1
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I will be your lecturer this term. 2
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Make sure you have a handout and make sure you have read it. 3
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It tells you everything you want to know about the course. 4
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This course is about waves and vibrations. 5
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About oscillations,periodic and not so periodic events. 6
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When you look around in the world,you see them everywhere. 7
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For one thing,your heart beat. 8
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That's a periodic oscillation. 9
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At least,I hope,that for most of you,it is periodic. 10
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You breathing is some kind of periodic motion. 11
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The blinking of your eyes,your daily routines and your habits,you eating, 12
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you sleeping,taking a shower,your classes and occationally doing some work, 13
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all those are periodic actions. 14
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When you drink,I drink some orange juice. 15
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The notice as I try to move the liquid down into my stomach, 16
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that it's not a steady stream but it's a periodic motion. 17
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Look at my throat. 18
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In fact,even if I don't want to swallow the liquid, 19
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there simply have a bottle with liquid and I turn it over, 20
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that we all know that the water doesn't come out like a steady stream but it goes "cro cro …." 21
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That's some kind of a periodic motion. 22
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I have here a toy which I use to entertain my dinner guests, 23
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particularly physics is interesting. 24
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There is liquid here and the idea is to get the liquid there. 25
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And then the problem is how can you do it in the fastest possible way. 26
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Well,if you turn it over and you will see that phenomenon that I just mentioned 27
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which is that "cro cro..." 28
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It is not a steady stream it's almost pathetic the way that it runs from one side to the other. 29
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And it will take minutes before it is there. 30
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But it can't be done in 17secs. 31
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And during the 5 minutes into mission that we have, 32
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you may give that a try and I hope you won't break it. 33
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And see whether any of you can think of a way that you can transfer the liquid in 17secs. 34
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You have breakfast in the morning and you casually put your breakfast plate on the table. 35
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What you hear?Some kind of a periodic motion. 36
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And two things can happen to this plate. 37
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It can move as a mucilage as I call this as a mucilage because I am an astronomer. 38
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But it can also wobble. 39
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In fact,something can wobble without moving as a mucilage. 40
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And something can move as a mucilage without wobbling. 41
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In this case it does both. 42
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And a fabulous example of that is what called the Euler's disc. 43
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Which is a metal disc you will see it shortly there,and this metal disc, 44
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we are going to wobble in the similar way that I wobble the plate. 45
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And then we will follow its motion as the mucily and the wobbling frequency. 46
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What is interesting as you will see that the as mucile motion which has a certain period, 47
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that period gets longer in time. 48
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But the wobble motion,the frequency goes up. 49
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So I will start it here. 50
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And then I will show it to you in the way that is more appealing. 51
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And you can follow that,it's an amazing toy to work out to physics. 52
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Very very difficult. 53
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I was told that professor WILL CHECK in the MIT once gave a one-hour lecture 54
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exclusively on the explanation of this Euler's disc. 55
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So try to see the as mucile motion it will be clearer as it slows down further. 56
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You may be able to hear the wobble motion I'll hold my microphone close up. 57
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Can you hear it? 58
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Very high frequency already. 59
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Did you hear it? 60
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It's quite amazing,isn't it when you look at this? 61
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The wobbling frequency increases quite rapidly. 62
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Look how the mucile motion slows down. 63
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And how the frequency of the wobble goes up. 64
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Ah….Now comes to stop. 65
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That's a very difficult piece of physics right there. 66
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If you take a tennis ball,this is a super ball and you bounce it 67
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Oh,hehe…. 68
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This gots total involvement. Thank you. 69
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Then you also get some kind of a periodic motion will by again the frequency increases 70
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just like the case of the Euler's disc and the breakfast plate. 71
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Here is an object that is floating in the liquid in water. 72
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and if I push that a little farther in and I let it go. 73
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It wobbles and there's a very unique frequency you will be able to calculate in 803(Class Number). 74
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A very unique period of one complete oscillation as this object goes up and down. 75
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Even winds,steady winds can generate periodic or almost periodic motion 76
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which all of you have experienced you walk outside,it's windy and your hair goes like this. 77
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Your hair doesn't go flat like this, 78
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always has this tendency just like a flag does the same thing. 79
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If I generate wind here,and I have here some aluminum, 80
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and you will see that this wind doesn't make the aluminum just go straight out but it wobbles. 81
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There is a certain period to that. 82
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After work,if you wanna have some fun, 83
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what is more fun than riding your own walking horse? 84
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That's a periodic motion. 85
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Falling in love can be a periodic event. 86
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Now don't do it too often because as most of you know,quite exhausting. 87
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The motion of electrons,atoms,molecules periodic and oscillatory. 88
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The motion of the moon,the planets and the stars,periodic,oscillatory. 89
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Sound is a beautiful example. 90
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I produce sounds,I produce sound by oscillating my vocal cords. 91
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I produce there by pressure waves. 92
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My vocal cords push on the air,suck on the air,push on the air 93
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which produces a pressure wave and the pressure wave propagates out in the lecture hall, 94
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reaches your ear drum,your ear drums start to move back and forth. 95
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And your brains tell you that you hear a sound. 96
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I have here a toning fork which is designed so that if I give it a hit, 97
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that the prongs move 256 times per sec. 98
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We call that 256Hz. A Hz is one oscillation per sec. 99
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And all of you can hear that. 100
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Pressure waves. 101
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I generated it.We will deal with them in 8.03 102
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They travel to the air,reach your ear drum and your ear drum starts to shake. 103
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This is a higher frequency 440Hz. 104
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Most human beings can hear in the range from 20Hz to 20kHz. 105
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And animals who can go away beyond 20kHz. 106
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And to be nice to you for the first time this first lecture, 107
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I would like to test your hearing and that will be free of charge. 108
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I am not so much interested in knowing what your lowest frequency is 109
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but what your highest frequency is. 110
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So I'm going to generate here sound. 111
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I will start with 100Hz. 112
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Then we will go up higher and higher and then we will see where your hearing stops. 113
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So let's start with 100Hz. 114
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I'm not going to ask you who hears clearly because all of you can. 115
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Let's now go to a kilo Hz 1000Hz. 116
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Piece of cake,right? 117
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2000 no problem. 118
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I have to change now my scale.4000 119
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I am not saying that this will be going to be a pleasant test. 120
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Next,5000. 121
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This is what violins can make. 122
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6000 123
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Anyone of my audience who cannot hear 6000? 124
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7000 125
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Anyone of my audience cannot hear 7000? 126
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I cannot hear 7000 127
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I hear nothing. 128
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With age you lose your ability to hear high frequencies. 129
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You will experience that in your lifetime. 130
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You won't escape that. 131
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Now for some people,lose more than that,I can not hear when 6000Hz 132
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I hear nothing. 133
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10000 134
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12000 135
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14000 136
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Now I want to see hands if you can not hear any longer. 137
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Who can not hear 14000? 138
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Don't be ashamed,because it is not your fault. 139
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14000,all right,we're slowly going up. 15000 140
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Who can not hear.Raise your hands. 141
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Ah,Professor Mavalvala. You're also getting old,know this? 142
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16000. Who can not hear 16000? 143
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Of course,the ones who has already raised your hands you don't have to raise your hands again. 144
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Who can not 16000? 145
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Who can not? 146
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17000 18000 147
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Now we are going to change. 148
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Now I want you to raise your hands if you can hear it. 149
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So I first now go to 20000. 150
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19000. I'm going to 19000. 151
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Wo,oh sorry. 152
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I was only off by a factor of 10. 153
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19000. Who can hear it? 154
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Fantastic! 155
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20000 156
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21 wa. You see how it cut off. 157
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Very sharp. 22 158
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Very good. 22,23, 159
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25,27 160
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Some of you have amazing ears because I have already turned it off at 21000 161
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All right,key,absolutely key in this course will be simple harmonic oscillations. 162
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Because they are extremely common in nature. 163
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For simple harmonic oscillation, 164
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and you have seen this of course in 8.01 can be written as follows: 165
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(see the blackboard) you can write a sin here if you want to. 166
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Xo is the amplitude but the largest displacement from equilibrium 167
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Omega is the angular frequency,angular frequency 168
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Omega,which we express in terms of radius per sec 169
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The period T (see the blackboard) is then expressed in terms of seconds. 170
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And the frequency f which is (see the blackboard) is what we call Hz. 171
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Number of circles per sec. 172
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Do not confuse Omega with f,there is a factor of 2pi difference. 173
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If I have a uniform circular motion, 174
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and I project that uniform circular motion along to any line in the blackboard, 175
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then I get a simple harmonic motion. 176
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So I take,for simplicity just this is horizontal line that I could take any other line 177
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Let's call that the x direction and this point be Xo. 178
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And I take an object which is rotating around 179
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Here is the object. It's going around. 180
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Uniform circular motion 181
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If I project this onto the X axis and this angle is theta, 182
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then this position here is (see the blackboard) 183
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And if I make theta a function of pi,theta equals (see the blackboard) 184
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This Omega is what we call not angular frequency but we call it angular velocity. 185
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This is awkward in physics that 186
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we have the same symbol for angular velocity and for angular frequency. 187
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In this case they happen to be the same numerically 188
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because it's a uniform circular motion that's an accident. 189
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So now you can see that Xo then becomes(see the blackboard) 190
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because the two are the same. 191
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I do not have to call the position t equals zero here 192
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I can choose t equals zero anywhere along the circumference 193
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and that introduce then phase angle fi 194
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We call that initial condition. 195
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So Xo is amplitude,Omega is the angular frequency 196
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and fi has to be adjusted so that time t equals to zero you get the right angle. 197
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Get the right position. 198
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An easy example of a simple harmonic motion is a spring system. 199
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If I have here a spring and this is in its relax position. 200
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The spring constant is k,the mass is m and X equals zero here. 201
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And I bring it farther out,I bring it to position X 202
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Then there is a spring force that wants to drive it back to equilibrium. 203
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It's a restoring force that is the spring force. 204
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Let's arbitrarily call this direction plus. 205
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The spring force,we call minus kX,minus because if X is positive, 206
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then the figured forces is in the opposite direction. 207
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If the mass of the spring can be ignored, 208
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if it is negligibly small compared to the mass of the object, 209
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I can write down Newton Second Law F=ma 210
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You may remember that from your good old days. 211
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And so ma is mX double dot,this is now minus kX 212
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It is really a effective notation since this is one dimensional problem 213
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that the minus sign takes care of the directions. 214
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And so I can massage this a little further 215
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And I can write this as (see the blackboard) 216
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And what is the solution to this differential equation. 217
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This is a differential equation(see the blackboard) 218
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This is the solution. The simple harmonic motion provided that ω is (see the blackboard). 219
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So I advise you to take this function substitute in here and you will see 220
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the output. Yes,you can satisfied this equation provided that ω is 221
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the square root of K over m. Notice which is not so intuitive that is 222
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angle of frequency ω and therefore also the period of oscillation (see the blackboard) 223
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is independent of X。. 224
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So it's independent of how far I move it away from equilibrium. 225
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if I move it far out it will take the same amount of time to one oscillation 226
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and if I move it out teeny weeny little bit,not so intuitive. 227
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So ω is independent of my initial conditions; it's independent on how I start 228
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the experiment off; it's independent of φ ; it's independent on what I call 229
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t equal zero. Nature doesn't give it gambled. I called t equal zero 230
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Nature has one answer for the frequency and that's only determined 231
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by K and by m,not by my initial conditions,not so intuitive 232
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If I take the same spring and if I hang the spring vertically. 233
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there is the spring. Due to gravity,the object will come to a halt equilibrium 234
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a little lower obviously. If now I displace it from this equilibrium position 235
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and I let it oscillate. I get exactly the same frequency. Maybe that is not 236
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so intuitive either. And you can work that out for yourself. It's 8.01 problem. 237
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What that means is that you can define this as x equals zero 238
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ignore gravity completely. And set up your differential equation 239
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as if there was no gravity. And this is x equals zero, 240
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so you offset it over a distance x from that equilibrium position. 241
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You only allow for a spring force minus Kx and every thing works. 242
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and of course you should be able to prove that is correct. 243
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If the spring oscillates in the simple harmonic fashion, 244
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and we have such a spring here. 245
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Marcos if you can do me a favor and get it up here. 246
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Then I should be able to demonstrate that's a uniform circular motion 247
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projected on the wall,we called shadow projection,should be able... 248
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Thank you,Marcos. 249
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should be able to have the same motion as my spring. 250
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provided of course that we very carefully make the period 251
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of oscillation of the spring exactly the same 252
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as the time for this object to go around. 253
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We then shadow it projected on there. And then I will even try to 254
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release this one,it's very difficult. 255
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at the same time that this one was here. 256
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and what you will see then? 257
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You will see the uniform circular motion projected becomes 258
00:23:43,440 --> 00:23:49,060
a simple harmonic motion and you will see the spring simple harmonic . 259
00:23:49,060 --> 00:23:58,090
And so we try to do that in shadow projection,we'll make it a little darker. 260
00:24:00,260 --> 00:24:03,620
And for that I need some light here. 261
00:24:06,510 --> 00:24:08,910
It's here. 262
00:24:16,130 --> 00:24:19,100
OK,and we ...oh somone already turned it off 263
00:24:19,100 --> 00:24:22,560
So here you see the spring there you see this object 264
00:24:22,560 --> 00:24:27,250
which is rotating in the circle,but you think 265
00:24:27,250 --> 00:24:32,940
it is a simple harmonic motion that's of course my objection,my objective. 266
00:24:32,940 --> 00:24:38,040
So now that is difficult I will have to block you for a few seconds. 267
00:24:38,040 --> 00:24:42,370
I will try now to release this the same time 268
00:24:42,370 --> 00:24:45,760
and also at the same amplitude. 269
00:24:45,760 --> 00:24:55,110
Eh..Boy,that wasn't my best day,was it？ 270
00:24:55,110 --> 00:25:07,570
no. no. Oh this is perhaps the best I can do today. 271
00:25:07,570 --> 00:25:10,350
So they don't go exactly next to each other,but you see 272
00:25:10,350 --> 00:25:14,280
that they have the same period and they both represent 273
00:25:14,280 --> 00:25:17,390
simple harmonic oscillation. The spring because we just calculate 274
00:25:17,390 --> 00:25:26,770
that and the projection of the uniform circular motion. 275
00:25:31,810 --> 00:25:38,850
So if we return to this spring,maybe we should remove this, 276
00:25:38,850 --> 00:25:44,940
if we return to the spring,then we have a period for the spring system 277
00:25:44,940 --> 00:25:49,890
which is (see the blackboard) 278
00:25:49,890 --> 00:25:55,480
and I want to bring this to a test,to a quantitated test. 279
00:25:55,480 --> 00:26:00,710
how accurate is this? I'm going to double the mass 280
00:26:00,710 --> 00:26:02,910
that I'm going to hang on that spring,I'm going to 281
00:26:02,910 --> 00:26:09,480
measure the periods. Then I want the mass which is twice as high 282
00:26:09,480 --> 00:26:15,160
I want that period to be the square root of two times higher 283
00:26:15,160 --> 00:26:18,440
so that's what this equation predicts. 284
00:26:18,440 --> 00:26:21,800
Now whenever you want to do a measurement in physics, 285
00:26:21,800 --> 00:26:26,810
whereby you want to compare numbers,you have a certain goal in mind. 286
00:26:26,810 --> 00:26:30,700
a measurement without the uncertainty in the measurement is completely 287
00:26:30,700 --> 00:26:37,770
meaningless. You must know the accuracy of your measurement. 288
00:26:37,770 --> 00:26:50,990
So M1 is (see the blackboard) and M2 is (see the blackboard) 289
00:26:50,990 --> 00:26:56,520
And that's the best we can do. That's an extremely small error 290
00:26:56,520 --> 00:27:05,190
this is the error of only 0.04 percents and this is only half as large 291
00:27:05,190 --> 00:27:10,040
And now comes the question if I measure the periods of oscillation 292
00:27:10,040 --> 00:27:17,100
with the 500 grams hanging on the spring,how accurately can I do that? 293
00:27:17,100 --> 00:27:20,560
On the good day I can do it to 0.1 second accuracy 294
00:27:20,560 --> 00:27:22,330
I have to start it then have to stop it 295
00:27:22,330 --> 00:27:26,590
if I do that ten times obiviously you will get different answers 296
00:27:26,590 --> 00:27:30,680
and they vary by about a tenth of a second on a good day. 297
00:27:30,680 --> 00:27:35,360
On a bad day,two-tenths of a second. 298
00:27:35,360 --> 00:27:38,500
I don't know whether today is a good or what it's a bad day. 299
00:27:38,500 --> 00:27:40,450
Let's say it's in between. 300
00:27:40,450 --> 00:27:46,300
So let's say I can do it to 0.15 seconds which I cannot guarantee, 301
00:27:46,300 --> 00:27:55,870
but I will try. So I can measure the period to (see the blackboard) 302
00:27:55,870 --> 00:28:01,480
this is with M1. However,I can get a very accurate measurement 303
00:28:01,480 --> 00:28:07,310
for the time for the period if I make ten oscillations. Because 304
00:28:07,310 --> 00:28:12,660
if I make ten oscillations,the error in T goes down by the fact of ten. 305
00:28:12,660 --> 00:28:17,270
because the 0.15 is 0.15,that's not going to change. 306
00:28:17,270 --> 00:28:21,900
So I am going to oscillate ten times,and then we are going to 307
00:28:21,900 --> 00:28:28,120
make a prediction about what we should measure for the higher mass. 308
00:28:28,120 --> 00:28:36,490
Let's first measure the period of... this is the spring 309
00:28:36,490 --> 00:28:42,920
and here is the 500 gram plus or minus 0.2 310
00:28:42,920 --> 00:28:46,720
I'm going to oscillate it. We already know that it's independent of 311
00:28:46,720 --> 00:28:53,300
amplitude and then I'm going to start it when it is down that it is easiest 312
00:28:53,300 --> 00:28:58,390
for me. And I will count to ten and you will count to ten and then we'll stop. 313
00:28:58,390 --> 00:29:21,320
So let's give it an oscillation. Yeah,1,2,3,4,5,6,7,8,9,10. 314
00:29:21,320 --> 00:29:30,850
14.96. That's right. It is done. 315
00:29:30,850 --> 00:29:40,380
14.96. If you want to see whether this is a good day or whether this is a bad day 316
00:29:40,380 --> 00:29:47,550
We can measure it again. 317
00:29:47,550 --> 00:30:04,860
1,2,3,4,5,6,7,8,9,10.14.98 318
00:30:04,860 --> 00:30:12,340
So this is not a bad day. but it's luck that comes out so close of course. 319
00:30:12,340 --> 00:30:20,260
So now we can make a prediction that 10 times Tm2 must be 1.414 320
00:30:20,260 --> 00:30:29,510
which is the square root of two,times 14.97 321
00:30:29,510 --> 00:30:42,490
OK,so I take the 1.414 and multiplie that by 14.97 and I find 21.17 322
00:30:42,490 --> 00:30:44,930
21.17 323
00:30:44,930 --> 00:30:49,030
and of course this has to multiplied also by 1.414 324
00:30:49,030 --> 00:30:53,500
so that becomes plus or minus 0.2 seconds. 325
00:30:53,500 --> 00:31:05,570
That is a prediction. This is predicted and now comes the observation. 326
00:31:05,570 --> 00:31:16,260
This is a thrilling moment for you. Because what is at stake is the integrity of physics. 327
00:31:16,260 --> 00:31:20,710
And this is going to be measured plus or minus 0.15. Right? 328
00:31:20,710 --> 00:31:28,420
Everytime I make a measurement... plus or minus 0.15 329
00:31:28,420 --> 00:31:47,490
I'm nervous. So I'm going to add 500 grams. Period will indeed increase. 330
00:31:47,490 --> 00:32:22,380
I'm going to oscillate it. Yeah,1,2,3,4,5,6,7,8,9,10. 331
00:32:22,380 --> 00:32:32,300
Oh,boy. Oh,boy. What have I done. What have I done. 332
00:32:32,300 --> 00:32:49,640
20.52. We have a problem. Physics is not working. 333
00:32:49,640 --> 00:32:54,050
Anyone of you have an idea whether there are something wrong 334
00:32:54,050 --> 00:33:00,240
with the equation or whether there are something wrong with Walter Lewin? 335
00:33:00,240 --> 00:33:09,950
Any idea? Come on. 336
00:33:09,950 --> 00:33:17,080
Give it a try,in the worst case your suggestion is not correct. 337
00:33:17,080 --> 00:33:26,800
Yeah. Ur,you accuse me right? Ur,What's your name? 338
00:33:26,800 --> 00:33:33,080
Questioning my 0.15,you say man you couldn't do even better than 0.4 seconds maybe. 339
00:33:33,080 --> 00:33:35,640
Then of course the two will be consistent with each other. 340
00:33:35,640 --> 00:33:46,000
Thank you,very nice of you. 341
00:33:46,000 --> 00:33:50,480
Friction,ok,now that's a very good suggestion. 342
00:33:50,480 --> 00:33:51,710
I know what you begin to think like 343
00:33:51,710 --> 00:33:55,730
a physicist and you also thought like a physicsist,because indeed if my uncertainty 344
00:33:55,730 --> 00:34:00,050
is higher than 0.15 you could be right. 345
00:34:00,050 --> 00:34:05,430
Friction,in this case,we will deal with friction later in the course 346
00:34:05,430 --> 00:34:10,620
has such a negligibly small effect that couldn't be measure in either one 347
00:34:10,620 --> 00:34:15,550
of these two.In any case,it's almost the same for both,because 348
00:34:15,550 --> 00:34:19,790
the shape has no change. It's a very good suggestion. Friction doesn't 349
00:34:19,790 --> 00:34:25,720
come near the proper explanation,but you tried. And that's good. 350
00:34:25,720 --> 00:34:30,550
One more try. Mass of the spring! 351
00:34:30,550 --> 00:34:37,300
We have said earlier. Ur,you may know how early,but I did say it. 352
00:34:37,300 --> 00:34:39,780
You can replay the tape. I can prove it to you. 353
00:34:39,780 --> 00:34:44,750
I said if the mass of the spring is negligible. 354
00:34:44,750 --> 00:34:49,160
and that is the equation. Now what we do 355
00:34:49,160 --> 00:34:52,840
when the mass cannot be ignored. That's not so easy. 356
00:34:52,840 --> 00:34:57,840
But I request it some mandatory reading, 357
00:34:57,840 --> 00:35:01,450
and I'm sure all of you have done that before this lecture. 358
00:35:01,450 --> 00:35:05,880
And the mandatory reading was French P60 to P61 among others. 359
00:35:05,880 --> 00:35:12,680
And French says that if the mass of the spring itself is M, 360
00:35:12,680 --> 00:35:18,060
and M divided by 3 is substantially less than the mass 361
00:35:18,060 --> 00:35:23,410
at the end of the spring,then a very very good approximation is 362
00:35:23,410 --> 00:35:28,130
that the period of oscillation is then this and he actually 363
00:35:28,130 --> 00:35:40,910
derives it. So the period is higher. So we can bring this to a test now. 364
00:35:40,910 --> 00:35:54,150
In other words,the mass of the spring we have weighed that in our case is (see the blackboard) 365
00:35:54,150 --> 00:36:08,880
and so M divided by 3 is (see the blackboard). That's a very small error by the way. 366
00:36:08,880 --> 00:36:15,490
So 0.1 percent error,0.1 percent error. 367
00:36:15,490 --> 00:36:25,770
And so we can now do the following test. We can now take the ratio of these two. 368
00:36:25,770 --> 00:36:49,240
and eliminate there by K,so we can write now (see the blackboard) 369
00:36:49,240 --> 00:36:56,940
And that number is easy to calculate because you know m2,you know m1, 370
00:36:56,940 --> 00:37:05,220
you know these numbers. And I have calculated it for you and it is 1.377. 371
00:37:05,220 --> 00:37:10,550
And the uncertainty it's so small compare to my timing uncertainty that I don't 372
00:37:10,550 --> 00:37:15,070
even have to allow for any uncertainty in that number. Because we remember 373
00:37:15,070 --> 00:37:19,740
the uncertainty in these masses is also the order of 0.1 percent compared that with the 374
00:37:19,740 --> 00:37:26,470
uncertainty in the observations of the time which was close to 1 percent. 375
00:37:26,470 --> 00:37:32,160
So we can bring this now to a test,and all I must do now is multiply if I want to 376
00:37:32,160 --> 00:37:48,100
find out 10Tm2 and I take 1.377,and I multiply it by Tm1,times 10Tm1. 377
00:37:48,100 --> 00:38:02,720
So I take this number . Now I'm really getting nervous,not joking. 14.96 multiplied by 1.377 378
00:38:02,720 --> 00:38:11,570
that is 20.61 and the uncertainty would be the same uncertainty as in there 379
00:38:11,570 --> 00:38:18,120
which is one percent uncertainty. So that is 0.2 seconds. 380
00:38:18,120 --> 00:38:24,020
This number you can now compare with this number. 381
00:38:24,020 --> 00:38:29,680
On the bottom,within the error of measurements,they are now agree. 382
00:38:29,680 --> 00:38:33,090
This is what we observed,and this is what we predicted. 383
00:38:33,090 --> 00:38:39,500
If we apply the proper relation and take the mass of the spring into account. 384
00:38:39,500 --> 00:38:45,190
So you see that physics works,except that this equation was too simple 385
00:38:45,190 --> 00:38:53,740
to be used for our observation. Notice by the way that this 1.414 386
00:38:53,740 --> 00:38:58,430
in our case is lower. 387
00:38:58,430 --> 00:39:03,210
All right. 388
00:39:03,210 --> 00:39:16,100
In 8.03,we will often do not always use complex notation. 389
00:39:16,100 --> 00:39:20,180
And the reason why we do that is that it can a time simplify 390
00:39:20,180 --> 00:39:24,130
your life. You can completely free to choose when you want to 391
00:39:24,130 --> 00:39:30,800
use it or when you don't want to use it. You can be the judge. 392
00:39:30,800 --> 00:39:36,750
So let's talk a little bit about complex numbers. 393
00:39:36,750 --> 00:39:48,440
I start with a circle,and this is the complex plain. 394
00:39:49,140 --> 00:39:53,250
The blackboard is a complex plane,it's quite promotion for the blackboard. 395
00:39:53,250 --> 00:40:02,850
And here,I call this axis "the real axis",so all the real numbers are on this axis, 396
00:40:02,850 --> 00:40:07,830
and this be +1,and this be -1. 397
00:40:07,830 --> 00:40:19,980
I call this axis "the imaginary axis",so this one is +j,this one is -j, 398
00:40:19,980 --> 00:40:23,450
and j is the square root of -1. 399
00:40:24,000 --> 00:40:32,380
We don't call it "i" in general beacause "i" is sense for current so we pick "j". 400
00:40:32,380 --> 00:40:43,760
I now pick a position here which now represents a complex number,call this angle θ, 401
00:40:43,760 --> 00:40:49,290
and I project this,this is position Z,a complex number. 402
00:40:49,290 --> 00:40:54,720
This is the real part of that complex number, 403
00:40:54,720 --> 00:40:58,690
and this is the imaginary part of that complex number. 404
00:40:58,690 --> 00:41:08,770
So you can see that indeed,z can be written since this length is one is the cosine of θ, 405
00:41:08,770 --> 00:41:12,170
plus j times the sinθ. 406
00:41:12,170 --> 00:41:17,140
So that this part which is real,and this is sinθ because this is one. 407
00:41:17,140 --> 00:41:21,490
I have to multiply that by j. 408
00:41:21,490 --> 00:41:38,230
And this now,according to Euler,great mathematician Euler,after whom is this also mentioned,already in 1748, 409
00:41:38,230 --> 00:41:50,210
he proved that this is the same as e to the power jθ. 410
00:41:50,210 --> 00:41:58,670
This equality is mind-boggling,and when I saw this equality for the first time, 411
00:41:58,670 --> 00:42:06,700
I didn't believe it,number 1,and I could hardly sleep at night because I couln't prove it. 412
00:42:06,760 --> 00:42:10,990
See I haven't had any Taylor Expansion yet,so i couldn't prove it. 413
00:42:10,990 --> 00:42:14,410
Just my teacher in high school said "This is the case." 414
00:42:14,410 --> 00:42:18,610
And I said why,he said "This is the way please." 415
00:42:18,610 --> 00:42:23,110
But we now can prove this,you can do the Taylor Expansion of the cosθ, 416
00:42:23,110 --> 00:42:28,550
Taylor Expansion of the sinθ and Taylor Expansion of the e(jθ), 417
00:42:28,550 --> 00:42:35,990
and it's exactly correct,not under approximation. 418
00:42:36,510 --> 00:42:38,700
So why would we ever want to use this? 419
00:42:38,700 --> 00:42:45,840
Well,if you make these things go around,going back to my uniform circular motion here. 420
00:42:45,840 --> 00:42:52,340
If I make that point go around,and i only look at the real part,I have a simple harmonic motion. 421
00:42:52,340 --> 00:43:07,920
So if I change θ into wt,then I get that(see the blackboard) 422
00:43:07,920 --> 00:43:14,150
The real part of which is a simple harmonic motion,and of course I am not stuck to an amplitude of one, 423
00:43:17,230 --> 00:43:22,240
I can easily make the amplitude A times larger. 424
00:43:22,240 --> 00:43:34,720
And of course there is nothing wrong depending upon my initial conditions to have here a phase angle Ψ. 425
00:43:34,720 --> 00:43:46,500
And this then is (see the blackboard) according to Euler. 426
00:43:46,500 --> 00:43:56,150
So what that means is that if you use this as you try your function to solve a differential equation, 427
00:43:56,150 --> 00:44:02,380
you can manipulate this very easily,you can take first derivative,second derivative of exponentials, 428
00:44:02,380 --> 00:44:08,140
it's extremely easy. And then when you done,you take the real part of Z, 429
00:44:08,140 --> 00:44:13,540
and outputs x as a function of time and you done. 430
00:44:13,540 --> 00:44:17,660
As I said,it's up to you when you want use it next lecture, 431
00:44:17,660 --> 00:44:20,420
I'll give you an example of why it's clearly the way to go, 432
00:44:20,420 --> 00:44:27,480
I wouldn't even know how to do it in any other way,but often you do have a choice. 433
00:44:27,480 --> 00:44:36,080
So we are interested in the real part of that which is then our exceptable solution. 434
00:44:36,080 --> 00:44:54,390
So if we have a complex number Z=a+jb,then we should always be able to write it as(see the blackboard) 435
00:44:54,390 --> 00:45:01,600
The amplitude A is the square root of a squares plus b squares. 436
00:45:01,600 --> 00:45:12,510
And tgθ is b over a,that follows immediately from that figure. 437
00:45:12,510 --> 00:45:18,660
So in problem set 1,you will get some chance to practise. 438
00:45:18,660 --> 00:45:23,840
I'll give you a few interesting cases and classic case that 439
00:45:23,840 --> 00:45:36,690
all of you in your life time have to be able to do once is the very nonintuitive problem j to the power j 440
00:45:36,690 --> 00:45:42,650
When I saw for the first time j to the power j,I said to myself "What on earth can be more complex than 441
00:45:42,650 --> 00:45:50,000
j to the power j?" But it's real,it's not complex. 442
00:45:50,000 --> 00:45:56,540
And you will rassle with this,there is infinite number of solutions,not one,all of them are correct. 443
00:45:56,540 --> 00:46:06,350
And I will help you a little,it's the first time i want to be nice to you,it's only the first time. 444
00:46:06,350 --> 00:46:14,570
I can also write (see the blackboard) Do you agree? 445
00:46:14,570 --> 00:46:24,440
Because it simply means that the angle θ is pi over 2,it's here,so end up here. So that's j. 446
00:46:24,440 --> 00:46:29,240
I'm not saying it is a very nice way of expressing j,but it is j. 447
00:46:29,240 --> 00:46:45,010
But not only is this j,I can also rotate an integral number times 360 degrees n=0,1,2,3, 448
00:46:45,010 --> 00:46:49,990
rotate either clockwise or counterclockwise and it's again j. 449
00:46:49,990 --> 00:46:55,130
Becuase if I rotate 90 degrees,it's j. 450
00:46:55,130 --> 00:47:00,620
But if I rotate another 360 degrees,it's again j,or if I rotate back 360 degrees. 451
00:47:00,620 --> 00:47:17,160
So you see that this is also a way to write j,and that will help you,believe me. 452
00:47:17,160 --> 00:47:22,400
I will always have a five-minute break during this 85-minute lecture, 453
00:47:22,400 --> 00:47:27,550
so that you can stretch your legs if you can manage to make it back and forth to the bathroom, 454
00:47:27,550 --> 00:47:29,320
that's fine,but that's your problem. 455
00:47:29,320 --> 00:47:32,340
I will start exactly after 5 minutes. 456
00:47:32,340 --> 00:47:40,220
However,every Tuesday,during part of these five minutes,we will have a mini quiz. 457
00:47:40,220 --> 00:47:47,590
It's really mini,it's this small. And we will collect it after the lecture, 458
00:47:47,590 --> 00:47:54,560
and you will even get some credits for that. But only on Tuesdays. But not today. 459
00:47:54,560 --> 00:48:00,830
Before we go into this five-minute break today, 460
00:48:00,830 --> 00:48:05,950
I want you to see something so that you have something to think about. 461
00:48:05,950 --> 00:48:10,900
Believe me it's healthy,MIT student to sleep. 462
00:48:10,900 --> 00:48:14,190
But it is also healthy sometimes to not sleep. 463
00:48:14,190 --> 00:48:22,640
Sleep is nice and worry just the way that I have sleepless nights in high school about all these equations. 464
00:48:22,640 --> 00:48:28,070
It's healthy,the reason why it's healthy is because once you see the solution, 465
00:48:28,070 --> 00:48:32,050
you say:"Ahhhh!" Of course,and you never forget it. 466
00:48:32,050 --> 00:48:36,660
But as if someone tells you from the start,you say:"Yeah,of course." 467
00:48:36,660 --> 00:48:39,680
And you forget it the next day and don't remember. 468
00:48:39,680 --> 00:48:44,650
So what I want you to see is a remarkable example. 469
00:48:44,650 --> 00:48:52,820
Often oscillations that can be produced not by winds as we have seen,but by heats and by cooling. 470
00:48:55,450 --> 00:49:05,180
I have here a nice pipe,and there is a grid here I can touch it. I'm touching it now. 471
00:49:05,220 --> 00:49:08,490
That's all the risks. It's a open pipe and there is grid here. 472
00:49:08,490 --> 00:49:12,840
When I heat that grid and cool it,somehow, 473
00:49:12,840 --> 00:49:22,310
it generates a 110Hz oscillation of pressure wave which you will be able to hear. 474
00:49:22,310 --> 00:49:41,430
And I will give you until the end of December maybe mid-December to come up with a solution why it's doing that. 475
00:49:41,430 --> 00:50:01,720
I'm heating the grid now. 476
00:50:01,720 --> 00:50:05,740
110Hz roughly,you want to play with this,don't break it, 477
00:50:07,470 --> 00:50:11,060
try to transfer the liquid in few seventeen seconds. 478
00:50:11,060 --> 00:50:20,770
I will resume the lecture exactly 5 minutes from now. 479
00:50:20,770 --> 00:50:29,320
If you turn this in the tornado,you rotate it and then you open up a funnel of air 480
00:50:29,320 --> 00:50:34,730
and so it's never the problem that the liquid can not go through. 481
00:50:34,730 --> 00:50:39,150
There's always pressurely calibrium and I don't remember how long it takes 482
00:50:39,150 --> 00:50:45,950
but I thought it was seventeen seconds. But if you want to,we can time that. May even be less. 483
00:50:45,950 --> 00:50:55,350
I now what to address the issue of simple harmonic oscillation of a pendulum. 484
00:50:55,350 --> 00:51:05,190
As you will remember from age of one,if you have a pendulum,length l,mass m. 485
00:51:05,190 --> 00:51:12,560
And if the mass of the string is neglectably small compared to the mass that is hanging here. 486
00:51:12,560 --> 00:51:20,600
Then the period of oscillation is (see the blackboard) 487
00:51:20,600 --> 00:51:31,810
g in the Boston area being to high degree of accuracy 9.80m/s2. 488
00:51:31,810 --> 00:51:40,920
If you simply take l approximately 1 meter,then you can see that you get the period of about 2 seconds. 489
00:51:40,920 --> 00:51:48,770
And if you make the length about 25cm,that is 4 times shorter,then you will expect this period 490
00:51:48,770 --> 00:51:54,200
which is 2 times shorter,which is about 1 second. 491
00:51:54,200 --> 00:52:02,980
And without any pretence of accuracy,just eyeballing,look,really testing. 492
00:52:02,980 --> 00:52:08,290
If I just eyeball this to be about a meter,if I oscillate this back and forth, 493
00:52:08,290 --> 00:52:10,990
it's about 2 seconds for one oscillation. 494
00:52:10,990 --> 00:52:17,050
One,two,one,two,one,two. 495
00:52:17,050 --> 00:52:29,810
If however make it 25cm,4 times shorter,then it is very close to one second. No interference here. 496
00:52:29,810 --> 00:52:37,860
One,one,one,one,one,one. 497
00:52:37,860 --> 00:52:45,410
Remarkable when you look at this equation is that just like in the case of the spring. 498
00:52:45,410 --> 00:52:48,370
It is independent of the amplitude, 499
00:52:48,370 --> 00:52:52,640
you know the words whether I have a large amplitude or small amplitude, 500
00:52:52,640 --> 00:52:56,140
it will take the same amount of time to go back and forth. 501
00:52:56,140 --> 00:53:04,720
Well,not quite for pendulum. When we derive this period, 502
00:53:04,720 --> 00:53:10,300
you remember that you have to assume what we called "small angle approximations". 503
00:53:10,300 --> 00:53:19,130
And you will see that again and again with called "small angle approximations". 504
00:53:19,130 --> 00:53:27,920
With small angle approximations,the sinθ is always the same as θ in radians. 505
00:53:27,920 --> 00:53:36,020
Now if you ask me how small is small,it's a matter of taste. 506
00:53:36,020 --> 00:53:41,030
In 26100(classroom?),we have the mother of all pendulums. 507
00:53:41,030 --> 00:53:56,080
5.18m long,quite impressive. So we have a pendulum with l is 5.18±0.05m. 508
00:53:56,080 --> 00:54:02,300
We cannot measure it any better than 5cm. It has to be under stretch when we measure it, 509
00:54:02,300 --> 00:54:06,090
and then you have to go all the way to the ceiling and all the way down. 510
00:54:06,090 --> 00:54:11,720
Marcos does that to risk his life and he claims that the best he can do is 5cm. 511
00:54:11,720 --> 00:54:16,280
We have 31 pounds hanging under there. 512
00:54:16,280 --> 00:54:24,380
We try doing the same,believe me,we try with technicians of MIT to have that pendulum here. 513
00:54:24,380 --> 00:54:27,630
And that one day it look good,but finally they said:"No,we can't do it." 514
00:54:27,630 --> 00:54:29,600
They can't install it here,it's a safety issue. 515
00:54:29,600 --> 00:54:36,210
So unfortunately we don't have the mother of all pendulums here. 516
00:54:36,210 --> 00:54:45,920
In 26100,when I lecture neutronian mechanics,I demonstrated that the period that this pendulum 517
00:54:45,920 --> 00:54:51,610
produces is extremely close within the era of measurement which you predict. 518
00:54:51,610 --> 00:54:55,920
Another words,the mass of the string is indeed negelectably small 519
00:54:55,920 --> 00:55:03,210
compared to the mass of the object,we even omit the string once,I don't remember what it was, 520
00:55:03,210 --> 00:55:09,340
but it was such a small fraction of m that indeed could be ignored. 521
00:55:09,340 --> 00:55:14,360
So the prediction that is,if you simply put this l in there. 522
00:55:14,360 --> 00:55:29,050
T predicted purely on the base of that simple equation equals 4.57±0.02s. 523
00:55:29,050 --> 00:55:35,570
And this 0.02 is the result of this 0.05,there's 1% error in here. 524
00:55:35,570 --> 00:55:45,970
Right,five out of 518 is 1%. So the error in T is half a percent because it's the square root. 525
00:55:45,970 --> 00:55:49,950
And so we got a half percent error,and I round that off. 526
00:55:49,950 --> 00:55:53,370
So that is the prediction. 527
00:55:53,370 --> 00:56:01,200
And then i make two measurements,one at 5 degree angle and one at 10 degree angle, 528
00:56:01,200 --> 00:56:10,670
and I did that 10 times,so 10T(5o) and 10T(10o), 529
00:56:10,670 --> 00:56:16,250
it was in 1999,those were my good days,they were my good times,right? 530
00:56:16,250 --> 00:56:18,930
Past is always the good. 531
00:56:18,930 --> 00:56:25,030
And so by then claims that i could do this to an accuracy of 0.1s. 532
00:56:25,030 --> 00:56:29,660
I had a lot of currency in those days and I measured the 5 degree, 533
00:56:29,660 --> 00:56:35,840
and what did I find,unbelievable,truly unbelievable,purely lucky, 534
00:56:35,840 --> 00:56:42,870
I found exactly that number,which of course is an accident because my accuracy was no better than 0.1s. 535
00:56:42,870 --> 00:56:48,890
And then I did it at 10 degree angle and then I found this, 536
00:56:48,890 --> 00:56:58,490
so I demonstrated that indeeds 5 and 10 degrees are still considered small angles for that approximation. 537
00:56:58,490 --> 00:57:05,650
And it's within the uncertainty of my measurement what you expect. 538
00:57:05,650 --> 00:57:15,110
Then I wanted to demonstrate which is not so intuitive that the period is independent of mass, 539
00:57:15,110 --> 00:57:18,670
which is not the case for the spring. 540
00:57:18,670 --> 00:57:25,570
So now if you change the mass and you don't change l,you expect no change in period. 541
00:57:25,570 --> 00:57:31,250
And that's what I really wanted to show you here but I can't. 542
00:57:31,250 --> 00:57:38,710
Therefore I've decided to show you what I did in 1999 if you can show with that 2-minute version 543
00:57:38,710 --> 00:57:46,250
of my video lectures,then you can judge for yourself to all the extents 544
00:57:46,250 --> 00:57:56,220
the mass does not influence the..... 545
00:57:56,220 --> 00:58:02,740
One of the most remarkable things I just mention to you is that the period of the oscillations 546
00:58:02,740 --> 00:58:07,080
is independent of the mass of the object. 547
00:58:07,080 --> 00:58:13,820
That would mean if I join in the bob and I swing down with the bob, 548
00:58:13,820 --> 00:58:17,650
that you should get the same period or should you not? 549
00:58:17,650 --> 00:58:24,790
I'm asking you a question before we do this awful experiment. 550
00:58:24,790 --> 00:58:28,880
Would the period come out to be the same or not? 551
00:58:28,880 --> 00:58:36,690
Some of you think it's the same,have you thought about it that I'm a little bit taller than this object? 552
00:58:36,690 --> 00:58:43,140
And therefore maybe effectively the length of the string has become a little less, 553
00:58:43,140 --> 00:58:47,170
if I sit up like this,and if the length of the string is a little less, 554
00:58:47,170 --> 00:58:52,330
the period would be a little shorter. Yeah? Be prepared for that. 555
00:58:52,330 --> 00:58:57,790
On the other hand...Well,I'm not quite prepared for it... 556
00:58:57,790 --> 00:59:06,140
I will try to hold my body as horizontal as I possibly can in order to be at the same level as the bob, 557
00:59:06,140 --> 00:59:14,270
I will start when I come to a hold here. There we go. 558
00:59:14,270 --> 00:59:28,300
Now,you count. This hurts. 559
00:59:28,300 --> 00:59:40,270
I want to hear you out. 560
00:59:40,270 --> 01:00:11,210
Thank you. 561
01:00:11,210 --> 01:00:23,750
(see the blackboard) Physics works,I am telling you. 562
01:00:23,750 --> 01:00:30,460
Alright,so,I think this most convincing at least for the freshment that indeed the period of a 563
01:00:30,460 --> 01:00:38,080
pendulum is independent of the mass provided that you can ignore the mass of the 564
01:00:38,080 --> 01:00:43,390
string itself which is the case for that pendulum. 565
01:00:43,390 --> 01:00:47,830
Many pendulum some we will see in a 566
01:00:47,830 --> 01:00:54,340
are more complex more complicated than simply a massless string with an object at the end. 567
01:00:54,340 --> 01:00:58,550
And those pendulum we call physical pendulum. 568
01:00:58,550 --> 01:01:04,660
For instance,I could have this pair of compasses and just let it oscillate like this. 569
01:01:04,660 --> 01:01:06,840
That is not just a simple pendulum. 570
01:01:06,840 --> 01:01:13,180
or I could have a ruler like this.Put a pole through here,it has a pin,and have it osillate like this. 571
01:01:13,180 --> 01:01:16,800
But I can also have it oscillate here,it has a different period. 572
01:01:16,800 --> 01:01:21,760
If I oscillate it right in the middle,then it doesn't oscillate at all. 573
01:01:21,760 --> 01:01:27,640
Now comes the question how do we deal with that and most of you must have seen that in 8.01. 574
01:01:27,640 --> 01:01:31,480
but I do want to address that in quite some detail. 575
01:01:31,480 --> 01:01:36,400
So physical pendulum then looks like this. 576
01:01:36,400 --> 01:01:42,380
This is an object. and I drill an hole in here point P, 577
01:01:42,380 --> 01:01:45,870
and I put a pin in the wall and it can without friction, 578
01:01:45,870 --> 01:01:54,230
it can oscillate back and forth and that the center of mass be here,positon O. 579
01:01:54,230 --> 01:02:01,040
And this seperation between P and O is b. And O is the center of mass. 580
01:02:01,040 --> 01:02:06,020
You can choose p anywhere you want to,there is no restriction of p. 581
01:02:06,020 --> 01:02:11,070
So you see that this pendulum is offset over the angle 582
01:02:11,070 --> 01:02:14,680
θ and it will start to oscillate back and forth. 583
01:02:14,680 --> 01:02:18,720
And the question is what is the period. 584
01:02:18,720 --> 01:02:26,390
So clearly,we put the entire gravitational force at point O,in the center of mass. 585
01:02:26,390 --> 01:02:32,260
This is the force acting at that point. 586
01:02:32,260 --> 01:02:36,410
Now comes the question are there any other forces acting at this object? 587
01:02:36,410 --> 01:02:39,080
Or is this the only? 588
01:02:39,080 --> 01:02:43,190
If when you study it,you gonna take all forces into account. 589
01:02:43,190 --> 01:02:49,210
Who is happy that we have taken all forces into account.Raise your hands? 590
01:02:49,210 --> 01:02:51,600
And most of you are getting scared,right? 591
01:02:51,600 --> 01:02:54,530
who says "no,there has to be at least one another force?" 592
01:02:54,530 --> 01:03:03,710
And which force is that?....Where does it active?location? 593
01:03:03,710 --> 01:03:07,990
yeah.so there must be somehow a force at P to hold it up otherwise 594
01:03:07,990 --> 01:03:10,340
it would start to accelerate down. 595
01:03:10,340 --> 01:03:13,460
I am not even sure that this trade up,I doubt that. 596
01:03:13,460 --> 01:03:21,330
It may simply be a direction I don't want to think about that.Surly there has to be a force up. 597
01:03:21,330 --> 01:03:33,330
Now remember F=ma,when you deal with rotation of objects,and this is going to be rotational. 598
01:03:33,330 --> 01:03:51,830
Then this equation changes into (see the blackboard).The Angular acceleration 599
01:03:56,860 --> 01:04:01,840
Until if I pick P as my point of origin, 600
01:04:01,840 --> 01:04:08,010
then the torque,due to this force,does not contribute to my torque equation, 601
01:04:08,010 --> 01:04:16,620
because the torque is r×F between the position vector and the force. 602
01:04:16,620 --> 01:04:26,340
This is the position vector to the center of mass. And the position vector from P to P is zero. 603
01:04:26,340 --> 01:04:35,730
So if we deal with the torque relative to point P,that force is of no consequence. 604
01:04:35,730 --> 01:04:43,690
I am going to take P as my origin,and so now is the question what's the torque relative to point P? 605
01:04:43,690 --> 01:04:53,780
Well,it's r×F. r is this distance which is b. F is mg. 606
01:04:53,780 --> 01:05:00,750
but have a cross product,so I have to take the sin of this angular into account. 607
01:05:00,750 --> 01:05:04,470
That is the magnitude of the torque. 608
01:05:04,470 --> 01:05:13,380
And the magnitude of that torque then according to my rotational equivalent of f=ma 609
01:05:13,380 --> 01:05:22,410
equals the moment of inertia for rotation of that point P times θ double dot. 610
01:05:22,410 --> 01:05:29,610
However,it is a restoring torque. 611
01:05:29,610 --> 01:05:36,020
The torque you can do that with your right-hand whenever you have learned how to do that. 612
01:05:36,020 --> 01:05:40,730
The torque is in the blackboard. Perpendicular to the blackboard in the blackboard. 613
01:05:40,730 --> 01:05:43,960
R cross f is in the blackboard. 614
01:05:43,960 --> 01:05:50,260
I have rotated it counterclockwise which is a vector out of the blackboard. 615
01:05:50,260 --> 01:05:52,860
So one is like this,the other is like this. 616
01:05:52,860 --> 01:05:55,710
That is like saying the torque is restoring. 617
01:05:55,710 --> 01:06:04,670
The same reason why we wrote down f=-kx with this spring is why we now write this equals minors this, 618
01:06:04,670 --> 01:06:10,940
taking into account the direction of the vectors. 619
01:06:10,940 --> 01:06:16,770
So this is the differential equation that you will have to solve. 620
01:06:16,770 --> 01:06:20,300
And if now we go to small angular approximation, 621
01:06:20,300 --> 01:06:40,600
then the sinθ goes to θ if θ is radius. So I can rewrite down this(see the blackboard) 622
01:06:40,600 --> 01:06:48,910
And now,small angular approximation,we have a differential equation which is again a piece of cake. 623
01:06:48,910 --> 01:06:53,240
Simple harmonic oscillation. 624
01:06:53,240 --> 01:07:07,920
So the simple harmonic oscillation,solution must be(see the blackboard) 625
01:07:07,920 --> 01:07:14,580
And this ω is the square root of this number just like we earlier have the square root of k over m. 626
01:07:14,580 --> 01:07:38,780
ω now must be this.(see the blackboard) 627
01:07:38,780 --> 01:07:45,950
I want to repeat what I said earlier this ω is called angular frequency. 628
01:07:45,950 --> 01:07:51,030
The angular frequency is given,that's the angular frequency. 629
01:07:51,030 --> 01:08:00,120
Don't confuse that with θ dot which we also call ω,which is called angular velocity. 630
01:08:00,120 --> 01:08:07,150
And the angular velocity in this case is a strong function of time when the object comes to a halt 631
01:08:07,150 --> 01:08:11,960
the angular velocity is zero,because θ dot is zero. 632
01:08:11,960 --> 01:08:17,180
It is unfortunately that we give them the same symbol. 633
01:08:17,180 --> 01:08:23,090
So this is independent of time,but θ dot does depend on time. 634
01:08:23,090 --> 01:08:27,520
θ dot is the angular velocity. 635
01:08:27,520 --> 01:08:37,550
In the case of uniform circular motion,the two ωs are the same. 636
01:08:37,550 --> 01:08:45,140
So now we have all the ingrediants in hands to calculate for absurd-looking objects what the period of 637
01:08:45,140 --> 01:08:50,620
oscillation is,provided that we are able to calculate the moment of 638
01:08:50,620 --> 01:09:01,620
inertia about the point of rotation and of course we have to know b and the mass of the object. 639
01:09:01,620 --> 01:09:06,070
We have a wonderful example in your problem set. 640
01:09:06,070 --> 01:09:13,840
I will solve that equation,calculate this T. 641
01:09:13,840 --> 01:09:21,530
For a hoop,this is the hoop,all the mass is at the circumference, 642
01:09:21,530 --> 01:09:26,170
so it will be very easy to calculate the moment of inertial. 643
01:09:26,170 --> 01:09:34,070
And,we have a holding here. until we are going to oscillate it right at the ring. 644
01:09:34,070 --> 01:09:41,430
And so our geometry is easy but we should be able to bring this equation 645
01:09:41,430 --> 01:09:49,870
to our rigid test provided that we take into account the uncertainty of our measurement. 646
01:09:49,870 --> 01:09:58,390
So let me put here this. Circle,this is the hoop. 647
01:09:58,390 --> 01:10:07,570
So all the mass to very good approximation is at the circumference 648
01:10:07,570 --> 01:10:13,540
and the oscillation is about the axis perpendicular to the blackboard. 649
01:10:13,540 --> 01:10:18,730
Point P,this is the center of mass O. 650
01:10:18,730 --> 01:10:21,990
And I am going to offset this hoop, 651
01:10:21,990 --> 01:10:38,140
so this is when it is equilibrium and this is offset over angle θ and point O is now here,we call it point O'. 652
01:10:38,140 --> 01:10:41,120
And so,in the knowledge we work with it there, 653
01:10:41,120 --> 01:10:49,950
we have here force mg,and derivation is identical,we don't have to go over that. 654
01:10:49,950 --> 01:10:58,460
Again and the radius is R.And it is given we need it,R is given. 655
01:10:58,460 --> 01:11:03,180
I will show you what these numbers are later. 656
01:11:03,180 --> 01:11:10,620
So all I have to do now is to go to this equation and calculate the moment of inertia for rotation 657
01:11:10,620 --> 01:11:14,960
of an axis like this who point P. 658
01:11:14,960 --> 01:11:19,140
Who remembers how to do that? 659
01:11:19,140 --> 01:11:23,380
8.01 660
01:11:23,380 --> 01:11:26,140
Come on,the worst case is wrong. 661
01:11:26,140 --> 01:11:30,070
see one hand there.Who remembers?Let me ask you this: 662
01:11:30,070 --> 01:11:33,810
suppose it was rotating through an axis right through the center of mass. 663
01:11:33,810 --> 01:11:36,380
It's difficult because there is nothing to hold on to.Would you know then 664
01:11:36,380 --> 01:11:38,980
what's the moment of inertia? 665
01:11:38,980 --> 01:11:41,900
What is it then? 666
01:11:41,900 --> 01:11:45,420
Ah?You say yes but now you cry... 667
01:11:46,320 --> 01:11:48,340
ok? 668
01:11:48,340 --> 01:11:51,260
oh,moment of inertial is never mR,it's never,it's dimensionally 669
01:11:51,260 --> 01:11:56,290
wrong.But you try,it is better than not trying. 670
01:11:56,290 --> 01:12:01,730
yeah,m R2(R square) is what the moment of inertial would be if the axis was straight to O. 671
01:12:01,730 --> 01:12:12,570
I'm slowly working you up now. Now we move the axis from O to P. What happens now? 672
01:12:12,570 --> 01:12:19,540
How do we call this? What we call this theorem? Parallel Axis Theorem. 673
01:12:19,540 --> 01:12:29,300
Now we have to add the mass times the distance between the center of mass and that point square. 674
01:12:29,300 --> 01:12:34,460
That's the Parallel Axis Theorem.And so,(see the blackboard) 675
01:12:34,460 --> 01:12:43,530
the moment of inertia about point P is m R squared for rotation about this point we take the same axis 676
01:12:43,530 --> 01:12:55,610
we move it to P and we have to add m distance squared,so we have to add,plus m R squared,so we get two m R squared. 677
01:12:55,610 --> 01:13:04,420
And then we have b,what is b,what is the distance from P tothe center of mass? 678
01:13:04,420 --> 01:13:26,200
That's R. So we could now with the prediction that (see the blackboard),m goes,m always goes with pendulum. 679
01:13:26,200 --> 01:13:29,830
You never have to worry about m if you do right. 680
01:13:29,830 --> 01:13:33,000
m always goes not with springs but with pendulums. 681
01:13:33,000 --> 01:13:43,690
One R goes,and so you get (see the blackboard). 682
01:13:43,690 --> 01:13:52,740
Before we bring this to a test,see this remarkable answer,what does that make you think of ? 683
01:13:52,740 --> 01:13:56,460
Excuse me? 684
01:13:56,460 --> 01:14:02,200
It makes you think of a single Pendulum we are by the length is two R, 685
01:14:02,200 --> 01:14:04,020
which by no means is obvious,isn't it? 686
01:14:04,020 --> 01:14:10,310
In a another word,if we have a pendulum here,and I will hang here an object m, 687
01:14:10,310 --> 01:14:16,050
double have the same period,because it has a length 2R. 688
01:14:27,330 --> 01:14:33,540
Till now comes the essay test,so we don't have to measure the mass, 689
01:14:33,540 --> 01:14:41,350
but we did measure as accurately as we can,the radius,that's really all we have to do. 690
01:14:41,350 --> 01:14:46,390
And the measurement of the radius is a little uncertain, 691
01:14:46,390 --> 01:14:49,200
because it is not a perfect circle. 692
01:14:49,200 --> 01:15:01,090
So we measure it at various places,and we find that (see the blackboard) 693
01:15:01,090 --> 01:15:14,320
That is one percent uncertainty.And so we make a prediction out T,(see the blackboard) 694
01:15:14,320 --> 01:15:19,980
we get the square root of R.So the one percent of uncertainty becomes half a percent because of the square root. 695
01:15:19,980 --> 01:15:25,060
We take 2R unified by g,and you will find that the prediction, 696
01:15:25,060 --> 01:15:46,560
this is the prediction,is that T is(see the blackboard),that is because of the square-root and this becomes half a percent error. 697
01:15:46,560 --> 01:15:54,300
Now we do the observation,(see the blackboard) and you guess that of course we are going to do 10T, 698
01:15:54,300 --> 01:16:01,230
and if it is a good day,0.1 that will give myself a little bit extras. 699
01:16:01,230 --> 01:16:07,560
Leeway today 0.15 I'm fairly sure I should be able to do that. 700
01:16:07,560 --> 01:16:12,730
And so we bring this now to a test. 701
01:16:12,730 --> 01:16:24,600
Ready for this? I always like to start the timer when the object close to a halt, 702
01:16:24,600 --> 01:16:30,180
that is a better criteria when it goes through equilibrium. 703
01:16:30,180 --> 01:16:34,430
And I will not look at it,even if I did look at it,there is no way, 704
01:16:34,430 --> 01:16:44,700
I can stop that when I want to,so we give a offset I first want to swing in way that is not wobbling 705
01:16:44,700 --> 01:16:50,900
I make a very strong prediction I want to get that number 17.95. 706
01:16:50,900 --> 01:16:58,790
So I'd better make sure that it is. Oh,that is osillating happily,no,this is not happy. 707
01:16:58,790 --> 01:17:01,800
I don't want any wobbling like this. 708
01:17:01,800 --> 01:17:07,880
Ok. 709
01:17:07,880 --> 01:17:12,840
I think this looks good. If you are ready,I'm ready. 710
01:17:12,840 --> 01:17:33,580
Start. 1 2 3 4 5 6 7 8 9 now,yeah. 711
01:17:33,580 --> 01:17:35,250
17.80 712
01:17:35,250 --> 01:17:42,430
Ah,man,0.15. 713
01:17:42,430 --> 01:17:45,440
So wasn't it such a bad day after all for me? 714
01:17:45,440 --> 01:17:49,510
Ok,see you Tuesday and work on your problem sets.
| Name | Version | Size | Date | User |
| 01_a.srt | 1 | 70283 | 2/17/06 4:15 AM | OOPSSJTU |
Last Modified 2/21/06 8:30 AM
|
