1 00:00:00,000 --> 00:00:06,767 Gravity as we've said, is the force, that mostly governs the 2 00:00:06,767 --> 00:00:11,395 behavior an the motions of celestial objects. 3 00:00:11,395 --> 00:00:16,473 And so Newton's theory is going to be the cornerstone of everything we do from now 4 00:00:16,473 --> 00:00:20,993 on, and it behooves us to go back and fill in some details and see what more we 5 00:00:20,993 --> 00:00:23,426 can get out of this wonderful formulation. 6 00:00:23,426 --> 00:00:27,888 And so, in this clip we're going to work through some more detailed calculations, 7 00:00:27,888 --> 00:00:32,060 applying Newton's universal law of gravity and along the way learn some 8 00:00:32,060 --> 00:00:35,826 things, both about astronomy and about gravity and even some math. 9 00:00:35,826 --> 00:00:38,260 And so, let's proceed, we have things to do. 10 00:00:38,260 --> 00:00:42,995 First thing you have to do is reconcile our story about the universality of 11 00:00:42,995 --> 00:00:47,233 gravity with the rather simple picture that we've been using so far. 12 00:00:47,233 --> 00:00:51,891 So let's understand, when I am standing in the vicinity of the earth, of the 13 00:00:51,891 --> 00:00:56,840 actual gravitational force that the earth exerts on me tells me Newton's theory, is 14 00:00:56,840 --> 00:01:01,073 the sum total of gravitational attraction of every little bit of earth. 15 00:01:01,073 --> 00:01:06,081 because gravity is universal, every atom of earth attracts every atom of me and I 16 00:01:06,081 --> 00:01:08,525 need to add them all up. Now I'm pretty small, 17 00:01:08,525 --> 00:01:12,758 but I need to figure out, how to add up the some total of gravitational 18 00:01:12,758 --> 00:01:15,680 attraction from all different parts of the earth. 19 00:01:15,680 --> 00:01:20,940 And so Newton showed us the way to do this along the way inventing, theory of 20 00:01:20,940 --> 00:01:25,946 multiple integrals to do it, but what he showed and we're not going to repeat that 21 00:01:25,946 --> 00:01:30,573 calculation for that reason, is that if you just take a round spherically 22 00:01:30,573 --> 00:01:35,326 symmetric shell of mass, so imagine just a crust of the earth, with an empty 23 00:01:35,326 --> 00:01:39,955 inside. Then the total force it exerts on someone 24 00:01:39,955 --> 00:01:47,302 standing over here at some arbitrary point outside well, if this mass of the 25 00:01:47,302 --> 00:01:54,370 shell is N, and my distance from the center of the shell is R, then the force 26 00:01:54,370 --> 00:02:00,610 that the shell exerts on a person over here is GMM over R squared, and it's 27 00:02:00,610 --> 00:02:04,831 directed towards the center of the shell as it had to be by symmetry. 28 00:02:04,831 --> 00:02:09,603 The force is the same, therefore as would have been created by eliminating the 29 00:02:09,603 --> 00:02:14,070 shell and replacing it with a mass N located at the center of the circle. 30 00:02:14,070 --> 00:02:18,853 Now if I on the other hand ask the same question, and this will become important 31 00:02:18,853 --> 00:02:22,202 to us later, about what is the force on an observer here? 32 00:02:22,202 --> 00:02:26,029 Well here again, some parts of the shell, attract you this way. 33 00:02:26,029 --> 00:02:30,275 Some parts attract you this way. The result for the forces striking, the 34 00:02:30,275 --> 00:02:33,325 results for the force is that it's absolutely zero. 35 00:02:33,325 --> 00:02:37,870 There is no force applied by a spherical shell anywhere on the inside of it. 36 00:02:37,870 --> 00:02:43,393 We don't tend to make too many measurements on the inside of earth but 37 00:02:43,393 --> 00:02:47,194 we are on the outside. And what all of this tells me is that if 38 00:02:47,194 --> 00:02:51,900 I consider earth as a collection of concentric shells and I consider the full 39 00:02:51,900 --> 00:02:57,109 gravitational effect of the earth to be the sum of all the gravitational effects 40 00:02:57,109 --> 00:03:02,061 of each concentric shell, then this has the effect, as far as I am concerned of 41 00:03:02,061 --> 00:03:07,078 replacing the whole earth by an object with a mass equal to mass of the earth 42 00:03:07,078 --> 00:03:10,937 located at the center. And, this simplifies our calculation a 43 00:03:10,937 --> 00:03:14,217 great deal. Let's see an example of how we use this. 44 00:03:14,217 --> 00:03:18,398 So, let's go back to figuring out my weight. My mass you will recall, 45 00:03:18,398 --> 00:03:21,163 hopefully hasn't changed much, was 59 Kilos. 46 00:03:21,163 --> 00:03:26,040 And so to compute the force on me, standing on the surface of the earth That 47 00:03:26,040 --> 00:03:30,816 means my distance from that imaginary mass at the center of the earth is the 48 00:03:30,816 --> 00:03:34,476 radius of the earth. I will use the astrological symbol for 49 00:03:34,476 --> 00:03:39,376 earth to signify things like the mass of the earth and the radius of the earth 50 00:03:39,376 --> 00:03:44,104 because it's simple to produce in tech. And so the force on me is given by 51 00:03:44,104 --> 00:03:49,326 Newton's, expression, GMM over R squared, where I've replaced the Earth by an 52 00:03:49,326 --> 00:03:54,213 object with the same mass as Earth sitting at its centre, plugging in the 53 00:03:54,213 --> 00:03:59,367 numbers for G for the mass of the Earth and for its radius in metres squared. 54 00:03:59,367 --> 00:04:03,920 I find not surprisingly that when I multiply all of these things, 55 00:04:03,920 --> 00:04:07,270 guess what? I found our old friend, 56 00:04:07,270 --> 00:04:12,340 MG This object is what we call the gravitational acceleration on the surface 57 00:04:12,340 --> 00:04:15,129 of earth. And if you wish, then measuring the 58 00:04:15,129 --> 00:04:20,199 gravitational constant as Cavendish did, measuring the radius of the earth, which 59 00:04:20,199 --> 00:04:24,700 you can do geometrically, this is the way to determine the earth's mass. 60 00:04:25,820 --> 00:04:31,276 We've computed the gravitational acceleration on earth from Newton's 61 00:04:31,276 --> 00:04:36,700 formula We know, that as I get farther away, from 62 00:04:36,700 --> 00:04:41,030 the center of earth, or as I get higher up beyond, into and beyond the 63 00:04:41,030 --> 00:04:45,174 atmosphere, then the denominator in Newton's formula becomes larger. 64 00:04:45,174 --> 00:04:49,813 That R squared in the denominator, my distance from the center of the Earth 65 00:04:49,813 --> 00:04:53,896 grows, and gravity weakens. It's not that gravity does not exist of 66 00:04:53,896 --> 00:04:57,174 course beyond Earth, but it is weakened with distance. 67 00:04:57,174 --> 00:05:00,020 I want to get a quantitative estimate of this. 68 00:05:00,020 --> 00:05:04,968 So let H be my altitude above the ground. This is what Newton's formula predicts 69 00:05:04,968 --> 00:05:08,432 for the force of gravity on an object of mass MR, say, me. 70 00:05:08,432 --> 00:05:11,216 I'm going to rewrite this in a useful way. 71 00:05:11,216 --> 00:05:16,596 I'm going to rewrite this as. G times the mass of earth divided by R of 72 00:05:16,596 --> 00:05:21,429 radius of the earth squared. Well I've divided and multiplied, by the 73 00:05:21,429 --> 00:05:26,971 radius of the earth squared so here I need to put, radius of the earth squared, 74 00:05:26,971 --> 00:05:32,301 over, my actual distance, from the center of the earth squared, and then I'll 75 00:05:32,301 --> 00:05:36,422 remember MR. And then I recognize this combination, as 76 00:05:36,422 --> 00:05:41,254 what we previously called G. When H is zero this, quantity is one and 77 00:05:41,254 --> 00:05:45,518 I'm back to the surface. So, in the second line, I'm, rewriting 78 00:05:45,518 --> 00:05:48,863 this, as, my weight on the surface of the Earth 79 00:05:48,863 --> 00:05:54,191 times dimensionless number that indicates how much its decreased and then I 80 00:05:54,191 --> 00:05:59,520 simplify this dividing numerator and denominator by the radius of the Earth to 81 00:05:59,520 --> 00:06:03,432 get this expression. And the point I want to apply here is 82 00:06:03,432 --> 00:06:06,735 that I want to see, This is an exact expression. 83 00:06:06,735 --> 00:06:12,074 I want to see if I can simplify for cases where my height above the earth is not 84 00:06:12,074 --> 00:06:15,172 very large compared to the radius of the earth. 85 00:06:15,172 --> 00:06:20,313 And to the rescue comes of course none another than Newton who derived the 86 00:06:20,313 --> 00:06:25,059 following important expression. It is perhaps the only application of 87 00:06:25,059 --> 00:06:27,893 calculus we will use broadly in this class. 88 00:06:27,893 --> 00:06:31,979 So, I won't derive it, but it's a fact that if you take this 89 00:06:31,979 --> 00:06:36,560 combination, one plus x to the power, and if x is much smaller than one in 90 00:06:36,560 --> 00:06:42,390 absolute value, X can be negative or positive, then one plus X to the power A 91 00:06:42,390 --> 00:06:48,526 is, approximately given by one plus AX, the smaller X and the smaller A, the 92 00:06:48,526 --> 00:06:54,586 closer the approximation is, and because I'm not proving it, here's a few graphs 93 00:06:54,586 --> 00:06:59,342 that demonstrate this. this is the X axis, X is running from 94 00:06:59,342 --> 00:07:03,370 zero to a tenth. This is one plus x to the a for various 95 00:07:03,370 --> 00:07:08,480 values of a indicated here running from five to minus four, and you see that as 96 00:07:08,480 --> 00:07:13,520 long as x is not too large, this approximation is valid and this will 97 00:07:13,520 --> 00:07:17,510 allow us to simplify many, many calculations in the class. 98 00:07:17,510 --> 00:07:21,582 So, it's one, result of calculus that I want you to 99 00:07:21,582 --> 00:07:25,120 accept from me, and you're welcome to test it, 100 00:07:25,120 --> 00:07:29,916 Just as I have shown you in this graph you can get some more examples. 101 00:07:29,916 --> 00:07:33,204 So at the moment, let's apply it to this example. 102 00:07:33,204 --> 00:07:38,343 So here, H over R, the radius of the earth is X, which I'm going to assume is 103 00:07:38,343 --> 00:07:42,112 not too large. Remember a tenth, gives me an altitude of 104 00:07:42,112 --> 00:07:47,730 about, 700 kilometers, and A is negative two. And so the approximation yields one 105 00:07:47,730 --> 00:07:50,882 plus X to the minus two is about one minus 2X. 106 00:07:50,882 --> 00:07:56,431 And this gives me the approximate amount by which the, force of gravity decreases 107 00:07:56,431 --> 00:07:59,477 as I elevate myself to an altitude H above the 108 00:07:59,477 --> 00:08:03,164 surface of the ground. Of course when H is zero I reproduce my 109 00:08:03,164 --> 00:08:07,862 weight on the ground, and this is valid so long as your height is negligible or 110 00:08:07,862 --> 00:08:10,360 small compared to the radius of the earth. 111 00:08:10,360 --> 00:08:14,060 It supplies what we've learned to understanding potential energy and 112 00:08:14,060 --> 00:08:16,816 gravity. We've already had an expression for 113 00:08:16,816 --> 00:08:22,116 potential energy, mgh, but this was valid as long as we assume the force of gravity 114 00:08:22,116 --> 00:08:25,865 was constantly mg. We saw that as you get farther from the 115 00:08:25,865 --> 00:08:30,778 center of earth, the force decreases every height gain costs less energy and 116 00:08:30,778 --> 00:08:34,850 therefore taking that into account, we find an exact expression. 117 00:08:34,850 --> 00:08:39,299 This is the exact expression for gravitational potential energy, Newtonian 118 00:08:39,299 --> 00:08:42,004 gravity. These two do not look the same, and I 119 00:08:42,004 --> 00:08:45,190 didn't derive this because it requires some calculus. 120 00:08:45,190 --> 00:08:49,098 Let me convince you that they're not as different as they appear. 121 00:08:49,098 --> 00:08:52,405 Another opportunity to use some skills we've developed. 122 00:08:52,405 --> 00:08:56,674 So, let's write the potential energy using the correct expression at an 123 00:08:56,674 --> 00:09:00,460 altitude h, above the earth's surface. So I write. 124 00:09:00,460 --> 00:09:05,020 For distance from the earth's center, r plus h. 125 00:09:05,020 --> 00:09:11,548 And, as always, I rewrite this by scaling it to a known quantity. 126 00:09:11,548 --> 00:09:20,046 And, that would be this one. This is your potential energy on the surface of the 127 00:09:20,046 --> 00:09:23,776 earth. And, then I need to write this 128 00:09:23,776 --> 00:09:28,958 dimensionless quantity, which gives me the scaling. 129 00:09:28,958 --> 00:09:37,662 And again, I factored, I divide numerator and denominator by r to get -g and m over 130 00:09:37,662 --> 00:09:45,093 r times one plus h over r inverse. And another opportunity to apply Newton's 131 00:09:45,093 --> 00:09:50,731 formula, A is negative one, X is, your altitude, if your altitude is smaller 132 00:09:50,731 --> 00:09:56,138 than the earth's radius significantly, then Newton's formula applies. 133 00:09:56,138 --> 00:10:02,471 That replaces this by one minus H over R, and I will do the calculation, in my head 134 00:10:02,471 --> 00:10:06,950 and on that one, it's easy to do. It's this constant, number. 135 00:10:06,950 --> 00:10:12,665 And then minus H over R gives me a positive contribution here, an extra 136 00:10:12,665 --> 00:10:19,870 factor of R in the denominator, GM over R squared. 137 00:10:19,870 --> 00:10:26,917 and H, and I recognize two things here. One is this is independent of your 138 00:10:26,917 --> 00:10:31,688 altitude. It's just a constant, and this object here is G. 139 00:10:31,688 --> 00:10:37,502 So the difference from, between this exact ob, quantity and our approximation 140 00:10:37,502 --> 00:10:43,545 for H that is significantly smaller than the earth's radius is just a constant 141 00:10:43,545 --> 00:10:46,528 shift. We said that a constant shift is 142 00:10:46,528 --> 00:10:52,571 irrelevant and so this formula is as valid as this for H much smaller than the 143 00:10:52,571 --> 00:10:58,308 radius of earth and then at higher altitudes this becomes valid we can see 144 00:10:58,308 --> 00:11:01,355 that Mathematical calculation graphically 145 00:11:01,355 --> 00:11:05,172 here, this would be mgh. This is the exact potential energy. 146 00:11:05,172 --> 00:11:08,134 the graph shows us two salient properties. 147 00:11:08,134 --> 00:11:13,268 One is the potential energy is laways negative, that's just a choice of our 148 00:11:13,268 --> 00:11:17,349 constant that we added. It becomes closer to zero, it increases 149 00:11:17,349 --> 00:11:20,245 towards zero as one gets farther from earth. 150 00:11:20,245 --> 00:11:24,849 So, it approaches zero at infinite distance from earth near Earth it 151 00:11:24,849 --> 00:11:28,243 increasesm mgh and minus GMM over R are not the same. 152 00:11:28,243 --> 00:11:32,810 But if one shifts this down by an appropriate constant, then one sees that 153 00:11:32,810 --> 00:11:37,315 as long as H is much smaller than the radius of the Earth you get a good 154 00:11:37,315 --> 00:11:41,675 approximation. The statement oft heard that there is no 155 00:11:41,675 --> 00:11:44,926 gravity in space, you've seen is patently false. 156 00:11:44,926 --> 00:11:49,699 The earth's gravitational attraction extents throughout the universe. 157 00:11:49,699 --> 00:11:54,471 But it does weaken with distance. And so, perhaps the correct aversion 158 00:11:54,471 --> 00:11:59,175 that's times one here's, is that astronauts in the space station are 159 00:11:59,175 --> 00:12:04,155 weightless because the gravitational force of earth is weaker at larger 160 00:12:04,155 --> 00:12:10,414 distances from the center of the Earth. here we see Stephen Hawking weightless. 161 00:12:10,414 --> 00:12:16,607 no, no, no, the space station. as we have said, orbits at an altitude of 162 00:12:16,607 --> 00:12:20,564 400 kilometers. So, we can compute the force of 163 00:12:20,564 --> 00:12:27,273 gravitation, we can compute the weight of an object, say, myself, or even Hawking 164 00:12:27,273 --> 00:12:32,706 at an altitude of H above the earth. We found that the approximate version for 165 00:12:32,706 --> 00:12:37,753 the force of gravity for altitudes H smaller than the radius of the earth is 166 00:12:37,753 --> 00:12:41,031 this formula. This makes it easy to plug in the 167 00:12:41,031 --> 00:12:46,209 numbers, and you find that in the space station your wait is in fact 87 percent. 168 00:12:46,209 --> 00:12:51,453 In other words, the force with which the earth attracts to is 87 percent of what 169 00:12:51,453 --> 00:12:55,387 it would be on earth. Why then are astronauts weightless in the 170 00:12:55,387 --> 00:12:58,310 space station? Why is Hawking weightless in this 171 00:12:58,310 --> 00:13:01,234 picture? Stephen Hawking is far too important to 172 00:13:01,234 --> 00:13:06,227 humanity to allow him to travel to space and take the risks of real space travel. 173 00:13:06,227 --> 00:13:11,160 Hawking is weightless because, in fact he is accelerating towards the ground with 174 00:13:11,160 --> 00:13:15,788 an acceleration that is essentially g. When you are in free fall, there is no 175 00:13:15,788 --> 00:13:19,260 gravity, and there is a beautiful demo that explains this. 176 00:13:22,360 --> 00:13:26,174 We said that all you have to do is to be weightless is be in free fall. 177 00:13:26,174 --> 00:13:30,095 So what I'm going to do in this demonstration is I'm jumping off the box. 178 00:13:30,095 --> 00:13:34,232 I'm jumping in ultra slow motion, because we form a video over the high speed 179 00:13:34,232 --> 00:13:38,369 camera, and I've slowed it down further. And as I fall, to demonstrate that I'm 180 00:13:38,369 --> 00:13:41,270 really weightless, I'm dropping this little white ball. 181 00:13:41,270 --> 00:13:45,282 And what you see is that as I drop the ball, of course the ball's falling. 182 00:13:45,282 --> 00:13:47,976 So am I. To my eyes the ball is hovering in front 183 00:13:47,976 --> 00:13:51,989 of my face and it's not moving. And it won't stop falling in concert with 184 00:13:51,989 --> 00:13:55,947 me until my feet hit the ground. You can tell exactly the moment when my 185 00:13:55,947 --> 00:14:00,345 feet hit the ground because that is the moment when the ball is no longer level 186 00:14:00,345 --> 00:14:04,240 with my nose. Well, I hope you found that amusing. 187 00:14:04,240 --> 00:14:07,380 What we saw is that when you're freely falling, 188 00:14:07,380 --> 00:14:11,469 you, you experience no gravity. A ball you drop well however in front of 189 00:14:11,469 --> 00:14:14,309 your face. Had I thrown that ball up it would have 190 00:14:14,309 --> 00:14:18,625 relatively to my nose, moved up at a constant velocity, and this is important 191 00:14:18,625 --> 00:14:23,283 because in space there are no floors for your feet to hit and so everything is in 192 00:14:23,283 --> 00:14:27,770 free fall. For example, the earth as it orbits the sun in its circular motion is 193 00:14:27,770 --> 00:14:32,029 in free fall under the gravitational influence of the sun which means that 194 00:14:32,029 --> 00:14:36,573 when studying the motions of objects near if we can completely ignore the sun's 195 00:14:36,573 --> 00:14:39,300 gravity. We are all in free fall towards the sun. 196 00:14:39,300 --> 00:14:40,430 Really? Almost. 197 00:14:40,430 --> 00:14:47,133 The reason we're only almost all in free fall towards the sun is because we're all 198 00:14:47,133 --> 00:14:49,831 actually held together by staying on earth. 199 00:14:49,831 --> 00:14:54,469 So we're all falling towards the sun with, one constant acceleration, and that 200 00:14:54,469 --> 00:14:58,875 could not possibly be the free fall acceleration towards the sun, at all the 201 00:14:58,875 --> 00:15:03,224 positions we are because at different positions, the free fall acceleration 202 00:15:03,224 --> 00:15:07,746 towards the sun is different, depending on your distance from and direction to 203 00:15:07,746 --> 00:15:10,471 the sun. These differences, in the gravitational 204 00:15:10,471 --> 00:15:15,243 free fall acceleration at different points of an extended object, will lead 205 00:15:15,243 --> 00:15:22,245 to forces, tidal forces, we'll see why we call them tidal forces, that account for 206 00:15:22,245 --> 00:15:26,291 the differences in gravitational acceleration between different parts of 207 00:15:26,291 --> 00:15:29,117 an object. These differences manifest themselves in 208 00:15:29,117 --> 00:15:32,997 left over gravitational forces. Like all gravitational forces, they are 209 00:15:32,997 --> 00:15:36,655 proportional to your mass. So the tidal force on a refrigerator is 210 00:15:36,655 --> 00:15:39,038 more than the tidal force on a tennis ball. 211 00:15:39,038 --> 00:15:43,583 And they are given by your mass times the difference between the gravitational 212 00:15:43,583 --> 00:15:46,188 acceleration towards, say the sun, where you are, 213 00:15:46,188 --> 00:15:49,965 and the acceleration with which as a denizen of Earth you have to the R 214 00:15:49,965 --> 00:15:52,736 falling onto the Sun. That would be the gravitational 215 00:15:52,736 --> 00:15:56,553 acceleration towards the Sun, as it applies at the centre of the Earth. 216 00:15:56,553 --> 00:16:00,631 This sounds a little bit confusing and like many things can be clarified by a 217 00:16:00,631 --> 00:16:02,940 good calculation. Let's turn to that. 218 00:16:02,940 --> 00:16:06,224 Let's do this and let's do the calculation. 219 00:16:06,224 --> 00:16:10,121 so in this image we have the earth over here. 220 00:16:10,121 --> 00:16:16,079 The sun I'm going to assume is way over there in the left hand corner and way 221 00:16:16,079 --> 00:16:20,892 outside the image. And because the sun's gravitation applies 222 00:16:20,892 --> 00:16:25,400 at force to the earth given by FG equals G times the mass of the sun, 223 00:16:25,400 --> 00:16:30,160 divided by the distance to the sun squared, times the mass of the earth. 224 00:16:30,160 --> 00:16:35,408 The entire Earth is accelerating to the left towards the sun with an acceleration 225 00:16:35,408 --> 00:16:40,592 given by, this is equal to the mass of the Earth times the acceleration of the 226 00:16:40,592 --> 00:16:45,520 Earth and I can, as always with gravity, cancel the factors of the mass of the 227 00:16:45,520 --> 00:16:49,505 Earth and so what I find. Is that the entire earth is accelerating, 228 00:16:49,505 --> 00:16:54,100 towards the sun with, this acceleration, where, the distance to the sun I'm going 229 00:16:54,100 --> 00:16:58,521 to take as the, always, as the center to center distance, because that is what 230 00:16:58,521 --> 00:17:01,896 Newton taught us to do, with spherically symmetric objects. 231 00:17:01,896 --> 00:17:06,375 Now, the important thing is, the entire earth is accelerating to the left with 232 00:17:06,375 --> 00:17:10,796 this acceleration, and because this is the acceleration of free fall, gravity 233 00:17:10,796 --> 00:17:14,396 would cancel except. But if, if you are sitting at this point 234 00:17:14,396 --> 00:17:19,391 on earth, you are a little bit nearer the sun at the center of the earth by a very 235 00:17:19,391 --> 00:17:22,681 small amount. This is 6,400 kilometers, the distance of 236 00:17:22,681 --> 00:17:27,250 the sun is 150,000,000 kilometers, but it's a difference nonetheless so the 237 00:17:27,250 --> 00:17:31,575 gravitational acceleration towards the sun here is a little bit larger. 238 00:17:31,575 --> 00:17:36,327 And so the gravitational acceleration towards the sun at this point is larger 239 00:17:36,327 --> 00:17:40,652 because you are closer to the sun. Indeed, the denominator for computing 240 00:17:40,652 --> 00:17:43,820 here should be decreased by the radius of the earth. 241 00:17:43,820 --> 00:17:47,724 Similarly, the gravitational acceleration on the far side of the sun will be 242 00:17:47,724 --> 00:17:51,680 smaller than the value in the center of the earth because you're at a larger 243 00:17:51,680 --> 00:17:56,354 distance by the radius of the earth. Now, the radius of the earth is of course 244 00:17:56,354 --> 00:17:59,247 much smaller than the distance to the sun. 245 00:17:59,247 --> 00:18:05,033 This gives us a wonderful opportunity to apply our approximation that we learned 246 00:18:05,033 --> 00:18:07,375 from Newton. We do the usual steps. 247 00:18:07,375 --> 00:18:12,156 We write here. Pull out d squared of shortcut. 248 00:18:12,156 --> 00:18:18,982 The next step we will get one minus r, radius of the earth over distance to the 249 00:18:18,982 --> 00:18:24,597 sun to the power minus two. X is the small quantity radius of the 250 00:18:24,597 --> 00:18:31,596 earth divided by distance to the sun, a is negative two and the result is gn over 251 00:18:31,596 --> 00:18:36,100 d squared, which is, simply A. 252 00:18:36,100 --> 00:18:46,158 And then times one, plus two, R over D. And so, this is our approximate value, 253 00:18:46,158 --> 00:18:51,378 valid because the earth is much smaller, than the distance to the sun, for, the 254 00:18:51,378 --> 00:18:55,785 value of the acceleration, of gravity on the near side to the sun, 255 00:18:55,785 --> 00:18:59,040 and we can make a similar, calculation over here. 256 00:18:59,040 --> 00:19:04,057 The difference will be, this plus sign will turn into a minus sign when we 257 00:19:04,057 --> 00:19:08,328 multiply it by minus two. So, we can write, the acceleration over 258 00:19:08,328 --> 00:19:11,921 here, in a similar form. Now this allows us, that suggests, 259 00:19:11,921 --> 00:19:17,486 rewriting, this is a sum of two terms. The acceleration of with which the Earth 260 00:19:17,486 --> 00:19:23,377 approaches, and then an extra little bit so an object over here has a free-fall 261 00:19:23,377 --> 00:19:28,969 acceleration towards the sun that is larger than this one by the amount of 262 00:19:28,969 --> 00:19:34,925 this little yellow arrow which signifies A times 2R over D, and similarly an 263 00:19:34,925 --> 00:19:39,996 object on the far side of the sun has a gravitational acceleration toward the sun 264 00:19:39,996 --> 00:19:44,944 that is less than the acceleration with which the earth moves toward the sun by 265 00:19:44,944 --> 00:19:48,563 approximately the same amount. To this order of approximation. 266 00:19:48,563 --> 00:19:52,090 Now, one can apply a similar logic to these positions on Earth. 267 00:19:52,090 --> 00:19:56,469 Their distance from the Sun to first order, is the same as that of the center 268 00:19:56,469 --> 00:19:59,314 of the Earth. So, one asks, why is the gravitational 269 00:19:59,314 --> 00:20:03,011 acceleration here, different from that at the center of the Earth? 270 00:20:03,011 --> 00:20:07,675 Well, that is because to a very small extent the direction from here to the Sun 271 00:20:07,675 --> 00:20:11,884 is not the same as the direction from here to the Sun, as these arrows and 272 00:20:11,884 --> 00:20:15,980 again, a grossly exaggerated version demonstrate, this would have the Sun 273 00:20:15,980 --> 00:20:18,949 about here. The Sun should be very far, so this angle 274 00:20:18,949 --> 00:20:23,483 should be tiny, but again I can rewrite these as the gravitational acceleration 275 00:20:23,483 --> 00:20:26,410 at the center of the earth, plus a small correction. 276 00:20:26,410 --> 00:20:31,921 And so, the net result is that the gravitational acceleration with which the 277 00:20:31,921 --> 00:20:36,926 earth is moving towards the sun, that part of the sun's gravity is cancelled by 278 00:20:36,926 --> 00:20:40,157 the earth's motion. But the leftovers, 279 00:20:40,157 --> 00:20:43,444 these yellow little differences, are not cancelled. 280 00:20:43,444 --> 00:20:48,637 So if I put those differences back in their place, I see that the net result is 281 00:20:48,637 --> 00:20:52,384 that the Sun exerts a small force upward away from Earth, 282 00:20:52,384 --> 00:20:56,656 on this side of the Earth, and a small force upward away from the 283 00:20:56,656 --> 00:21:01,389 Earth on this side of the Earth. And a small force downward on this side 284 00:21:01,389 --> 00:21:04,610 of the Earth. In other words, the Sun, is doing its 285 00:21:04,610 --> 00:21:11,528 best to smoosh the Earth into this, oval that is elongated in the direction of the 286 00:21:11,528 --> 00:21:14,427 sun. And so of course the Earth is kind of a 287 00:21:14,427 --> 00:21:17,510 rigid object. The sun cannot manage to deform the 288 00:21:17,510 --> 00:21:20,971 Earth, but the Earth is under some kind of tidal stress. 289 00:21:20,971 --> 00:21:25,816 Now before we get too excited, let's figure out how strong this force is that 290 00:21:25,816 --> 00:21:29,466 we've calculated. Again, the magnitude of the force depends 291 00:21:29,466 --> 00:21:33,493 on what it's acting on. These are all accelerations, so imagine a 292 00:21:33,493 --> 00:21:37,269 one kilo rock and figure out the net force that this has. 293 00:21:37,269 --> 00:21:40,954 Or better yet. Compute an acceleration and compare it to 294 00:21:40,954 --> 00:21:46,230 the acceleration of gravity so we know, is this rock essentially rendered, 295 00:21:46,230 --> 00:21:50,200 Weightless as it lifted off the earth by these tidal forces. 296 00:21:50,200 --> 00:21:53,300 So we can do the calculation. We have here the. 297 00:21:53,300 --> 00:21:57,019 All the quantities we need. The title acceleration due to the sun is 298 00:21:57,019 --> 00:22:00,718 the difference. That little yellow difference, which was 299 00:22:00,718 --> 00:22:05,941 2GM over D squared times R over D. The salient point is that because I have 300 00:22:05,941 --> 00:22:11,297 2GM over D square times R over D, this decreases with distance from the sun like 301 00:22:11,297 --> 00:22:15,850 the third power of distance. Remember the gravitational force itself 302 00:22:15,850 --> 00:22:20,670 decreases like the second power. And I want to compare this acceleration 303 00:22:20,670 --> 00:22:25,156 to something that I know. So I will do the usual scaling trick that 304 00:22:25,156 --> 00:22:30,000 I like to use. Try to pull out a factor over here of 305 00:22:30,000 --> 00:22:33,710 The mass of the earth divided by the radius of the earth squared. 306 00:22:33,710 --> 00:22:38,411 I need didn't have the mass of the earth I had the mass of the sun so I get one 307 00:22:38,411 --> 00:22:40,790 factor here. I had the radius of the earth. 308 00:22:40,790 --> 00:22:46,015 Divided by the distance to the Sun cubed. So, I get a scaling factor, notice I have 309 00:22:46,015 --> 00:22:49,306 an acceleration that is known, this is our friend g. 310 00:22:49,306 --> 00:22:53,886 These are dimensionless factors, so I am relating an acceleration to an 311 00:22:53,886 --> 00:22:57,435 acceleration. And, plugging in the ratio of the Sun's 312 00:22:57,435 --> 00:23:02,144 mass to the earth, and the cube of the ratio of the earth's radius to the 313 00:23:02,144 --> 00:23:07,048 distance to the Sun, I find that the acceleration, the tidal acceleration due 314 00:23:07,048 --> 00:23:10,790 to the Sun is very small. It's five parts in a 100 million, 315 00:23:10,790 --> 00:23:15,207 the acceleration of gravity. So no it is not about to rip the earth 316 00:23:15,207 --> 00:23:16,130 apart. This is. 317 00:23:16,130 --> 00:23:19,526 The, the grav, tidal acceleration due to the sun. 318 00:23:19,526 --> 00:23:23,356 What else might exert a tidal influence on the earth? 319 00:23:23,356 --> 00:23:27,375 Well, the second Object that we might imagine would be the 320 00:23:27,375 --> 00:23:30,146 moon. Now the moon is much less massive than 321 00:23:30,146 --> 00:23:35,373 the sun but it is closer and so because of this, third power of the distance one 322 00:23:35,373 --> 00:23:40,223 might imagine that the tidal force generated by the moon might be comparable 323 00:23:40,223 --> 00:23:44,380 to that generated by the sun and indeed a simple scaling argument. 324 00:23:44,380 --> 00:23:48,814 Tells us that, to get the tidal acceleration due to this moon relative to 325 00:23:48,814 --> 00:23:53,492 that from the sun, I multiply by the ratio of the masses, and the ratio of the 326 00:23:53,492 --> 00:23:58,230 distances to the, power negative three. Plugging all the numbers in, I find, that 327 00:23:58,230 --> 00:24:03,029 in fact, the tidal acceleration due to the moon is approximately twice, that due 328 00:24:03,029 --> 00:24:06,067 to the sun. So the dominant tidal effects on earth, 329 00:24:06,067 --> 00:24:08,807 are. Lunar tides, rather than solar tides. 330 00:24:08,807 --> 00:24:11,443 but. Solar tides are not negligible. 331 00:24:11,443 --> 00:24:15,605 They are a 50% effect. The net result is a combination of the 332 00:24:15,605 --> 00:24:18,380 two. Let's take a look at what that does. 333 00:24:18,380 --> 00:24:24,535 So here we see the earth and the moon of course, drawn way too close to earth to 334 00:24:24,535 --> 00:24:28,762 be to scale. And we see the elongated shape into which 335 00:24:28,762 --> 00:24:32,470 the moon is attempting to draw the earth again, 336 00:24:32,470 --> 00:24:37,753 Exaggerated way out of proportion. Remember though that while the Earth is 337 00:24:37,753 --> 00:24:43,678 rigid and unable to respond to this tidal pull except very slightly, because it's a 338 00:24:43,678 --> 00:24:48,747 rigid rocky object, there is a soft component of Earth, namely the water. 339 00:24:48,747 --> 00:24:52,960 So the surface of the oceans will deform in this direction. 340 00:24:52,960 --> 00:24:57,043 Ever so slightly. again not to this extent but there will 341 00:24:57,043 --> 00:25:02,624 be high tide on this side of the earth and a low, high tide on this side of the 342 00:25:02,624 --> 00:25:05,754 earth. And low tide at these points, because of 343 00:25:05,754 --> 00:25:10,995 the, tidal information that we discussed. And of course, as the moon orbits the 344 00:25:10,995 --> 00:25:15,014 earth, the tidal bulge will point towards the Moon 345 00:25:15,014 --> 00:25:21,283 and as the Earth rotates, once a day, each point will, each point on Earth will 346 00:25:21,283 --> 00:25:24,809 appear, have two high tides and two low tides. 347 00:25:24,809 --> 00:25:30,529 And these because you pass through these two points and these once a day. 348 00:25:30,529 --> 00:25:36,876 And, because of the Moon's motion along the celestial sphere, tides will repeat 349 00:25:36,876 --> 00:25:41,030 not every 24 hours, but every 24 hours and 44 minutes, 350 00:25:41,030 --> 00:25:46,010 because the moon advances by 44 minutes relative to the sun, remember. 351 00:25:46,010 --> 00:25:51,568 And so, indeed if you look up the table of tides you'll find the tide repeats 352 00:25:51,568 --> 00:25:56,549 every 24 hours and 44 minutes. and, and this is because they are 353 00:25:56,549 --> 00:26:01,829 associated to the moon. But, there, there is another effect, that 354 00:26:01,829 --> 00:26:05,190 two effects that come into play. One is. 355 00:26:05,190 --> 00:26:09,032 We can include the sun. So, in this picture, the sun is way off 356 00:26:09,032 --> 00:26:12,685 to the right. The sun is attempting to extend the earth 357 00:26:12,685 --> 00:26:17,409 as the horizontally extended oblong. Where as the moon, attempts to draw the 358 00:26:17,409 --> 00:26:20,684 earth to whatever direction points towards the moon. 359 00:26:20,684 --> 00:26:24,618 When the moon is full. Earth sun and moon are in line and the 360 00:26:24,618 --> 00:26:29,520 sun's deformation, the moon's deformation are in exactly the same direction. 361 00:26:29,520 --> 00:26:33,698 We get very, very exaggeratedly high tides, they are called spring tides. 362 00:26:33,698 --> 00:26:38,287 When we have a quarter moon, the moon is elongating the earth this way, and the 363 00:26:38,287 --> 00:26:42,171 sun this way, and the tides are diminished, we call them neap tides. 364 00:26:42,171 --> 00:26:46,702 Tides are very weak at quarter moon, and of course when we get a new moon, the 365 00:26:46,702 --> 00:26:51,351 earth and the sun are elongating the earth in the same direction and tides are 366 00:26:51,351 --> 00:26:54,176 stronger. So, tides are strongest at new moon and 367 00:26:54,176 --> 00:26:57,221 at full moon. Those are the spring tides, Tides are 368 00:26:57,221 --> 00:27:01,510 weakest at quarter moon. Those are the neap tides, and Newton's 369 00:27:01,510 --> 00:27:06,148 theory makes easy sense of all that. And then there's another effect that we 370 00:27:06,148 --> 00:27:08,519 can take into account. The earth rotates. 371 00:27:08,519 --> 00:27:13,408 What the Earths rotation does is as the moon creates a bulge of water the 372 00:27:13,408 --> 00:27:18,429 rotation of the Earth, which is much faster than the moons orbit by a factor 373 00:27:18,429 --> 00:27:23,912 of 30 remember, drives this bulge so that water that was raised here is actually, 374 00:27:23,912 --> 00:27:25,300 over here. This means. 375 00:27:25,300 --> 00:27:29,201 farther to the east. This means that the high tide bulge is a 376 00:27:29,201 --> 00:27:34,139 little bit to the east of the moon, which means that high tide where you are is a 377 00:27:34,139 --> 00:27:37,919 little bit later than the time the moon crosses your meridian. 378 00:27:37,919 --> 00:27:41,942 it also has other implications which we'll discuss momentarily. 379 00:27:41,942 --> 00:27:46,820 But these are the three effects that this demonstration shows very convincingly. 380 00:27:46,820 --> 00:27:52,990 To summarize what we've seen, the moon, Attempts to deform, the earth. 381 00:27:52,990 --> 00:27:58,248 It succeeds in deforming, the old water in the oceans, so there's a bulge facing 382 00:27:58,248 --> 00:28:01,309 the moon. As the earth rotates the bulge moves 383 00:28:01,309 --> 00:28:06,168 around the earth, so, everybody gets two high tides, and two low tides, once 384 00:28:06,168 --> 00:28:11,492 every, 24 hours and 44 minutes so the time between, one high tide and the next, 385 00:28:11,492 --> 00:28:15,486 is in fact about twelve and a half hours rather than twelve. 386 00:28:15,486 --> 00:28:20,943 The earth's rotation drags the bulge to the east, so therefore it lags the moon 387 00:28:20,943 --> 00:28:24,419 the. Like everything else in that image the 388 00:28:24,419 --> 00:28:29,233 lag was way exaggerated. In fact it turns out to be about three 389 00:28:29,233 --> 00:28:33,293 degrees or twelve minutes. Now in addition to the Moon the Sun 390 00:28:33,293 --> 00:28:37,959 exerts a tidal force directed towards the Sun with about half as, the strength as 391 00:28:37,959 --> 00:28:41,300 that of the Moon. At full or new Moon, these act in concert 392 00:28:41,300 --> 00:28:45,851 creating, creating the intense tides we call spring tides, which have nothing to 393 00:28:45,851 --> 00:28:48,673 do with spring. Whereas at quarter Moon in either 394 00:28:48,673 --> 00:28:52,763 direction, the Sun's deformation counteracts the Moon's deformation and 395 00:28:52,763 --> 00:28:55,393 you get the relatively weak leap tides. Pull. 396 00:28:55,393 --> 00:29:00,223 Something we understand, even more, consequences of tidal forces. 397 00:29:00,223 --> 00:29:06,203 the moon is not able to deformed the earth, but when the moon formed a long 398 00:29:06,203 --> 00:29:10,420 time ago, much closer to the earth, we'll see, and molten. 399 00:29:10,420 --> 00:29:13,291 it was soft. And closer to earth. 400 00:29:13,291 --> 00:29:18,163 And the tidal force of earth. Remember at a closer distance with a mass 401 00:29:18,163 --> 00:29:22,141 bigger than the mass of the Moon you get more intense tidal forces. 402 00:29:22,141 --> 00:29:27,068 The tidal forces of Earth deform the Moon so that when it solidified it froze with 403 00:29:27,068 --> 00:29:29,800 a permanent bulge. The Moon is not quite round. 404 00:29:29,800 --> 00:29:34,666 Now the tidal forces act to keep this bulge always aligned in the direction of 405 00:29:34,666 --> 00:29:36,699 earth. We call this tidal locking. 406 00:29:36,699 --> 00:29:41,628 And this is the reason why, as I told you a long time ago, when we look up at the 407 00:29:41,628 --> 00:29:44,400 moon, we always see the same side of the moon. 408 00:29:44,400 --> 00:29:49,205 There is a near side and a far side to the moon, where the locks, the moon's 409 00:29:49,205 --> 00:29:54,442 spin period about its axis to its orbital period about the earth is tidal locking, 410 00:29:54,442 --> 00:29:57,030 and this is a phenomenon we'll see before. 411 00:29:57,030 --> 00:30:03,182 Moreover, since the tidal bulge on earth is being dragged to the east of the moon, 412 00:30:03,182 --> 00:30:09,260 the tidal force from the moon attempting to align the tidal bulge with the moon, 413 00:30:09,260 --> 00:30:12,841 is actually trying to slow down the rotation of the earth. 414 00:30:12,841 --> 00:30:16,891 Think about it differently, there is friction between the title bulge 415 00:30:16,891 --> 00:30:21,705 and the rotating earth, because the title bulge cannot keep pace with the rotating 416 00:30:21,705 --> 00:30:24,464 earth. It's held, locked three degrees away from 417 00:30:24,464 --> 00:30:27,341 the moon. What this is doing is it's slowing down 418 00:30:27,341 --> 00:30:28,867 of the earth rotation? Yes. 419 00:30:28,867 --> 00:30:34,863 The Earth's rotation is slowing down. But, and, but, but, Roland, what about 420 00:30:34,863 --> 00:30:39,949 conservation of angular momentum? Well angular momentum is conserved. 421 00:30:39,949 --> 00:30:45,348 The way it's conserved is that, that same bulge acts on the moon, to in in fact, 422 00:30:45,348 --> 00:30:50,637 sling it faster along its trajectory, so that the moon is being accelerated, and 423 00:30:50,637 --> 00:30:55,123 as it gains angular momentum, it's actually moving to higher orbits. 424 00:30:55,123 --> 00:30:58,070 So the moon is actually receding from earth. 425 00:30:58,070 --> 00:31:02,888 This is an observation first argued strongly by George Darwin, son of the 426 00:31:02,888 --> 00:31:07,509 famous Charles Darwin in 1898, and was the reasoning behind his theory 427 00:31:07,509 --> 00:31:12,460 that, when the moon formed, it was indeed much nearer earth, and as we will see 428 00:31:12,460 --> 00:31:17,807 that coincides with modern theories we think that the moon formed much closer to 429 00:31:17,807 --> 00:31:20,975 the earth. And, these were conjunctures until the 430 00:31:20,975 --> 00:31:25,956 Apollo missions left, reflectors on the moon enabling us to make a very precise 431 00:31:25,956 --> 00:31:29,997 measurement of the distance to the moon. And indeed we can now verify 432 00:31:29,997 --> 00:31:33,042 experimentally that the moon is receding form earth. 433 00:31:33,042 --> 00:31:37,083 So remember it is now exactly the angular size of the sun in the sky. 434 00:31:37,083 --> 00:31:41,124 In a few millennia total solar eclipses maybe a thing of the past. 435 00:31:41,124 --> 00:31:44,893 So we worked pretty hard, but we're getting somewhere. 436 00:31:44,893 --> 00:31:50,316 Understanding universal laws has given us a unifying understanding of many, many 437 00:31:50,316 --> 00:31:53,028 phenomena. So, we understand tidal forces. 438 00:31:53,028 --> 00:31:55,704 And remember, in space. Nothing is held up. 439 00:31:55,704 --> 00:32:00,253 Everything then is in free fall. Everything is an orbit around everything 440 00:32:00,253 --> 00:32:02,683 else. And so tidal forces are extremely 441 00:32:02,683 --> 00:32:05,737 important. this is very nicely evident in this 442 00:32:05,737 --> 00:32:08,666 beautiful image of M51, the whirlpool nebula. 443 00:32:08,666 --> 00:32:13,091 And what we see is that these are in fact, two galaxies that are in the 444 00:32:13,091 --> 00:32:16,975 process of colliding. Galactic collisions are slow and rather 445 00:32:16,975 --> 00:32:21,874 delicate but we see that there's a tail. Of stars from this galaxy being dragged 446 00:32:21,874 --> 00:32:24,954 out towards it's neighbor by the gravitational force. 447 00:32:24,954 --> 00:32:29,602 And now that we understand tidal forces, it's not surprising that there is an 448 00:32:29,602 --> 00:32:34,251 equal tail being dragged out by the same tidal forces because these are in free 449 00:32:34,251 --> 00:32:37,795 fall about each other. And the other side of the galaxy, tidal 450 00:32:37,795 --> 00:32:41,747 forces are approximately symmetric. These are falling in because the 451 00:32:41,747 --> 00:32:46,395 gravitational acceleration is larger than that with which the galaxy is falling, 452 00:32:46,395 --> 00:32:50,114 these because it's smaller. The net result is the galaxies being 453 00:32:50,114 --> 00:32:53,955 stretched in both directions. So understanding the uses and 454 00:32:53,955 --> 00:32:59,011 applications of F equals MA is a very powerful tool and we're going to continue 455 00:32:59,011 --> 00:33:03,246 to apply it as we go along. And what we're going to turn to next is 456 00:33:03,246 --> 00:33:07,922 understanding a little bit more about the forces in nature beyond gravity. 457 00:33:07,922 --> 00:33:11,398 What else is there? And something about the M, something 458 00:33:11,398 --> 00:33:15,316 about the structure of the matter upon which these forces act. 459 00:33:15,316 --> 00:33:20,309 This will be important for understanding the structure and workings of various 460 00:33:20,309 --> 00:33:24,291 astronomical phenomena. And so that's what we'll turn to in the 461 00:33:24,291 --> 00:33:24,923 next clip.