1 00:00:02,620 --> 00:00:08,933 Brahe, Kepler, and then Galileo have made great strides in taking these celestial 2 00:00:08,933 --> 00:00:14,535 objects and studying them as physical concepts that we can measure, and 3 00:00:14,535 --> 00:00:19,734 discuss, and hopefully understand. But the real understanding waited a few 4 00:00:19,734 --> 00:00:22,588 more decades for, as I promised, Issac Newton. 5 00:00:22,588 --> 00:00:27,439 We're going to spend the next few clips following in the steps the, rather large 6 00:00:27,439 --> 00:00:30,827 steps of Sir Isaac, and seeing what it was that he taught us. 7 00:00:30,827 --> 00:00:35,287 And we're going to start at the beginning with mechanics, the science of motion, 8 00:00:35,287 --> 00:00:37,715 which is going to underlie everything else. 9 00:00:37,715 --> 00:00:40,200 We're going to do some physics, bear with me. 10 00:00:40,200 --> 00:00:42,840 It's going to be intense. To study motion, 11 00:00:42,840 --> 00:00:46,567 the first thing we need to do is make mathematically precise what it is that 12 00:00:46,567 --> 00:00:49,665 Galileo said when he said, an object retains its state of motion. 13 00:00:49,665 --> 00:00:53,344 So mathematically, an object's state of motion is represented by an object's 14 00:00:53,344 --> 00:00:55,914 velocity. Velocity is a, physics concept. 15 00:00:55,914 --> 00:01:01,250 It means the, speed with which something is moving, along with its direction. 16 00:01:01,250 --> 00:01:05,234 So, we denote it, by this V with an arrow on top of it. 17 00:01:05,234 --> 00:01:10,642 V stands for velocity and the arrow reminds us that this is a number, the 18 00:01:10,642 --> 00:01:16,191 speed at which something is moving, along with the direction, so that, this ball 19 00:01:16,191 --> 00:01:21,812 can be moving at one meter per second, to your, left or my left and at one meter 20 00:01:21,812 --> 00:01:25,370 per second to my right or up or down or in or out. 21 00:01:25,370 --> 00:01:29,587 And those, even if the speeds are the same, are different values of the 22 00:01:29,587 --> 00:01:33,033 velocity. Velocity can be thought of, as we know, 23 00:01:33,033 --> 00:01:38,840 speed is measured in meters per second because it is the change in position, 24 00:01:38,840 --> 00:01:42,126 where you are, the distance divided by time. 25 00:01:42,126 --> 00:01:46,940 And velocity can change. The ball can start slowly and speed up. 26 00:01:46,940 --> 00:01:49,377 And its change is called the acceleration. 27 00:01:49,377 --> 00:01:53,672 So acceleration is the rate at which velocity changes, and acceleration is 28 00:01:53,672 --> 00:01:58,258 measured therefore, in meters per second. The difference, in velocity between the 29 00:01:58,258 --> 00:02:02,611 beginning and the end of an interval, divided by the length of the interval. 30 00:02:02,611 --> 00:02:06,732 The same way that velocity is distance divided by time, acceleration is 31 00:02:06,732 --> 00:02:10,331 velocity, per time. And so the rate of change of velocity is 32 00:02:10,331 --> 00:02:14,394 an acceleration, and is measured in meters per second squared, and like 33 00:02:14,394 --> 00:02:19,242 velocity, acceleration has a direction. So that, if an object is moving to the 34 00:02:19,242 --> 00:02:23,553 left and speeding up, then its acceleration points to the left. 35 00:02:23,553 --> 00:02:27,170 If an object is moving to the left and slowing down. 36 00:02:27,170 --> 00:02:31,919 That means its velocity points to the left but its acceleration points to the 37 00:02:31,919 --> 00:02:36,608 right, because the velocity is less to the left later in the interval than it 38 00:02:36,608 --> 00:02:39,471 was before. So, acceleration, like velocity, has a 39 00:02:39,471 --> 00:02:42,631 direction. And, in addition, to expressing the fact 40 00:02:42,631 --> 00:02:47,186 that an object can be speeding up or slowing down, there can be a change of 41 00:02:47,186 --> 00:02:51,499 velocity that is associated with maintaining the same speed, but merely 42 00:02:51,499 --> 00:02:55,083 change in direction. If an object is moving to the left and 43 00:02:55,083 --> 00:02:59,942 starts moving towards your camera, then this is change in velocity even if speed 44 00:02:59,942 --> 00:03:04,680 never changed and particularly useful example for us in studying astronomy is 45 00:03:04,680 --> 00:03:08,142 going to be in fact, of motion without changing the speed. 46 00:03:08,142 --> 00:03:12,820 Let's look at it and may be it will help clarify these concepts a little bit. 47 00:03:12,820 --> 00:03:18,784 Example you're going to want to study is the example of an object, say maybe a 48 00:03:18,784 --> 00:03:23,360 planet, moving in a circle about some center, say the sun. 49 00:03:23,360 --> 00:03:29,777 At a constant speed, so in this first pane, you see the, black dot which is our 50 00:03:29,777 --> 00:03:33,708 object. And it is set so that it is moving around 51 00:03:33,708 --> 00:03:38,522 the circle, at a constant speed. Its speed is not changing. 52 00:03:38,522 --> 00:03:45,180 Its velocity however is clearly changing, because if I freeze the animation here 53 00:03:45,180 --> 00:03:48,470 the velocity of the object, manifestly, is. 54 00:03:48,470 --> 00:03:51,870 Pointed in this direction, it's moving that way. 55 00:03:51,870 --> 00:03:56,488 Whereas, if I let the animation go a little farther, and freeze it again here, 56 00:03:56,488 --> 00:04:00,620 the object is moving at the same speed but in a different direction. 57 00:04:00,620 --> 00:04:04,960 And therefore, there is definitely a change in speed, and, to, quantify and 58 00:04:04,960 --> 00:04:09,542 understand that change in speed, we will move down, to this pane, where, I have 59 00:04:09,542 --> 00:04:14,003 put in, the object's velocity. It is as an object with a direction so 60 00:04:14,003 --> 00:04:19,188 we, we draw it with an arrow, and the arrow at the beginning points up because 61 00:04:19,188 --> 00:04:22,262 that's the direction in which the object is moving. 62 00:04:22,262 --> 00:04:26,784 And as the object moves, the velocity changes because, while the arrow is of 63 00:04:26,784 --> 00:04:30,040 the same length its direction is continually changing. 64 00:04:30,040 --> 00:04:35,001 So this is the way your velocity changes when you move around a circle at constant 65 00:04:35,001 --> 00:04:38,933 speed. Coming down here I've got two circles. 66 00:04:38,933 --> 00:04:43,973 The circle along which the object is moving and then this imaginary circle. 67 00:04:43,973 --> 00:04:49,350 And if you look what the imaginary circle does is it I've copied the blue arrow 68 00:04:49,350 --> 00:04:53,852 over to here and drawn it so that its tail is always at the origin. 69 00:04:53,852 --> 00:04:58,857 So the edge of this arrow is giving me the direction and, of course, the 70 00:04:58,857 --> 00:05:02,911 constant magnitude, constant size, of the velocity vector. 71 00:05:02,911 --> 00:05:08,485 when the arrow points to the right the object is here and it's moving to the 72 00:05:08,485 --> 00:05:11,960 right and so on. So this is my velocity vector V. 73 00:05:11,960 --> 00:05:14,891 Now I can understand how to measure acceleration. 74 00:05:14,891 --> 00:05:19,798 Acceleration is going to be the rate of change of this vector so that if I freeze 75 00:05:19,798 --> 00:05:23,328 the animation at any given point I can see what's going on. 76 00:05:23,328 --> 00:05:28,115 The object is over here, it is moving in this direction, that's why this velocity 77 00:05:28,115 --> 00:05:32,243 vector points this way, but I know where the velocity vector is going. 78 00:05:32,243 --> 00:05:37,209 It will next move over here to the right, so the vector that measures the change in 79 00:05:37,209 --> 00:05:42,235 velocity will point along the circle just as the vector that measures the change in 80 00:05:42,235 --> 00:05:46,927 position points along this circle. To make that clear, here's the same two 81 00:05:46,927 --> 00:05:52,107 circles and on this here on the velocity circle I've added a green arrow to 82 00:05:52,107 --> 00:05:57,013 represent the change and velocity. And as they play you can see that the 83 00:05:57,013 --> 00:06:02,125 green arrow is constant in magnitude because the velocity is rotating at a 84 00:06:02,125 --> 00:06:06,827 constant rate just because this thing is rotating at a constant rate. 85 00:06:06,827 --> 00:06:12,211 And the green arrow points along the velocity circle and what this green arrow 86 00:06:12,211 --> 00:06:17,050 is in fact is our acceleration. It's the rate of change of the velocity. 87 00:06:17,050 --> 00:06:20,585 And so to make that explicit, I've here, done three circles. 88 00:06:20,585 --> 00:06:25,339 This is the position circle, this is the velocity circle, and over here we have 89 00:06:25,339 --> 00:06:30,154 the acceleration circle or we have taken the green arrow the rate of change of 90 00:06:30,154 --> 00:06:34,543 velocity, and move its tail to the center, so that at any time, this gives 91 00:06:34,543 --> 00:06:37,347 me, the rate at which, the velocity is changing. 92 00:06:37,347 --> 00:06:42,467 So you can play with all three arrows and I'll post this file so you can try to run 93 00:06:42,467 --> 00:06:45,258 it. The main thing I want you to notice here 94 00:06:45,258 --> 00:06:50,287 is that if you look there's a relation between the direction of the acceleration 95 00:06:50,287 --> 00:06:54,820 arrow and of the position arrow. Shen the position arrow points here into 96 00:06:54,820 --> 00:06:58,235 the upper left the acceleration is to the bottom right. 97 00:06:58,235 --> 00:07:02,210 And that's not a coincidence velocity is 90 degrees to position. 98 00:07:02,210 --> 00:07:06,547 Acceleration is 90 degrees to velocity adding up to 180 degrees away from 99 00:07:06,547 --> 00:07:09,420 position. So when your out here your acceleration 100 00:07:09,420 --> 00:07:13,934 points that way and if I move it and freeze it again you'll see that this is 101 00:07:13,934 --> 00:07:16,865 always maintained. When your at this position your 102 00:07:16,865 --> 00:07:21,496 acceleration points in that position. So the acceleration at any given point is 103 00:07:21,496 --> 00:07:25,951 in fact at any given time is in fact pointing directly to the center of the 104 00:07:25,951 --> 00:07:28,823 circle. If you ask in which direction this object 105 00:07:28,823 --> 00:07:33,278 is moving it's moving along the circle. In what direction is it accelerating? 106 00:07:33,278 --> 00:07:37,562 That way towards the center. So we found that moving in a circular, in 107 00:07:37,562 --> 00:07:42,331 circular motion at a uniform speed, you're experiencing an acceleration at 108 00:07:42,331 --> 00:07:46,714 constant magnitude always directed towards the centre of the circle. 109 00:07:46,714 --> 00:07:51,355 Now if, it's important for us to understand this circular example better, 110 00:07:51,355 --> 00:07:55,222 so let's go through it. If the radius is R, and the speed V, I 111 00:07:55,222 --> 00:07:58,064 want to know. We know the direction of your 112 00:07:58,064 --> 00:08:01,337 acceleration. What is its magnitude in meters per 113 00:08:01,337 --> 00:08:04,878 second squared? So I'm looking for an acceleration in 114 00:08:04,878 --> 00:08:08,820 meters per second squared and it needs to be determined by. 115 00:08:08,820 --> 00:08:12,443 What is there to determine? The fact that you're moving around a 116 00:08:12,443 --> 00:08:14,821 circle of a given radius at a given speed. 117 00:08:14,821 --> 00:08:19,294 And we all have an intuition, that if you're going faster around a curve you'll 118 00:08:19,294 --> 00:08:21,672 need to accelerate more, to take the curve. 119 00:08:21,672 --> 00:08:25,352 And if a curve is a tight curve, you need more of an acceleration. 120 00:08:25,352 --> 00:08:29,372 So let's see if this is borne out. And we'll use a trick physicists use, 121 00:08:29,372 --> 00:08:32,259 it comes from units. We're going to do what we call 122 00:08:32,259 --> 00:08:35,826 Dimensional Analysis. We need to find some combination of R and 123 00:08:35,826 --> 00:08:38,135 of v. That could in principle be an 124 00:08:38,135 --> 00:08:41,849 acceleration. And it turns out there's only one because 125 00:08:41,849 --> 00:08:46,690 if you think about it R is measured in meters, it's a length. 126 00:08:46,690 --> 00:08:51,827 v being a velocity or a speed, is measured in meters per second. 127 00:08:51,827 --> 00:08:57,780 We need to fashion out of these something that could potentially equal a. 128 00:08:57,780 --> 00:09:02,678 And it's clear that the only way we're going to get seconds squared in here is 129 00:09:02,678 --> 00:09:07,391 to take the square of the velocity. So write the square of the velocity, this 130 00:09:07,391 --> 00:09:12,042 is a good idea but that is, has units of meters squared per seconds squared 131 00:09:12,042 --> 00:09:16,879 because it's the product of two things with units meters per second, so that's 132 00:09:16,879 --> 00:09:20,352 not quite right. This does not have the right dimensions 133 00:09:20,352 --> 00:09:25,189 but we need to clean out the meters, and to clean out the meters, we can simply 134 00:09:25,189 --> 00:09:30,103 divide by R, which is measured in meters. Now dimensional analysis doesn't always 135 00:09:30,103 --> 00:09:34,421 give the exactly right answer. Eh, if the acceleration could easily have 136 00:09:34,421 --> 00:09:39,713 turned out to be 3 Pi * v^2 / R. And the 3 Pi would not show up in 137 00:09:39,713 --> 00:09:43,302 dimensional analysis. In the case at hand it turns out that 138 00:09:43,302 --> 00:09:47,082 there is no constant, you can do the calculation, takes little 139 00:09:47,082 --> 00:09:49,648 bit of effort, but, we're not going to do it. 140 00:09:49,648 --> 00:09:54,397 The answer, is that we got the right answer for, precise, correct, expression, 141 00:09:54,397 --> 00:09:59,466 using just this dimensional analysis. So the acceleration towards the center of 142 00:09:59,466 --> 00:10:04,085 a circle, is given by v^2 / R. It has the fancy name, centripetal 143 00:10:04,085 --> 00:10:07,308 acceleration. Because it is towards the center. 144 00:10:07,308 --> 00:10:11,724 And this will show up often in what comes, in what comes next. 145 00:10:11,724 --> 00:10:16,285 We now have a nice way to state Galileo's principle of inertia. 146 00:10:16,285 --> 00:10:21,207 An object upon which no external influence acts, moves with constant 147 00:10:21,207 --> 00:10:24,610 velocity or, equivalently, has zero acceleration. 148 00:10:24,610 --> 00:10:28,333 Now we step it up. What happens when an external influence 149 00:10:28,333 --> 00:10:31,351 acts? And Newton introduces a brilliant concept 150 00:10:31,351 --> 00:10:36,423 here which is the external influence is quantified by something called a force. 151 00:10:36,423 --> 00:10:38,863 What is a force? It's a tricky concept. 152 00:10:38,863 --> 00:10:41,560 A force is that which causes acceleration. 153 00:10:41,560 --> 00:10:46,328 The idea that Newton introduces that is that you can imagine applying the same 154 00:10:46,328 --> 00:10:50,520 force to two different objects. And, we all have that intuit of idea. 155 00:10:50,520 --> 00:10:54,902 I can give an identical push to a tennis ball and to a refrigerator. 156 00:10:54,902 --> 00:10:59,993 And, we both, all have the intuit of idea that we'll get more of a motion out of 157 00:10:59,993 --> 00:11:04,762 the tennis ball that, that of the refrigerator, because the refrigerator is 158 00:11:04,762 --> 00:11:09,403 heavier or technically the refrigerator has a greater, has a larger mass. 159 00:11:09,403 --> 00:11:14,494 And so, Newton quantifies this intuition in the idea that indeed there's such a 160 00:11:14,494 --> 00:11:19,598 thing as a force The, something that measures the amount of external influence 161 00:11:19,598 --> 00:11:24,592 being applied to an object, and the object's acceleration is related to this 162 00:11:24,592 --> 00:11:28,470 force by a property of the object itself, known as its mass. 163 00:11:28,470 --> 00:11:32,865 And the mass is roughly the amount of stuff in the object, all counted, 164 00:11:32,865 --> 00:11:36,005 weighted appropriately. It's measured in kilograms. 165 00:11:36,005 --> 00:11:40,401 And it roughly, an object that is massive is an object that is heavier. 166 00:11:40,401 --> 00:11:44,797 We'll make that precise in a second. And force like acceleration has a 167 00:11:44,797 --> 00:11:47,811 direction. You can push something to the left or 168 00:11:47,811 --> 00:11:51,640 push it to the right. And so we have this new physical concept 169 00:11:51,640 --> 00:11:54,396 of force. We need to understand what units we 170 00:11:54,396 --> 00:11:57,642 measure force in. And that follows directly from this 171 00:11:57,642 --> 00:12:02,174 equation, a force of one unit is a force that causes a mass of one kilo to 172 00:12:02,174 --> 00:12:04,807 accelerate at one metre per second squared. 173 00:12:04,807 --> 00:12:09,278 So the unit of force is this unit, kilogram times metre per second square. 174 00:12:09,278 --> 00:12:13,687 And it is designated of course with the name newton, and denoted by an N. 175 00:12:13,687 --> 00:12:18,362 So that would be a force of one N. Before we go on to clarify some details 176 00:12:18,362 --> 00:12:22,705 here let's go onto Newton's third law. This was Newton's second law. 177 00:12:22,705 --> 00:12:27,825 Newton's third law explains why there has to be an external agent applying the 178 00:12:27,825 --> 00:12:29,511 force. And this is crucial. 179 00:12:29,511 --> 00:12:34,794 Newton tells us that universally, when an object A is applying the force to 180 00:12:34,794 --> 00:12:39,660 B, and the force is given by F. Which means B is being accelerated. 181 00:12:39,660 --> 00:12:44,975 Something is applying the force. Go find the thing that is applying the 182 00:12:44,975 --> 00:12:48,344 force. Then B, automatically, it reacts on A, we 183 00:12:48,344 --> 00:12:51,189 call it. And that means B applies to A. 184 00:12:51,189 --> 00:12:55,980 A force, that is equal in magnitude and opposite in direction. 185 00:12:55,980 --> 00:13:01,324 We write it mathematically as negative F. Negative a vector means a vector of the 186 00:13:01,324 --> 00:13:03,883 same size pointing in the opposite direction. 187 00:13:03,883 --> 00:13:08,546 What this means is that when you push on the refrigerator the refrigerator pushes 188 00:13:08,546 --> 00:13:10,878 back on you. This is not an action by the 189 00:13:10,878 --> 00:13:13,266 refrigerator. This is not some act of will. 190 00:13:13,266 --> 00:13:18,082 It's Newton's third law in action. Now, let's do an example of this. 191 00:13:18,082 --> 00:13:23,387 it was known already to Galileo that, if we let loose some object, we all know 192 00:13:23,387 --> 00:13:28,427 this, if I let something go, this ball right here, then an external force will 193 00:13:28,427 --> 00:13:30,019 act on it. How do I know? 194 00:13:30,019 --> 00:13:35,457 Because, the ball has an initial velocity of zero and clearly it does not maintain 195 00:13:35,457 --> 00:13:40,033 the initial velocity of zero. It begins to move downwards, so it has a 196 00:13:40,033 --> 00:13:44,343 downward pointing acceleration. And, in fact this acceleration was 197 00:13:44,343 --> 00:13:49,451 measured by Galileo Is that people after him and it is, has the interesting 198 00:13:49,451 --> 00:13:52,477 property that it is uniform for all objects. 199 00:13:52,477 --> 00:13:57,703 So any object in the vicinity of earth accelerates down, the velocity of the 200 00:13:57,703 --> 00:14:03,135 acceleration is called the acceleration of gravity and in our fancy units, it's 201 00:14:03,135 --> 00:14:08,430 about 9.8 meters per seconds squared. This is my son's ten kilo shot put ball. 202 00:14:08,430 --> 00:14:12,676 And when I hold this I'm preventing it from falling down with the standard 203 00:14:12,676 --> 00:14:16,810 acceleration of gravity by applying a force directed upwards to the ball. 204 00:14:16,810 --> 00:14:21,570 Newtons third law then tells us that if I'm applying a force to the ball point 205 00:14:21,570 --> 00:14:26,571 directed upward then the ball is clearly applying a force to my hand directed down 206 00:14:26,571 --> 00:14:30,428 and believe me I can feel it. Of course gravity is going to play an 207 00:14:30,428 --> 00:14:34,826 important role in what goes on. It's also a force that acts all around us 208 00:14:34,826 --> 00:14:38,117 so let's Quantify what we said, the weight of an 209 00:14:38,117 --> 00:14:41,803 object is defined to be the force gravity applies to it. 210 00:14:41,803 --> 00:14:46,871 We know that objects on earth fall with an acceleration, that I told you, 9.82 211 00:14:46,871 --> 00:14:51,215 meters per second squared. So, the force of gravity on any object is 212 00:14:51,215 --> 00:14:55,231 directed always down. It's, given by the mass of the object 213 00:14:55,231 --> 00:15:00,365 times this constant, which is why we confuse mass with weight but separate the 214 00:15:00,365 --> 00:15:05,690 two and there is a property of the object g is a property of earth. 215 00:15:05,690 --> 00:15:13,475 We often say that I weigh 59 kilos. A more correct statement is that my mass 216 00:15:13,475 --> 00:15:20,600 is 59 kilos. My weight here on earth is mg which is 59 217 00:15:20,600 --> 00:15:29,848 kilos times 9.82 meter per second squared or doing the calculation, I believe I did 218 00:15:29,848 --> 00:15:37,418 it right, 579 kilos times meter per second squared which is 579 Newtons. 219 00:15:37,418 --> 00:15:42,080 My weight is 579 Newtons. We'll play more with, 220 00:15:42,080 --> 00:15:47,189 The specifics of Newton's laws, but the first thing we need is to understand 221 00:15:47,189 --> 00:15:52,030 some, very deep and important, consequences that are mathematically 222 00:15:52,030 --> 00:15:55,459 derived, and Newton derived them from Newton's laws. 223 00:15:55,459 --> 00:16:01,040 And, the most important of these take the form of thing called conservation laws. 224 00:16:01,040 --> 00:16:06,754 There are quantities that you can define for a physical system that do not change. 225 00:16:06,754 --> 00:16:11,159 So, let's look at this. And, the first conservation law Involves 226 00:16:11,159 --> 00:16:15,981 this quantity called momentum. We define another vector called momentum. 227 00:16:15,981 --> 00:16:20,840 It's essentially the velocity vector, but multiplied by mass so, roughly it 228 00:16:20,840 --> 00:16:25,764 measures, the amount of umph, the amount of inertia some people say that the 229 00:16:25,764 --> 00:16:30,820 object pass a, refrigerator moving at two meters per second, has, more momentum, 230 00:16:30,820 --> 00:16:34,168 than a tennis ball, moving at two meters per second. 231 00:16:34,168 --> 00:16:37,320 The reason this is important, is because, F being, 232 00:16:37,320 --> 00:16:40,916 satisfying Newton's law, let me rewrite it again here. 233 00:16:40,916 --> 00:16:45,734 Newton's second law, F is mass times the rate of change of the velocity. 234 00:16:45,734 --> 00:16:50,348 Since mass does not change, it's a property of the object, F could be 235 00:16:50,348 --> 00:16:54,216 thought of as the rate of change of this momentum thingy. 236 00:16:54,216 --> 00:16:59,510 And so, if F is the rate of change of the momentum, we have something very deep. 237 00:16:59,510 --> 00:17:04,610 Imagine that there are only two objects in the universe but there are forces 238 00:17:04,610 --> 00:17:07,922 between them. So A acts on B and the velocity of B 239 00:17:07,922 --> 00:17:11,500 changes and B acts on A and the velocity of A changes. 240 00:17:11,500 --> 00:17:16,120 What we know, though, is that the force that A applies to B, whatever it is, is 241 00:17:16,120 --> 00:17:20,740 equal in magnitude and opposite in direction to the force that B applies to 242 00:17:20,740 --> 00:17:23,232 A. Which means that the rate of change of 243 00:17:23,232 --> 00:17:28,096 A's momentum is equal in magnitude and opposite direction to the rate of change 244 00:17:28,096 --> 00:17:33,141 of B's momentum with the consequence that the rate of change of the total momentum, 245 00:17:33,141 --> 00:17:37,032 add up these two vector object, the total momentum cannot change. 246 00:17:37,032 --> 00:17:40,193 Now add more objects and the same principle extends. 247 00:17:40,193 --> 00:17:44,581 There is in the universe a total. Conserved object, the total momentum of 248 00:17:44,581 --> 00:17:48,701 the universe and that cannot change. Moreover, if you could isolate some 249 00:17:48,701 --> 00:17:53,203 subset of the universe so that it doesn't interact, there are no external force on 250 00:17:53,203 --> 00:17:57,431 it, then the momentum of that little group of objects is completely conserved. 251 00:17:57,431 --> 00:18:01,878 The other forces can do is allow objects to exchange some momentum, some momentum 252 00:18:01,878 --> 00:18:06,270 can be transferred from one object to the other so that the momentum of the two 253 00:18:06,270 --> 00:18:10,223 objects change but their total is unchanged and we say that momentum is 254 00:18:10,223 --> 00:18:13,023 conserved. We call this a conservation law and such 255 00:18:13,023 --> 00:18:15,384 conservation laws are critically important. 256 00:18:15,384 --> 00:18:17,690 Let's take a look at what this looks like. 257 00:18:17,690 --> 00:18:22,240 This is a toy appropriately enough called Newton's Cradle. 258 00:18:22,240 --> 00:18:26,845 It has, a row of five steel balls hanging from, 259 00:18:26,845 --> 00:18:32,559 framework and we will pull one ball off to our left so that when I release it, it 260 00:18:32,559 --> 00:18:38,666 will impact the real balls with the horizontal momentum pointing to the right 261 00:18:38,666 --> 00:18:43,151 and let's see what happens. What happens is that the incoming ball 262 00:18:43,151 --> 00:18:47,617 comes to a complete halt, transferring all of its horizontal momentum to the 263 00:18:47,617 --> 00:18:52,318 next ball down the line, which transfers it to the next, and so on, until the last 264 00:18:52,318 --> 00:18:56,961 one leaves the pile at essentially the same momentum as the one with which the 265 00:18:56,961 --> 00:19:00,545 initial ball came in. We can repeat the same process with two 266 00:19:00,545 --> 00:19:03,307 balls and see a similar interesting process. 267 00:19:03,307 --> 00:19:07,949 We can do three, and if we are ambitious, even four balls, and reproduce the same 268 00:19:07,949 --> 00:19:11,970 results. I hope I convinced you that we can see 269 00:19:11,970 --> 00:19:17,213 mathematically where Newton's laws lead to momentum conservation very directly. 270 00:19:17,213 --> 00:19:22,324 A little more math, that we're not going to follow through, shows that there's 271 00:19:22,324 --> 00:19:28,824 another way to find a conserved quantity that is associated deeply with circular 272 00:19:28,824 --> 00:19:31,664 motion. So it involves picking a center for 273 00:19:31,664 --> 00:19:36,592 motion and it's called angular momentum because circles are parameterized by 274 00:19:36,592 --> 00:19:39,536 angles. And it's a quantity that is given, it's 275 00:19:39,536 --> 00:19:44,364 related to momentum, here's our friend momentum, but it's the momentum times the 276 00:19:44,364 --> 00:19:48,666 radius of a circle in which one moves. And what one finds is if you have a 277 00:19:48,666 --> 00:19:52,857 collection of objects that all are moving together, you sum up, just as with 278 00:19:52,857 --> 00:19:56,824 momentum, the total angular momentum. The mass of each of them times the 279 00:19:56,824 --> 00:20:01,350 velocity of each of them times the radius at which each of them is moving around 280 00:20:01,350 --> 00:20:04,312 the circle. This is assuming everybody is moving in a 281 00:20:04,312 --> 00:20:07,217 circle. There are more complicated expressions in 282 00:20:07,217 --> 00:20:09,954 another cases. Then this angular momentum is also 283 00:20:09,954 --> 00:20:14,144 conserved in that it can be traded between different parts of a system but 284 00:20:14,144 --> 00:20:17,662 the total is conserved. This is going to be extremely important 285 00:20:17,662 --> 00:20:20,064 to us because things in space tend to spin. 286 00:20:20,064 --> 00:20:24,300 So let's demonstrate that with our valiant demonstrator. 287 00:20:24,300 --> 00:20:29,283 Standing on a platform and holding out some weights, I have my friend Derek 288 00:20:29,283 --> 00:20:34,696 start me spinning slowly, and what we see is that as I pull the weights in they 289 00:20:34,696 --> 00:20:38,542 have a significant factor of m and I'm making their r smaller. 290 00:20:38,542 --> 00:20:43,195 My angular momentum is conserved by making all of me spin faster and I can 291 00:20:43,195 --> 00:20:48,219 control my speed when I pull my arms out I slow down, when I pull them in, I speed 292 00:20:48,219 --> 00:20:51,073 up again. This is a great demonstration of the 293 00:20:51,073 --> 00:20:55,912 conservation of angular momentum, if not perhaps of exceptional physical grace. 294 00:20:55,912 --> 00:20:59,200 There's another conservational law that you can show, 295 00:20:59,200 --> 00:21:03,775 follows from Newton's equations. Imagine an object, upon which the only 296 00:21:03,775 --> 00:21:08,446 force that acts is gravity like this, rubber ball when I throw it up and down. 297 00:21:08,446 --> 00:21:12,996 So as long as the ball is in the air essentially gravity is the only force 298 00:21:12,996 --> 00:21:15,909 acting. And what we know is that when I throw it, 299 00:21:15,909 --> 00:21:20,823 as it moves up it'll slow down, and then as it comes down it'll accelerate, moving 300 00:21:20,823 --> 00:21:23,310 down. This is formulized in the following 301 00:21:23,310 --> 00:21:26,161 statement. It turns out, that if you form the 302 00:21:26,161 --> 00:21:31,211 combination m times g times h where m times g is the force that gravity applies 303 00:21:31,211 --> 00:21:34,389 and h is the height of the object above something. 304 00:21:34,389 --> 00:21:38,393 My hand, the floor, it doesn't really matter what, we'll show why. 305 00:21:38,393 --> 00:21:43,798 And if you take this quantity which we call gravitational potential energy and 306 00:21:43,798 --> 00:21:51,845 add to it this combination of speed and mass m * v^2 / 2, this combination is 307 00:21:51,845 --> 00:21:56,713 called the object's kinetic energy. Then the sum of both of these is 308 00:21:56,713 --> 00:21:59,367 constant. As h increases the object slows down 309 00:21:59,367 --> 00:22:03,751 because, that's how gravity works, and as it falls down, it speeds up, and this 310 00:22:03,751 --> 00:22:07,731 mathematically expresses this. Notice that it's clear from here why I 311 00:22:07,731 --> 00:22:12,346 could have measured the height from the floor or my hand, or the bottom floor of 312 00:22:12,346 --> 00:22:16,730 the building it doesn't matter, that just adds a constant that never changes. 313 00:22:16,730 --> 00:22:20,940 So, adding a constant to the energy, until very late in this class will be 314 00:22:20,940 --> 00:22:24,593 completely irrelevant. Now this is very nice but it's only true 315 00:22:24,593 --> 00:22:29,185 if the only force acting is gravity and in general there are other forces that 316 00:22:29,185 --> 00:22:31,685 act. And so in general this conservation of 317 00:22:31,685 --> 00:22:33,733 energy is violated. But not really. 318 00:22:33,733 --> 00:22:38,704 What really goes on is that whenever there is another force other than gravity 319 00:22:38,704 --> 00:22:41,536 that is acting, a typical example is friction. 320 00:22:41,536 --> 00:22:46,443 If I roll my chair back, it slows down. I move it, it had kinetic energy and the 321 00:22:46,443 --> 00:22:49,338 energy disappeared. Where did the energy go to? 322 00:22:49,338 --> 00:22:54,246 Well, if you listened closely, you could have heard that some of it turned into 323 00:22:54,246 --> 00:22:57,643 sound energy. It's also true that there is friction in 324 00:22:57,643 --> 00:23:02,174 the bearings of the wheels, and that converts some of the energy to heat. 325 00:23:02,174 --> 00:23:06,390 Heat is a form of energy and friction is the force that translates, 326 00:23:06,390 --> 00:23:10,749 converts kinetic energy very happily in the heat. 327 00:23:10,749 --> 00:23:14,946 So there are many different forms of energy: sound, light, heat, chemical 328 00:23:14,946 --> 00:23:19,563 energy, electric energy, nuclear energy. We will talk about all of them in turn 329 00:23:19,563 --> 00:23:22,981 they'll all show up. When you add them all up together it 330 00:23:22,981 --> 00:23:25,500 turns out, in any process, the total energy. 331 00:23:25,500 --> 00:23:29,837 Is in the universe is conserved. And again, if you isolate a chunk of the 332 00:23:29,837 --> 00:23:34,174 universe from the rest, then the total energy in that chunk is conserved. 333 00:23:34,174 --> 00:23:39,114 Energy is neither produced nor destroyed. this is, might come as surprise to all 334 00:23:39,114 --> 00:23:43,752 the politicians who talk about the need to produce energy or conserve energy. 335 00:23:43,752 --> 00:23:48,270 Both of those are political terms, but scientifically energy can neither be 336 00:23:48,270 --> 00:23:52,005 produced nor destroyed. And it can be conserved, in fact that's 337 00:23:52,005 --> 00:23:54,776 always is. It's a very important concept to us. 338 00:23:54,776 --> 00:23:57,909 Let's remember the units in which we measure energy. 339 00:23:57,909 --> 00:24:02,436 So They follow from this equation, energy is measured in units of kg * m^2 / s^2. 340 00:24:05,713 --> 00:24:11,264 If you plug it in you'll see that both of these terms have the same units, which is 341 00:24:11,264 --> 00:24:15,143 good because otherwise adding them up would make no sense. 342 00:24:15,143 --> 00:24:19,423 And, this is dignified by the name joule, and indicated by a J. 343 00:24:19,423 --> 00:24:23,460 So this is joule. And, it's hard to have exactly a sense 344 00:24:23,460 --> 00:24:28,059 for how much a Joule is. So, let me help you by perhaps suggesting 345 00:24:28,059 --> 00:24:32,234 the following idea. our body, to push this chair back and 346 00:24:32,234 --> 00:24:37,540 forth to move my arms I am producing kinetic energy by converting chemical 347 00:24:37,540 --> 00:24:42,069 energy from food I ate. And, we measure the energy content of off 348 00:24:42,069 --> 00:24:46,031 food in calories. We all have some sense of what calorie 349 00:24:46,031 --> 00:24:48,720 is. So one Joule is about 4.2 calories. 350 00:24:48,720 --> 00:24:52,566 Fun demo of conservation of energy is this bullying bar pendulum. 351 00:24:52,566 --> 00:24:57,181 I am holding it up against my face, its got potential energy because the angle 352 00:24:57,181 --> 00:25:00,850 elevates it from the floor. I release it, it requires potential 353 00:25:00,850 --> 00:25:05,288 energy swinging to the other side converting it back to potential, and when 354 00:25:05,288 --> 00:25:10,141 it comes back, if energy is conserved, it will not go any higher than it did before 355 00:25:10,141 --> 00:25:13,940 and therefore will not smash my face. Let's try it. 356 00:25:13,940 --> 00:25:16,611 This is it. Those are Newton's laws. 357 00:25:16,611 --> 00:25:22,717 I don't know if you appreciate what I mean by this is it but by the end of this 358 00:25:22,717 --> 00:25:26,915 class I think you'll have a deep understanding of this. 359 00:25:26,915 --> 00:25:30,274 In a very real sense, this is all of science. 360 00:25:30,274 --> 00:25:36,380 Certainly all of physical science. the equation that governs science is F = 361 00:25:36,380 --> 00:25:39,280 ma. The rest of, certainly a century of 362 00:25:39,280 --> 00:25:45,063 physics, is figuring out the details in the sense of figuring out m, what kind of 363 00:25:45,063 --> 00:25:48,300 objects are in the universe? What do forces act on? 364 00:25:48,300 --> 00:25:52,380 And figuring out F, what are the forces between various objects? 365 00:25:52,380 --> 00:25:56,814 One way I want you to think about this equation, fancy name is a differential 366 00:25:56,814 --> 00:26:01,480 equation, but what this really means is it's a prescription for figuring out what 367 00:26:01,480 --> 00:26:04,129 the universe will do next at any given moment. 368 00:26:04,129 --> 00:26:08,518 And the way it works is this. If we know the forces that act on things, 369 00:26:08,518 --> 00:26:13,756 and we happen to know at some instant where all of the relevant objects in our 370 00:26:13,756 --> 00:26:16,938 system are. And because this only determines the 371 00:26:16,938 --> 00:26:21,844 acceleration, we need to know how they're moving, because if a ball is here, 372 00:26:21,844 --> 00:26:26,949 clearly there are several different motions it can do under the influence of 373 00:26:26,949 --> 00:26:28,740 gravity. It could just fall. 374 00:26:28,740 --> 00:26:32,327 Or if it is initially moving up, it will go up and then go down. 375 00:26:32,327 --> 00:26:36,597 But once you know where it is to begin with and which way it's moving, then 376 00:26:36,597 --> 00:26:38,989 gravity takes over and tells you the rest. 377 00:26:38,989 --> 00:26:43,544 In other words you can take the positions and velocities in a given instant, use 378 00:26:43,544 --> 00:26:48,213 those to figure out where the forces are, perhaps where it is might influence what 379 00:26:48,213 --> 00:26:51,687 forces act, is it touching my hand, is it not touching my hand. 380 00:26:51,687 --> 00:26:54,420 That allows you to figure out the accelerations. 381 00:26:54,420 --> 00:26:57,495 Those tell you in turn how the velocities will change. 382 00:26:57,495 --> 00:27:00,570 So you can figure out the velocities an instant later. 383 00:27:00,570 --> 00:27:04,820 Using those velocities, you can now figure out the new positions, figure out 384 00:27:04,820 --> 00:27:09,410 the new forces and repeat this process is called solving a differential equation 385 00:27:09,410 --> 00:27:14,056 and in essence what the universe does is that it solves Newton's equation all the 386 00:27:14,056 --> 00:27:16,493 time. What this allows you do, do is that if 387 00:27:16,493 --> 00:27:20,913 you have some knowledge of what we call the initial data, all of the position and 388 00:27:20,913 --> 00:27:25,446 velocities of the parts of the system at any given instant, you can predict what 389 00:27:25,446 --> 00:27:30,229 everything is going to do into the infinite future and also you can roll the 390 00:27:30,229 --> 00:27:35,291 clock backwards and figure out where everything was at any given time all the 391 00:27:35,291 --> 00:27:39,444 way into the indefinite past. And as I said, most of a century of 392 00:27:39,444 --> 00:27:44,765 physics goes into filling in the details of F and m and applying Newton's laws in 393 00:27:44,765 --> 00:27:47,880 various situations and finding the consequences. 394 00:27:47,880 --> 00:27:52,877 Let's start that process with the force that's most important to us, which is 395 00:27:52,877 --> 00:27:54,500 gravity, in the next clip.