Abstract
Galileo had shown by experimentation and theory how things fall, and documented his findings in his 1638 Dialog Concerning Two New Sciences. He asked simple questions about everyday phenomena and tested them with experiment until he understood the general laws behind the phenomena of motion. Isaac Newton (1642–1727) applied Galileo’s and Kepler’s laws of motion to the moon and planets and their satellites, and, with brilliant and subtle geometrical reasoning, developed the universal theory of gravitation that united earthly and heavenly phenomena. This he laid out with astonishing rigor in the first part of Newton’s Mathematical Principles of Natural Philosophy (first published in Latin in 1687, and commonly known today by the first word of its Latin title, the Principia), perhaps the greatest masterwork of human thought in the history of science. Newton showed that the same force (gravitation) that caused the fall of an apple from a tree caused Jupiter’s moons to orbit Jupiter, Earth’s Moon to orbit Earth, and all the planets and moons together to orbit the Sun.
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Notes
- 1.
Sir Isaac Newton [1]. (Short title) Principia. Book I (On the Motion of Bodies) and Book III (The System of the World) treat the motions of bodies and the consequences of gravitation. Book II of the Principia, however, deals with the motion of bodies in resisting mediums, of little concern in contemporary celestial mechanics. It may have been written chiefly to rebut Descartes theory of vortices. See, for example, the last Scholium in Book II, after Proposition LIII: “Hence it is manifest that the planets are not carried round in corporeal vortices; for, according to the Copernican hypothesis, the planets going round the sun revolve in ellipses, having the sun in their common focus…But the parts of a vortex can never revolve with such a motion.” From Sir Isaac Newton [1]. All excerpts from the Principia discussed here are drawn from this translation.
- 2.
Being at “rest” in space was shown by Einstein in 1905 to be an entirely relative concept. For now we will use the terms “rest” and “motion” in the sense of absolute rest and motion relative to an absolute frame of reference, as Newton intended.
- 3.
Principia, 13.
- 4.
Principia, Book III, 551–52.
- 5.
Principia, Axioms, or Laws of Motion, 13.
- 6.
If the Moon’s orbit were perfectly circular, its velocity at all points would be constant. It is assumed so for our instructional purposes, but any object in an elliptical orbit will move with velocities that vary depending on the nature of the ellipse, as it draws nearer or farther from the body orbited; its mean motion, however, will be constant. In the case of the Moon, too, the irregularities in the distribution of the mass of the Earth, the effects of the Sun’s gravity, etc. all influence the shape of its orbit.
- 7.
A “change of motion” in Law II is a change in [mass × velocity]. In dealing with fixed masses, as is the case with most classical problems of celestial mechanics, Law II means that force = mass × change in velocity. Since a change in velocity is acceleration, then force = mass × acceleration. It can be said that the measure of mass is the force required to accelerate it. The equation also suits those situations where mass does change, such as in rockets whose expenditures of fuel continually reduce the mass of the rocket as it ascends.
- 8.
Sometimes acceleration and force will appear to be used interchangeably; this is because acceleration is force per unit mass.
- 9.
Principia, Definition V, 2.
- 10.
Principia, Book I, Proposition I, Corollary 4, 42.
- 11.
While we cannot now take to time to prove this corollary to the proposition, it is an important one. It holds even for non-circular orbits. Newton creates parallelograms on the arcs of each orbit, with the chords forming the long axis of each parallelogram. For large arcs that are not circular, the line on each from A to the center of force may not be congruent (not match up) with the bisector of the chord. But as we take smaller and smaller units of time, the arcs too are diminished, the bisector lines converge with the force lines, and both point to the center of force. Thus, as we imagine the increments of time approaching zero (as Δt → 0) the direction of the force vectors in each orbit ultimately coincide with the lines bisecting the chords.
- 12.
The Moon’s orbit is not too far from circular, and so revolves around the Earth at a fairly constant velocity, running through its phases predictably in a cycle of a little more than 27 days. If an object is moving uniformly in a circular orbit, it will journey equal distances along any arc in equal times. It will also, according to Newton’s very first proposition (and consistent with Kepler’s Second Law discussed in Chap. 4), sweep out equal areas in equal times:
The areas which revolving bodies describe by radii drawn to an immovable centre of force, do lie in the same immovable planes, and are proportional to the times in which they are described. (Principia, Book I, Proposition I, Theorem I.)
This too makes intuitive sense. Visualize each orbit composed of little triangles drawn from the center, where the sides are the radii and the bases are the distances traveled by the object in equal times. The length of each base thus depends on the velocity of the body. Since the area of each triangle is proportional to its base times height, where height – the radius – is constant, the areas of all triangles in a given orbit will be the same. The areas, too, will be proportional to the velocity, since, as we said, the object’s motion is uniform along the arc. Now increase the number of triangles by making them smaller and smaller, until the triangles effectively merge into a circle, and the bases merge into the arc, and the conclusion is the same. In Newton’s words: “Now let the number of those triangles be augmented, and their breadth be diminished ad infinitum; and… their ultimate perimeter …will be a curved line: and therefore the centripetal force, by which the body is continually drawn back from the tangent to this curve, will act continually…” (Principia, Book I, Proposition I, Theorem I, 41.) This line of argument is how Newton demonstrated his first proposition.
- 13.
Principia, Book I, Proposition IV, 45.
- 14.
Ibid., Corollary 1.
- 15.
We use the symbol f to represent centripetal acceleration. We will maintain the reference here to forces, as Newton did, in describing the proportions, since acceleration is force per unit mass.
- 16.
Principia, Book I, Proposition IV, Corollary 9, 46.
- 17.
For example, if s = .01, then s2 = .0001; if s = .0001, then s2 = .0000001.
- 18.
The phrase centrifugal force appears to have originated with Christiaan Huygens who stated his Theorems on Centrifugal Force Arising from Circular Motion as early as 1659, having developed them even earlier. He did not publish them, however, until 1673, where they appeared appended to his book on pendulums. See Pendulum Clock, 176–8.
- 19.
John Herival [2].
- 20.
Ibid.
- 21.
Newton developed this concept in the Scholium to his Proposition IV. Principia, Book I, Proposition IV, 47.
References
Newton I (1949) Mathematical principles of natural philosophy (trans: Motte A, 1729, revised by Cajori F). University of California Press, Berkeley, p 395
Herival J (1965) The background to Newton’s principia, vol 7. Oxford University Press, London
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MacDougal, D.W. (2012). How the Moon Falls Toward the Earth (but Keeps Missing It). In: Newton's Gravity. Undergraduate Lecture Notes in Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5444-1_5
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