| Isaac Newton's Physics:
Meet Isaac Newton.
The year of Galileo's death was also the birth year of Isaac Newton, arguably the greatest mind in the history of physics (notwithstanding the example of Albert Einstein). His story is told briefly on pages 133-137 of the text, but in class I added a few details for general interest. For instance, Newton was born very prematurely, and scarcely survived childhood; his mother is quoted as saying that the infant would have fit easily into a pint jar.
Newton was the first of the great mathematical physicists, and stories abound about his brilliance. One interesting example concerns the practice, common in his day, by which mathematicians would set each other difficult problems, which they would communicate by mail for others to solve -- if they could! One European scientist sent such a problem to England, and received an anonymous but complete solution almost right away. At the time, Newton was in a reclusive or actively anti-social mood, so claimed no credit for the solution; but the creator of the problem is reported to have said: ``I recognize the lion by his claws,'' by which he meant that no one but Newton could have completed so elegant a solution so quickly.
Late in his life, Newton became paranoid and sometimes quite irrational, an aspect which may be attributable to his alchemical experiments - for although he was a great intellect and scientist, he also worked in areas which today we would call pseudo-science or 'fringe' science. Alchemy, the search for a method to turn base metal (lead) into pure gold, was one of the great research goals in Newton's day, and lots of mysterious mumbo-jumbo and incancations often accompanied this kind of work. It had a semi-empirical basis in chemistry too, and Newton apparently used a lot of mercury in his experiments, with the result that he may have been slowly poisoning himself (the dangers of heavy metals were not known in his day) in a way which manifested itself in his aberrant behaviour.
Newton had a fascinating career, with long affiliations to Cambridge University and its famous Trinity College. But he held other posts as well, including that of Master of the Royal Mint. Partly for that reason, but principally for his scientific accomplishments, in the 1970's he was commemorated on the back of the British one-pound note, now replaced by a coin. We remember him more, however, for his contributions to physics, contributions which were summarised in the `Principia' (see p. 134). In this section of the notes, we will consider one particular aspect: Newton's contributions to mechanics (the study of how bodies move when they are acted on by forces). The theory of universal gravitation, the basis for all astronomical investigation, will be the subject of the next section.
The First Law: Inertia Formalised.As we noted before, there seem coincidentally to be many examples of physical laws `in threes.' Here is another example: Newton's famous three laws of motion. While these warrant careful consideration, and while they can be expressed in technical and mathematical terms, my earnest wish is that you will develop a complete intuitive understanding of what they mean qualitatively. To clarify your understanding, therefore, let us consider them in simple conversational terms. The first law is merely a restatement, in technical terms, of the notion of inertia, a concept introduced by Galileo. Newton now makes explicit the understanding that an object in any state of motion (including rest) will remain unchanged in that state (which means that those at rest will remain at rest) unless some unbalanced force is acting. The word `unbalanced' merely acknowledges that we don't expect any motion to result from balanced forces. If you and your friend both push on a car, one at the front and one at the back, the forces will balance each other and nothing will happen. That is, all kinds of forces can be present, but unless there is an excess force in some direction, there will be no change in the state of motion of the body being pushed or pulled. Let us consider an immediate implication which follows from the First Law. Think of the space shuttle and its astronauts orbiting the Earth. The shuttle does not move in a straight line, but rather follows a curved path around the Earth. This must mean that some force is acting on it! The force is gravity, as we will see: if the space shuttle did not feel the gravitational force of the Earth, it would simply move in a straight line, and gradually leave the Earth behind. In other words, the shuttle is most emphatically not beyond the Earth's gravity, as is commonly believed, despite the fact that the astronauts experience weightlessness. (I will return to this point later.) This consideration also makes clear the incorrectness of Galileo's thinking: he believed that the circular motion of the moon around the Earth was a natural 'coasting' which was related to inertia without the requirement of any forces at play.
The Second Law: When Forces Are Unbalanced.The second law, perhaps the most famous in all of physics, merely makes quantitative something which we know already by everyday experience. If you kick a tennis ball, it moves away quickly. If you kick a cannon ball, it does not, even if you use the same amount of force. The difference is that massive objects (those containing many atoms and lots of material) are not easily set it motion. This is because they have lots of inertia (resistance to being accelerated from one state of motion, such as rest, into another state of motion). The law is usually given in the form of a very simple equation: F = ma. Let us think in words about what this means! Suppose first of all that you have an object of a given mass (m) -- that is, an object containing some total number of atoms, some total amount of matter. If a tiny force F acts on it, it will accelerate at a given rate a, which will be rather small (since F is small). If you increase the force F to some larger value, the acceleration a will be larger. In other words, bigger forces make objects accelerate more rapidly than small forces do. A powerful engine can get your car up to highway speed more quickly than a weak engine would. Please note, though, that even a small force can produce a large final speed if it is allowed to act for long enough! The small force generates a small acceleration, so the object gains speed only very slowly; but if the force is applied for many minutes (or hours, or years) the body may wind up moving quite quickly after all. This will become important later on, when we consider interstellar travel. We will have to consider the alternate merits of accelerating to high speed very quickly, using powerful rockets for a brief time, or accelerating rather slowly, using feeble rockets which are allowed to burn for a very long time. Now consider a situation in which you have a force of fixed size (F) at your disposal - say, all the strength you can muster with your two arms in trying to clear stones and boulders from your garden. If you apply the full force to a stone of small mass, (m) you can really send it flying (and of course in practice you would not bother applying all your force to it, but rather conserve your energy for the more taxing jobs). But a very massive stone, with large m, would be accelerated only a little, and you would really have to struggle to make it move perceptibly. By the way, this is an opportune moment to warn you about a very common misuse of a word. In physics, a massive body is one which contains a lot of matter in total -- many atoms, or atoms which themselves are particularly massive because they contain many protons and neutrons. This may have nothing to do with the size of the object. A small lead block may be much more massive -- much more resistant to being set in motion by a push, for instance -- than a much larger beach ball. My experience is that students often use the word "massive" to mean nothing more than "big" (often in the sense of "awesome"). Later in the course, we will learn, for example, that as the sun uses up its nuclear fuel, it will expand enormously, becoming a red giant star of such large size that the Earth may wind up inside its outermost parts. But the sun will be no more massive at that time than it is now -- it will contain as many atoms as it ever did, and will not have 'put on weight.' Of course, it will be much less densely packed on average: the atoms will be more widely spread, but the total mass will not have changed. Be very careful about how you use this word, which has a very precise physical meaning! A warning: many students misunderstand the meaning of the force 'F' in the equation above. It is a measure of the external force which has to be applied to a body of mass m to make it accelerate at a rate a. It is not an indication of the force which that body applies if it runs into something else! It is true that a massive moving body can cause a lot of damage -- imagine an ocean liner running into a small dock, for instance -- but that is not what we are talking about above. The correct understanding is as follows: Newton's second law says that a massive body (like the ocean liner) would have to have a large external force F applied to it if we are to change its state of motion to any perceptible extent. (To prevent the moving ocean liner from hitting the dock, you have to apply some pretty powerful brakes or give it a really hard push with a tugboat!)
The Third Law: Action and Reaction.The third law, also known as the "action-reaction" law, is one that causes many people a lot of confusion. Partly this is because it is used in some situations as a kind of vague metaphor for human behaviour. (If you get mad at me, I'll react by getting mad at you.) But in physics it has a very clearcut meaning. When one body acts on another (as when I use my finger to push a book across the table), then there is a reaction of equal size acting the other way (so my finger feels a force which we register as the resistance of the book to being moved). One reason for confusion is that a lot of people think that if two forces are "equal and opposite" then nothing will happen -- they must cancel out. Why then does anything ever move at all? The answer, of course, is that the forces do not cancel! They act on different bodies, and can have an effect. When you do a pushup, for instance, you are pressing down on the ground, pushing it away from you. The equal and opposite force (the "reaction" of the Earth acting on you) pushes you up and away from the ground. In fact, under the influence of these two forces of equal size, both you and the Earth move, but the Earth is so huge and massive that it budges an immeasurably small amount (remember Newton's second law!), while your body moves perceptibly. By the way, it is worth thinking for a moment about how these forces are transmitted. When you do a pushup, you are flexing your arms in such a way as to push your constituent atoms into the ground, or at least try to. If the ground had no structural rigidity -- imagine doing a pushup on water! -- your hands would merely slide seamlessly into it. As it is, though, the material is held in a rigid configuration by the electric forces between all the constituent atoms, molecules, and crystals. You are trying to force your constituent atoms (those in your hands and fingers) into these already crowded regions. As your atoms are pushed ever closer to those of the ground, the electric repulsion between the various particles resists the motion and stops your progress. This force, applied to the Earth, pushes it away from you; and the reaction force pushes you away from the Earth. Again, the reason that you are lifted bodily is because you have some rigidity of your own. If you had arms like cooked spaghetti, you would merely flop onto the floor. (Have you ever tried to push a car with a rope?) It does not take volition or conscious intent to make a force act on a body, or to generate a reaction force. Consider a brick sitting on the floor, for instance. I will anticipate Newton's introduction of gravitation to point out what you already know: the gravity of the Earth is pulling down on the brick, and if there were no floor there it would merely accelerate downwards, or fall, in accordance with Newton's second law. But, just as with you and the pushup, the gravitational tug downwards has the effect of trying to intermingle the atoms of the brick with those of the floor. The repulsive force between the electrically-charged constituents of the atoms resists that action (Newton's third law) and is strong enough to hold the brick up.
Deterministic Physics.With the three laws of mechanics in place, along with the new law of universal gravitation which we will explore in the next section, Newton felt able to figure out in complete detail the way bodies and particles move, at least in principle. For instance, if you were given a problem in which you were told where all the particles are -- every atom, every speck of matter -- and how fast and in what direction they are moving, then you could hope to figure out exactly where they will be at any later time, because we know how they will subsequently interact when they collide and rebound, or when they tug on each other by electrical or gravitational forces. A simple analogy is to the billiard table. When you plan and make a shot, it is with the expectation that the balls will hit and rebound in a predictable way. Indeed, we saw a visualisation of this in the lecture, with a ``three-cushion billiard'' expert making some very impressive shots using this predictive ability. If this were to be possible for all of nature, it would imply that we live in what is known as a deterministic universe, one in which all future motions, accelerations, actions and interactions are preordained by the present position and motion of all the particles. Needless to say, this is unsatisfactory from the human viewpoint, because it seems to negate the notion of free will. Am I typing up these notes and making these remarks because it was preordained by the location of all the atoms in the universe in the time of the dinosaurs? Modern physics, you may be reassured to learn, has slipped the bonds of determinism in ways we will explore later. Perhaps you already know that quantum mechanics, as it is called, has an inbuilt indeterminacy that there is no getting around. How this relates to ``free will'' is, however, a very difficult and perhaps metaphysical question.
The Great Conservation Laws.A lot of very profound physics is encapsulated in the so-called conservation laws, which are statements that certain quantities are `conserved' (unchanging in total) in isolated systems. We have encountered this twice before, once in my discussion of how the explains the fact that stars are hot, and again when I explained how the explained the stability of the spin of the Earth. The time is now right, however, to explore the issue a little more deeply. In Newton's time, the concept of the conservation laws was not as developed as nowadays, so this perspective is not one that Newton had fully available to him. In modern terminology, we believe that the following quantities are conserved in a closed system (that is, one in which no external influences or forces intrude): the total linear momentum (to be defined below) the total angular momentum the total electrical charge the total energy the total mass I demonstrated some of this in class. Consider, for instance, the conservation of energy. If I lift a piece of chalk above the table, I have done some work against gravity (my muscles have expended some stored chemical energy by burning up sugars and other fuels). My virtue of its new position, the chalk now possesses some "gravitational potential energy." When I drop it, the potential energy vanishes, or rather is converted to kinetic energy, the energy of motion. When the chalk hits the table, its directed motion stops but the total energy is still conserved; it goes into the heating of the table (the impact makes the atoms jiggle around more vigorously) and the noise which you hear (the impact jiggles the atoms in the air, and this disturbance spreads out as a sound and rattles your eardrums). Let us turn now to linear momentum, which is, as the name implies, a measure of the momentum (a word which may have some intuitive meaning for you) carried by an object moving along some particular direction, or line, of motion. In fact, the amount of linear momentum an object carries is given by its total mass times its speed of motion. Again, this equation is sterile on its own, so let us think of some applications. In a football game, stopping a fullback from crossing the goal line is more difficult if he is moving at speed, with a good deal of momentum, and there is an obvious advantage if the fullback is a large (massive) player rather than someone of very slight build. Likewise, we all know that a baseball thrown at ninety miles an hour carries more ``punch'' than a ping-pong ball moving at the same speed. In a sense, it is as though you were to ask how much damage a moving object could do if it should be involved in a collision. Interestingly, there are two ways to quantify this ``punch'' (or, if you prefer, this "ability to do significant damage"). One is to consider the kinetic energy (the energy of motion) of the body; the other, which is not the same, is to consider the linear momentum. Why two different ways? It turns out that the conservation of linear momentum is intimately related to Newton's Second Law, while the conservation of energy has very broad-reaching implications in a host of physical situations. Those of you who have taken physics courses know that some types of problems are more easily solved by considering the energetics; in other problems, considering the momentum may be the key to a quick solution. (No matter what your approach, of course, you should get the same answer.) In general, some thoughtful consideration of both these components together -- the energy and the momentum -- provides deep insights into physical behaviour. I demonstrated this in the lecture with a device which has a row of billiard balls hanging on strings, a desk-top executive toy often seen in stores and offices. The simplest way to use such a toy is to pull back and then release a single ball at one end. It swings forward and hits the remaining unmoving balls, of which there are typically half-a-dozen. A single ball immediately flies off the other end, and rises up to about the same height as the original ball was pulled back (which implies that it flies off with just about the same speed as the original ball had). That seems, in some intuitive sense, to be just what you would expect. So far so good. But what happens when two balls are dropped together at one end? The answer is that we always get two balls coming off the other end. Why? Why not one ball with twice the speed? Or four balls with half the speed? As I demonstrated, with the use of some very simple equations, the behaviour of the swinging balls before and after impact can be completely understood in terms of the conservation of energy and the conservation of linear momentum. This is a beautiful example of the complete and quantitative predictability inherent in Newtonian physics: there is one uniquely correct way events will unfold from a given starting situation. Things do not always work out that neatly, by the way! Some physical situations are horrendously more complicated. Think, for instance, of a row of stopped cars at an intersection. If an inattentive motorist runs into the back of the row at high speed, you will not generally see a single car pop off the front of the row with the other cars left sitting there unscathed! The difference is that the cars are designed to absorb some of the energy of the collision by crumpling and breaking apart. (This absorption of energy, seen in its extreme example in racing car accidents, actually protects the occupants by soaking up much of the energy of the collision.) Any calculation of the behaviour would have to take into account all these effects. The billiard-ball executive toy is especially simple in that the resilient balls collide elastically, which means that essentially all of the kinetic energy stays in that form. The balls escape the collision unscathed. A fine example of the conservation of linear momentum is to imagine yourself standing in an unmoving canoe on a placid lake, with a heavy stone held against your chest. Since everything is motionless, there is no linear momentum associated with this system of people and objects. Now fling the stone away from you as fast as you can! Since it is now moving to the right (let us say) with some speed, something else must be moving to the left if the total linear momentum is to be conserved (i.e. if it is still to add up to zero, as it did before). This, of course, is accomplished by the sudden backward motion of you and the canoe -- with the likely result that you lose your balance and fall into the water. This can equally well be considered from the point of view of Newton's Laws. The First Law reminds us that when you are standing still, you and the stone don't change your state of motion since no unbalanced forces are at play. Suddenly your muscles twitch and apply an unbalanced force to the stone, which is accelerated to the right, in accordance with Newton's Second Law. (The stone is accelerated only as long as you keep pushing on it, applying the unbalanced force; as soon as it leaves your grip, the force vanishes and the stone flies freely through the air at an unchanging speed - expect for air resistance and the effects of gravity, that is.) Meanwhile, Newton's Third Law reminds us that the unbalanced force you applied to the stone is matched by one of equal size acting on yourself but pointing in the opposite direction. This will set you in motion, according to Newton's Second Law. Of course, if you are very massive (or if the canoe is filled with other people and stones), you will not accelerate very much -- but move you will!
How Rockets Work.As we will see in the next section, Newton's Laws plus the Law of Universal Gravitation explain how and why the planets orbit the sun as they do. In considering these matters, Newton imagined the way in which various objects would move if they were to be launched horizontally at high speed from a mountaintop. When we consider such objects orbiting the Earth, one tends inevitably to visualise rockets, such as those used in the space program. This visualisation can be a little misleading, so I want to comment on it in a couple of different ways. The first of these is to consider how rockets actually work. Earlier, we considered an implication of Newton's Third Law: when you do a pushup you actually move the Earth! You push on it, and the reaction force pushes on you, so that you lift up from the surface. The forces are of the same size (``equal and opposite''), so you move more than the Earth itself does (since you are so much less massive: see Newton's Second Law), but in principle both move, even if the Earth's motion is immeasurably small. Now consider a rocket. When most people think of such devices, they visualize: 1 a huge flame and hot gas pouring out the back; and 2 the rocket pushing against the ground or the air, rather in the way that you push the blade of your paddle against the water when you propel a canoe. The first of these aspects is misleading; the second is just plain wrong. The rocket actually works by virtue of Newton's Third Law (or alternatively and equivalently through the Conservation of Linear Momentum). Within the rocket engine, the burning of the fuel heats the gases; this raises the pressure so that the gases try to expand in all directions. Since there is a nozzle at the back, the gases rush out that way at high speed. The equal and opposite reaction force pushes the rocket the other way. (Alternatively, we can just recognize that the total linear momentum has to be conserved.) The important point is that this would work regardless of the nature of the stuff thrown out the back. You could, for instance, build a little treadmill device to throw bricks out the back and thereby accelerate the rocket the other direction! The reason we use a hot flame is simply that the rocket is accelerated most efficiently if the ejected material moves at high speed, and the burning of liquid fuel heats it so much that the gases come out very fast indeed. So it is merely a matter of efficiency. The rocket needs nothing to `push against' and will function in the vacuum of space perfectly well. (In the 1920s, by the way, the New York Times published a strident editorial in which they criticised a physics professor who, they said, had completely forgotten his basic physics in even discussing the prospects of future space travel. According to the Times, rockets would never function in the vacuum of space! Events have proven them wrong, of course.) Indeed, rockets benefit from the lack of air resistance, which merely retards their acceleration as they climb away from the Earth's surface. Nor do rockets need to be streamlined, except insofar as it helps get them up out of the Earth's atmosphere with minimal drag. The Space Shuttle looks like an airplane because that is what it turns into on return to the Earth: it has to use its aerodynamics to glide to a safe landing. But a hundred years from now, it is possible that we may see a manned interstellar spacecraft, built in and launched from the vacuum of near-Earth orbit, which could be shaped like a cauliflower, for all that it matters.
What is Missing?Newton has provided us with laws which describe how objects will move when they are affected by various forces -- pushed by hands, pulled by strings, collided with by billiard balls, and so on. What was missing, up until Newton's day, was the important concept of a force which could reach out across empty space, in the complete absence of strings or obvious contact between bodies, to influence the ways in which the moons, planets and stars themselves move. That force, of course, was gravity, the subject of the next section of the notes. Previous chapter:Next chapter
0: Physics 015: The Course Notes, Fall 2004 1: Opening Remarks: Setting the Scene. 2: The Science of Astronomy: 3: The Importance of Scale: A First Conservation Law. 4: The Dominance of Gravity. 5: Looking Up: 6: The Seasons: 7: The Spin of the Earth: Another Conservation Law. 8: The Earth: Shape, Size, and State of Rotation. 9: The Moon: Shape, Size, Nature. 10: The Relative Distances and Sizes of the Sun and Moon: 11: Further Considerations: Planets and Stars. 12: The Moving Earth: 13: Stellar Parallax: The Astronomical Chicken 14: Greek Cosmology: 15: Stonehenge: 16: The Pyramids: 17: Copernicus Suggests a Heliocentric Cosmology: 18: Tycho Brahe, the Master Observer: 19: Kepler the Mystic. 20: Galileo Provides the Proof: 21: Light: Introductory Remarks. 22: Light as a Wave: 23: Light as Particles. 24: Full Spectrum of Light: 25: Interpreting the Emitted Light: 26: Kirchhoff's Laws and Stellar Spectra. 27: Understanding Kirchhoff's Laws. 28: The Doppler Effect: 29: Astronomical Telescopes: 30: The Great Observatories: 31: Making the Most of Optical Astronomy: 32: Adaptive Optics: Beating the Sky. 33: Radio Astronomy: 34: Observing at Other Wavelengths: 35: Isaac Newton's Physics: 36: Newtonian Gravity Explains It All: 37: Weight: 38: The Success of Newtonian Gravity: 39: The Ultimate Failure of Newtonian Gravity: 40: Tsunamis and Tides: 41: The Organization of the Solar System: 42: Solar System Formation: 43: The Age of the Solar System: 44: Planetary Structure: The Earth. 45: Solar System Leftovers: 46: The Vulnerability of the Earth: 47: Venus: 48: Mars: 49: The Search for Martian Life: 50: Physics 015 - Parallel Readings.
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(Friday, 27 November, 2020.)