Page images
PDF
EPUB

608. Now, since it is each individual atom of the Earth that attracts each individual atom of the weight, we might expect, from our definition of gravity as well as from the well-known law that every action has a reaction, that the Earth, when the weight is dropped, at the end of one second rises upwards to the weight with the same momentum that the weight moves downwards to the Earth. No doubt it does; but as the Earth is a very large mass, this momentum represents a velocity infinitesimally small.

609. Again, were the Earth twice as large as it is, it would produce in one second of time a double velocity, or 64 feet per second; and were it only half as large, we should have only half the velocity, or 16 feet per second produced.

610. Hence we see that at the surface of the Moon the gravity is very small, whereas at the surface of the Sun it is enormous. There remains to consider the element of

distance.

A body at the surface of the Earth, or 4,000 miles from its centre, acquires, as we have seen, by virtue of the Earth's attraction, the velocity of 32 feet per second at the end of one second. During this one second it has not, however, fallen 32 feet; for, as it started with no velocity at all, and only acquired the velocity of 32 feet at the end, it will have gone through the first second with the mean velocity of 16 feet; it will, in fact, have fallen 16 feet from rest in one second. Now this body, at the distance of the Moon, or sixty times as far off, would only fall in one second towards the Earth a distance of

1612 or 1612 of a foot. Let us look into this a little 60X60 3600

closer.

611. Experiment shows, as we have seen, that attraction, or gravity, at the Earth's surface causes a body to fall 16 feet in the first second of fall, after which it has acquired a velocity of 2 × 16 = 32 feet during the second

second, and so on, according to the square of the time

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

stop

612. The Moon's curved path is an exact representation of what the path of our cannon-ball (Art. 604) would be at the Moon's distance from the Earth; in fact, the Moon's path MM', in Fig. 78, is compounded of an original impulse in the direction at right angles to EM, and

M'

FIG. 78.-Action of Gravity on the Moon's path.

therefore in the direction MB, and a constant pull towards the Earth-the amount of pull being represented for any arc by the line MA (Fig. 78). To find the value of MA, let us take the arc described by the Moon in one minute, the length of which is found by the following propor

tion :

=

MM'.

27d. 7h. 43m.: Im.:: 360°: 33′′ nearly From this value of the arc, the length of the line MA is found to be 16, feet when ME =

=

240,000 miles. That

is, a body at the moon's distance falls as far in one rninute as it would do on the Earth's surface in one second-that is, it falls a distance 60 times less. A body on the Earth's surface is 4,000 miles from the Earth's centre, whereas the Moon lies at a distance of 240,000 from that centre-that is, exactly (or exactly enough for our present purpose) 60 times more distant.

3600

613. It is found, therefore, that the deflection produced in the Moon's orbit from the tangent to its path in one second is precisely of 16 a foot. Here we see that, as the Moon is sixty times further from the Earth's centre than a stone at the Earth's surface, it is attracted to the Earth 60 X 60, or 3600 times less. In fact, the force is seen experimentally to vary inversely as the square of the distance of the falling body from the surface. It was this calculation that revealed to Newton the law of universal gravitation.

614. Long before Newton's discovery, Kepler, from observations of the planets merely, had detected certain laws of their motion, which bear his name. They are as follows:

I. Each planet describes round the Sun an orbit of elliptic form, and the centre of the Sun occupies one of the foci.

II. The areas described by the radius-vector of a planet are proportional to the time taken in describing them.

III. If the squares of the times of revolution of the planets round the Sun be divided by the cubes

of their mean distances, the quotient will be the same for all the planets.

615. We have already in many places referred to the first law II and III. require special explanation, which/ we will give in this place. We stated in Art. 293 that the

planets moved faster as they approached the Sun; II. tells how much faster. The radius-vector of a planet is the line joining the planet and the Sun. If the planet were always at the same distance from the Sun, the radius-vector would not vary in length; but in elliptic orbits its length varies; and the shorter it becomes, the more rapidly does the planet progress. This law gives the exact measure of the increase or decrease of the rapidity.

616. In Fig. 79 are given the orbit of a planet and the Sun situated in one of the foci, the ellipticity of the

P

FIG. 79.-Explanation of Kepler's second law.

planet's orbit being exaggerated to make the explanation clearer. The areas of the three shaded portions are equal to each other. It is readily seen that where the radiusvector is longest, the path of the planet intercepted is shortest, and vice versa. This, of course, is necessary to produce the equal areas. In the figure, the arcs P P1, P2 P3, and P4Ps, are those described at mean distances perihelion and aphelion respectively, in equal times; therefore, as a greater distance has to be got over at perihelion

and a less one at aphelion than when the planet is situated at its mean distance, the motion in the former case must be more rapid, and in the latter case slower, than in other parts of the orbit.

617. The third law shows that the periodic time of a planet and its distance from the Sun are in some way bound together, so that if we represent the Earth's distance and periodic time by 1, we can at once determine the distance of, say, Jupiter from the Sun, by a simple proportion; thus

[blocks in formation]

That is, whatever the distance of the Earth from the Sun may be, the distance of Jupiter is 3/140 times greater. 618. The following table shows the truth of the law we are considering :

[blocks in formation]

Time squared.

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

Distance cubed.

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

LESSON XLIX.-Kepler's Second Law proved. Centrifugal Tendency. Centripetal Force. Kepler's Third Law proved.

Ellipse.

The Conic Sections.

Movement in an

619. As these laws were given to the world by Kepler, they simply represented facts; for, owing to the backward

« PreviousContinue »