Concepts in Physics. Mike Hobson. Lent Term Aims and Objectives. Non-examinable, but... Consolidation of Core Material. How Physics Really Works

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1 Concepts in Physics Mike Hobson Lent Term 2007 Aims and Objectives Non-examinable, but... Consolidation of Core Material How Physics Really Works What the text-books don t tell you

2 The Case Studies Galileo and the Nature of the Physical Sciences The Origin of Maxwell s Equations Thermodynamics and Statistical Physics Dimensional Analysis Chaos and Self-organised Criticality The Origins of the Concept of Quanta (1). Up to The Origins of the Concept of Quanta (2). After A Brief History of 20th Century Cosmology Relativity Special and General

3 The Books of the Course

4 Case Study 1 The Galileo Case and the Nature of the Physical Sciences 127 BC: Hipparchus, the first of the great astronomers Catalogue of 850 stars in the Northern Sky. 2nd century AD: Claudius Ptolemeaus, or Ptolemy, wrote the 13 volumes of The Great Composition, which became known as the Almagest. The Ptolemaic system of the world dominated astronomical thinking until the 16th century. The system had to account for the observed motions of the planets on the sky. The concept of epicyclic motion was needed to explain the motions of the planets.

5 Epicyclic Motion The motion of Saturn in AD 133 as observed by Ptolemy The system was based upon two key concepts from Greek mathematics. The only allowable motions were uniform motion in a straight line uniform circular motion Circular epicyclic motion Although complicated, the system worked well in order to predict the positions of the Sun, Moon and Planets, which were needed for the preparation of almanacs and to determine the correct dates for religious festivals.

6 It was obvious from the motions of the stars that their motions defined perfect circles on the sky. Evidence for Circular Motion

7 Evidence for Circular Motion There was an alternative view: 3rd century BC: Letter from Archimedes to King Gelon. It was obvious from the motions of the stars that their motions defined perfect circles on the sky. Aristarchos of Samos has published certain writings on the (astronomical) hypotheses. The presuppositions found in these writings imply that the universe is much greater than we mentioned above. Actually, he begins with the hypothesis that the fixed stars and the Sun remain without motion. As for the Earth, it moves around the Sun on the circumference of a circle with centre in the Sun.

8 The Rejection of the Heliocentric Hypothesis This picture was rejected by the followers of Aristotle for two physical reasons: If the earth rotated, why do you not observe an object thrown vertically upwards to land at a different spot? If objects are not supported, they fall under gravity. Therefore, if the Sun were the centre of the Universe rather than the Earth, everything ought to fall towards that centre. But, if objects are dropped, they fall towards the centre of the Earth and not towards the Sun. Thus, religious belief was supported by scientific rationale. Hence the need to develop the picture of the epicyclic motion of the heavenly bodies about the stationary earth.

9 The Ptolemaic System of the World The basic Ptolemaic system with circular orbits. With the passage of time, more complex epicyclic motions were needed to account for the details of the planetary orbits.

10 The Copernican System of the World Up till the time of Copernicus, the motions of the Moon and the planets were taken from the Alphonsine Tables prepared by the Rabbi Isaac Ben Sid of Toledo under the patronage of Alfonso the Wise in In the early 16th century, Aristarchus s model was revived by Copernicus as providing a simpler description of the motions of the Sun, Moon and planets. The orbits were taken to be circular about the Sun. Copernicus s work was presented to Pope Clement VII in 1533, who made a formal request for their publication. The first edition of De Revolutionibus Orbium Coelestium (On the Revolutions of the Celestial Spheres) was brought to Copernicus on his deathbed in 1543.

11 The Copernican Model of the Solar System In his forward to De Revolutionibus, Osiander remarked that the Copernican model was simply a calculating device, but, in the text, it is clear that Copernicus really believed that the planets orbit the Sun. This had two immediate implications: The Copernican Universe from Copernicus s De Revolutionibus Orbium Celestium The size of the universe is increased since no parallax of the fixed stars had been observed. Shouldn t all objects fall to the centre of the Universe?

12 Thomas Digges Model of the Universe In England, the Copernican picture was enthusiastically adopted by Thomas Digges. In Thomas Digges version of the Copernican picture, the Solar System is embedded in an infinite distribution of stars, which were assumed to be objects like the Sun.

13 Tycho Brahe ( ) Tycho Brahe was inspired to begin his great series of observations of the stars and planets by the inaccuracies in the Alphonsine Tables. With the sponsorship of Frederick II of Denmark, he constructed his great observatory on the island of Hven. The instruments were the most advanced available, resulting in an improvement in accuracy of a factor of 10 over previous measurements. Technical innovations: Systematic and random errors. Effects of the bending of the instruments. Effects of atmospheric refraction. Importance of the independence of the astronomical data.

14 Tycho s Observatory at Hven Tycho built two observatories, Uraniborg and Stjerneborg. This picture shows the Uraniborg Observatory with some of the instruments he had constructed, including the Great Mural Transit Circle and the 7-foot globe on which the locations of the stars were plotted. Note the use to numerous clocks to ensure the precise measurement of the time of transit of the stars and planets.

15 Tycho Brahe s Observatory Uraniborg on Hven

16 Tycho Brahe s Observatory Stjerneborg on Hven

17 Tycho s Achievement Over a period of 20 years, Tycho measured positions accurate to about 1-2 arcmin for the Sun, Moon, planets and 777 stars. He developed his own Tychonic Cosmology, a compromise between the Copernican and Ptolemaic pictures. The planets orbit the Sun, but the Sun and planets orbit the Earth which is stationary at the centre of the Universe. According to Tycho, the Copernican system, although not in accord with physical principles, was mathematically admirable. In 1600, the year before his death, he employed Kepler to analyse his observations of the motion of Mars.

Johannes Kepler (1571 1630) Kepler was a mathematician of genius, who had the technical skill to analyse Tycho s magnificant set of data on the planetary orbits.

18 Johannes Kepler ( ) Kepler was a mathematician of genius, who had the technical skill to analyse Tycho s magnificant set of data on the planetary orbits. Unlike Tycho, he was a convinced Copernican from the beginning. In 1597, he published his model for the Solar System, involving the embedding of the 5 regular platonic solids within one another. Remarkably, the model could account for the orbits to within an accuracy of about 5%.

From Kepler s Mysterium Cosmographicum The Earth s orbit is the measure of all things: circumscribe around it a dodecahedron and the circle containing it will be Mars; circumscribe around Mars a

19 From Kepler s Mysterium Cosmographicum The Earth s orbit is the measure of all things: circumscribe around it a dodecahedron and the circle containing it will be Mars; circumscribe around Mars a tetrahedron, and the circle containing this will be Jupiter; circumscribe about Jupiter a cube and the circle containing this will be Saturn. Now inscribe within the Earth an icosahedron and the circle contained in it will be Venus; inscribe within Venus an octahedron, and the circle contained in it will be Mercury. You now have the reason for the number of planets.

20 Kepler s Chronology The publication of the Mysterium Cosmographicum brought Kepler European fame. Copies were sent to Tycho and Galileo. Central to the story is Kepler s discovery of his Three Laws of Planetary Motion Succeeds Tycho as Imperial Mathematician at Prague Discovery of the Second (Areal) law from the Earth s motion about the Sun Discovery of First law from study of the orbit of Mars 1609 Second law published in The New Astronomy 1610 Galileo s Sidereus Nuncius published Discovery of the Third Law while writing Harmonices Mundi Notice that Kepler s results came from a very deep understanding of geometry. All the computations were done by hand using geometric techniques.

21 Kepler s Second Law Divine Providence granted to us such a diligent observer in Tycho Brahe that his observations convicted this Ptolemaic calculation of an error of 8 minutes of arc; it is only right that we should accept God s gift with a grateful mind... Because these 8 minutes of arc could not be ignored, they alone have led to a total reformation of astronomy. Kepler s Second Law Equal areas are swept out by the line from the Sun to a planet in equal times.

22 Kepler s First Law Kepler attempted to fit ovoids to the orbits of the planets, but still could not obtain a good enough fit. He needed a figure between the circle and the ovoid - the ellipse. The New Astronomy, Based on Causes, or Celestial Physics was published in 1609, four years after he had made the discovery of the second law. Kepler s First Law The planetary orbits are ellipses with the Sun in one focus.

Galileo s Observations of the Satellites of Jupiter Two pages from Galileo s Sidereus Nuncius Galileo observed the motions of the satellites of Jupiter each clear night from 7 January to 2 March 1610.

23 Galileo s Observations of the Satellites of Jupiter Two pages from Galileo s Sidereus Nuncius Galileo observed the motions of the satellites of Jupiter each clear night from 7 January to 2 March Kepler was elated by the discovery of this mini-solar system: The conclusion is quite clear. Our moon exists for us on Earth, not for the other globes. Those four little moons exist for Jupiter, not for us. Each planet in turn, together with its occupants, is served by its own satellites. From this line of reason we deduce with the highest degree of probability that Jupiter is inhabited.

The Harmony of the World In 1618, Kepler began writing a summation of all his previous work in the influential treatise entitled Harmony of the World.

24 The Harmony of the World In 1618, Kepler began writing a summation of all his previous work in the influential treatise entitled Harmony of the World. This may be considered the Grand Unified Theory of its day, the subjects including: Geometry Architecture Harmony Metaphysics Psychology Astrology Astronomy Metaphysics Macrocosmos Microcosmos

25 Discovery of Kepler s Third Law of Planetary Motion Kepler had reached the writing of Book V, Chapter III, 8th Division of the Harmony of the World, when he discovered the third law of planetary motion. Kepler s Third Law The period of a planetary orbit is proportional to the three-halves power of the mean distance of the planet from the Sun. This was the crucial discovery which eventually led to Newton s law of gravity.

26 Galileo Galilei ( ) Galileo was a physicist in the sense we would recognise today. He was strongly opposed to Aristotelian physics.

Galileo Galilei (1564 1642) Galileo was a physicist in the sense we would recognise today. He was strongly opposed to Aristotelian physics.

27 Galileo Galilei ( ) Galileo was a physicist in the sense we would recognise today. He was strongly opposed to Aristotelian physics. For example, according to Aristotle, If a certain weight move a certain distance in a certain time, a greater weight will move the same distance in a shorter time, and the proportion which the weights bear to each other, the times too will bear to one another; for example, if the half weight cover the distance in x, the whole weight will cover it in x/2. This is just wrong. Galileo may well have tested the idea by dropping weights from the leaning tower of Pisa.

28 Galileo Galilei as Physicist Galileo began to take the Copernican theory seriously in order to explain the origin of tides in the Adriatic. A huge force is needed to raise the tide by 5 feet. He began a magnificent set of experiments to eludicate the nature of motion. The three great achievements were: The law of acceleration, x = 1 2 at2. Galileo s law the time for free fall down the diameter of a circle equals the time to roll down a chord. The period of the swing of a long pendulum is independent of its amplitude.

29 Galileo s Physics Experiments The law of acceleration, x = 1 2 at2 The period of the swing of a long pendulum is independent of its amplitude. Galileo s law

Galileo s Telescopic Discoveries 1609 1612 The invention of the telescope by Hans Lipperhey was announced in 1608.

30 Galileo s Telescopic Discoveries The invention of the telescope by Hans Lipperhey was announced in Galileo constructed his own telescopes which had magnifying power up to a factor 30 by early The Moon is mountainous rather than a perfectly smooth sphere. The Milky Way was resolved into stars, rather than a uniform distribution of light.

Galileo s Telescopic Discoveries 1609 1612 The rings of Saturn, which he took to be close satellites of the planet.

31 Galileo s Telescopic Discoveries The rings of Saturn, which he took to be close satellites of the planet. The phases of the planet Venus were consistent with the Copernican picture. Jupiter has four satellites orbiting the planet, like a miniature Copernican Solar System (see before).

32 The Galileo Affair 1615 Galileo s letter to the Grand Duchess Christina Galileo accused of suspicion of heresy Galileo acquited, but the Congregation of the Index ruled that Copernicanism was philosophically and scientifically untenable and theologically heretical Urban VIII concluded that Copernicanism could be discussed, provided it was only considered hypothetically Galileo s Dialogue on the Two Chief World Systems, Ptolemaic and Copernican 1632 Galileo condemned by the Inquisition.

33 The Galileo Affair Problems with the Geokinetic Hypothesis 1. The Deception of the Senses 2. Astronomical Problems 3. Physical Arguments 4. The Authority of the Bible 5. The Hypothetical Nature of the Copernican theory

34 Physical Problems 1. If the Earth moves, falling bodies should not fall vertically. 2. The speed of projectiles fired in the direction of motion of the Earth and in the opposite direction should be different if the Earth were rotating. 3. The Extruding Power of Whirling. Objects placed on a rotating potter s wheel are flung off if they are not held down. Why are we not flung off the Earth if it is rotating? 4. Copernican picture was inconsistent with Aristotelian physics. 5. If Aristotelian physics was to be rejected, what was going to replace it?

35 Hypothetical Nature of the Copernican Picture In 1616, Cardinal Roberto Bellarmine wrote... it appears to me that Your Reverence (Foscarini) and Signor Galileo did prudently to content yourself with speaking hypothetically and not positively, as I have always believed Copernicus did. For to say that, assuming the Earth moves and the Sun stands still serves all appearances better than eccentrics and epicycles, is to speak well. This has no danger in it, and it suffices for mathematicians. But to wish to affirm that the Sun is really fixed in the centre of the heavens... is a dangerous thing, not only by irritating all the theologians and scholastic philosophers, but also by injuring our holy faith....

36 Deduction Deduction and Induction If it is raining, the streets are wet. It is raining. Therefore, the streets are wet.

37 Deduction and Induction Deduction If it is raining, the streets are wet. It is raining. Therefore, the streets are wet. Induction If it is raining, the streets are wet. The streets are wet. Therefore, it is raining.

38 The Nature of the Physical Sciences Galileo could not prove that the Copernican model was correct on the basis of the observations he had available. For example, his observations of Venus would have been consistent with Tycho s picture of the world. Even a sufficiently complicated Ptolemaic model could be devised to explain the observations. The key point is that physics is a hypothetico-deductive process. We make hypotheses and see how economically we can explain observed physical phenomena. The best theories are those which can explain large amounts of independent data quantitatively and make predictions to new circumstances.

39 The Nature of Physics A scientifically satisfactory model has the capability of making predictions about apparently unrelated phenomena. I use the word model in describing this process rather than asserting that it is in any sense truth. Galileo s enormous achievement was to realise that the models to describe nature could be put on a rigorous mathematical basis. In perhaps his most famous remark, he stated in his treatise Il Saggiatore (The Assayer) of 1624: Book of Nature... is written in mathematical characters This was the great achievement of the Galilean revolution. Notice that even the apparently elementary facts established by Galileo required a remarkable degree of imaginative abstraction.

40 Two New Sciences Following his condemnation in 1632, Galileo was placed under house arrest for the rest of his life. He wrote Two New Sciences, summarising all the work he had accomplished on the laws of motion. The relativity of motion Galilean Relativity. x y z t = x vt = y = z = t The Law of Inertia this was to become Newton s First Law of Motion. The application of the laws of acceleration and inertia to the motion of projectiles.

Isaac Newton (1641 1727) At the age of 22, while at home at Woolsthorpe because of the Great Plague, Newton made the following discoveries: The Binomial Series The Differential

41 Isaac Newton ( ) At the age of 22, while at home at Woolsthorpe because of the Great Plague, Newton made the following discoveries: The Binomial Series The Differential Calculus The Integral Calculus The Theory of Colour in Optics The Unification of Celestial Mechanics and the Law of Gravity Note: Newton was also a brilliant experimental physicist.

42 Newton and the Law of Gravity Kepler s Harmony of the World was in Trinity Library and the laws of planetary motion were probably brought to Newton s attention by Isaac Barrow. While at Woolsthorpe, he wrote that The notion of gravitation [came to me] as I sat in contemplative mood [and] was occasioned by the fall of an apple. First, he rederived the expression for the centripetal acceleration of an object moving in a circle of radius r at speed v a = dv dt = v2 r.

43 The Law of Gravity There must therefore be a centripetal force, f = ma, holding the planets in their orbits. Kepler s third law states that the period T of the planet s orbit is proportional to r 3/2. The speed of a planet in its orbit is v = 2πr/T and so v 1 r 1/2 Therefore, the force which keeps the planets in their orbits must be f = ma = mv2 1 r r 2 This is the primitive form of Newton s inverse square law of gravity. Now, if gravity is universal, the same force which binds the planets in their orbits, should also cause apples to fall to the ground.

44 Universal Gravity The Moon is 60 times further away from the centre of the Earth than the apple is. Hence the centripetal acceleration of the Moon is only 1/60 2 = 1/3600 times that due to gravity at the surface of the Earth. The acceleration due to gravity at the surface of the Earth is m s 2. The Moon s centripetal acceleration is v 2 /r. The orbital period of the Moon about the Earth is days. Its mean distance from the Earth is r = 384,408,000 m. Therefore, the mean speed of the Moon is v = 1,023 m s 1 and v 2 /r = m s 2. The ratio of the Moon s acceleration to the local acceleration due to gravity is therefore = 3,

45 This is the origin of Newton s law of gravity Universal Gravitation There were, however, problems f = GM 1M 2 r 2 i r The orbits of the planets are ellipses, not circles What is the influence of the other planets? Is it correct to place all the mass of the Earth at its centre? Newton could not account for the details of the Moon s motion. By 1684, he had solved these problems.

46 Newton s Principia Mathematica The results of Newton s researches were eventually published in 1687 as his Philosophiae Naturalis Principia Mathematica. Newton s Laws of Motion

47 After Newton s Principia Mathematica This was only one side of Newton s character. Newton as Master of the Mint Newton the Alchemist Newton and the Interpretation of the Scriptures His writings on Alchemy and the Scriptures were each as extensive as his researches into mathematics and natural philosophy.

48 Alchemy and the Scriptures Newton and the Interpretation of the Scriptures Newton s Alchemy Newton s classification of chemical substances. Between 1733 and 1922, there were twelve editions of Newton s biblical studies.

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