foreign [Music] [Applause] [Music] thank you when a torque acts on a spinning object the angular momentum changes the rate of change of angular momentum is equal to the applied torque under the influence of a torque a spinning object precesses we have now mastered the three great conservation laws of mechanics in a world that seems full of change those three things are the same now as they've always been and always will be forever energy momentum and angular momentum but there's something strange about those laws Isaac Newton who gave us all of mechanics never himself thought of a conservation laws they all come directly out of his great gift to us f equals m a but he himself never extracted them it's almost as if he purposely left something over for somebody else to do there's always been another small mystery about Newton and his laws he based his mechanics on three laws but we only seem to use one of them everything we do seems to come from f equals m a well I suspect those two little Mysteries are actually related to each other I think Newton may have had an Inkling about conservation and that's what his third law is really all about let me try to explain take the case of energy is never created or destroyed but it can flow from one place to another that's called work work remember is a force acting through a distance but if there's a force on one body there must always be the reverse force on another body that's Newton's third law and that means when work is going into one body it's always coming out of something else and that's the way energy is conserved oh momentum forces the rate of change of momentum but if there's a force causing the momentum of one body to change there must be the reverse force acting on another body causing its momentum to change in the opposite direction and that's why momentum is conserved and the same is true once again of angular momentum torque is equal to the rate of change of angular momentum but a torque on one body always implies a reverse torque on another and so angular momentum flows from one body to the other and angular momentum is conserved and so you see all three conservation laws can be thought of as coming out of Newton's third law well a subject for today once again has to do with torques and angular momentum as a matter of fact I think it's time we had a little torque hi mister hi I got a flat tire you sure do can you fix it sure come back in about a half an hour long before there was the bicycle there was the wheel itself and long before that there was simply the foot feet transport their Bearer under normal conditions and without too heavy load about five kilometers per hour feet operate by man woman or child power the advance of culture can be traced along the path of rapidly improving systems of Transportation before humans let their fingers do the walking they called on beasts of Burden beasts and burdens differed in different cultures but from ancient times onward while those who rode might have been beastly humans had a way of steadily advancing culture and increasingly ways to save time in their Endeavors and wear and tear on their foot gear but no matter how and how far human Destiny is bound to travel it really got rolling with the wheel about four thousand BC the Sumerians of Mesopotamia probably invented the wheel theirs might have been a tree slice or maybe flat pieces of wood tied together in a circle more or less but there's one thing about wheels that hasn't changed since then wheels are designed to spin there's nothing special about a wheel that's not spinning forget to hold it up and it'll fall over like anything else the force of gravity naturally tips the wheel down in other words if the wheel doesn't spin it has no place to go but down but a spinning wheel is quite another matter a spinning wheel has angular momentum and remember there's that strange twist when momentum's angular [Music] foreign it's not the force that changes angular momentum it's the twist if R cross f equals zero the angular momentum is conserved [Music] obviously not all forces behave that way for example to open a valve or to get a wheel rolling twist is just what's needed torque is the twisting force that causes rotation and it's equal to R cross f but a spinning wheel is already rotating it has angular momentum to start with now torque can make that angular momentum change but in a different way in fact now gravity can play a strange trick its force is downward but the torque is sideways according to the equation Tau equals DL DT the angular momentum will rotate sideways and rather than fall down so will the wheel itself so in a roundabout way it's easy to illustrate the fact that a spinning wheel doesn't fall but beyond this example of Life copying animation what's really going on here what keeps the Wheel from falling When Gravity tries to make a non-spinning wheel fall s push the top of the wheel outward rather than downward which creates a component of velocity in the outward Direction [Music] the same thing happens at the bottom but inward instead of outward and the same thing happens when the wheel spins now however when the outward velocity builds up it's at the side of the wheel outward on one side [Music] inward on the other [Music] each spoke adds its own push at the top causing a jog at the side [Music] so the net result isn't downward motion [Music] sideways once the wheel starts going around that way it's called precessing spinning bicycle wheel and a spinning top have much in common a non-spinning top is an inclined to balance on its point on the other hand a spinning top doesn't tip over onto its side instead gravity produces a different kind of motion precession whether it's spinning or not the top acts as if all its mass were concentrated at its Center a point called the center of mass the top is tilted even a little bit the result is a torque when this happens to a spinning top the torque nudges the angular momentum of the top around in a little circle round and round she goes with a little bit of friction coming along to cause heating at the point of the top and of course whenever Heat's produced a certain amount of energy is lost that's because heat which is the random motion of molecules is itself another form of energy everything absolutely everything on Earth has a certain amount of heat and therefore a certain amount of energy and no matter where that energy resides within a crackling fire or a crackling flow of ice or the potential energy of a gyroscope that seems to resist gravity all energy no matter its formed is always conserved that's the conservation of energy it's a profound law of physics and so Universal it can be seen even in the precession of a toy here too Heats being produced some of that energy comes from lowering the center of mass of the toy which sinks lower as the object's precession gets larger but despite the loss of energy rotation continues with remarkable stability frowned and round she goes and round and round she'll continue to go until something interrupts the precession of course before something interrupts their precession tops and toys can precess in the most amazing and amusing ways but a toy gyroscope perhaps like the child who plays with one can grow up to become a mature instrument of Science and Technology while this gyroscope may be a lot of fun to watch it's no toy and it's not designed to play games in fact it's designed not to precess if a gyroscope is balanced at its own Center of mass the force of gravity won't make it precess even if it supports swivel in all directions the axis will remain stable and keep the gyroscope pointed in the same fixed Direction the idea here is to design a gyroscope that hardly precesses at all and even in a toy store the idea of a sophisticated gyroscope can have serious implications in the real world aided by gyroscopes any deviation from a predetermined heading in an aircraft or submarine can be detected and corrected automatically but in the old days if a plane were to risk a polar root it'd have a snowball's chance of finally landing on even the right continent close to the pole a magnetic compass is no use at all set straight by gyroscopes Airlines routinely fly over the poles the value of the gyroscope isn't limited to the poles nor even to the atmosphere of the Earth in outer space where precious time and Rocket Fuel must be used wisely gyroscopes are more than ever vital elements of navigation whether in the water of the Earth [Applause] clouds above it or in outer space a gyroscope's value depends on its ability to spin in a fixed Direction without precessing gyroscope designers reduce precession by eliminating torques and improving design [Music] when an object precesses tip of its angular momentum Vector moves in uniform circular motion with a constant angular speed called Capital Omega Capital Omega is the rate of precession the equations of uniform circular motion say that V the rate of change of R is Omega times r in the same way Tau the rate of change of L is capital Omega times l so Omega is just torque divided by L [Music] since a good gyroscope's one that hardly precesses at all the bike wheel isn't good for navigation it precesses too much why torque is R cross F since they're perpendicular for the bicycle wheel the magnitude is R times f the force comes from Gravity angular momentum is M times R cross B all the Mass is at The Rim so the magnitude of L is M times the radius of the wheel times the velocity of the rim since V is Omega R the precession rate is the distance to the pivot times G divided by the rotation rate of the wheel times the radius of the wheel squared the gyroscope will precess least if it's spinning fast if the wheel is large and if it's carefully balanced if all those factors are at work a gyroscope holds its course extremely well exactly how well they're not perfect of course because anything made by human hands is subject to Tiny imbalances and slight imperfection mechanical or otherwise but the best gyroscopic guidance systems come pretty close to perfection in fact they feature gyroscopes that drift as little as 100 seconds of art per day but even that precise as it is isn't the best gyroscope around today what is what is the best gyroscope on Earth Isaac Newton had the answer in fact he was the first to discover it the best gyroscope on Earth is the Earth itself the Earth as Newton knew spins on an axis that's tilted at 23 and one-half degrees with respect to the perpendicular to the plane of the ecliptic that is the plane of the Earth's orbit around the sun in other words as seen from the Earth the Sun seems to be orbiting in a plane tilted 23 and one-half degrees to the plane of the Equator and twice a year the Sun Passes directly through the plane of the Equator these points where the sun intersects the plane of the equator are called the equinoxes meaning day and night have equal length and these points have long been vital to astronomers [Music] from the earliest nights astronomers have observed the phenomenon of the equinoxes the Chinese the Greeks the Babylonians and the Turks across time and within the greatest civilizations they shared a Fascination for the equinoxes which were used to Mark the exact moments of seasonal change the Egyptians the Mayans all the ancient world however different their lands might have been they shared the same sky and whatever differences their religions might have had ancient astronomers look toward the heavens with something in common [Music] in one way or another they calculated the precise position of the equinoxes but even Heavenly looks were deceiving the equinoxes are not fixed in the heavens they drift a bit each year very slowly to the West and in the precise perpendicular Direction though this went unnoticed for thousands of years the pole star seems to drift as well nonetheless ancient astronomers viewed the pole star differently they saw it as fixed in place absolutely and eternally like a cosmic myth in China it was respected as a goddess with the power to prolong life itself [Music] Arabic Scholars viewed the star as an evil one they believed the North Star had slain the great warrior of the sky who Forever After lay in a cosmic coffin outlined in sparkling detail by the Big Dipper but for all the movement and mythology surrounding it Polaris has had its practical aspects for centuries known as the lodestar the steering star and the ship star it helps sailors navigate All The Shining Seas when the Egyptians built the Pyramid of Khufu they seemed to have carried out their plan in light of the pole star itself which was visible from the bottom of one of the pyramid's deep shafts three thousand years before Christ however the pole star wasn't the Polaris it is today the Egyptians called it thuben and it was a different star across the sands of time and careful charts of astronomy Polaris is inching even closer to the pole where it will come closest in the 22nd century the drift of the pole star is related to the precession of the vernal equinox tonight the vernal equinox is in the constellation Pisces and it's slowly moving toward the constellation of Aquarius which means that contrary to certain revolutionary Notions of the 1960s the age of Aquarius won't come into play for 400 years in any case 400 years ago the most revolutionary man on the scene was Nicholas Copernicus in 1543 de revolucionibus presented an explanation for the phenomenon of migration in the heavens the precession of the equinoxes he wrote is due to the axis of the earth slowly tracing out a circle understood Equinox precession better than anyone before him it took Isaac Newton to explain it fully his explanation for the drift of the equinoxes came from his discovery that the Earth is a gyroscope and from his knowledge that the Earth isn't perfectly spherical if the Earth were perfectly spherical the sun would exert no torque on it but it's not so like any other gyroscope the Earth precesses rotating about its axis causes the Earth to bulge slightly at the equator [Music] the Sun's gravity acting most strongly on the nearest bulge exerts a small torque about the Earth's center when the sun is on one side of the Earth when the sun is on the other side a torque in the same direction [Music] the torque is tiny but it's enough to make the Earth's axis of rotation precess very slowly very slowly indeed one complete Loop takes about 26 000 years [Music] compare that to this gyroscope which precesses through a complete cycle every few seconds compared to the earth a bike is no match as a gyroscope to imagine a better gyroscope than the Earth but harder still to imagine riding at home for lunch [Music] the gyroscope is a strange and beguiling instrument but what I always find most amazing is the fact that Newton understood it and was able to explain the procession of the equinoxes anyone else who had done that and only that would be remembered today but for Newton it's lost in the Welter of his accomplishments Newton had a younger friend named Edmund Halley and one day in 1684 he casually told his friend that he had discovered that an inverse r-squared law of gravity would give Orbits for the planets in the form of conic sections it's hard to imagine how his reaction to that revelation being Newton's friend couldn't have been very relaxing in the best of circumstances but Newton had just told them that he had discovered the key to the universe how he prevailed upon his friend to publish his discovery which Newton ultimately did and it turned out to be true Newton had discovered the key to the universe and we'll turn that key next time precession is the result of a torque applied to a spinning object the rate of recession is determined by the angular momentum of the object and the toy the Earth itself is a gyroscope precessing under the torque applied by the Sun thank you I thought it would be overcome oh my God [Music] Annenberg media for information about this and other Annenberg media programs call 1-800 learner and visit us at www.learner.org