Laws of Motion

The Law of Gravitation.

The application of this theory of motion depends on the discovery of a classification of forces, or laws of force. We have experience of mutual actions between terrestrial bodies, such as pressure between bodies in contact, and magnetic attractions and repulsions. To this experience Newton added the law of gravitation, namely that between any two particles there is a force of attraction, proportional to the product of their masses, and inversely proportional to the square of the distance between them. It might be supposed that this law could be tested at once by laboratory experiments, but the force between bodies that can be handled is so small that it is difficult even to detect it ; thus Newton depended mainly on astronomical verification. Laboratory measurements have been made since, from 1798 onwards, in particular cases; the principal result being that the mass of the earth in pounds is known. New ton made an estimate of this which proved to be nearly correct, but it was not essential for his work.

In the Principia we have Galileo's acceleration attributed to gravitation, and shown to be consistent with the moon's orbit about the earth; there are also calculations, according to the law of gravitation, extending to the whole solar system, treated as isolated, including the theory of precession of the equinoxes due to motion of the earth's axis. The orbits of comets, and the theory of tides, and a variety of other problems, are also con sidered. The verifications of the Newtonian theory, thus ob tained, were overwhelming. Kepler's laws, with a slight modifica tion introduced by Newton in the third law, were shown to agree with the orbit each planet would perform if it and the sun alone existed; and the perturbations of these orbits, so far as they had been detected, to be such as might be attributed to the coex istence of all the planets. With regard to the perturbations of an orbit of one body about another, due to a third body, the best data at Newton's disposal were the irregularities of the orbit of the moon about the earth. Attributing these to the disturbing influence of the sun, Newton showed how they could be accounted for—a remarkable mathematical feat for a first attempt at a thing of this kind.

Relativity.

Further verifications of the Newtonian scheme for the solar system were carried on during the next two centuries. They were not absolutely perfect; but no definite establishment of a flaw in the theory occurred till 1915, when a small error in the motion of the planet Mercury, as calculated from known data by Newton's methods, was shown by Einstein to be accounted for by his theory of relativity. Seventy years earlier, when Uranus seemed to go astray, the errors were accounted for by perturba tions due to the planet Neptune, previously unknown. For Mer cury a similar explanation had been diligently sought, but had not been found. The theory of relativity is held to have estab lished that, for a planet so near the sun, and possessing, as Mercury does, an orbit whose eccentricity is not very small, Kepler's elliptic orbit is at fault to a perceptible extent, and will be replaced by a slightly different fundamental orbit. The solar eclipses of 1917 and 1922 confirmed this by showing apparent dis placements of stars seen near the sun which were predicted by the theory of relativity; a new property of bodies, associated with mass, being thus verified. The result is that the Newtonian

theories have become an approximately correct fragment of a wider and more fundamental theory, and their scope and degree of accuracy have been to some extent ascertained.

Every scientific theory is provisional, and is subject to correc tions and limitations ; and the discovery of these is an important step in the direction of progress. E. Mach's Science of Mechanics may be consulted with reference to difficulties that have been felt in the past with regard to a presumed universal validity of the Newtonian apparatus. Such difficulties are now happily at an end, and the measures of mass by inertia and by gravitation are rationally harmonized. (See RELATIVITY.) Terrestrial Dynamics.—Let us return now to terrestrial dynamics, dealing with the motion of material bodies relative to the earth. Here the Newtonian theories are found to be sound; but all exact calculations are rather complicated, chiefly on account of the earth not being a Galilean base. Practically it is necessary to discover, for each class of problem, what sort of approximation gives a sufficiently correct result. The simplest approximate device, in many cases sufficient, though no feature of it is accurate, is the obvious one of treating the earth as if it were a Galilean base, and applying to each particle its weight by way of external force, treated as constant. Here weight is not technically a force, but is employed to represent the joint effect of gravitation and the earth's motion. It is Galileo's acceleration multiplied by the mass of the particle. A closer approximation to exact conditions gives a falling body a divergence from the plumb line vertical, which would amount to about half an inch in a fall of 23oft. at the equator. This sort of effect is shown clearly by the trade winds. The theory of such an apparatus as Foucault's pendulum, or the gyro-compass (see GYROSCOPE), requires more exact accourit to be taken of the earth's motion. And in the theory of ocean tides, the main factor is the diurnal variation of gravitational force, due to changes in the positions of the sun and moon relative to the earth as it rotates. Here we have a thing which should affect the regularity of a pendulum clock; but, when the magnitude of the effect on a clock is calculated, it is found to be too small to be observed. (See MECHANICS and TIDES.) The i9th century saw the important development of the theory of energy, which linked the dynamics of material bodies with other branches of physics and with chemistry, and thus stimulated wider applications of dynamical theory. But the quantum theory (q.v.) has now demonstrated limitations of this procedure in the case of atomic systems. Thus we see that, during the first quarter of the loth century, a definite thing has been accom plished with regard to the Newtonian theories, in that the range over which they are practically valid, which had remained uncer tain for more than two centuries, has been to a considerable extent marked out.