| The Spin of the Earth: Another Conservation Law.
Another Example of a Conservation Law: Angular Momentum.
In my discussion of the causes of the seasons, I remarked that the axis of the Earth's rotation points in a particular direction in space and maintains that orientation more or less unchanging. The regular progression of the seasons is in fact completely dependent upon this stability, which arises because of the spin of the Earth. It is a consequence of a physical principle called the Conservation of Angular Momentum.
What, exactly, does this mean? Well, since this is a Conservation Law, it is analogous to the earlier example we looked at (the Conservation of Energy ). You may remember that I introduced the concept in a straightforward explanation of why stars are formed as hot bodies; you might like to revisit the relevant section, entitled
to reconsider the far-reaching consequences of that simple notion. In essence, I explained that energy cannot merely vanish: it simply appears in some other form. A book held high above the floor has potential energy; when it falls, the energy becomes kinetic (the energy of motion); and it can be made to fall onto, say, a turning paddle wheel to make it do some work. In like fashion, when atoms fall together under the influence of gravity, their inward rush leads to vigourous collisions and subsequent jostling about: the resultant body is hot!
The Conservation of Angular Momentum is similar, and worth understanding well in a qualitative way since we will be meeting it in various important contexts as the course progresses. It is really a statement that a body which is spinning around an axis within itself (like the Earth turning once a day) or moving in circular motion around some other point (as the moon moves around the Earth) possesses some amount of an attribute we call `angular momentum,' a quantity which has a precise mathematical definition. This angular momentum cannot be removed or disappear at a whim; it must show up in some other form so that the total amount of angular momentum (spinning and turning motion) is conserved (stays the same).
This may strike you as impossible, since it seems to be violated all the time! Think about a bicycle, for instance. You stand beside it at rest, but then jump on and pedal forwards. Soon the wheels are spinning around rapidly as you travel. Where did those spinning motions come from? We started with nothing spinning; now we have two wheels spinning!
The answer, believe it or not, is that your action in getting the bicycle wheels turning has also imparted a tiny bit of spin to the Earth itself in the opposite direction, thanks to the friction between the wheels and the ground! Since the Earth is enormously more massive than the bicycle, its state of motion is imperceptibly different to our eyes from what it had been, but I can assure you that that is where the angular momentum in the bicycle wheels `comes from.'
We see many other examples of this. When you throw a football and snap your wrist to set it spinning in a perfect spiral pass, the angular momentum it gains is exactly compensated by a tiny bit of turning motion directed through your arm and body, and eventually into the ground through your planted feet - or through the impact as you land, if you threw the ball while in the air. In class, we saw video examples of my son throwing a spinning 'Diabolo', otherwise known as a 'Chinese yo-yo', high into the air (and catching it on a string). You may also have noted that Olympic divers who leap off the platform whip their arms to one side across their bodies to impart a certain amount of spin to themselves, just as a cat can reorient itself when falling by moving its tail and legs about to allow the rest of its body to turn the other way.
Why the Stability?Now that you know that angular momentum must be conserved, you can understand why spinning motions impart stability. If the Earth is spinning with its axis pointing in a particular direction in space, it has a certain amount of angular momentum. The only way it can turn so that the axis points in some other direction is if some other body or set of bodies `takes up' some new or changed spin so that the total angular momentum remains just what it was before. This is easy to accomplish when you are riding a bicycle, even though the spinning wheels have a lot of angular momentum, because the bicycle and the Earth are in actual contact, with friction acting between the tires and the pavement. But the Earth is a chunk of rock in the vacuum of space, spinning without any obvious frictional forces to slow it or change its direction of spin. (We will see later, however, that there are some external forces -- the small gravitational tugs from the planet Jupiter, for instance -- which lead to a very slow change of the rate and direction of the spin, while still conserving total angular momentum.) I demonstrated the sense of this in class by sitting on a freely-turning piano stool while holding a spinning bicycle wheel in my hands. (Please remember that I had to get the wheel spinning while bracing my feet on the floor, for the reason given before. I had to get the wheel's new angular momentum from somewhere, and in effect I wind up moving the Earth a little in exchange!) Now, with my feet off the floor, I am isolated and cannot `unload' the angular momentum of the wheel. The consequence is dramatic: when I turn the wheel so that its spin axis (the axle of the wheel) points in some new direction, my whole body and the piano stool start to spin in a compensatory way, so that the total angular momentum remains the same. When I return the wheel to its original orientation, my own spin stops; and I am able to repeat this start-stop effect again and again. Why does the wheel ever slow down? The answer, of course, is friction. The wheel is passing through the air, bumping into molecules and setting them moving in a way which `carries off' some of the angular momentum. A more important consideration is the fact that the axle of the wheel, although well-oiled, is not free of friction, and neither is the pivot of the piano stool. The frictional rubbing leads to a gradual transmission of the angular momentum from the wheel back to the surroundings. We see many manifestations of the stability imparted by spinning motions: a Frisbee maintains its stable flight because it is spinning: if it is thrown without spin, it is very floppy in flight. the stability of a bicycle is better when you are moving fast than when you are moving slowly because the spinning wheels have a tendency to keep spinning with the already-determined orientation. If you try sitting on a bicycle at rest, you will certainly find that it is very unstable! (One of the tricks in teaching children to ride a bicycle is to persuade them that, at least to some extent, `faster is better.') gyroscopes (which are rapidly-spinning flywheels) in rocket ships help to maintain the direction in which the rockets are pointed. Similarly, on board aircraft and ships, gyroscopes provide a useful reference frame. As the boat turns, the gyroscope - which is mounted in extremely low-friction bearings - keeps pointing the same direction in space. One look at it tells you that the ship's orientation has changed. rifle bullets are set spinning by the "rifling," or grooves, in the barrel of the gun. This helps keep the bullets pointed in the wanted direction. Diabolos, like the one in the film shown in class, rely on this law. Once set spinning, they can be thrown into the air, where they maintain their orientation thanks to their spins. This allows the hourglass-shaped Diabolos to be caught neatly on their strings as they fall back to Earth. You may remember that in the film clip I showed you, my son Alex (who is a very skillful user of a Diabolo) was able to fling his a hundred feet or more into the air and then catch it. As noted, the Earth provides a good example. It spins as it orbits the sun, but, in accordance with this particular conservation law, there should be no significant change either in the direction of the rotation axis as the Earth moves or in its speed of rotation. The Earth's constancy in this respect explains why the Pole Star, which appears to hover almost directly over the Earth's North Pole, stays there year after year, and why the days are of such predictable and reliable length. (Of course the number of hours of sunlight changes, because of our changing perspective as we move around the sun, but the rate of spin and duration of the actual day - 24 hours - does not, except for a tiny slowing down for reasons to be explained later.)
Matters of Definition: Rotation and RevolutionJust for future clarity, I should point out here that a body is said to rotate when it spins about an axis which is inside itself. Examples include tops, Diabolos, merry-go-rounds, bicycle wheels, LPs on a record player, etc. A body is said to revolve about something when it moves bodily around a point outside itself. Examples include a runner going around a track, a planet going around the sun, and a satellite going around the Earth. The Earth, of course, does both: it spins (rotates) on its axis, and it orbits (or revolves about) the Sun. 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.)