Thus let D (Fig. 13) be the centre of the earth, and let a be the place of the moon at any moment, and c be its place one second later. When the moon was at A, it was moving in the direction A B, and would have continued to move in this line, by virtue of the first law of motion, had not the earth interfered. It would thus have been at B at the end of the first second, but by the attraction of the been pulled from B to C in one second of time. corresponds to the 49 metres through which a the surface of the earth through the force of gravity in one second; so that we have the following proportion :

A

D

FIG. 13.

B

earth it has

B C, in fact, body falls at

Force of earth's attraction at the moon is to force of same at the earth's surface as BC is to 4'9 metres.

Now AC being the arc of a circle, of which D is the centre, and A B a tangent at A, we have, by a well-known proposition in geometry, A B2 = 2 CD. B C nearly, or (since A B is nearly equal to AC) AC2 = = diameter of orbit X BC A C2 nearly; hence B C = diameter of orbit.

Now,

the moon is, in round numbers, 240,000 miles from the earth's centre, or sixty times as far as the earth's surface is, and hence the circumference of the moon's orbit is 1,508,000 miles. This distance it describes in 27d. 7h. 14m., and hence the length described in one second, or A C, will be 1,508,000 2,360,580

Again, the diameter of the orbit, or double the radius, is 480,000 miles; hence B C

=

(0.639)2
480,000

miles = 00137 metres.

Hence, also, the proportion previously stated becomes-Force of earth's attraction at the moon is to force of same at earth's surface as 00137 metres is to 4'9 metres,

or as

I

3,600

is to 1; that is to say, as a matter of fact, the force of the earth's attraction at the moon's surface is

3 of what it is at the earth's surface. Now, the moon is sixty times farther away from the earth's centre than the earth's surface is; and hence if the law of gravitation, as announced by Newton, holds good, the earth's attraction at

the moon's surface should only be

I

(60)2

of what it is at the

: this, therefore, is the

proportion which ought to hold if Newton's law be true; but this we have seen is also the actual proportion between the two attractions. The law, therefore, holds good in the case of the moon, and in like manner it might be shown to hold for the various members of the solar system, regarding the sun as the gravitating body.

38. We are thus led to the grand law of universal gravitation, which may be stated as follows:-Every substance in the universe attracts every other substance with a force jointly proportional to the mass of the attracting and of the attracted body, and varying inversely as the square of the distance.

To illustrate this: suppose that we have two bodies of mass equal to unity and distance from each other equal also to unity, and suppose (for the occasion) that the attraction between them is also unity.

(1) Were the one body increased six times, the whole attraction would become 6.

(2) Were both bodies increased six times, each unit of the one body would attract each unit of the other with a force equal to unity; hence the whole attraction would be 6 × 6, or 36.

(3) In like manner if one body had a mass equal to 6, and the other a mass equal to 4, the whole attraction would

be 24.

(4) If in addition to the masses in (3) the distance were

doubled, the attraction would now be

24

2 X

=

6.

(5) Let the mass of the one be 9, of the other 7, and the

39. We cannot well make use of falling bodies to illustrate the laws of motion, because they fall too rapidly for us; but there is a machine, called Atwood's Machine, which so modifies the velocities of falling bodies as to make them suitable for our purpose. The essential part of this machine is a fixed pulley over which a fine silk thread passes. The two ends of the axle of this pulley (Fig. 14) are made to lie on the circumferences of two wheels, by which means the friction of the axle is reduced to a minimum. A thread passes over the circumference of the pulley, and to the extremities of this thread two hollow boxes of equal weight are attached; a clock connected with the machine shows seconds, and beats in an audible and distinct manner. Besides this, we have a graduated rod to which are attached several plates and several rings; the plates, as in the figure, will arrest either of the boxes in its descent, but the rings will allow it to pass.

40. Experiment A.—Suppose now that each of the boxes weighs 100 grammes, and that we put 400 grammes into the one and 450 into the other: also let us take into account the pulley in our estimation of the weight of the boxes. It is clear that the heavier box will begin to descend, and that the lighter will mount up.

The mass to be moved is in all (including the weight of the boxes) 1,050 grammes, while the force is that caused by the 50 grammes of excess in one of the boxes.

=

=

50 X 9'8
490, which
On the other hand, the

Now, a gramme being unit of mass, this force will be denoted by 50 X force of gravity · will therefore be the moving force. whole mass to be moved is 1,050, and hence the velocity acquired in one second (represented by the moving force

divided by the mass to be moved) will be 490

1,050

= 0'467

nearly, while the space

passed through will be

half of this (Art. 21), or O'2335 metres. Suppose now we place on our graduated rod a plate o°2335 of a metre below the box containing the larger weight, and allow this box to fall just when the clock is beating a second, it ought to strike against the plate exactly one second after it began its descent, and just when the clock is beating the next second.

Experiment B.-Having made Experiment A, and ascertained that the agrees with our

result

calculations, let us now

double our moving force,

keeping however the same total mass.

FIG. 14.

Thus let there be

375 grammes put into the one box, and 475 into the other, so that the total mass, including that of the boxes, will be 475 +375 + 200 = 1,050, or the same as before, while the excess of weight will be 100 grains, and the moving force 100 × 9.8 = 980, being double of that in Experiment A. The 980 velocity acquired in one second will therefore be

1,050

=

o'934 metres nearly, while the space passed through will be half of this, or o'467 metres.

Let us therefore place a plate so as to arrest the box with the larger weight when it has passed through 0'467 metres,

and we shall find as before that it will strike against the plate exactly one second after it began its descent.

Let us now realize what we have proved by means of these two experiments. In Experiment A we had a mass equal to 1,050, and a moving force equal to 50 × 9·8, and we obtained as the result the velocity of 0467 metres in one second.

In Experiment B we have the same mass, namely, 1,050, and a double moving force, or 100 X 98, and we obtained in one second a velocity equal to o‘934 metres.

We thus see that while the mass remains the same, the velocity generated in unit of time varies as the force.

41. Experiment C.—In this experiment let us keep the same force we had in Experiment A,—namely, that produced by an excess of 50 grammes, but let us diminish the whole mass by one-half. This will be done by putting 137 5 grammes into the one box, and 187.5 grammes into the other, for the excess will be 50 grammes, the same as before, while the whole mass, including that of the boxes, will be 1375 + 1875 + 200 = 525, or the half of 1,050, which was the mass in Experiment A. The velocity acquired in one second will here be 50 X 9'8

525

= 0.934, and the space passed

over in one second half of this, or o'467 metres.

Placing the stage at this distance below the heavier box, it will be found to reach it at the expiration of exactly one second.

Comparing together Eperiments A and C, we find that in both we had the same moving force, or that due to an excess of 50 grammes; but in Experiment A we had a mass of 1,050, while in Experiment C the mass was only half of this. The result obtained was a velocity in Experiment C twice as great as that in Experiment A. We thus see that while the moving force remains the same, the velocity generated in unit of time varies inversely as the mass. Taking the results of all these experiments, we perceive that the correct measure of a force is that already given, namely, the momentum, or product of mass into velocity generated in unit of time.