32. Centripetal Acceleration
Acceleration is the rate of change of velocity, but the velocity of any body has both a magnitude, known as the speed, and a direction. The velocity of a body moving on a circle is continually changing its direction, turning toward the circle’s center, so even at constant speed it undergoes a continual acceleration toward the center, known as its centripetal acceleration.
Let us calculate the centripetal acceleration of a body traveling on a circle of radius r, with constant speed v. During a short time interval from t1 to t2, the body will move along the circle by a small distance vΔt, where Δt (delta t) = t2 – t1, and the radial vector (the arrow from the center of the circle to the body) will swivel around by a small angle Δθ (delta theta). The velocity vector (an arrow with magnitude v in the direction of the body’s motion) is always tangent to the circle, and hence at right angles to the radial vector, so while the radial vector’s direction changes by an angle Δθ, the velocity vector’s direction will change by the same small angle. So we have two triangles: one whose sides are the radial vectors at times t1 and t2 and the chord connecting the positions of the body at these two times; and the other whose sides are the velocity vectors at times t1 and t2, and the change Δv in the velocity between these two times. (See Figure 24.) For small angles Δθ, the difference in length between the chord and the arc connecting the positions of the bodies at times t1 and t2, is negligible, so we can take the length of the chord as vΔt.
Now, these triangles are similar (that is, they differ in size but not in shape) because they are both isosceles triangles (each has two equal sides) with the same small angle Δθ between the two equal sides. So the ratios of the short and long sides of each triangle should be the same. That is
and therefore
This is Huygens’ formula for the centripetal acceleration.
Figure 24. Calculation of centripetal acceleration. Top: Velocities of a particle moving on a circle at two times, separated by a short time interval Δt. Bottom: These two velocities, brought together into a triangle whose short side is the change of velocity in this time interval.
33. Comparing the Moon with a Falling Body
The ancient supposed distinction between phenomena in the heavens and on Earth was decisively challenged by Newton’s comparison of the centripetal acceleration of the Moon in its orbit with the downward acceleration of a falling body near the surface of the Earth.
From measurements of the Moon’s diurnal parallax, its average distance from the Earth was accurately known in Newton’s time to be 60 times the radius of the Earth. (The actual ratio is 60.27.) To calculate the Earth’s radius Newton took 1' (minute of arc) at the equator to be a mile of 5,000 feet, so with 360° for a circle and 60’ to 1°, the earth’s radius was
(The mean radius is actually 20,926,300 feet. This was the greatest source of error in Newton’s calculation.) The orbital period of the Moon (the sidereal month) was accurately known to be 27.3 days, or 2,360,000 seconds. The velocity of the Moon in its orbit was then
This gives a centripetal acceleration of
According to the inverse square law, this should have equaled the acceleration of falling bodies on the surface of the Earth, 32 feet/second per second, divided by the square of the ratio of the radius of the Moon’s orbit to the radius of the earth:
It is this comparison, of an “observed” lunar centripetal acceleration of 0.0073 foot/second per second with the value expected from the inverse square law, 0.0089 foot/second per second, to which Newton was referring when he said that they “answer pretty nearly.” He did better later.
34. Conservation of Momentum
Suppose two moving objects with masses m1 and m2 collide head-on. If in a short time interval δt (delta t) object 1 exerts force F on object 2, then in this time interval object 2 will experience an acceleration a2 that according to Newton’s second law obeys the relation m2a2 = F. Its velocity v2 will then change by an amount
δv2 = a2 δt = F δt/m2
According to Newton’s third law, particle 2 will exert on particle 1 a force –F that is equal in magnitude but (as indicated by the minus sign) opposite in direction, so in the same time interval the velocity v1 of object 1 will undergo a change in the opposite direction to δv2, given by
δv1 = a1 δt = −F δt/m1
The net change in the total momentum m1v1 + m2v2 is then
m1δv1 + m2δv2 = 0
Of course, the two objects may be in contact for an extended period, during which the force may not be constant, but since the momentum is conserved in every short interval of time, it is conserved during the whole period.
35. Planetary Masses
In Newton’s time four bodies in the solar system were known to have satellites: Jupiter and Saturn as well as the Earth were known to have moons, and all the planets are satellites of the Sun. According to Newton’s law of gravitation, a body of mass M exerts a force F = GMm/r2 on a satellite of mass m at distance r (where G is a constant of nature), so according to Newton’s second law of motion the centripetal acceleration of the satellite will be a = F/m = GM/r2. The value of the constant G and the overall scale of the solar system were not known in Newton’s time, but these unknown quantities do not appear in the ratios of masses calculated from ratios of distances and ratios of centripetal accelerations. If two satellites of bodies with masses M1 and M2 are found to be at distances from these bodies with a known ratio r1/r2 and to have centripetal accelerations with a known ratio a1/a2, then the ratio of the masses can be found from the formula
In particular, for a satellite moving at constant speed v in a circular orbit of radius r the orbital period is T = 2πr/v, so the centripetal acceleration v2/r is a = 4π2r/T2, the ratio of accelerations is a1/a2 = (r1/r2)/(T2/T1)2, and the ratio of masses inferred from orbital periods and ratios of distances is
By 1687 all the ratios of the distances of the planets from the Sun were well known, and from the observation of the angular separation of Jupiter and Saturn from their moons Callisto and Titan (which Newton called the “Huygenian satellite”) it was also possible to work out the ratio of the distance of Callisto from Jupiter to the distance of Jupiter from the Sun, and the ratio of the distance of Titan from Saturn to the distance of Saturn from the Sun. The distance of the Moon from the Earth was quite well known as a multiple of the size of the Earth, but not as a fraction of the distance of the Earth from the Sun, which was then not known. Newton used a crude estimate of the ratio of the distance of the Moon from the Earth and the distance of the Earth from the Sun, which turned out to be badly in error. Aside from this problem, the ratios of velocities and centripetal accelerations could be calculated from the known orbital periods of planets and moons. (Newton actually used the period of Venus rather than of Jupiter or Saturn, but this was just as useful because the ratios of the distances of Venus, Jupiter, and Saturn from the Sun were all well known.) As reported in Chapter 14, Newton’s results for the ratios of the masses of Jupiter and Saturn to the mass of the Sun were reasonably accurate, while his result for the ratio of the mass of the Earth to the mass of the Sun was badly in error.
Endnotes
PART I: GREEK PHYSICS
1. Matter and Poetry
1. Aristotle, Metaphysics, Book I, Chapter 3, 983b 6, 20 (Oxford trans.). Here and below I follow the standard practice of citing passages from Aristotle by referring to their location in I. Bekker’s 1831 Greek edition. By “Oxford trans.,” I mean that the English language version is taken from The Complete Works of Aristotle—The Revised Oxford Translation, ed. J. Barnes (Princeton University Press, Princeton, N.J., 1984), which uses this convention in citing passages from Aristotle.
2. Diogenes Laertius, Lives of the Eminent Philosophers, Book I, trans. R. D. Hicks (Loeb Classical Library, Harvard University Press, Cambridge, Mass., 1972), p. 27.
3. From J. Barnes, The Presocratic Philosophers, rev. ed. (Routledge and Kegan Paul, London, 1982), p. 29. The quotations in this work, her
eafter cited as Presocratic Philosophers, are translations into English of the fragmentary quotations in the standard sourcebook by Hermann Diels and Walter Kranz, Die Fragmente der Vorsokratiker (10th ed., Berlin, 1952).
4. Presocratic Philosophers, p. 53.
5. From J. Barnes, Early Greek Philosophy (Penguin, London, 1987), p. 97. Hereafter cited as Early Greek Philosophy. As in Presocratic Philosophers, these quotations are taken from Diels and Kranz, 10th ed.
6. From K. Freeman, The Ancilla to the Pre-Socratic Philosophers (Harvard University Press, Cambridge, Mass., 1966), p. 26. Hereafter cited as Ancilla. This is a translation into English of the quotations in Diels, Fragmente der Vorsokratiker, 5th ed.
7. Ancilla, p. 59.
8. Early Greek Philosophy, p. 166.
9. Ibid., p. 243.
10. Ancilla, p. 93.
11. Aristotle, Physics, Book VI, Chapter 9, 239b 5 (Oxford trans.).
12. Plato, Phaedo, 97C–98C. Here and below I follow the standard practice of citing passages from Plato’s works by giving page numbers in the 1578 Stephanos Greek edition.
13. Plato, Timaeus, 54 A–B, from Desmond Lee, trans., Timaeus and Critias (Penguin Books, London, 1965).
14. For instance, in the Oxford translation of Aristotle’s Physics, Book IV, Chapter 6, 213b 1–2.
15. Ancilla, p. 24.
16. Early Greek Philosophy, p. 253.
17. I have written about this point at greater length in the chapter “Beautiful Theories” in Dreams of a Final Theory (Pantheon, New York, 1992; reprinted with a new afterword, Vintage, New York, 1994).
2. Music and Mathematics
1. For the provenance of these stories, see Alberto A. Martínez, The Cult of Pythagoras—Man and Myth (University of Pittsburgh Press, Pittsburgh, Pa., 2012).
2. Aristotle, Metaphysics, Book I, Chapter 5, 985b 23–26 (Oxford trans.).
3. Ibid., 986a 2 (Oxford trans.).
4. Aristotle, Prior Analytics, Book I, Chapter 23, 41a 23–30.
5. Plato, Theaetetus, 147 D–E (Oxford trans.).
6. Aristotle, Physics, 215p 1–5 (Oxford trans.).
7. Plato, The Republic, 529E, trans. Robin Wakefield (Oxford University Press, Oxford, 1993), p. 261.
8. E. P. Wigner, “The Unreasonable Effectiveness of Mathematics,” Communications in Pure and Applied Mathematics 13 (1960): 1–14.
3. Motion and Philosophy
1. J. Barnes, in The Complete Works of Aristotle—The Revised Oxford Translation (Princeton University Press, Princeton, N.J., 1984).
2. R. J. Hankinson, in The Cambridge Companion to Aristotle, ed. J. Barnes (Cambridge University Press, Cambridge, 1995), p. 165.
3. Aristotle, Physics, Book II, Chapter 2, 194a 29–31 (Oxford trans., p. 331).
4. Ibid., Chapter 1, 192a 9 (Oxford trans., p. 329).
5. Aristotle, Meteorology, Book II, Chapter 9, 396b 7–11 (Oxford trans., p. 596).
6. Aristotle, On the Heavens, Book I, Chapter 6, 273b 30–31, 274a, 1 (Oxford trans., p. 455).
7. Aristotle, Physics, Book IV, Chapter 8, 214b 12–13 (Oxford trans., p. 365).
8. Ibid., 214b 32–34 (Oxford trans., p. 365).
9. Ibid., Book VII, Chapter 1, 242a 50–54 (Oxford trans., p. 408).
10. Aristotle, On the Heavens, Book III, Chapter 3, 301b 25–26 (Oxford trans., p. 494).
11. Thomas Kuhn, “Remarks on Receiving the Laurea,” in L’Anno Galileiano (Edizioni LINT, Trieste, 1995).
12. David C. Lindberg, in The Beginnings of Western Science (University of Chicago Press, Chicago, Ill., 1992), pp. 53–54.
13. David C. Lindberg, in The Beginnings of Western Science, 2nd ed. (University of Chicago Press, Chicago, Ill., 2007), p. 65.
14. Michael R. Matthews, in Introduction to The Scientific Background to Modern Philosophy (Hackett, Indianapolis, Ind., 1989).
4. Hellenistic Physics and Technology
1. Here I borrow the title of the leading modern treatise on this age: Peter Green, Alexander to Actium (University of California Press, Berkeley, 1990).
2. I believe that this remark is originally due to George Sarton.
3. The description of Strato’s work by Simplicius is presented in an English translation by M. R. Cohen and I. E. Drabkin, A Source Book in Greek Science (Harvard University Press, Cambridge, Mass., 1948), pp. 211–12.
4. H. Floris Cohen, How Modern Science Came into the World (Amsterdam University Press, Amsterdam, 2010), p. 17.
5. For the interaction of technology with physics research in modern times, see Bruce J. Hunt, Pursuing Power and Light: Technology and Physics from James Watt to Albert Einstein (Johns Hopkins University Press, Baltimore, Md., 2010).
6. Philo’s experiments are described in a letter quoted by G. I. Ibry-Massie and P. T. Keyser, Greek Science of the Hellenistic Era (Routledge, London, 2002), pp. 216–19.
7. The standard translation into English is Euclid, The Thirteen Books of the Elements, 2nd ed., trans. Thomas L. Heath (Cambridge University Press, Cambridge, 1925).
8. This is quoted in a Greek manuscript of the sixth century AD, and given in an English translation in Ibry-Massie and Keyser, Greek Science of the Hellenistic Era.
9. See Table V.1, p. 233, of the translation of Ptolemy’s Optics by A. Mark Smith in “Ptolemy’s Theory of Visual Perception,” Transactions of the American Philosophical Society 86, Part 2 (1996).
10. Quotes here are from T. L. Heath, trans., The Works of Archimedes (Cambridge University Press, Cambridge, 1897).
5. Ancient Science and Religion
1. Plato, Timaeus, 30A, trans. R. G. Bury, in Plato, Volume 9 (Loeb Classical Library, Harvard University Press, Cambridge, Mass., 1929), p. 55.
2. Erwin Schrödinger, Shearman Lectures at University College London, May 1948, published as Nature and the Greeks (Cambridge University Press, Cambridge, 1954).
3. Alexandre Koyré, From the Closed World to the Infinite Universe (Johns Hopkins University Press, Baltimore, Md., 1957), p. 159.
4. Ancilla, p. 22.
5. Thucydides, History of the Peloponnesian War, trans. Rex Warner (Penguin, New York, 1954, 1972), p. 511.
6. S. Greenblatt, “The Answer Man: An Ancient Poem Was Rediscovered and the World Swerved,” New Yorker, August 8, 2011, pp. 28–33.
7. Edward Gibbon, The Decline and Fall of the Roman Empire, Chapter 23 (Everyman’s Library, New York, 1991), p. 412. Hereafter cited as Gibbon, Decline and Fall.
8. Ibid., Chapter 2, p. 34.
9. Nicolaus Copernicus, On the Revolutions of Heavenly Spheres, trans. Charles Glenn Wallis (Prometheus, Amherst, N.Y., 1995), p. 7.
10. Lactantius, Divine Institutes, Book 3, Section 24, trans. A. Bowen and P. Garnsey (Liverpool University Press, Liverpool, 2003).
11. Paul, Epistle to the Colossians 2:8 (King James translation).
12. Augustine, Confessions, Book IV, trans. A. C. Outler (Dover, New York, 2002), p. 63.
13. Augustine, Retractions, Book I, Chapter 1, trans. M. I. Bogan (Catholic University of America Press, Washington, D.C., 1968), p. 10.
14. Gibbon, Decline and Fall, Chapter XL, p. 231.
PART II: GREEK ASTRONOMY
6. The Uses of Astronomy
1. This chapter is based in part on my article “The Missions of Astronomy,” New York Review of Books 56, 16 (October 22, 2009): 19–22; reprinted in The Best American Science and Nature Writing, ed. Freeman Dyson (Houghton Mifflin Harcourt, Boston, Mass., 2010), pp. 23–31, and in The Best American Science Writing, ed. Jerome Groopman (HarperCollins, New York, 2010), pp. 272–81.
2. Homer, Iliad, Book 22, 26–29. Quotation from Richmond Lattimore, trans., The Iliad of Homer (University of Chicago Press, Chicago, Ill., 1951), p. 458.
3. Homer, Odyssey, Book V, 280–87. Quotations from Robert Fitzgerald, trans., The Odyssey (Farrar, Straus and Giroux, New York, 1961), p. 89.
4. Diogenes Laertius, Lives of the Eminent Philosophers, Book I, 23.
5. This is the interpretation of some lin
es of Heraclitus argued by D. R. Dicks, Early Greek Astronomy to Aristotle (Cornell University Press, Ithaca, N.Y., 1970).
6. Plato’s Republic, 527 D–E, trans. Robin Wakefield (Oxford University Press, Oxford, 1993).
7. Philo, On the Eternity of the World, I (1). Quotation from C. D. Yonge, trans., The Works of Philo (Hendrickson Peabody, Mass., 1993), 707.
7. Measuring the Sun, Moon, and Earth
1. The importance of Parmenides and Anaxagoras as founders of Greek scientific astronomy is emphasized by Daniel W. Graham, Science Before Socrates—Parmenides, Anaxagoras, and the New Astronomy (Oxford University Press, Oxford, 2013).
2. Ancilla, p. 18.
3. Aristotle, On the Heavens, Book II, Chapter 14, 297b 26–298a 5 (Oxford trans., pp. 488–89).
4. Ancilla, p. 23.
5. Aristotle, On the Heavens, Book II, Chapter 11.
6. Archimedes, On Floating Bodies, in T. L. Heath, trans., The Works of Archimedes (Cambridge University Press, Cambridge, 1897), p. 254. Hereafter cited as Archimedes, Heath trans.
7. A translation is given by Thomas Heath in Aristarchus of Samos (Clarendon, Oxford, 1923).
8. Archimedes, The Sand Reckoner, Heath trans., p. 222.
To Explain the World: The Discovery of Modern Science Page 35
