whereas no projectile has ever been thrown further than 5 miles.

In reality, the parabolic theory of projectiles holds only for a vacuum ; and the atmospheric air, exerting its resisting agency, totally changes all the phenomena-not only changing the path, but, whatever may have been the initial velocity, bringing it speedily down below 1,280 feet per second. The cause of this phenomenon may be understood from Fig. 105. Let B be a cannon-ball, moving from A to C with a velocity of more than 2,000 feet per second. In its flight it removes a column of air between A and B, and as the air flows into a vacuum only at the rate of 1,280 feet per second, the ball leaves a vacuum behind it. In the same manner it powerfully compresses the air in front. This, therefore, steadily presses it into the vacuum behind, or, in other words, retards it, and soon brings its velocity down to such a point that the ball moves no faster than the air moves- that is, 1,2-0 feet per second.

A

B

Fig. 105.

A shot, thrown with a high initial velocity, not only deviates from the parabolic path, but also to the right and left of it, perhaps several times. A

A

30 40 50

ball striking on the earth or water at a small angle bounds forward or ricochets, doing this again and again until its motion

ceases.

The initial velocity given by gunpowder to a ball, and, therefore, the explosive force of that material, may be determined by the Ballistic pendulum. This consists of a heavy mass, A, Fig. 106, suspended as a pendulum, so as to move over a graduated Into this, at the centre of percussion, the ball is fired. The pendulum moves to a corresponding extent over the graduated arc, with a velocity which is less according as the weight of the ball and pendulum is greater than the

Fig. 106.

arc.

weight of the ball alone.

It

The explosive force of gunpowder is equal to 2,000 atmospheres. expands with a velocity of 5,000 feet per second, and can communicate to a ball a velocity of 2,000 feet per second. The velocity is greater with long than short guns, because the influence of the powder on the ball is longer continued.

CHAPTER XXI.

OF MOTION ROUND A CENTRE.

Peculiarity of Motion on a Curve-Centrifugal Force-Conditions of Free Curvilinear Motion-Motion of the Planets-Motion in a Circle-Motion in an Ellipse-Rotation on an Axis-Figure of Revolution—Stability of the Axis of Rotation.

IN considering the motion of bodies down inclined planes, we have shown that the action of gravity upon them may be divided into two portions—one producing pressure upon the plane, and therefore acting perpendicularly to its surface; the other acting parallel to the plane, and therefore producing motion down it.

It has also been shown that, in some respects, there is an analogy between movements over inclined planes and over curved lines, but a further consideration proves that between the two there is also a very important difference. A pressure occurs in the case of a body moving on a curve, which is not found in the case of one moving on a plane. It arises from the inertia of a moving body. Thus, if a body commences to move down an inclined plane, the force producing the motion is, as we have seen, parallel to the plane. From the first moment of motion to the last, the direction is the same, and inasmuch as the inertia of the body, when in motion, tends to continue that motion in the same straight line, no deflecting agency is encountered.

A

But it is very different with motion on a curve. descent from A to B is perpetually changing; the curve from its form resists, and therefore deflects the falling body. At any point its inertia tends to continue its motion in a straight line; thus, at A, were it not for the curve, it would move in the line A a, at B in the line B b, these lines being tangent to the curve at the points A and B. The curve, therefore, continually deflecting the falling body, experiences a pressure itself—a pressure which obviously does not occur in the case of an inclined plane. This pressure is

66

Here the direction of

Fig. 107.

denominated centrifugal force," "because the moving body tends to fly from the centre of the curve.

In the foregoing explanation we have regarded the body as being

H

W

50

Fig. 108.

compelled to move in a curvilinear path, by means of an inflexible and resisting surface. But it may easily be shown that the same kind of motion will ensue without any such compelling or resisting surface, provided the body be under the control of two forces, one of which continually tends to draw it to the centre of the curve in which it moves, while the other, as a momentary impulse, tends to carry it in a different direction.

Thus, let there be a body, A, Fig. 108, attached by another body, S, and also subjected to a projectile force tending to carry it in the direction A H. Under the conjoined influence of the two forces it will describe a curvilinear orbit, A T W.

The point to which the first force solicits the body to move is termed the centre of gravity-that force itself is designated the centripetal force, and the momentary force passes under the name of tangential force.

The following experiment clearly shows how, under the action of such forces, curvilinear motion arises. Let there be placed upon a table a ball, A, and from the top of the room, by a long thread, let there be suspended a second ball, B, the point of suspension being vertically over A. If now we remove B a short distance from A, and let it go, it falls at once on A, as though it were attracted. It may be regarded, therefore, as under the influence of a centripetal force emanating from A. But if, instead of simply letting B drop upon A, we give it an impulse in a direction at right angles to the line in which it would have fallen, it at once pursues a curvilinear path, and may be made to describe a circle or an ellipse, according to the relative intensity of the tangential force given it.

Fig. 109.

This revolving ball imitates the motion of the planetary bodies round the sun.

To understand how these curvilinear motions arise, let C be the centre of gravity, and suppose a body at the point a. Let a tangential force act on it in such a manner as to drive it from a to b in the same time as it would have fallen from a to d. By the parallelogram of forces it will move to f When at this point, f, its inertia would tend to carry it in the direction fg, a distance equal to a f, in a time equal to that occupied in passing from a to f; but the constant attractive force still operating tends to bring it to h; by the parellologram of forces it therefore is carried to k and by similar reasoning we might show that it will next be found at n, and so on. But when we consider that the centripetal force acts continually, and not by small interrupted impulses, it is obvious that, instead of a crooked line, the path which the body pursues will be a continuous curve. The planets move in their orbits round the sun, and the satellites round

;

Fig. 110.