1 00:00:01,160 --> 00:00:05,767 So Newton has this beginnings of an understanding of the science of 2 00:00:05,767 --> 00:00:10,698 mechanics, and, the first problem to which he's going to apply it, is Kepler's 3 00:00:10,698 --> 00:00:16,019 recent results about the motion of the planets around the sun, and Newton has, a 4 00:00:16,019 --> 00:00:20,691 beautiful chain of logic that we can actually follow, that allows him to 5 00:00:20,691 --> 00:00:26,076 derive the fundamental physical principle that we suggest that might be underlying 6 00:00:26,076 --> 00:00:29,710 Kepler's answer. Newton's mechanical, understanding tells 7 00:00:29,710 --> 00:00:35,066 him that as we saw, if an object of mass m is moving around in a circle of radius 8 00:00:35,066 --> 00:00:39,766 R with uniform speed v, which is a simplified version of the elliptical 9 00:00:39,766 --> 00:00:44,400 motion that Kepler actually studied, Newton was doing the general case. 10 00:00:44,400 --> 00:00:48,724 We know that moving around in a circle involves a centripetal acceleration 11 00:00:48,724 --> 00:00:53,048 towards the center of the circle. So if you see anywhere in the universe an 12 00:00:53,048 --> 00:00:57,603 object that is governed by Newton's laws, like everything is, moving around in a 13 00:00:57,603 --> 00:01:01,984 circle of radius R with speed V then there is a force acting on that object 14 00:01:01,984 --> 00:01:06,193 and attracting it towards the center of the circle in which it is moving. 15 00:01:06,193 --> 00:01:10,344 And the magnitude of that force is the centripetal acceleration that we 16 00:01:10,344 --> 00:01:14,726 computed, v^2 / R times the mass of the object because F = ma. 17 00:01:14,726 --> 00:01:17,840 If you know the acceleration you can deduce the force. 18 00:01:17,840 --> 00:01:21,858 Now Newton follows. The moon orbits the earth. 19 00:01:21,858 --> 00:01:24,697 The moon is actually moving in a circle about the earth. 20 00:01:24,697 --> 00:01:28,702 So we know that there is a force acting on the moon, which we can compute if we 21 00:01:28,702 --> 00:01:32,556 know the radius of the moon's orbit. And thence because you know how long it 22 00:01:32,556 --> 00:01:36,510 takes to go around you know its velocity. If you knew the mass of the moon you 23 00:01:36,510 --> 00:01:40,516 could compute the force that the earth applies to the moon in order to keep it 24 00:01:40,516 --> 00:01:43,304 moving in a circle. Why do I think the earth applies it? 25 00:01:43,304 --> 00:01:46,752 Well because it's directed towards the earth, that's a natural guess. 26 00:01:46,752 --> 00:01:50,200 But there's another ingredient here. We know that the earth attracts 27 00:01:50,200 --> 00:01:53,789 everything. Little rubber balls fall onto the earth 28 00:01:53,789 --> 00:01:57,074 because there's this attractive force we call gravity. 29 00:01:57,074 --> 00:01:59,690 Apples fall on people's heads apocryphally. 30 00:01:59,690 --> 00:02:04,906 Could it be that the force that keeps the moon in orbit around the earth is the 31 00:02:04,906 --> 00:02:07,841 same as the one that makes tennis balls fall? 32 00:02:07,841 --> 00:02:11,319 Well Newton says, probably let's take this seriously. 33 00:02:11,319 --> 00:02:14,568 Extrapolating that, let's look at another system. 34 00:02:14,568 --> 00:02:19,983 we know Kepler found that the planets orbit the sun in, well almost, perfectly 35 00:02:19,983 --> 00:02:25,399 circular orbits at almost uniform speed. again, that means that acting on each 36 00:02:25,399 --> 00:02:30,679 planet, is a force directed towards the center of the orbit, in other words, the 37 00:02:30,679 --> 00:02:33,048 sun. And we can compute that force. 38 00:02:33,048 --> 00:02:37,651 So it makes sense to guess that the sun will be applying this force. 39 00:02:37,651 --> 00:02:41,510 And the sun, therefore, applies a force on all the planets. 40 00:02:41,510 --> 00:02:43,921 Uh-huh. Then by Newton's third law, it would 41 00:02:43,921 --> 00:02:48,070 follow that since the Sun must be applying force to all the planets, among 42 00:02:48,070 --> 00:02:52,388 them the Earth, the Earth also applies an equal and opposite force to the Sun. 43 00:02:52,388 --> 00:02:56,649 In other words, if the Sun applies a force to the Earth directly towards the 44 00:02:56,649 --> 00:03:01,079 Sun, it follows that the Earth applies a force to the Sun directly back towards 45 00:03:01,079 --> 00:03:04,458 the Earth. And now, we have lots of cases where the 46 00:03:04,458 --> 00:03:08,959 earth is applying a force to things directed towards the center of the earth. 47 00:03:08,959 --> 00:03:13,694 Perhaps that force too is the same force that makes the moon orbit the earth and 48 00:03:13,694 --> 00:03:17,610 apples fall on people's heads. It all hangs together in a beautiful 49 00:03:17,610 --> 00:03:23,083 framework and we can do the math. So let's start with the idea of something 50 00:03:23,083 --> 00:03:26,919 orbiting the Sun. So we'll have a planet, we'll call it 51 00:03:26,919 --> 00:03:32,673 mass mp, p stands for planet, any planet. And if it orbits the Sun at radius R with 52 00:03:32,673 --> 00:03:38,427 speed v then Newton's understanding of mechanics tells us that the force the Sun 53 00:03:38,427 --> 00:03:44,962 applies to that particular planet is the mass of the planet times v^2 / R for that 54 00:03:44,962 --> 00:03:48,737 planet. two clips ago when we studied Kepler's 55 00:03:48,737 --> 00:03:54,055 laws, we found, that in fact for any planet that orbits the sun, there is a 56 00:03:54,055 --> 00:03:59,081 relation between v and R, that is independent of which planet, you are 57 00:03:59,081 --> 00:04:02,650 considering. For all of the planets, the square of 58 00:04:02,650 --> 00:04:09,060 their speed v^2, is 4 pi squared divided by this interesting constant, that I 59 00:04:09,060 --> 00:04:13,358 called K for Kepler, times, the radius of the planet's orbit. 60 00:04:13,358 --> 00:04:20,768 So I can, put this expression into that, and I can find, F is equal to the mass of 61 00:04:20,768 --> 00:04:25,807 the planet times the square of its velocity. 62 00:04:25,807 --> 00:04:32,384 4 pi squared over KR. Divided by the radius of its orbit. 63 00:04:32,384 --> 00:04:40,874 Or rearranging things a moment for a bit four pi squared divided by K times the 64 00:04:40,874 --> 00:04:46,955 mass of the planet divided by R^2. Why is this a way to write it? 65 00:04:46,955 --> 00:04:50,565 Because the mass of the planet is a property of the planet. 66 00:04:50,565 --> 00:04:53,869 R is something that changes between different planets. 67 00:04:53,869 --> 00:04:56,989 4 Pi squared over K is the same for all planets. 68 00:04:56,989 --> 00:05:01,534 So, let's see what we do with this. Well, remember the idea was that it's all 69 00:05:01,534 --> 00:05:04,720 going to hang together it's all going to be the same force. 70 00:05:04,720 --> 00:05:09,650 The sun applies the force given by this expression we found to each planet and to 71 00:05:09,650 --> 00:05:14,519 all the planets it applies the same the, the, the force given by the same constant 72 00:05:14,519 --> 00:05:16,920 K. By Newton's third law, each planet 73 00:05:16,920 --> 00:05:21,482 applies a force of that magnitude, this precise magnitude, to the sun. 74 00:05:21,482 --> 00:05:27,185 Now universality says that the force that the planet applies to the sun, the force 75 00:05:27,185 --> 00:05:31,747 that the sun applies to the planet, should follow from some physical 76 00:05:31,747 --> 00:05:33,760 principle. It doesn't say well, 77 00:05:33,760 --> 00:05:37,181 the sun is one thing and the planet is another. 78 00:05:37,181 --> 00:05:42,860 They're both physical object, and so symmetry between the two objects tells 79 00:05:42,860 --> 00:05:48,757 us, that if it's proportional to the mass of the planet, it should be proportional 80 00:05:48,757 --> 00:05:54,800 to the mass of the sun, and of course I can, do that without any problem I can, 81 00:05:54,800 --> 00:06:00,406 say that there is some constant which I will call G, and I, if I write that 82 00:06:00,406 --> 00:06:04,996 constant such that it's related to K. And to the mass of the sun. 83 00:06:04,996 --> 00:06:08,273 By this relation. Then the force e, expression that I'm 84 00:06:08,273 --> 00:06:12,779 writing is equivalent to F is G times the mass of the Sun times the mass of the 85 00:06:12,779 --> 00:06:16,552 planet divided by R^2. The advantage of this formulation over 86 00:06:16,552 --> 00:06:19,818 this one is that here it's clear that there is a symmetry. 87 00:06:19,818 --> 00:06:23,141 What the planet does to the Sun, the Sun does to the planet. 88 00:06:23,141 --> 00:06:27,139 The force between them is equal in magnitude and opposite in direction. 89 00:06:27,139 --> 00:06:29,730 And so there's this symmetric way to write it. 90 00:06:29,730 --> 00:06:34,292 But it was clear already that this form, object here doesn't depend on the planet. 91 00:06:34,292 --> 00:06:36,770 So it has to be in, some property of the Sun. 92 00:06:36,770 --> 00:06:41,098 The only wild leap here is that the property of the sun it depends on is the 93 00:06:41,098 --> 00:06:44,134 same property. The mass and the dependence is the same 94 00:06:44,134 --> 00:06:48,800 as the property of the planet on which it depends since it determines the force in 95 00:06:48,800 --> 00:06:52,380 a symmetric way. And now, 96 00:06:52,380 --> 00:06:55,758 we can jump to the fact that it's much more universal. 97 00:06:55,758 --> 00:06:59,886 Remember, the sun attracts the planets. The planets attract the sun. 98 00:06:59,886 --> 00:07:03,390 Earth attracts the moon and the moon pulls on the Earth. 99 00:07:03,390 --> 00:07:08,385 And in fact, the only way to get a consistent, understanding of this is to 100 00:07:08,385 --> 00:07:11,868 imagine that the force of gravity is truly universal. 101 00:07:11,868 --> 00:07:16,470 Any object applies a force to any other object in the entire universe. 102 00:07:16,470 --> 00:07:20,419 And the force is proportional to the mass of two objects. 103 00:07:20,419 --> 00:07:25,338 It's inversely proportional to the square of the distance between them. 104 00:07:25,338 --> 00:07:30,257 And, there's this universal constant, that gravitation, that defines what 105 00:07:30,257 --> 00:07:35,731 gravitation is and how powerful it is, which very appropriately is given the 106 00:07:35,731 --> 00:07:40,647 name Newton's constant, the, First measurement we have of Newton's constant 107 00:07:40,647 --> 00:07:45,174 is in 1798 by Cavendish, who actually measures the gravitational attraction 108 00:07:45,174 --> 00:07:49,399 between two metal balls, whose masses he carefully measured separately. 109 00:07:49,399 --> 00:07:52,720 So he can actually measure the distance and the masses. 110 00:07:52,720 --> 00:07:56,152 And find the force, and extract a value for G. 111 00:07:56,152 --> 00:08:02,712 And G is in the units that are convenient to us, given by this number of 6.67 * 112 00:08:02,712 --> 00:08:08,357 10^-11 Newtons times meters squared per kilogram squared. 113 00:08:08,357 --> 00:08:13,849 So that when you multiply it by two masses, kilograms squared divided by 114 00:08:13,849 --> 00:08:18,350 meters, you get a force in Newtons. So we have this equation. 115 00:08:18,350 --> 00:08:23,265 And, the way in which it is superior to everything that we have seen so far 116 00:08:23,265 --> 00:08:26,858 cannot be overstated. This is a universal statement about 117 00:08:26,858 --> 00:08:31,143 anything attracting anything else. And from this follows, for example, 118 00:08:31,143 --> 00:08:35,996 Kepler's Third Law, and of course, by a little more math, all of his other laws, 119 00:08:35,996 --> 00:08:41,038 describing the motion of the planets, the motion of the Moon, and as we shall see, 120 00:08:41,038 --> 00:08:44,756 plenty more things. So we found the deep underlying physics 121 00:08:44,756 --> 00:08:48,537 that we were looking for. Newton found it, and we have gotten 122 00:08:48,537 --> 00:08:51,500 there. So let's think a little bit about orbits. 123 00:08:51,500 --> 00:08:55,100 the moon, as we said, has a force applied to it. 124 00:08:55,100 --> 00:08:58,700 The earth is attracting the moon gravitationally. 125 00:08:58,700 --> 00:09:02,125 And one might ask, why is does not the moon fall on earth? 126 00:09:02,125 --> 00:09:05,249 And the answer is that the moon is falling on earth. 127 00:09:05,249 --> 00:09:09,997 Remember moving in a circle is motion with a constant acceleration towards the 128 00:09:09,997 --> 00:09:12,400 earth. Just as my little rubber ball was 129 00:09:12,400 --> 00:09:17,147 accelerated down towards earth, so the moon is accelerated towards earth with a 130 00:09:17,147 --> 00:09:20,767 constant acceleration. To demonstrate this idea, we go back to 131 00:09:20,767 --> 00:09:24,538 our pink bowling ball. You'll notice that when I hold it away 132 00:09:24,538 --> 00:09:28,957 from me and let it go, the Earth's gravity will always make it fall towards 133 00:09:28,957 --> 00:09:33,376 my head since I'm standing under the suspension point, whatever direction I 134 00:09:33,376 --> 00:09:36,793 let it go in. But if I give it an initial velocity that 135 00:09:36,793 --> 00:09:41,625 is just right, it will, happily circle about my head in a roughly circular orbit 136 00:09:41,625 --> 00:09:44,630 while accelerating towards the center at all times. 137 00:09:44,630 --> 00:09:49,397 And we also see that by giving it either a larger or smaller initial velocity, I 138 00:09:49,397 --> 00:09:53,330 get elongated orbits. They're the analogue of elliptical orbits 139 00:09:53,330 --> 00:09:57,860 in the case of actually gravitational motion around the earth or the sun. 140 00:09:57,860 --> 00:10:02,424 We'll do more detailed calculations of all this later, but for the moment we 141 00:10:02,424 --> 00:10:07,348 understand that orbiting is just a way of falling without ever hitting the ground, 142 00:10:07,348 --> 00:10:11,448 to paraphrase Douglas Adams. What we saw is that uniform circular 143 00:10:11,448 --> 00:10:15,740 motion about the center of gravity is a solution to Newton's equations. 144 00:10:15,740 --> 00:10:20,395 And if you think back this would require that I launch the bowling ball with 145 00:10:20,395 --> 00:10:25,171 precisely the speed predicted by the Kepler relation and its distance from the 146 00:10:25,171 --> 00:10:28,406 center of gravity. What happens if the initial velocity is 147 00:10:28,406 --> 00:10:32,217 either too large or too small? Well as with the bowling ball you get at, 148 00:10:32,217 --> 00:10:36,189 elongated orbits, and you will not be shocked to learn that in the case of 149 00:10:36,189 --> 00:10:40,858 Newtonian gravity and Newton solved these cases the result you get is precisely the 150 00:10:40,858 --> 00:10:44,830 elliptical orbits with the, center of gravity at one focus that Kepler had 151 00:10:44,830 --> 00:10:47,728 observed for the motion of the planets around the sun. 152 00:10:47,728 --> 00:10:51,700 And so, you find that there are all these elliptical orbits, and in fact. 153 00:10:51,700 --> 00:10:56,266 There are also open orbits like this hyperbolic one we see here which 154 00:10:56,266 --> 00:11:01,334 describes the motion of an object that comes in from infinity and recedes off to 155 00:11:01,334 --> 00:11:04,900 infinite distance. Of course, if you start along any point 156 00:11:04,900 --> 00:11:09,842 of this orbit with the correct velocity then you will pre- reproduce the later 157 00:11:09,842 --> 00:11:13,533 part of the orbit. So these describe the motions of objects 158 00:11:13,533 --> 00:11:18,726 but are not bound to the gravitating blue ball over there but rather are deflected 159 00:11:18,726 --> 00:11:21,604 by it. We'll return to orbits like this in the 160 00:11:21,604 --> 00:11:22,230 future. So, 161 00:11:22,230 --> 00:11:25,839 this is very nice. But, it gets even more general than that. 162 00:11:25,839 --> 00:11:30,879 Newton's law is universal, so it'll apply to any two objects that are orbiting 163 00:11:30,879 --> 00:11:34,799 under the mutual gravity. A little more work shows that we've a 164 00:11:34,799 --> 00:11:39,653 little bit been cavalier, we sort of assumed that Sun was stationary and earth 165 00:11:39,653 --> 00:11:43,200 orbiting it or the earth stationary and moon orbiting it. 166 00:11:43,200 --> 00:11:45,918 You should have known that there's something wrong there. 167 00:11:45,918 --> 00:11:49,210 Because if the moon is accelerating towards earth due to the earth's 168 00:11:49,210 --> 00:11:52,930 gravitational attraction, well then the earth must be accelerating towards the 169 00:11:52,930 --> 00:11:55,220 moon due to the moon's gravitational attraction. 170 00:11:55,220 --> 00:11:59,297 Of course, the acceleration of the earth is less than that of the moon, because 171 00:11:59,297 --> 00:12:02,642 the earth is more massive. And, this is the sort of the situation 172 00:12:02,642 --> 00:12:06,877 here, a very massive object and a lighter object are orbiting under the influence 173 00:12:06,877 --> 00:12:09,386 of gravity. The cross that is fixed there is the 174 00:12:09,386 --> 00:12:12,888 center of mass of the system. It's distance from the center of each 175 00:12:12,888 --> 00:12:15,973 object is inversely proportional to the mass of the object. 176 00:12:15,973 --> 00:12:19,684 So, it's closer to the mass of the object, it's the point at which these 177 00:12:19,684 --> 00:12:24,257 things will balance And but they both orbit each other, eat the center of each 178 00:12:24,257 --> 00:12:29,244 object describes the circle so they're always on opposite sides of this circle. 179 00:12:29,244 --> 00:12:34,292 Taking into account this recoil you find that in fact the distance between the two 180 00:12:34,292 --> 00:12:38,975 the radius or if its an elliptical orbit the semi major axis still satisfies 181 00:12:38,975 --> 00:12:43,780 Kepler's law no matter what the object is as it has nothing to do with the sun. 182 00:12:43,780 --> 00:12:48,292 The correct statement it turns out when you take into account the motion of the 183 00:12:48,292 --> 00:12:52,523 heavy object, is that the count, the constant that relates the square of the 184 00:12:52,523 --> 00:12:57,543 period to the cube of the semi-major axis is in fact, 4 pi squared over G times the 185 00:12:57,543 --> 00:13:01,041 total mass of the system. The sum of the two masses, now in the 186 00:13:01,041 --> 00:13:04,200 case of the sun and the Earth or the Earth and the moon. 187 00:13:04,200 --> 00:13:08,795 One of them is so much more massive than the other that you can approximate ma- 188 00:13:08,795 --> 00:13:12,644 the earth sun mass as the sun and the earth moon mass as the earth. 189 00:13:12,644 --> 00:13:16,838 This is tantamount to making the approximation that the sun, respectively 190 00:13:16,838 --> 00:13:20,170 the earth, does not move and only the lighter object moves. 191 00:13:20,170 --> 00:13:24,382 In, in cases like binary stars or maybe when you are completing the orbit of 192 00:13:24,382 --> 00:13:27,930 Jupiter around the sun. When the masses are closer to each other, 193 00:13:27,930 --> 00:13:31,580 then you need to apply this more careful calculation. 194 00:13:31,580 --> 00:13:36,225 This will be a very useful result to us. Because it means that if we see any two 195 00:13:36,225 --> 00:13:40,407 objects orbiting each other in the universe, and we can somehow find the 196 00:13:40,407 --> 00:13:44,995 period and the semi-major axis, we can compute their mass and, if you look in 197 00:13:44,995 --> 00:13:49,466 your texts and you find the mass of planets and galaxies and stars, you might 198 00:13:49,466 --> 00:13:53,938 ask yourself, how did somebody compute the mass of a distant star or for that 199 00:13:53,938 --> 00:13:56,958 matter Jupiter? And the answer is, you find something 200 00:13:56,958 --> 00:13:59,513 orbiting it, and you apply Kepler's relation. 201 00:13:59,513 --> 00:14:02,940 And so this will be an extremely powerful tool as we go on. 202 00:14:02,940 --> 00:14:08,468 All this theory, let's do an example and let's consider the International Space 203 00:14:08,468 --> 00:14:12,176 Station. The International Space Station orbits at 204 00:14:12,176 --> 00:14:17,774 an altitude above the ground of about 370 kilometers, adding that to the earth's 205 00:14:17,774 --> 00:14:23,302 radius, we find that it's the radius of its orbit about the center of the earth, 206 00:14:23,302 --> 00:14:27,221 is about 6,700 kilometers. And so we can find its period. 207 00:14:27,221 --> 00:14:34,882 Its period is given by our relation P^2 = K * R^3 and plugging in our value for 208 00:14:34,882 --> 00:14:40,853 pay, K, we find 4 Pi squared over GM earth times R^3. 209 00:14:40,853 --> 00:14:47,216 Notice that I've neglected the mass of the international space station relative 210 00:14:47,216 --> 00:14:52,046 to the mass of the earth which is a very fair approximation. 211 00:14:52,046 --> 00:14:57,030 And when we plug in the numbers we find that the period is 212 00:14:57,030 --> 00:14:59,971 taking the square root, you get this expression. 213 00:14:59,971 --> 00:15:03,100 Extracting the 2 Pi, plugging in all the numbers. 214 00:15:03,100 --> 00:15:07,710 Newton's constant the mass of the Earth and remembering to convert the radius of 215 00:15:07,710 --> 00:15:11,980 the Earth to meters to match the units in which we wrote Newton's constant. 216 00:15:11,980 --> 00:15:16,946 We find a period of 55 hundred seconds or about 91 minutes, the International Space 217 00:15:16,946 --> 00:15:21,856 Station and all lower satellites, I mean nothing was special to the, International 218 00:15:21,856 --> 00:15:25,128 Space Station. Anything that orbits at about the Earth's 219 00:15:25,128 --> 00:15:27,758 surface, orbits the Earth every 90 minutes. 220 00:15:27,758 --> 00:15:32,259 When you see satellites crossing the heavens, they all appear to move without 221 00:15:32,259 --> 00:15:36,233 uniform angular velocity. This is the reason, Kepler is telling us 222 00:15:36,233 --> 00:15:39,240 that they all orbit once every hour and a half. 223 00:15:39,240 --> 00:15:44,618 I hope that you're as impressed as I am with what we've managed to achieve by 224 00:15:44,618 --> 00:15:47,740 following Newton. We have found the fundamental, 225 00:15:47,740 --> 00:15:51,967 universal laws that underlay the regularities of Kepler's laws. 226 00:15:51,967 --> 00:15:55,542 in Newton's time, what to help people appreciate this was 227 00:15:55,542 --> 00:15:59,841 the understanding by Edmund Haley that comets, these weird objects that would 228 00:15:59,841 --> 00:16:04,027 appear in the sky move along the celestial sphere in some random direction 229 00:16:04,027 --> 00:16:07,798 then disappear again. Were actually objects that were at highly 230 00:16:07,798 --> 00:16:12,746 elliptical, highly eccentric elliptical orbits about the sun, spent most of their 231 00:16:12,746 --> 00:16:17,633 time far from the sun and invisible and were visible to us when they came near. 232 00:16:17,633 --> 00:16:22,334 He used data that had been collected to predict the appearance in 1705 of a 233 00:16:22,334 --> 00:16:25,798 particular comet in the sky over London to within a day. 234 00:16:25,798 --> 00:16:31,056 And luckily he got it just right, people stepped out on the day that Haley's Comet 235 00:16:31,056 --> 00:16:36,827 as it has come to be called showed up and the, the validity and the importance of 236 00:16:36,827 --> 00:16:42,305 Newton's result was understood. here's a beautiful image of the passage 237 00:16:42,305 --> 00:16:47,356 of comet Haley in 1986, it has a 75 year period as it orbits the sun. 238 00:16:47,356 --> 00:16:51,871 Try to figure out its semi major axis. Gravity works, we'll come back and do 239 00:16:51,871 --> 00:16:54,711 some more details. We have some more things to learn about 240 00:16:54,711 --> 00:16:57,404 how much information we can get out of Newton's theory. 241 00:16:57,404 --> 00:16:59,167 We'll turn to that in the next clip.