How Much Do You Weigh? A Thought Experiment
The everyday manifestation of gravity is our sensation of weight. Most people know that astronauts in orbit around the Earth feel weightless, and assume that this is because they are "outside the Earth's gravitational field" or "beyond the Earth's gravity." This is a very common but quite incorrect interpretation.
You can discuss or define weight in two interrelated ways, one technical and one more-or-less physiological or psychological.. Let us consider the differences.
The technical definition expresses weight in terms of the downward force of the Earth's gravity acting on one's body. The problem is that you may not be aware of that force: if you are falling freely, you may feel like you are floating weightlessly, but of course the force is acting on you none the less, and you are bound eventually to hit the ground with a thump. (Parachutists come close to feeling this, but they experience the wind as they plummet through the air, so it is not as peaceful as you might think. Skin divers with carefully selected weight belts are in neutral buoyancy, so feel the sensation more correctly, but they are surrounded by water and need to breathe artificially, which distorts the perception.)
One very important point is that even if you are up in the Space Shuttle orbiting the Earth, the effect of gravity is not very different from what it is here on the surface of the planet! Reconsider the equation on page 138, which states that the force of gravity falls off like the inverse square of the distance; but also remember that Newton showed that the spherical Earth produces a gravitational force just as though all its mass were located at the center, 6000 kilometers below you. Consequently, when you go up in the shuttle to an altitude of (say) 100 kilometers above the ground, your distance from the centre of the Earth has increased from 6000 to 6100 kilometers -- a change of 2.5%. The force of gravity up there is somewhat weaker, it is true, but only by about 5% or so.
In other words, the distance above the ground is not the critical factor; it is how far we are from the centre that really matters. The force acting on the astronauts is not very different from what you or I experience here at sea level. But why then do the astronauts feel weightless and float about inside the Shuttle? Before answering that question, let me re-emphasise that there is abundant proof that the shuttle is not beyond Earth's gravity -- if it were, it would simply move off in a straight line. The very fact that it is orbiting the Earth tells us that a force is at work. No, the weightlessness of the astronauts needs a different explanation.
As I said, that explanation has its roots in the psychology and physiology of perceptions . It is best understood by appeal to an experiment which you can imagine but which you would certainly not want to carry out in real life -- a so-called `thought' or Gedanken experiment. (The word 'Gedanken' is German, and reflects the fact that Einstein, a German, used many of these thought experiments in his ruminations.) Imagine standing in an elevator at the top of a very tall building. Suppose further that you are holding a heavy object in your arms -- a bag of potatoes, say. If someone were now to cut the elevator cable, what would happen? (Don't try this at home.) I think you will agree that the elevator, you, and the potatoes will now start to fall rapidly, and indeed to accelerate, downwards. Moreover, if you let go of the bag of potatoes, it will not fall to the floor of the elevator, but will appear to hover in front of you. In other words, from your point of view the bag is weightless. If you put your hand underneath it, it does not press down on you or require muscle power to hold it up. (But note that it still has mass and inertia. If you push sideways on the bag, it will move a bit, but still resist your efforts. It will not absolutely fly off to the side of the elevator in response to a tiny tap of the finger.)
What else will you notice? Well, your feet no longer press down on the floor of the falling elevator, and you `float' within it. (A hint of this effect can be experienced when an elevator starts downward abruptly and you feel a momentary decrease in your weight.) I'm sure you can see the point: you and the bag appear weightless because you and it are falling together under gravity, in a state of what we call free fall. This is exactly analogous to the astronauts in the shuttle. The shuttle is falling freely as it orbits the Earth, and the astronauts are merely passengers floating freely within it (or outside it on a space walk; their precise position makes no difference). They are clearly not unaffected by the Earth's gravity, though! It determines their path through space.
In short, what we usually call weight is our perception of the forces which are required to hold us steady -- those which keep us from falling freely in the presence of the Earth's gravitational field. This is different from a physicist's definition of weight as a force, but the ideas are clearly related through Newton's Third Law. You must remember, however, that the removal of the supports (the dropping of the elevator floor beneath your feet, for instance) does not eliminate the force itself.
Space Stations.The moment has come to clarify a common misapprehension. No matter how logical it seems, the astronauts building any future space station will be quite unable to fling girders around with great abandon, although you may imagine them as looking rather as ants do when they handle long sticks or grasses. The problem is that objects orbiting the Earth are not massless. They contain just as many atoms as they ever did, and need a healthy push to get them moving at all. In other words, they have just as much inertia as ever, even if they seem weightless. If an astronaut pushes on a girder, he may get the impression that it moves briskly away from him, but in fact what happens (remember Newton's third law!) is that the force and its reaction lead to him and the girder both moving, but in opposite directions. The problem is that he moves rather quickly, the girder very slowly! The fact that the girder still contains all its mass will be brought home to you if you imagine yourself in a spacesuit and suddenly noticing that a runaway steel girder is moving towards you at high speed. It will still pack an enormous wallop, because of its mass; the fact that it is `weightless' doesn't save you. This does not mean that there are not advantages to working in space. The first is that one has complete freedom to move around, on, and under the object under construction. You can move `underneath' a girder with no fear that it will fall and squash you like a bug. It is already `falling' in its orbit around the Earth, and you in parallel with it! Moreover, the space station will not need to be built with the structural rigidity needed on the ground. (Think, for instance, of the top of the CN Tower, which has to be held up by an enormous column of cement and structural steel.) But manipulating the girders and such is still going to be a problem.
A Remarkable Coincidence?Let us now turn to something very deep -- one of the deepest thoughts in all of modern physics. We begin with a return to the time of Aristotle. Aristotle argued, with an clear appeal to common sense, that heavier objects should fall more rapidly to the ground than light objects. This seems at first thought to be borne out in day-to-day life, where we tend to think of light things like a feather or a piece of snow, both of which drift and flutter to the ground as compared to, say, a stone. But it took centuries, until the experiments of Galileo, to show that this common-sense view is simply wrong. Indeed, as precisely as we can measure it, all objects fall with identical accelerations under the influence of gravity. Measurements of this sort have demonstrated, for instance, that any difference in the acceleration of two vastly different bodies (say, a small lump of gold and a big lump of cheese -- although that particular combination has never been tested!) must be less than a billionth of a percent. The real problem with the feather and the snowflake, of course, is that they are so disproportionately affected by air resistance. To gauge the sort of difference this makes, drop two identical pieces of paper side by side. They flutter down quite irregularly, and indeed may not land at exactly the same moment. But if you crumple one into a ball, it will now fall much faster: its reduced cross-sectional area vastly reduces the air resistance it feels. True ballistic motion, of course, requires the complete elimination of irrelevant external influences like this. When that is accomplished, we find that there are no measureable differences. But why should we find it puzzling that there is this excellent agreement? To understand this, imagine holding two lumps of lead, a big one and a small one. Consider the following: Every atom in the big lump of lead pulls on every atom within the Earth, and every atom in the Earth pulls on every atom in the big lump. The result is that there is a strong gravitational force pulling them towards each other, and we have to strain to prevent the lump falling to the ground. Meanwhile, every atom in the small lump of lead pulls on every atom within the Earth, and every atom in the Earth pulls on every atom in the small lump. The net effect, given that there are far fewer atoms in the small lump than in the big lump, is that there is a modest gravitational force pulling them towards each other. To hold the lump above the ground is not particularly difficult. Let us suppose, just to be specific, that the big lump has one hundred times as many atoms as the small lump does. (The actual number does not matter.) That means that the force of the Earth on the big lump is one hundred times as great as that on the small lump: in everyday parlance, it weighs one hundred times as much. From Newton's Second Law, then, you might expect the big lump to accelerate quite rapidly, since it feels a much stronger force, a force which your senses tell you about when you strain to hold the lump up. But the two lumps also have inertia, a resistance to being accelerated. The inertia is, by coincidence, also exactly proportional to the total mass present. In other words , the big lump is one hundred times as sluggish (or as massive, or has one hundred times as much inertia) as the small lump, exactly the factor needed to compensate for the stronger force it feels. Consequently, the two objects fall precisely side by side. Now I would not be surprised in the slightest to hear you say that this is a non-issue, that it seems like no mystery at all. In fact it is profoundly puzzling, for the following reason. Matter has the ability to act in two quite different ways. First of all, it attracts other matter to it, as though every atom is reaching out with grappling hooks to grab every other nearby atom. In this respect, the body has an outwardly effective `active' mass. But matter also resists being accelerated, according to Newton's Second Law, in a way which is also directly proportional to the total mass. In this respect, matter has a sort of `passive' mass, which we call its inertia. There is no obvious reason why these should be equivalent for every kind of matter. To see this slightly differently, think about how electrically charged particles behave. Imagine putting a negatively-charged electron and a positively-charged proton side-by-side in a powerful electric field. (Ignore the relatively small pull they exert on each other, just as we ignored the feeble mutual gravitational pull between the big and small lead balls.) Each of these particles feels a force. (Because they are oppositely charged, the forces act in opposite directions, but that doesn't matter here. I don't care about the direction, just the rate at which the particles will be accelerated by the forces.) The interesting thing is that the particles accelerate at very different rates even though the forces are exactly equal in size. The forces are the same, of course, because the positive and negative charges are equal in size (although opposite in sign), but the very light electron is sent flying, while the sluggish proton, two thousand times as massive, scarcely budges. This is because the force each one feels depends on its electric charge, while its acceleration depends on its inertia. Since charge and inertia (or mass) are not the same thing, it is no surprise that they move differently. But in the context of gravity there is likewise no obvious a priori reason why `active mass' and `passive mass' should be the same! It could be, for instance, that the active mass is determined by the total content of an atom (adding up all the protons, neutrons and electrons together), while the passive mass might depend in some way on the relative proportions of protons and neutrons. If that were true, atoms of different elements would fall to the ground at different rates. How can we explain the wonderful coincidence that, where this one force is concerned, the active and passive aspects exactly compensate? Even Isaac Newton could not; the answer came with Einstein. Before we turn to that, let us note one of the consequences of this amazing behaviour. If you were to launch a battleship and a feather side-by-side into identical orbits around the Earth, at some very high altitude so that atmospheric resistance could be made completely negligible, then they would `fall' side-by-side around the Earth as they orbit, and do so for aeons to come . Likewise, the astronauts inside the Shuttle `fall' around the Earth at exactly the same pace as the Shuttle itself, apparently weightless within or beside it.
The Principle of Equivalence.I will not be able to explain the complete resolution of this fundamental question about the nature of gravity now; it will arise again later in the year in Phys 016 when I discuss the very nature of space, time, and the universe itself. For the moment, though, let me merely tell you that the explanation came with Einstein's statement of what is now called the Principle of Equivalence, which is in essence a restatement of the observation that all objects move in identical paths when in free-fall under the influence of gravity. The difference, with Einstein, is that he removed the need to think of forces reaching out across space to `grab onto' objects passing by. Rather, he hypothesised that lumps of matter (like the sun, or indeed any body) distort the very geometry of space and time, putting ripples and wrinkles into it. Free-moving bodies then merely follow these distortions in the geometry, just as a golf ball rolling across a hilly green curves and changes direction as it moves. We will explore the consequences of this completely new way of looking at gravity - no longer `action at a distance' - late in the Winter Term.
Determining Masses in Astronomy: What Not To Do.How do we work out the mass of a planet -- Mars, for instance? Many students make a very common mistake: they assume that astronomers act rather like the midway huckster who guesses your weight. You know the sort of showman I mean: for two dollars, this person claims to be able to estimate your weight to within a couple of kilos. If he fails, you win a prize -- probably worth about fifty cents! More often than not, however, he succeeds. How does he do this? The answer, of course, is that he relies on the fact that people are much the same in composition and structure, with comparable proportions of muscle, fat and bone. We vary in stature, of course, but a little experience allows him to compensate for that; he does not have to worry that one of us has a dense, pure platinum skeleton. (If you want to win that fifty-cent prize, carry some thin lead plates in your pockets and under your shirt.) The astronomical equivalent of the huckster's exercise -- what NOT to do -- would be as follows: Determine the distance to a particular planet, or perhaps the moon, by bouncing radar signals off it and seeing how long it takes the signal to return. In this way, for instance, we can determine that the moon is about 240,000 miles away. Using a telescope, see how big across the planet or moon looks in angular size. For instance, the moon is half a degree across, which means that you can hide it twice over behind your thumb held out at arm's length. Combine these pieces of information to determine the true size of the object. (In effect, you are answering the question "How big must the moon be to look as large as it does given that it is so far away?") The moon, for example, is about 2000 miles across. So far so good: this is all correct, and will tell you the real size of the object. But here comes the stumbling block. To determine the mass, we have to know what the object is made of. To mimic the midway showman, we would have to make some assumption about the composition of the object - perhaps assuming that the moon is made of rock very like that on the Earth - and then calculate how massive such a big chunk of rock must be. But what if we're wrong? What if the moon is hollow, or has a gaseous central part, is made of un-Earthlike low-density rocks? You might think that there is independent evidence we could use. Instead of merely assuming that a planet is like the Earth, why not try to estimate its composition in some way? (We have seen that there are ways of analysing the light which gives us some such information.) In this way, for instance, we find that Jupiter and Saturn consist principally of the lightest elements (hydrogen and helium), as does the sun. Can we not then calculate the actual mass of the object? Well, the short answer is still ``no'', although you will probably come closer than you would have otherwise. The procedure is deeply flawed, but not merely in the sense that it is unlikely to give you the precisely correct answer. The real problem is that it requires you to assume as true the very fact that you most want to learn. We are interested in finding out whether the other planets are like the Earth, or different. Are they uniform throughout, or stratified in composition and/or structrual property? Perhaps, for instance, Mercury is pure iron throughout, below a thin rocky surface. To decide such matters, we need some completely independent estimate of the total mass, a value we can use as a starting point for drawing reasonable conclusions about the composition. But where will such independent estimates come from?
Weighing' the Earth.We can start very locally. How do we know the total mass of the Earth? It is worth re-emphasising that we do not determine this by examining the surface rocks and then simply assuming that it is of uniform composition throughout. Nor can we drill down far enough to directly explore the nature of the deeper rocks and materials which make it up. No, our understanding of the composition, and the way in which the Earth's interior is stratified, comes from the indirect measurements and observations which you will hear about in lectures to come. But although the composition and physical state are hard to determine, and rely on indirect methods, one of the very simplest things to determine is the total mass of the Earth. All we need to do is drop a `test particle' from our hand, and see how rapidly it accelerates downward. A small ball will do, for instance, or a piece of chalk. The argument is straightforward: We can measure the rate at which the particle accelerates toward the ground. By Newton's Second Law (F = m a), this tells us how strong a force must be pulling on the particle. The force is nothing other than the gravitational tug provided by the Earth itself. That pull is given by the formula on page 138 of your text, but can be summarised briefly in words as follows: the force depends on how far you are from the center of the Earth (which is known), the size of the `gravitational constant' G (which can be determined experimentally), the mass m of the small particle (which is known), and the mass of the Earth itself - the only thing we don't know. We can turn the equation around and work out the one unknown! In essence, we ask how much mass the Earth must contain to make the piece of chalk fall at the rate we observe. Knowing the total mass of the Earth, we can now work out its mean (average) density. When we do so, we discover that it is about five times as dense as water, whereas the surface rocks are about three times as dense as water. This suggests that the central parts are denser than the outer parts (as if we had made a snowball with a hard-packed icy core) or perhaps even of different composition (as if we had hidden a rock in that snowball!). Indeed, we will see that both these things are true: the Earth has a dense core of molten iron, whereas the outer rocks are mostly silicates.
The Masses of the Other Planets and the Sun.How do we work out the masses of other planets and the sun? How do we do the equivalent of dropping a `test particle' onto Saturn? Conveniently, nature often comes to our rescue by providing moons. They move in orbits which are completely controlled by the gravitation of the parent planet, so if we know how far out the moon is and how fast it is moving (or equivalently its period of revolution), both of which are easily determined things, we can determine the mass of the planet. If nature is not so accommodating -- Mercury and Venus have no moons, for instance -- we can try to evaluate their masses more indirectly, by working out how such planets perturb the orbits of other objects, like the Earth itself around the sun. Since the planets are far apart, these perturbations are fortunately small. (If not for that, the Earth's orbit would be quite unstable and we would not always be so comfortably located at our present distance from the sun.) Small though they may be, the effects are yet measurable, and have been used for this purpose. Indeed, this was once the only way in which we had any inkling of the mass of the planet Mercury. Alternatively, we can rely on other objects which occasionally pass somewhat closer to the problem planets -- comets and asteroids, for instance, or (these days) artificial space probes launched from the Earth. The paths of these bodies through the Solar System are dominated by the gravity of the sun, but suffer alterations if they approach the planets closely enough to feel their gravitation fairly strongly. To determine a planet's mass, then, you have to work out how much gravitational influence it has on things moving near it. Please note, however, that we cannot work out the mass of a planet by seeing how long it takes to go around the sun! Its own motion is in response to the sun's influence, and has nothing to do with its own mass. If the Earth were to be replaced by a randomly-chosen object -- a brick, say, or a feather, or a lump the size of Mars -- it would still orbit the sun in exactly a year. (As we said earlier, a feather and a battleship would orbit the Earth in identical fashion under the influence of gravity. Likewise, the way in which these different objects `fall' around the sun does not depend on their masses.) Obviously, though, we can analyse the motion of a planet to work out the mass of the sun itself, since that is the body which controls the motion of the planet. The argument is as before: the sun's gravitational influence determines why the Earth takes exactly a year to go around its path of known size. (If the sun were more massive, the Earth would have to be moving faster to resist the tendency to fall in, and would complete a full circuit in less time than it takes now. In general, the more massive the central object, the faster the test particles move under its gravitational influence.) I should emphasise that this simple treatment is valid only if the `test particles' are negligible in mass compared to the objects they orbit -- the feather and the battleship are both negligible relative to the Earth; the Earth and the brick are both negligible relative to the Sun. If the Earth were to be replaced by something extraordinarily more massive, like a second star, we would have to consider how the sun itself would move in response to the much larger gravitational force acting on it: the whole situation would change. As noted earlier, Newton showed that Kepler's laws really describe how the sun and any given planet move around their common center of mass, like the adult and child on the teeter-totter. The sun is so dominant, however, that it can effectively be considered at rest. (Replacing the Earth with a brick, or vice versa, will not disturb the sun in its placid location at the center of the solar system.) But to be rigorous we should always remember this restriction to small `test particles.' Even a battleship is a mere particle relative to the Earth itself!
The Last Frontier: Weighing The Stars.There exist binary stars, pairs of stars which are in mutual orbit around their common centre of mass. It is possible, in ways we will explore later in the course, to study the motions of the component stars in a binary system. This allows us to determine the masses of the stars themselves. You will probably recognize right away that this is no longer a case of a negligible `test particle' orbiting a massive object. Typically the two stars are comparable in mass, so one has to consider the way in which they both move. But this is mere bookkeeping: the problem is physically well understood. Of somewhat more interest is what this application entails. It requires us to assume that Newton's Law of Gravitation, and his Laws of Motion, apply out there, light years away, and that the strength of gravity (the constant G in the equation on page 138) has the same value out there as it does here in our laboratories. If we are wrong about this very bold assumption - if the laws of physics change from place to place in the universe - we will get the stellar masses terribly wrong. But there is at least one `consistency check' which reassures us. Some binary systems contain stars which are very much like the sun in colour, temperature, spectrum, and so forth; not unreasonably, we deduce that they are very much like the sun in all respects. Lo and behold, the masses we deduce for such stars are very like what we deduce for the sun itself. Thus reassured, we press on. By the way, despite the title of this sub-section, this is not really `the last frontier.' We will see later in Phys 016 that we use measurements of the speeds with which stars move within galaxies to work out the masses of the galaxies. Moreover, galaxies themselves are being slowed down (decelerated) in the outward rush which is the `expansion of the universe.' The magnitude of this deceleration provides a measurement of the amount of matter present in the universe -- if you will, the `mass of the visible universe' itself. On this scale, a galaxy consisting of a hundred billion stars can serve as a negligible `test particle.' 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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(Monday, 17 May, 2021.)