Изображения страниц
PDF
EPUB

ever, or for ever move on at the same rate and in the same straight line, unless it is acted upon by some external force.

2. If a body be acted upon by only one force, it will move in whatever direction that force impels it, and with a velocity proportional to the intensity of the force.

Thus, if the bodies composing the solar system were acted upon only by the force of gravitation, they would move directly towards each other, with velocities directly as their masses or quantities of matter, and inversely as the squares of their distances,—that is, the satellites or secondary planets would move toward their primaries, and the primaries toward the sun, until the whole were accumulated into one mass. From the distances of the planets, and their quantities of matter, we are enabled to calculate the precise times in which, supposing them at their mean distance, they would perform their journey.

[merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small]

When a body is acted upon by two forces, the force that results from their combination, and, consequently, has the direction and velocity of the motion of the body, will vary according to the mode in which the two forces

act.

Let us first suppose that the two forces are equal to each other, and that each of them tends to draw or drive the body in a straight line. Now, in this case, it is perfectly obvious that, if the two forces acted in the same direction, the resulting force would be equal to them both; if they acted in directions diametrically opposite, the one would completely overcome the other, so that there would remain no force to move the body, which would consequently remain at rest. The maximum or greatest effect of two equal forces, is when they act in the same direction, and this is equal to their sum; and the minimum, or least effect, is when they act directly opposite to each other, and this is equal to nothing. Hence

3. The effect produced by the joint action of two equal forces may be nothing, or their sum, or any thing intermediate.

this

Let us see how the amount is in any one case to be determined. For purpose, let B be any body acted upon by two equal forces, a, c, and a, d. If the angle, c, a, d, were opened

till a, c, and a, d, were in the same straight line, the one force would just balance the other; and if both forces were brought into the same direction, by narrowing the angle, c, a, d, till a, c, and a, d, had the same situation, the result would be their sum. Hence, generally, the resulting force increases as the angle is diminished, and diminishes as the angle is increased. The force,

e

C

B

in these cases, will always produce motion in a direction intermediate between those of the two forces-midway when they are equal, and nearer to the more powerful when they are not.

If the two forces are unequal, the result will still be their sum when in the same direction; their difference when in opposite directions; and something less than the former, but greater than the latter, when they form an angle, and it will increase as the angle is diminished. Both the extent and the direction of this force are susceptible of calculation; as it is found that the resulting force, a, e, is the diagonal of a parallelogram, of which the two forces, a, c, and a, d, are the sides. Hence,

4. When a body is acted upon by two forces, making an angle with each other, the resulting force is expressed by the diameter of a parallelogram, of which the two forces are the sides, and joining the angle made by the two forces.

Again, if a body is acted upon by two forces, the one of which tends to drive or draw it forwards in the same straight line, and the other to draw or drive it to the same fixed centre, the body will (if these forces are uniform, that is, act always with the same intensity, and are also at right angles to each other) move continually in the circumference of a circle, of which the point, whence the fixed force acts, will be the centre.

Thus, if the body, B, is impelled by b, c, the gravitation toward the body, C, and also by the force b, f, acting at right angles to that gravitation, it will move uniformly in the circumference of the circle, B, F, E, D.

B

F

C

D

E

The force by which the revolving body, B, tends toward the central body, C, is called a centripetal force; and in the case of gravitation, it is directly as the quantity of matter, and inversely as the square of the distance.

The force by which the revolving body, B, tends to move along the line, b, f,is called a centrifugal force; and it depends upon, or rather is measured by, the velocity of the revolving body, which, in order to produce a circular motion, must be perfectly uniform; and the rate depends upon the distance from the central body.

There is a familiar illustration of this: Take a ball with a thread attached. If you hang it by the thread without motion, the force of gravitation makes it stretch the thread and hang downwards to the earth. Swing it round, and it acquires a circular motion, stretching the thread outwards with greater force the more rapidly that it is swung; and if the thread breaks, or is let slip, the ball flies off, until the force which it had acquired by swinging be exhausted, and it falls to the earth by the force of gravity. The force which the ball acquires by swinging is a centrifugal force; the force of the thread which retains it is a centripetal force; and if both were continued, the ball would continue to swing round in a circle.

By comparing the periods of the planets with their mean distances from the sun, Kepler deduced the law of their comparative velocities, which has been confirmed and found to agree with the doctrine of gravitation, by Newton and other astronomers. This is as follows:

5. "The squares of the times of revolution in any two planets, vary in

proportion to the cubes of their mean distances from the sun,"-that is, if the square of the time of one be multiplied by the cube of the distance of another, the products will, in all cases, be equal.

Thus, if the mean distance of the earth be 95 millions of miles, and its period one year, and the distance of Mars 144 millions, the period of Mars may be found thus:

As 95: 144 = 19: the square of the period of Mars in years; 144 x 144 x 144

or

95 x 95 x 95

the period of Mars.

Multiplying the upper numbers together, dividing their product by that of the under, and taking the square root, we have the periodic time of Mars 1 year 318 days, which differs from the real period, taking all the variations of the motion into account, by only three or four days; and this is sufficiently near-for shewing that the principle is correct: that, if the times of revolution and distances of the planets thus agree, they are sustained in their orbits by the mutual operation of equal centrifugal and centripetal forces, the first tending to drive the planet forward in a straight line in the plane of its orbit, and the latter deflecting it from thatplane toward the sun.

The square of the velocity is thus always inversely as the cube of the distance, whether as applied to different planets or to the same planet in different parts of its orbit. Hence if a planet be one-fifth nearer to the sun at one time than at another, we can easily calculate the comparative rates of its motion: they will be

√5 × 5 × 5 when nearest; and

√4 × 4 × 4 when most distant:

that is, when one-fifth nearer, it will pass over rather more than eleven miles of absolute space, in the same time in which, at the greater distance, it takes to pass over eight miles.

SECTION II.

MOTION OF A PRIMARY PLANET.

From a careful comparison of the numerous observations made by Tycho Brahe, Kepler was enabled to establish two other laws of the planetary motions, besides that of the relation between the distance from the sun and the rate of motion. They are these:

1. All the planets revolve round the sun in such a way as that the radius vector, or line joining the centres of the sun and planet, (the one end of which is supposed to remain fixed in the place of the sun, and the other to move along with the planet) describes equal areas-that is, passes over equal portions of space-in equal periods of time.

2. That each planet moves in an ellipse, having the sun in one of the foci.?!

These laws, which were the result of observation and fact, and not of any assumed hypothesis, were found by Newton to agree in every respect with the theory of gravitation toward the sun. He demonstrated,

1. That the deflection of each planet is a force always directed toward the sun.

2. That the primary planets and comets are retained in their orbits round the sun, and the satellites round their respective primaries, by a force tending toward the central body.

3. That the force by which a planet describes areas proportional to the times, round the focus of its elliptic orbit, is inversely as the square of its distance from that focus. Thus, if the distance be one-fifth less, the comparative force will be, as 12 to 8; the motion in absolute space was already stated as being rather more than 11 to 8; this deflecting force follows the same law as gravitation, and consequently balances that.

4. That, if a planet, when at its mean distance from the sun, be projected, with the velocity which it then has, in a direction perpendicular to the radius vector, it will describe a circle round the sun in precisely the same time as it describes the ellipse.

These four propositions, for the demonstrations of which there is no room in this short essay, contain the whole elements of the planetary motions; and by keeping them in mind, as well as what has been said of the force resulting from two forces acting on the same body at an angle, and also referring to the following figure, a general idea may be formed of the rate of motion in a planet at every point of its orbit, as well as of the rea sons why it advances toward the sun, and again recedes from that luminary.

[ocr errors][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small]

In this figure, let the ellipse represent the orbit of any of the primary planets, as for instance that of the earth. S, is the upper focus, or place of

« ПредыдущаяПродолжить »