1
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We have discussed free oscillations, harmonic oscillators without damping, 2
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and then we introduce damping. 3
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But in each of those cases, we let the simple harm-oscillator do its own thing 4
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We did not interfere with it. 5
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Today that's going to change. 6
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Today we are going to impose our will onto the simple harmonic oscillator. 7
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And we can impose our will on it by driving it with a force. 8
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And then see what the net result is. 9
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And let's start with a simple examle 10
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that I have here a spring with spring constant k. 11
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And an object mass m. 12
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And this be equilibrium position x equal zero. 13
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There will be damping. 14
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I will introduce again b/m equals γ and ω zero squared equals k/m. 15
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We've seen this before this is the shorthand notation. 16
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So now in addition to the fact that when the object is away from equilibrium, 17
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that there's here a spring force. 18
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I am now going to apply on that object a force. 19
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May not be easy to do we'll get back to that how we do that. 20
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But I can apply a force on that. 21
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Maybe through magnetic fields. Maybe through electric fields. 22
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And I'm going to, this force is now going to have the character F zero times cosωt. 23
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I impose on that system now that frequency ω, 24
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and I can choose that anything I want to. 25
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So now I can write down the differential equation of motion. 26
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Newton's second law (see the blackboard). Nothing new. 27
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That's the spring force. 28
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Minus b x dot. Nothing new. That's the damping. 29
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But now comes this external force F zero cosωt. 30
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What I'm going to now? I'm going to move this to the complex plane. 31
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Not that that is absolutely necessary but I'm so used to that. 32
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So I'm going to write this now in terms of Z. 33
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And then we take the real part of Z later then this is back, goes back to x. 34
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So I'm going to write this now in terms of Z double dot. 35
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I divide m out. 36
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We got plus γ Z dot plus ω zero squared times Z, 37
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and that now becomes F zero divided by m. 38
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Remember I divided m out. 39
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And then we get cosωt for which I will write e to the power jwt 40
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because I work now in the complex plane. 41
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And through Euler, I can always convert that back to cosine. 42
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My trial function for Z, 43
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which is a complex notation, is some amplitude times e to the power jwt minus δ. 44
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Now crucial is that you understand why this ω and this ω are the same. 45
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This is the ω of my driving the system that is my will that I impose on that system. 46
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Clearly given enough time and in the beginning the system may be unhappy 47
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and it may do all kinds of nasty things 48
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which we will discuss next lecture. 49
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But ultimately I will come out to be the winner 50
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and ultimately that system is bound to start oscillating with the frequency that I imposed on it. 51
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If I start shaking you, in the beginning you may not like that, then you may oppose to that. 52
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But ultimately I will be the winner. 53
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And I will make you shake with that frequency ω. 54
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So clearly the ultimate solution must have the same ω as the driver. 55
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What's the meaning of this δ? 56
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Well, it is not so obvious that the object will have been in the same phase as the driver. 57
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It is possible that when the force is pointing in this direction, 58
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that the object may be going in the other direction. 59
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And you will see that. That indeed can happen. 60
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And so this δ is a phase angle which takes into account the possibilty 61
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that the driver and the object in their motion are not exactly inphase. 62
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We call this solution a steady state solution. 63
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Steady state that you must wait long enough for the system not to fight you any longer. 64
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That will be part of my next lecture: The fighting issue. 65
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This is when I ultimately come out to be the winner and when the system follows my will. 66
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So now I am going to take the second derivative. 67
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So I get minus ω squared. jw comes out twice. 68
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So I get minus ω squared. 69
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Then I get plus γjw. 70
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Then I get plus ω zero squared. 71
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And that whole thing multiplied by A to the power e jwt minus δ 72
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that now equals F zero divided by m times e to the power jwt. 73
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So the whole thing is now in the complex plane. 74
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And you see the e to the power jwt cancels on both sides. 75
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So I lose my e to the power jwt and I'm going to multiply both sides by e to the power j plus δ. 76
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So I lose my δ here but it appears then here. 77
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And so if I make that simple change, algebraic change. 78
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We are going to get (see the blackboard) 79
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that multiplied by A must now be equal F zero divided by m times e to the power jδ. 80
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ωt is gone. 81
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And I've moved the δ to the right side. 82
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Can we live with that? 83
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And this can be written (see the blackboard). 84
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I'm still in the complex plane but that's Euler. 85
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Now let's compare the apples with apples and the oranged with oranges. 86
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This is an apple and this is an apple that means it's real. 87
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And that means this is an apple. 88
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But there are also oranges. 89
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This is an orange has a j and this is an orange that has a j. 90
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And so for this equation to always hold at all moments in time. 91
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The apples must be equal to the apples on this side. 92
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And the oranges on this side must be equal to the oranges on that side. 93
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So it looks like one equation but it really is two equations. 94
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So we now get that (see the blackboard) 95
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Apples on both sides are equal. 96
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And now we get the oranges on both sides. 97
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(see the blackboard) 98
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Two equations with two unknows, A as unknown and δ as unknown. 99
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And they are easy to solve. 100
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If you square them, then you get sin squared δ 101
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and cosine squared δ and you add them up. That's one. 102
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And so that immediately gives you then what A is. 103
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So A is going to be (see the blackboard) 104
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So this is the amplitude of the object. 105
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I will massage that we will talk about this word at least in next ten minutes. 106
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It's a very complicated funtion. 107
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We want to see through that equation what that actually means. 108
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And tangent of δ is easy to find because you divide this equation by that one, 109
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you get immediately the tangent of δ. A disappears and F zero over m disappears. 110
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And so you get that tangent of that angle δ is (see the blackboard). 111
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We can now return to the real world. 112
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And if we return to the real world, we have to change x and Z back into x. 113
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And so our final solution which I will put in color will then be 114
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that x as a funcion of t is (see the blackboard) 115
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And that is the ω. That's my will that I impose on the system. 116
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Notice that there I know two adjustable constants which we were so used to in the past. 117
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In the past we said that well you can start the system at t equals zero. 118
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You can define the position. You can give it certain velocity. 119
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So you always expect that in your solution there are two adjustables 120
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in order to meet the initial conditions. 121
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There are none here. 122
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And the reason for that is that this is a steady state solution 123
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which means the system doesn't even remember anymore what the situation was at t equals zero. 124
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It has lost all its memory. 125
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And so A which is the amplitude of that object is not, not something that you may choose. 126
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A follows immediately from this equation which is a complex function, 127
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ω, ω zero F zero and so on. 128
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And δ is also non-negotiable. 129
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δ has nothing to do with your initial conditions. 130
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δ follows from γ, ω and ω zero. 131
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So now we are going to look at and try to understand the complexity of the amplitude. 132
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For one thing it is pleasing that F zero was upstairs. 133
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It is intuitively pleasing that if the force that you apply becomes larger that the amplitude will become larger. 134
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That's reasonable. 135
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It is also pleasing to see that it has a γ here downstairs. 136
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That means if there is a huge amount of damping, you don't expect A to be very large. 137
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So that's also pleasing that you see a γ downstairs. 138
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Now I want to evaluate in detail what is hidden in this very difficult equation. 139
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And let me try out your intuition, common sense. 140
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without looking at my solutions, without looking at differential equations, 141
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without looking at equation A. 142
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Just common sense now. 143
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Suppose I apply a force here on this object. 144
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with a frequency which is near zero. 145
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So it takes a hundred million years for it to reach its maximum. 146
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And then it takes another hundred million years for the force to go to zero and so on. 147
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When that force has a value F zero, what do you think will be the position of that object? 148
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If you know that position, that may tell you what A is, 149
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what the amplitude is of that object, 150
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without any differetial equations. 151
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Anyone of you able to immediately say, of course, A has to be this. 152
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Maybe that is a little tougher. I see some hands there, I know. 153
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Or you're just doing your hear. 154
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Yeah! Of course. 155
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If that force goes so slowly that all moments in time there must be equilibrium, 156
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which means the spring force, which is of course kx and the force that you apply which is your F. 157
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And so if you do it extremely slowly, the two must always cancel each other. 158
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And so I make the prediction now that when ω goes to zero 159
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that A should become F zero divided by k. 160
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That is that x. 161
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Now let's look at this equation, let's see whether that is true. 162
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We make ω zero. We make ω zero. So this equation tells us 163
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that this F zero divided by m and then downstairs we have ω zero squared. 164
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But ω zero squared is k/m. 165
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And you see indeed so that's exactly what you get. 166
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Now without looking at the equations, 167
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can you guess what the phase difference is between the driver and the follower? 168
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If it takes a hundred million years for that force to slowly reach its maximum, 169
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and a hundred million years to go back again. 170
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What do you think will be the phase difference between the two? 171
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It will be zero, of course. Plenty of time for that object to follow. 172
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So you expect that δ becomes zero. 173
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Well, if ω becomes zero, this ω this zero is ω squared. 174
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This is...So this goes away. Here is a zero upstairs so you see the tangent of δ is zero. 175
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And that indeed is what you see. 176
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So the two follow each other. It's extremely boring the whole thing to watch 177
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and the amplitude is exactly what you predicted. 178
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Let's now do something more interesting. 179
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And let us drive it at what we call the resonance frequency. We give it that word. 180
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That's the frequency that the system really would love to oscillate 181
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in the absence of any damping and in the absence of my doing the silly thing by driving it. 182
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So now we are at what we call at resonance. 183
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So this term goes away and this term now becomes ω0γ. 184
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So you now get that (see the blackboard). 185
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Well, if you remeber that we introduce a quailty factor ω0 divided by γ, 186
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which is a dimensionless number. 187
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Then you can also write this as (see the blackboard). 188
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So that's nice to remember that at resonance, if you define this as resonance, 189
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the amplitude of the object is Q times higher than what it would be at extremely low frequency. 190
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Interesting to remember. So this is the amplitude at very low frequency 191
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and when you drive it at resonance, it is Q times higher. 192
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And then I will put that here. When ω goes to infinity, 193
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everything goes so fast that the object has no time to follow the driver. 194
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The object goes nuts because this high frequency. It can't do anything. 195
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And so A would then go to zero. And let's check that. 196
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If ω goes to infinity, you see the downstairs here goes to infinity. 197
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So A goes indeed into zero. So you have no amplitude at all. 198
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What is not so obvious that δ here is π. 199
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And what is not so obvious here either, 200
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that δ goes to π/2 in case of resonance. 201
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In other words, at resonance the driver and the follower are 90 degrees out of phase. 202
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The follower is 90 degrees behind. 203
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Very hard to imagine what that is like. But I will demonstrate it. 204
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You will be able to see it. 205
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What it means here that at very high frequencies, the amplitude of the object goes to zero. 206
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But what I will be able to show you that if the driver goes 207
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in this direction that the object goes in this direction. 208
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So they are 180 degrees out of phase that I can show you. 209
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That's what it means when δ equals π. 210
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So now we can make a graph, a plot of A as a function of ω. 211
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So here is ω and here is A. 212
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And then this be the resonance frequency ω0. ω a little straighter. 213
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So you start at very low frequency. This is zero. 214
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We start here with F0 divided by K. 215
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We all agree that this was obvious. And then the amplitude will build up. 216
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Goes through a maximum, goes down and ultimately goes to zero. 217
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And at this value ω0, this value is Q times F0 divided by K. 218
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Now for those of you who look very carefully 219
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You may have noticed that the maximum here that I have drawn is not at ω equals ω0. 220
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which may go against to instinct. This maximum occurs at a frequency which 221
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we will call ωmax which is always a little bit below ω0 but for high Q systems 222
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as I will show you shortly. 223
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It is effectively the same. I will come back to this. 224
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The δ in phase as a function of ω. This is π and this is π/2 225
00:20:53,320 --> 00:21:03,740
and this is ω0. Then that δ will change in the following way 226
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which is a way hard to imagine that then what A is doing. 227
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You are inphase at very low frequencies. At resonance precisely at ω0 228
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you have the π/2, 90 degrees out of phase. 229
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And at very high frequencies, you will see that the two are out of phase 230
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and I will be able to demomstrate that to you. 231
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Coming back to this mysterious maximum, not so mysterious actually. 232
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Where is this? At what frequency do we have really the maximum amplitude? 233
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Well, to calculate that you would have to take the derivative of 234
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that the monstrous equation. You would have to take dA/dω. 235
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And you go, you ask that to be zero. 236
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So that's when the maximum occurs. 237
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And I will leave you with that exercise may take you a few minutes to do that. 238
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And you find then that ωmax. 239
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So where are the real maximum is located at the maximum in terms of 240
00:22:10,280 --> 00:22:22,480
the amplitude is (see the blackboard). 241
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Not so intuitive that it is there. 242
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And if you like to write that in terms of Q which is often done. 243
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Then ωmax so that is the frequency at which the amplitude reaches its maximum 244
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is (see the blackboard). 245
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And the reason why this is nice, you can immediately if you know Q, 246
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you can immediately evaluate it what the difference is percentage width 247
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between ωmax and ω0. 248
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If you want to know what the maximum amplitude itself is? So what Amax is? 249
00:23:13,100 --> 00:23:18,430
So that's really this value. It must be very close to Q times F0/Q(?K) 250
00:23:18,430 --> 00:23:20,350
but it is a little higher. 251
00:23:20,350 --> 00:23:26,250
Then you can write that in the following form and that's just 252
00:23:26,250 --> 00:23:32,930
a matter of algebraic manipulation. And you get Q here what you expect. 253
00:23:32,930 --> 00:23:45,660
And then downstair you get something like this (see the blackboard). 254
00:23:45,660 --> 00:23:53,590
And so now let's put in some numbers so that you get some feeling for the answers that we have. 255
00:23:53,590 --> 00:24:01,290
Suppose we have an example of a Q equals 5. 256
00:24:01,290 --> 00:24:03,140
Some modest value for Q. 257
00:24:03,140 --> 00:24:07,260
Most pendulums that we have, the Q is way higher than 5. 258
00:24:07,260 --> 00:24:14,030
So I take a modest number for Q. If I go to this equation here, 259
00:24:14,030 --> 00:24:21,030
2 squares 25, 2 times 25 is 50. That's 2% that I have to take the square root. 260
00:24:21,030 --> 00:24:23,410
So it's only 1% off. 261
00:24:23,410 --> 00:24:35,510
So ωmax divided by ω0 is 0.99. It's only one percent lower. 262
00:24:35,510 --> 00:24:38,840
It's only one percent below ω0. 263
00:24:38,840 --> 00:24:42,330
And then if you want to know now what Amax is. 264
00:24:42,330 --> 00:24:50,590
So you would think that Amax is very close to Q times F0/K, 265
00:24:50,590 --> 00:24:55,800
but it is not Q times. It is a little larger. 266
00:24:55,800 --> 00:25:03,310
And so if we defined Amax divided by A0. A0 now is meant to be the amplitude 267
00:25:03,310 --> 00:25:06,520
when ω equals 0, that is short hand notation. 268
00:25:06,520 --> 00:25:13,410
This number is not Q, a little higher, is now 5.03. 269
00:25:13,410 --> 00:25:18,320
And you can see it if Q is higher, then of course these numbers become 270
00:25:18,320 --> 00:25:23,610
even closer then ωmax becomes even closer to ω0. 271
00:25:23,610 --> 00:25:31,990
And then the maximum A becomes even closer to Q times F0/K. 272
00:25:31,990 --> 00:25:38,590
Rarely ever will we be bothered too much with the fact that the resonance frequency 273
00:25:38,590 --> 00:25:43,080
which we call ω0 is not exactly the frequency 274
00:25:43,080 --> 00:25:47,150
whereby the response of the object is a maximum. 275
00:25:47,150 --> 00:25:51,740
Very rarely ever will that become an issue. 276
00:25:51,740 --> 00:25:57,610
I want to show you now a transparency from your own book. 277
00:25:57,610 --> 00:26:03,440
Don't take notes. This is from French. You see here 278
00:26:03,440 --> 00:26:09,530
that the function, A but it is divided by A0 which is that F0 divided by K. 279
00:26:09,530 --> 00:26:16,590
So that is the amplitude for zero frequency. So when you start off its one. 280
00:26:16,590 --> 00:26:21,430
That ratio is one by definition, right? Because it's Aω divided by A0. 281
00:26:21,430 --> 00:26:25,170
And horizontally you see ω divdied ω0. 282
00:26:25,170 --> 00:26:29,660
So by definition, right here at the one sign is that point 283
00:26:29,660 --> 00:26:36,220
that I put ω0 there. And you see here these various curves 284
00:26:36,220 --> 00:26:42,150
for different values of Q, and the one that I made green is Q equals 10. 285
00:26:42,150 --> 00:26:46,570
And no surprise that height is at plus 10. 286
00:26:46,570 --> 00:26:53,000
Because we predicted that it's Q times higher than the amplitude when we have low frequency. 287
00:26:53,000 --> 00:26:57,620
And you see indeed that red one is very close to 10. 288
00:26:57,620 --> 00:27:03,530
If you look at the one that has a mark Q equals 3, which is this one. 289
00:27:03,530 --> 00:27:13,850
If you look very carefully, you may see that the maximum A shifted slightly below the value one, 290
00:27:13,850 --> 00:27:19,260
which is ω, ω is ω0. But even for Q equal 3, 291
00:27:19,260 --> 00:27:22,460
the difference is insignificantly small. 292
00:27:22,460 --> 00:27:28,540
And then at the bottom you see the δ function, the phase delay, 293
00:27:28,540 --> 00:27:33,970
the object follows the driver at very low frequency. 294
00:27:33,970 --> 00:27:40,320
Precisely, δ is zerp, at resonance is precisely π/2 90 degrees. 295
00:27:40,320 --> 00:27:43,960
Very hard to imagine but I will try to show it to you. 296
00:27:43,960 --> 00:27:47,710
And at very high frequencies, they go like this. 297
00:27:47,710 --> 00:27:54,830
They are 180 degrees out of phase. 298
00:27:54,830 --> 00:28:02,320
So now comes the question: how do you apply a force on a system. 299
00:28:02,320 --> 00:28:08,470
It's nice to say that is a force, but you have to think of a way that you can actually do that. 300
00:28:08,470 --> 00:28:12,150
And I will just discuss one case with you. 301
00:28:12,150 --> 00:28:15,650
And then I will try to demonstrate it also. 302
00:28:15,650 --> 00:28:22,960
If I have a pendulum and I want a force on this object. 303
00:28:22,960 --> 00:28:30,180
Then I can that as you will see in a indirect way by starting to move my hands here. 304
00:28:30,180 --> 00:28:35,360
You will see how that translate into a force on that object by moving my hands. 305
00:28:35,360 --> 00:28:45,070
I am now the driver. My displacement now is in inches, not a force but is in inches. 306
00:28:53,310 --> 00:29:06,220
And here is the object, but I am going to move my hand in a way η (see the blackboard). 307
00:29:06,220 --> 00:29:12,880
That is the frequency that I decide. I impose that frequency on the top of that pendulum. 308
00:29:12,880 --> 00:29:19,000
And the amplitude of my hands in terms of inches or miles or light years, 309
00:29:19,000 --> 00:29:24,910
that is a linear scale η0. This is not a force. 310
00:29:24,910 --> 00:29:28,660
This is not a force. It's displacement. 311
00:29:28,660 --> 00:29:33,380
While I take a picture at one moment in time, and what do I see. 312
00:29:33,380 --> 00:29:37,900
I see that this is what the pendulum looks like. 313
00:29:37,900 --> 00:29:46,330
This angle is θ. The pendulum is displaced over a distance x from equilibrium. 314
00:29:46,330 --> 00:29:52,090
And the top is displaced over a distance η. This is walter Lewin. 315
00:29:52,090 --> 00:29:55,380
I am doing that. I am there with my hands. I can't help it. 316
00:29:55,380 --> 00:30:01,370
This is where I am and this is where the object is. 317
00:30:01,370 --> 00:30:06,480
So now I want to put in all the forces that I have worked. 318
00:30:06,480 --> 00:30:11,740
I will move it up a little bit because I want to have a little bit room for my forces. 319
00:30:11,740 --> 00:30:19,360
So makes the length a little shorter. So here's the object and here's the object. 320
00:30:19,360 --> 00:30:23,530
There are only two forces on this object. 321
00:30:23,530 --> 00:30:35,960
And that is gravity which is mg. And that is the tension. There is nothing else. 322
00:30:35,960 --> 00:30:48,440
I call this x equal 0 and I call this displacement x away from equilibrium. 323
00:30:48,440 --> 00:30:56,410
For small angles, I want to argue that T is very close to mg. 324
00:30:56,410 --> 00:30:59,040
For one thing if you hold them vertically and you do nothing, 325
00:30:59,040 --> 00:31:02,280
and there is no motion. It's obvious that T is mg. 326
00:31:02,280 --> 00:31:06,430
Two forces have to cancel each other. That's clear. 327
00:31:06,430 --> 00:31:13,640
But I can show you that even if the angles are modest that that should also be the case. 328
00:31:13,640 --> 00:31:23,970
Suppose I decompose T in two directions: vertical direction, so this is T times 329
00:31:23,970 --> 00:31:33,620
the cosθ, and the horizontal direction, so this is T times the sinθ. 330
00:31:33,620 --> 00:31:38,660
If the angles are very small, the object is hardly moving at all in this direction. 331
00:31:38,660 --> 00:31:41,830
The motion is almost exclusively in this direction. 332
00:31:41,830 --> 00:31:45,530
So that is no acceleration in the Y direction 333
00:31:45,530 --> 00:31:50,630
or I should say the acceleration in the Y direction is negligibly small. 334
00:31:50,630 --> 00:31:57,720
So that means to high degree of accuracy, Tcosθ is always the same as mg. 335
00:31:57,720 --> 00:32:07,850
High degree of accuracy. But for small angles cosθ itself is one, therefore T equals mg. 336
00:32:07,310 --> 00:32:15,490
And so the force that is driving this object back to equilibrium is Tsinθ, 337
00:32:15,490 --> 00:32:22,690
and so that force is mgsinθ to a high degree of accuracy. 338
00:32:22,690 --> 00:32:28,610
I'm going to introduce again that γ is b/m thought of its damping. 339
00:32:28,610 --> 00:32:34,000
And I am going to introduce that ωo squared equals g/l, 340
00:32:34,000 --> 00:32:35,990
ωo squared is g/l, 341
00:32:35,990 --> 00:32:40,400
g/l being the ... the square root of g/l being the 342
00:32:40,400 --> 00:32:48,110
resonance frequency of a pendulum length l, independent of 343
00:32:48,110 --> 00:32:51,880
the mass of the object as we have seen before. 344
00:32:51,880 --> 00:32:55,320
So now I'm going to write down Newton's Second Law, 345
00:32:55,320 --> 00:33:00,650
so I get m x double dot, 346
00:33:00,650 --> 00:33:09,050
then I get minus b x dot, that is the damping, minus b x dot, 347
00:33:09,050 --> 00:33:13,420
and now comes this force which is the only one that 348
00:33:13,420 --> 00:33:16,800
wants to drive it back to equilibrium. 349
00:33:16,800 --> 00:33:18,840
It's the restoring force. 350
00:33:18,840 --> 00:33:22,370
And so that's force if you accept my t being mg, 351
00:33:22,370 --> 00:33:27,570
that is mg times the sin of θ. 352
00:33:27,570 --> 00:33:34,380
That's the differential equation that I now have to solve. 353
00:33:34,380 --> 00:33:40,010
That is a driven system. Now here I had a driven system 354
00:33:40,010 --> 00:33:47,530
and boy, I saw force here. I don't see anything like that there. 355
00:33:47,530 --> 00:33:50,970
Where on earth does Walter Lewin come into this picture? 356
00:33:50,970 --> 00:34:04,620
Who is doing something? Have I overlook myself perhaps? 357
00:34:04,620 --> 00:34:11,730
Excuse me...I changed nothing. Would I change anything? 358
00:34:11,730 --> 00:34:15,440
I've changed nothing, but I don't see myself anymore. 359
00:34:15,440 --> 00:34:18,260
So what's wrong? Is there anything wrong with this? 360
00:34:18,260 --> 00:34:23,610
Where do I show up in this equation? 361
00:34:23,610 --> 00:34:30,070
So where in that equation do I show up? 362
00:34:30,070 --> 00:34:39,410
What is the sin of θ? What is the sin of θ? 363
00:34:39,410 --> 00:34:43,960
What is the sin of this angle? 364
00:34:43,960 --> 00:34:54,370
x minus η, that's Walter Lewin. x minus η divided by l. 365
00:34:54,370 --> 00:34:56,920
There I am. 366
00:34:56,920 --> 00:34:59,090
And so I'm going to substitute that in here, 367
00:34:59,090 --> 00:35:02,510
and I'm going to divide that by m. 368
00:35:02,510 --> 00:35:07,020
Well, let's not divide by m yet. Let’s just say x double dot, 369
00:35:07,020 --> 00:35:17,100
minus b x dot, and now we get minus mg times x/l, 370
00:35:17,100 --> 00:35:19,990
and now we bring Walter Lewin to the other side. 371
00:35:19,990 --> 00:35:30,590
And so we get plus mg times η divided by l, 372
00:35:30,590 --> 00:35:34,510
and η is ηo times cosωt 373
00:35:34,510 --> 00:35:40,690
because I am moving my hand that η is a function of time. 374
00:35:40,690 --> 00:35:45,430
Let me write down an mg in here and then we will check this. 375
00:35:45,430 --> 00:35:50,680
so mg times sinθ has two terms that it has an mg times x/l, 376
00:35:50,680 --> 00:35:53,560
but it also has an mg times η/l. 377
00:35:53,560 --> 00:35:56,030
And I bring that η/l on this side, 378
00:35:56,030 --> 00:35:59,230
but I know that η is changing in time, 379
00:35:59,230 --> 00:36:03,820
and so you see now Walter Lewin is right there. 380
00:36:03,820 --> 00:36:08,660
And now I divide by m and I substitute ωo squared in here. 381
00:36:08,660 --> 00:36:12,030
Oh... I had an m here too. 382
00:36:12,030 --> 00:36:14,260
You should had screamed there wasn't an m there. 383
00:36:14,260 --> 00:36:16,130
Are you decided not to divide by m, remember? 384
00:36:16,130 --> 00:36:22,930
Now I am going to divide by m, so I get x double dot. 385
00:36:22,930 --> 00:36:27,200
There's an equal sign here. You should not be sleeping. 386
00:36:27,200 --> 00:36:29,240
You're not supposed to sleep. 387
00:36:29,240 --> 00:36:36,460
This is an equal sign, minus b x dot, right? Equals minus b, yeah. 388
00:36:36,460 --> 00:36:38,120
We on business now. 389
00:36:38,120 --> 00:36:44,500
And so I'm going to...This should also...You all sleeping. 390
00:36:44,500 --> 00:36:47,370
Why all of you are sleeping? My goodness. 391
00:36:47,370 --> 00:36:51,310
m x double dot, minus b x dot, minus mg kx/l, 392
00:36:51,310 --> 00:36:55,090
and then the minus and the minus becomes plus. 393
00:36:55,090 --> 00:36:58,220
Right? Try not to sleep. 394
00:36:58,220 --> 00:37:04,080
So x double dot, now we get plus γ times x dot, 395
00:37:04,080 --> 00:37:10,420
now we get plus ωo squared times because g / l is ωo squared, 396
00:37:10,420 --> 00:37:13,610
times x, so I divide the m out, 397
00:37:13,610 --> 00:37:24,030
and now I get equals ωo squared times ηo times cosωt. 398
00:37:24,030 --> 00:37:27,350
I will move this l up a teeny little bit, 399
00:37:27,350 --> 00:37:31,460
and I'm going to look now at that equation at the bottom. 400
00:37:31,460 --> 00:37:38,290
And I'm now overjoyed happiness, 401
00:37:38,290 --> 00:37:43,310
because this one looks almost like a carbon copy 402
00:37:43,310 --> 00:37:50,160
of the one that I had here with an Focosωt. 403
00:37:50,160 --> 00:37:59,380
And now instead of an Fo divided by m cosωt, I now have this. 404
00:37:59,380 --> 00:38:04,380
so this takes the place of my earlier Fo divided by m, 405
00:38:04,380 --> 00:38:07,980
Fo divided by m is an acceleration by the way. 406
00:38:07,980 --> 00:38:10,950
Now better be an acceleration because this is an acceleration, 407
00:38:10,950 --> 00:38:12,720
and apples have to be apples. 408
00:38:12,720 --> 00:38:15,400
so this is an acceleration, this is an acceleration, 409
00:38:15,400 --> 00:38:18,170
and this is also an acceleration, 410
00:38:18,170 --> 00:38:20,880
multiply ωo squared by distance, 411
00:38:20,880 --> 00:38:23,830
then you get distance divided by time squared. 412
00:38:23,830 --> 00:38:27,860
So you see now how the connection between the two go. 413
00:38:27,860 --> 00:38:33,320
Where originally I got an Fo/m times cosωt. 414
00:38:33,320 --> 00:38:35,610
Now because Walter Lewin's motion, 415
00:38:35,610 --> 00:38:40,890
I'm going to get an ωo squared times ηo. 416
00:38:40,890 --> 00:38:41,710
And so you see now 417
00:38:41,710 --> 00:38:49,500
how this motion of my hands indeed translate into a force on the object. 418
00:38:49,500 --> 00:38:52,600
Well, I have the solution. I don't have to do anything. 419
00:38:52,600 --> 00:38:57,500
All I have to do is change this by ωo squared times ηo, 420
00:38:57,500 --> 00:39:02,910
and I'm done. Differential equations are identical. 421
00:39:02,910 --> 00:39:06,300
I don't even have to change the tangent of δ. Nothing changes. 422
00:39:06,300 --> 00:39:08,940
This is the only thing that changes. 423
00:39:08,940 --> 00:39:15,780
So we are done. We can now make some predictions. 424
00:39:15,780 --> 00:39:19,990
The prediction is that if I'm going to shake this pendulum, 425
00:39:19,990 --> 00:39:25,570
and I'm going to do that very slowly, taking one hour to go to the left, 426
00:39:25,570 --> 00:39:29,240
and taking one hour to the right, and if my amplitude is ηo, 427
00:39:29,240 --> 00:39:34,810
what do you think that the amplitude A, 428
00:39:34,810 --> 00:39:37,760
the solution of my differential equation will be? 429
00:39:37,760 --> 00:39:40,920
Another words, I'm going to shake very slowly. 430
00:39:40,920 --> 00:39:45,410
What do you think A will be without looking at the differential equation? 431
00:39:45,410 --> 00:39:47,130
So I just go with my hands like this. 432
00:39:47,130 --> 00:39:52,760
Amplitude ηo, amplitude ηo, and I do it very slowly. 433
00:39:52,760 --> 00:39:57,890
What will be the amplitude of A? ηo. 434
00:39:57,890 --> 00:40:02,250
So you expect that this goes to ηo. 435
00:40:02,250 --> 00:40:05,120
If you don't believe it, go to this equation, 436
00:40:05,120 --> 00:40:08,210
substitute in here 0, in here 0, you get ωo squared, 437
00:40:08,210 --> 00:40:11,500
each of these ωo squared, you see ηo. 438
00:40:11,500 --> 00:40:14,060
Exactly what that equation predicts, 439
00:40:14,060 --> 00:40:16,810
but your common sense says the same thing. 440
00:40:16,810 --> 00:40:18,560
Now what do you think δ is 441
00:40:18,560 --> 00:40:23,030
if I'm going to move this pendulum very slowly to the left and to the right. 442
00:40:23,030 --> 00:40:25,940
Of course. Of course your object will follow me. 443
00:40:25,940 --> 00:40:28,770
We will be ridiculous if I take one week to go from here to here, 444
00:40:28,770 --> 00:40:32,510
the object will be there, right? Obviously the object is always here. 445
00:40:32,510 --> 00:40:36,350
So you also predict that δ is 0. 446
00:40:36,350 --> 00:40:41,670
And so now we can make a quick prediction that at resonance 447
00:40:41,670 --> 00:40:47,570
you probably get Q times ηo, and then δ will become π/2. 448
00:40:47,570 --> 00:40:52,850
And when you go to very high frequency, then A will go to zero, 449
00:40:52,850 --> 00:41:00,130
and then δ will go to pi. And this is what I want to demonstrate to you. 450
00:41:00,130 --> 00:41:06,100
So the final solution of this pendulum, which I will write down in red, 451
00:41:06,100 --> 00:41:13,250
is going to be that x equals A times cos of ωt minus δ, 452
00:41:13,250 --> 00:41:15,980
just as we had before A is none negotiable, 453
00:41:15,980 --> 00:41:17,940
has nothing to do with the initial conditions. 454
00:41:17,940 --> 00:41:21,820
δ is none negotiable, has nothing to do with initial conditions. 455
00:41:21,820 --> 00:41:26,770
This is the steady state solution. 456
00:41:26,770 --> 00:41:35,350
Alright, let me take my shoes off because then you can see it better. 457
00:41:35,350 --> 00:41:43,840
Alright. Here is a pendulum. 458
00:41:43,840 --> 00:41:46,790
It could be very exciting, I'm going to tell you. 459
00:41:46,790 --> 00:41:53,670
I'm going to move this with ω very close to 0. Very exciting. 460
00:41:53,670 --> 00:42:04,050
I'm doing it right now. Aren't you thrilled? No, you are not thrilled. 461
00:42:04,050 --> 00:42:11,860
But I'm moving and now I'm going to go back. 462
00:42:11,860 --> 00:42:15,190
Do you agree that A the amplitude of that object 463
00:42:15,190 --> 00:42:18,640
is exactly the same as the amplitude of my hand? 464
00:42:18,640 --> 00:42:21,230
Do you agree? Do you see that? That is why that A is ηo, 465
00:42:21,230 --> 00:42:24,960
and that follows from that rather complicated equation. 466
00:42:24,960 --> 00:42:27,670
Did you see that δ was 0? 467
00:42:27,670 --> 00:42:32,580
Did you see that we went hand in hand, so to speak no pun implied. 468
00:42:32,580 --> 00:42:34,310
We're going hand in hand, right? 469
00:42:34,310 --> 00:42:38,320
That one follows exactly my hands, for δ is 0. 470
00:42:38,320 --> 00:42:43,860
Let's now go to high frequency, very high frequency, way above resonance. 471
00:42:43,860 --> 00:42:47,250
And what you see now is that the object is not moving very much, 472
00:42:47,250 --> 00:42:48,690
but if you look very carefully, 473
00:42:48,690 --> 00:42:53,300
you'll see when my hand is here, the object tends to go there. 474
00:42:53,300 --> 00:42:55,130
And when my hand is here, the object tends to go there. 475
00:42:55,130 --> 00:43:04,800
That is that pi. Ready? You see that there is almost no motion, 476
00:43:04,800 --> 00:43:05,980
A is near zero, 477
00:43:05,980 --> 00:43:09,030
but can you really see that 478
00:43:09,030 --> 00:43:12,610
the phase differences by... can you that 180 degrees? 479
00:43:12,610 --> 00:43:15,740
You see if the A is exactly zero, of course that you can not tell, 480
00:43:15,740 --> 00:43:18,030
so I'm trying not to go infinitely fast, 481
00:43:18,030 --> 00:43:21,740
I go a little slower than infinitely fast. 482
00:43:21,740 --> 00:43:25,710
Can you see it? Ok, now comes the resonance. 483
00:43:25,710 --> 00:43:30,110
And now it will be very difficult to see this π/2. 484
00:43:30,110 --> 00:43:31,010
That's almost impossible. 485
00:43:31,010 --> 00:43:33,090
That's not my objective 486
00:43:33,090 --> 00:43:39,660
but my objective is to show you that enormously small, very small ηo here 487
00:43:39,660 --> 00:43:43,730
will give an amplitude there which is Q times higher. 488
00:43:43,730 --> 00:43:45,700
So you get a huge swing 489
00:43:45,700 --> 00:43:49,800
when my hand is hardly moving at all. That's the power of Q. 490
00:43:49,800 --> 00:43:54,710
There we go. First get it into it. There it is. 491
00:43:54,710 --> 00:43:57,810
Now this is resonance. Would you agree this is resonance? 492
00:43:57,810 --> 00:44:01,460
Now look at my hands. My hand is moving 493
00:44:01,460 --> 00:44:07,070
probably no more than with an amplitude of 3mm. No more. 494
00:44:07,070 --> 00:44:10,960
And yet I see an amplitude there of 60cm 495
00:44:10,960 --> 00:44:12,480
that would mean that very roughly 496
00:44:12,480 --> 00:44:19,760
this pendulum has a Q of 200, namely 60cm divided by 3mm. 497
00:44:19,760 --> 00:44:23,310
So this is even a way to make an extremely rough guess, 498
00:44:23,310 --> 00:44:26,450
admittedly very rough, of the Q value. 499
00:44:26,450 --> 00:44:28,260
You can not even see my hand move, 500
00:44:28,260 --> 00:44:31,620
we be honest. You can't even see my hand move 501
00:44:31,620 --> 00:44:34,330
but I know I am moving it a little. 502
00:44:34,330 --> 00:44:40,780
Oh...you're lying...No... you were not...No, you were not. 503
00:44:40,780 --> 00:44:45,570
Ok, so you see all the goodies that we have calculated actually 504
00:44:45,570 --> 00:44:51,300
can be demonstrated and show up quite dramatically. 505
00:44:51,300 --> 00:44:57,030
Suppose I have a spring system like this 506
00:44:57,030 --> 00:45:02,210
and I want the force on that object here. 507
00:45:02,210 --> 00:45:05,060
Well, what I can do is just to shake it here 508
00:45:05,060 --> 00:45:09,560
in the way extremely similar to what I did there. 509
00:45:09,560 --> 00:45:14,060
And when I shake it there, we can make certain predictions. 510
00:45:14,060 --> 00:45:18,830
We can make predictions now based on the knowledge that we have. 511
00:45:18,830 --> 00:45:24,800
Suppose I shake it with an amplitude ηo, 512
00:45:24,800 --> 00:45:27,280
no differential equations...no...nothing for now. 513
00:45:27,280 --> 00:45:31,800
But I know that somehow it will come out in terms of the force at the object. 514
00:45:31,800 --> 00:45:34,590
So I know that when I write down the differential equation, 515
00:45:34,590 --> 00:45:36,440
of course it shows up exactly this way, 516
00:45:36,440 --> 00:45:42,750
I get an ωo squared times ηo, except the ωo squared is now k over m. 517
00:45:42,750 --> 00:45:46,210
So what do you think if I shake it as ω equal 0, 518
00:45:46,210 --> 00:45:51,740
what is then the amplitude that this object will have 519
00:45:51,740 --> 00:45:56,590
relative to my motion ηo? 520
00:45:56,590 --> 00:46:03,360
I move my hands. ηo infinitely long. What will this object do? 521
00:46:03,360 --> 00:46:10,090
We just follow it. So you get this answer. What will be the δ? 522
00:46:10,090 --> 00:46:11,340
It will be 0. 523
00:46:11,340 --> 00:46:15,690
When I had the resonance, what will be the amplitude of that object 524
00:46:15,690 --> 00:46:19,620
hanging from the spring? Will be q times higher than my ηo. 525
00:46:19,620 --> 00:46:22,170
What will be the phase difference? 90 degrees. 526
00:46:22,170 --> 00:46:25,520
When I shake like crazy, A will go to 0. 527
00:46:25,520 --> 00:46:27,960
So with this spring, 528
00:46:27,960 --> 00:46:31,040
if you shake it like this, which is part of your problem set, 529
00:46:31,040 --> 00:46:36,480
you will see exactly the same result that we have there done for a pendulum. 530
00:46:36,480 --> 00:46:44,150
And this now, I want to independently demonstrate to you. 531
00:46:44,150 --> 00:46:46,780
I have here an air track. 532
00:46:46,780 --> 00:46:50,430
I can blow out air, so that the object here starts floating, 533
00:46:50,430 --> 00:46:55,260
so we can make the damping very small by making it float. 534
00:46:55,260 --> 00:47:00,930
But if we lower the air flow, the damping becomes a little higher. 535
00:47:00,930 --> 00:47:03,670
I have a spring here with spring constant k 536
00:47:03,670 --> 00:47:07,390
and I have another spring here with spring constant k. 537
00:47:07,390 --> 00:47:12,350
They both have spring constant k. 538
00:47:12,350 --> 00:47:14,880
And now I'm going to drive this here 539
00:47:14,880 --> 00:47:28,490
at an extremely low frequency over a distance ηo at maximum. ηocosωt. 540
00:47:28,490 --> 00:47:31,600
What do you think the amplitude of this object will be? 541
00:47:31,600 --> 00:47:42,100
At that very low...Yeah....very very good...very good 542
00:47:42,100 --> 00:47:46,740
Not ηo, but why is it half? Because we have 2 springs. 543
00:47:46,740 --> 00:47:50,840
So effectively the spring constant is twice that. Exactly. 544
00:47:50,840 --> 00:47:57,780
So if I go very slowly, you will see that this displacement here 545
00:47:57,780 --> 00:48:00,510
will be twice as high as this displacement. 546
00:48:00,510 --> 00:48:02,570
But what I really want to show you is 547
00:48:02,570 --> 00:48:04,510
they are in phase. 548
00:48:04,510 --> 00:48:09,280
This one will go to the right when this one goes to the right. 549
00:48:09,280 --> 00:48:14,720
Now comes the catch. I showed you earlier that this is steady state solution. 550
00:48:14,720 --> 00:48:21,380
In the beginning, the system doesn't like me. It hates me. It fights me. 551
00:48:21,380 --> 00:48:23,020
It doesn't like that ω. 552
00:48:23,020 --> 00:48:27,850
It wants to do something different which is part of next week's lecture. 553
00:48:27,850 --> 00:48:30,580
And you will see that in the beginning 554
00:48:30,580 --> 00:48:35,820
and so we have to be a little patient before my will survives. 555
00:48:35,820 --> 00:48:37,070
Ready for that? 556
00:48:37,070 --> 00:48:40,060
So I am going to start now to drive the system 557
00:48:40,060 --> 00:48:43,060
at the frequency which is below resonance. 558
00:48:43,060 --> 00:48:45,910
I want you to see two things that they go hand in hand, 559
00:48:45,910 --> 00:48:49,180
and I want you to see that the... 560
00:48:49,180 --> 00:48:53,370
you're going to very low frequency. 561
00:48:53,370 --> 00:48:54,500
I will give you the amplitude....here, 562
00:48:54,500 --> 00:48:56,860
This is twice the amplitude of the driver. 563
00:48:56,860 --> 00:49:00,610
Now it is here the spring and now the spring is here. 564
00:49:00,610 --> 00:49:04,020
So it's only... this much is 2ηo. 565
00:49:04,020 --> 00:49:07,340
So ηo is no more than 3 quarter of an inch. 566
00:49:07,340 --> 00:49:17,770
And now we are going to let that object be exposed to this driver, 567
00:49:17,770 --> 00:49:24,330
and we will give it a little bit of time to recognize me. 568
00:49:24,330 --> 00:49:29,440
It takes a little bit of time to reach the steady state solution. 569
00:49:29,440 --> 00:49:35,050
And next time we will learn how much time it actually takes. 570
00:49:35,050 --> 00:49:38,590
So if you wanna be a little bit patient, 571
00:49:38,590 --> 00:49:49,030
and you will see. If we give it too much damping, 572
00:49:49,030 --> 00:49:59,030
too little air, then of course it starts to get stuck. 573
00:49:59,030 --> 00:50:02,980
Yaaa, we close. We are close, for me, close enough. 574
00:50:02,980 --> 00:50:06,680
Now look at it. They are going both to the left for me. 575
00:50:06,680 --> 00:50:08,800
Both to the right for me. 576
00:50:08,800 --> 00:50:11,490
For you, they are going now both to the right, 577
00:50:11,490 --> 00:50:13,670
and going both to the left, 578
00:50:13,670 --> 00:50:17,080
going both to the right, and both to the 579
00:50:17,080 --> 00:50:19,280
er er, and both to the left. 580
00:50:19,280 --> 00:50:22,330
Now this was the amplitude, 581
00:50:22,330 --> 00:50:24,740
twice the amplitude of the driver. 582
00:50:24,740 --> 00:50:27,440
And when you look carefully here, it's less. 583
00:50:27,440 --> 00:50:32,170
This is that ηo zero over two, that this gentleman immediately notice, 584
00:50:32,170 --> 00:50:34,930
because we have two springs. 585
00:50:34,930 --> 00:50:40,060
So you see here, apart from the factor of two, you see the delta zero 586
00:50:40,060 --> 00:50:44,980
and you see that the amplitude indeed is half of the amplitude of the driver 587
00:50:44,980 --> 00:50:46,940
because of the two springs. 588
00:50:46,940 --> 00:50:59,040
Now we are going to resonance ωo and now nasty things may happen. 589
00:51:01,960 --> 00:51:11,820
It may break. We have to give it time. You see what funny things it's doing. 590
00:51:11,820 --> 00:51:18,350
Not the steady state yet, have to wait. 591
00:51:22,450 --> 00:51:24,940
Just a little patient. 592
00:51:40,940 --> 00:51:43,920
Give it more time. 593
00:51:53,840 --> 00:51:57,360
Now it is also that the... remember this is only moving this much. 594
00:51:57,360 --> 00:52:00,880
Look how much this is moving. 595
00:52:03,000 --> 00:52:06,090
I may even be exactly at resonance. 596
00:52:06,090 --> 00:52:18,920
We can only do the best we can here, may not be exactly at resonance. 597
00:52:20,630 --> 00:52:22,650
Oh, boy. Close to resonance now. 598
00:52:22,650 --> 00:52:28,030
Oh, yeah! Oh, ah, look at that! 599
00:52:28,030 --> 00:52:30,840
Oh! Am I at resonance? 600
00:52:30,840 --> 00:52:32,490
I think I got it, then you see? 601
00:52:32,490 --> 00:52:34,920
They're neither in phase nor out of phase. 602
00:52:34,920 --> 00:52:36,600
Now you see the 90 degrees. 603
00:52:36,600 --> 00:52:39,260
Look at this teeny weeny little displacement here 604
00:52:39,260 --> 00:52:41,720
and look what this man is doing. 605
00:52:41,720 --> 00:52:49,840
That is resonance. Markable! Ah! Marcos, where's Marcos? 606
00:52:49,840 --> 00:52:55,340
We hit it. Right on! 607
00:52:55,340 --> 00:53:00,530
Now I will oscillate it way over resonance. 608
00:53:00,530 --> 00:53:04,010
Not way but over resonance. 609
00:53:04,010 --> 00:53:08,500
First, but first have the system first come down. 610
00:53:08,500 --> 00:53:11,400
Now I will change the frequency above resonance 611
00:53:11,400 --> 00:53:14,670
so that now you will see the phenomenon that 612
00:53:14,670 --> 00:53:21,280
I discussed earlier that the amplitude is very small. 613
00:53:21,280 --> 00:53:24,260
Again we have to wait a little, look how fast it is going 614
00:53:24,260 --> 00:53:36,080
and that they will go 180 degrees out of phase. 615
00:53:36,080 --> 00:53:40,850
Now look, this is going this much. Huh? Back and forth. 616
00:53:40,850 --> 00:53:45,820
This one not doing very much. 617
00:53:45,820 --> 00:53:49,060
Now you can see it I can. It float like this 618
00:53:49,060 --> 00:53:50,840
chi chi chi chi... Can you see it? 619
00:53:50,840 --> 00:53:53,580
That is 180 degrees out of phases. 620
00:53:53,580 --> 00:53:58,510
Very clear. 5 minutes break. 621
00:53:58,510 --> 00:53:59,960
See you back here in exactly 5 minutes. 622
00:54:03,360 --> 00:54:07,170
Now so we have discussed today some simple systems: 623
00:54:07,170 --> 00:54:13,930
pendulum one object, springs one object, one resonance frequency. 624
00:54:13,930 --> 00:54:16,010
But soon in 803 625
00:54:16,010 --> 00:54:20,590
we will discuss systems with more than one objects. 626
00:54:20,590 --> 00:54:23,360
For instance, if I put three cars on here 627
00:54:23,360 --> 00:54:27,600
with four springs three resonance frequencies. 628
00:54:27,600 --> 00:54:31,050
If I have a triple pendulum which I'll demonstrate next week. 629
00:54:31,050 --> 00:54:33,750
One pendulum with all the other below, 630
00:54:33,750 --> 00:54:35,880
the other three resonance frequencies. 631
00:54:35,880 --> 00:54:39,840
Five cars on there, five resonance frequencies. 632
00:54:39,840 --> 00:54:45,910
So simple objects like a dinner plate or just a regular glass 633
00:54:45,910 --> 00:54:49,070
has enormous number of resonance frequencies. 634
00:54:49,070 --> 00:54:56,240
It can oscillate in many many different ways. 635
00:54:56,240 --> 00:55:00,320
If you drive your car, your wheels turn around. 636
00:55:00,320 --> 00:55:05,450
That's certain oscillation, a certain period underlying. 637
00:55:05,450 --> 00:55:06,500
And you may notice that 638
00:55:06,500 --> 00:55:09,560
at certain speed that something in your car begins to rattle. 639
00:55:09,560 --> 00:55:10,900
They're annoying. 640
00:55:10,900 --> 00:55:14,130
All you have to do is go a little slower 641
00:55:14,130 --> 00:55:16,090
or go a little faster and it stops. 642
00:55:16,090 --> 00:55:19,950
You go off resonance for that object. 643
00:55:19,950 --> 00:55:23,320
Now you may go on resonance for another object, of course. 644
00:55:23,320 --> 00:55:29,250
And some cars rattle at any speed. 645
00:55:29,250 --> 00:55:30,820
You have a radiator in your room 646
00:55:30,820 --> 00:55:34,660
which rotates that is also underlying oscillation and a period. 647
00:55:34,660 --> 00:55:40,100
That may start to cause resonance in the frame 648
00:55:40,100 --> 00:55:42,700
you may hear some awful noise sometimes. 649
00:55:42,700 --> 00:55:46,970
Unfortuntely these fans you cannot change this speed so easily, 650
00:55:46,970 --> 00:55:50,110
but you can go from state three to two to one 651
00:55:50,110 --> 00:55:52,820
and then this terrible noise will go away. 652
00:55:52,820 --> 00:55:55,330
You take a washing-machine or dryer. 653
00:55:55,330 --> 00:55:58,370
I used to remember a friend of mine in the Netherlands had a dryer 654
00:55:58,370 --> 00:56:00,870
and when he started the dryer, 655
00:56:00,870 --> 00:56:03,550
at very early phase when it works at the certain frequency 656
00:56:03,550 --> 00:56:06,800
the whole dryer would start to walk through the room. 657
00:56:06,800 --> 00:56:08,090
It walked. 658
00:56:08,090 --> 00:56:11,170
And then at higher frequency, it of course would stop. 659
00:56:11,170 --> 00:56:14,280
Resonance. Resonances are everywhere 660
00:56:14,280 --> 00:56:18,530
and they often occur when you don't expect them. 661
00:56:18,530 --> 00:56:20,530
You open a faucet. 662
00:56:20,530 --> 00:56:23,680
You think it is a steady stream of water which I am sure it is. 663
00:56:23,680 --> 00:56:24,910
But sometimes you hear, 664
00:56:24,910 --> 00:56:28,760
ahhhhh, an unbelievable sound that drives you almost nuts. 665
00:56:28,760 --> 00:56:30,640
I am sure all of you have heard that sometimes. 666
00:56:30,640 --> 00:56:33,700
If it isn't in dormitory may be at the hotels or at the home. 667
00:56:33,700 --> 00:56:36,850
All you have to do is open the faucet a little more or a little less, 668
00:56:36,850 --> 00:56:40,440
and it goes away. 669
00:56:40,440 --> 00:56:44,470
And it is really extremely loud at an annoying resonance. 670
00:56:44,470 --> 00:56:47,090
If you take something as simple as a wine glass 671
00:56:47,090 --> 00:56:50,630
which has a tremendous number of resonances, 672
00:56:50,630 --> 00:56:56,240
then I can make you listen to a well-known resonance 673
00:56:56,240 --> 00:56:59,500
which is by rubbing the rim of glass. 674
00:56:59,500 --> 00:57:00,900
When I rub the rim of the glass, 675
00:57:00,900 --> 00:57:04,700
I am not exciting it at one particular frequency. 676
00:57:04,700 --> 00:57:06,740
Sure we know that resonance frequency. 677
00:57:06,740 --> 00:57:09,760
I am exciting it at lots and lots of frequencies, 678
00:57:09,760 --> 00:57:13,240
I dump on it the whole spectrum of frequencies. 679
00:57:13,240 --> 00:57:14,880
But the glass is mean. 680
00:57:14,880 --> 00:57:18,080
It just picks out the one which is its resonance. 681
00:57:18,080 --> 00:57:22,260
That's where builds up a large value for A. It ignores all the others. 682
00:57:22,260 --> 00:57:27,360
And that's why I can make it resonate at that particular frequency. 683
00:57:27,360 --> 00:57:31,620
Listen to it. 684
00:57:34,640 --> 00:57:36,930
This is not one frequency what I am doing 685
00:57:36,930 --> 00:57:39,220
and has nothing to do with the time that for me to go around. 686
00:57:39,220 --> 00:57:44,480
It's a very high pitch about 420 Hz. 687
00:57:44,480 --> 00:57:49,430
So the rubbing is like dumping a spectrum of frequencies on it 688
00:57:49,430 --> 00:57:53,640
and it selects what it likes the most. 689
00:57:53,640 --> 00:57:59,050
When I was a student I remember we often had a after-dinner speaker. 690
00:57:59,050 --> 00:58:03,260
We had dinners at. For alternative, we don't have after-dinner speaker. 691
00:58:03,260 --> 00:58:06,470
And more often than not we didn't like the after-dinner speaker. 692
00:58:06,470 --> 00:58:08,510
We didn't like the speech. 693
00:58:08,510 --> 00:58:10,070
and so we make that very clear 694
00:58:10,070 --> 00:58:14,730
and the way we did that is all our wine glasses. 695
00:58:14,730 --> 00:58:18,280
An enormous sound in that dining-hall 696
00:58:18,280 --> 00:58:22,480
and the speaker very quickly got the message, of course. 697
00:58:22,480 --> 00:58:24,960
That's the enormous sound you can generate. 698
00:58:24,960 --> 00:58:28,280
And most of these wine glasses were roughly the same. 699
00:58:28,280 --> 00:58:32,910
So it was always a tone that was loud 700
00:58:32,910 --> 00:58:37,050
and clear and almost one frequency. 701
00:58:37,050 --> 00:58:42,310
You've seen Fuddys(?) lately of the storms, 3 storms in a row. 702
00:58:42,310 --> 00:58:45,240
And you must remember sometimes that you saw traffic sign. 703
00:58:45,240 --> 00:58:47,650
Here a pole then the traffic sign 704
00:58:47,650 --> 00:58:51,590
and then even though some kind of a crazy wind going, 705
00:58:51,590 --> 00:58:53,080
the traffic sigh goes like this 706
00:58:53,080 --> 00:58:58,600
"chi", "piii". All resonance frequencies. 707
00:58:58,600 --> 00:59:00,330
It can't even break. 708
00:59:00,330 --> 00:59:04,770
Even though the wind appears to be relatively steady. 709
00:59:04,770 --> 00:59:09,480
The wind then generates in a way a whole spectrum of frequencies 710
00:59:09,480 --> 00:59:13,160
and this traffic sigh picks out the one that it likes the most. 711
00:59:13,160 --> 00:59:16,470
And then it goes nuts at a frequency. 712
00:59:16,470 --> 00:59:17,550
That is a resonance frequency. 713
00:59:17,550 --> 00:59:21,080
And resonances can become destructive, of course 714
00:59:21,080 --> 00:59:23,590
if these amplitudes are too high, 715
00:59:23,590 --> 00:59:26,830
then things can break down you may have noticed I was worried here 716
00:59:26,830 --> 00:59:29,390
when the amplitude became so large that 717
00:59:29,390 --> 00:59:35,670
this spring might even break or the car might jump off that track. 718
00:59:35,670 --> 00:59:40,870
And of course, a classic example of this destructive resonance is 719
00:59:40,870 --> 00:59:46,210
the destruction of a very famous bridge in this country in 1940. 720
00:59:46,210 --> 00:59:52,340
Then the Tacoma Narrow's Bridge, washington state, was destroyed by wind. 721
00:59:52,340 --> 00:59:55,590
And this bridge has many different resonance frequencies, 722
00:59:55,590 --> 00:59:58,080
one like this and some like this, 723
00:59:58,080 --> 01:00:01,340
depending upon the wind's ranks, different resonances 724
01:00:01,340 --> 01:00:04,520
were excited at different moments in time. 725
01:00:04,520 --> 01:00:07,250
But it ultimately led to the destruction of the bridge. 726
01:00:07,250 --> 01:00:11,840
Now most of you have seen this movie, but just for the few who haven't, 727
01:00:11,840 --> 01:00:13,610
I really want you to see this movie. 728
01:00:13,610 --> 01:00:16,050
You cannot go through MIT 729
01:00:16,050 --> 01:00:22,660
and not having seen the destruction of the Tacoma Bridge movie. 730
01:00:29,370 --> 01:00:31,400
There are countries 731
01:00:31,400 --> 01:00:35,980
where soldiers are not allowed to cross a bridge 732
01:00:35,980 --> 01:00:39,410
when they are marching in step. 733
01:00:39,410 --> 01:00:41,910
That's the case in the Netherlands, my home country. 734
01:00:41,910 --> 01:00:44,650
It's also the case in many European countries. 735
01:00:44,650 --> 01:00:48,430
The story has it as in England, I think, more than 100 years ago 736
01:00:48,430 --> 01:00:51,360
when soldiers went over the bridge in step, 737
01:00:51,360 --> 01:00:54,490
that the bridge collapsed. 738
01:00:54,490 --> 01:00:56,190
Whether that was the result of the soiders 739
01:00:56,190 --> 01:01:00,400
we'll never know, but in any case from those they scan the order, 740
01:01:00,400 --> 01:01:07,610
that soldiers have to go out of step before they cross the bridge. 741
01:01:07,610 --> 01:01:11,780
There is a rumor that most of you have heard that 742
01:01:11,780 --> 01:01:16,560
there are women who are capable of singing 743
01:01:16,560 --> 01:01:21,390
with such a loud voice they can break a wine glass. 744
01:01:21,390 --> 01:01:23,520
They have to tone exactly at the frequency, 745
01:01:23,520 --> 01:01:26,130
the resonance frequency of the wine glass, 746
01:01:26,130 --> 01:01:33,760
and they come out huge volume and the glass breaks. 747
01:01:33,760 --> 01:01:38,440
I don't believe it, but it's a rumor. 748
01:01:38,440 --> 01:01:41,600
There was a commercial many years ago, 749
01:01:41,600 --> 01:01:43,510
some of you may never have seen it, 750
01:01:43,510 --> 01:01:46,150
for Memorax (brand name). 751
01:01:46,150 --> 01:01:48,570
Memorax was a tape, 752
01:01:48,570 --> 01:01:52,440
a special tape for audio tape recorders. 753
01:01:52,440 --> 01:01:55,170
This guy is going to a concert 754
01:01:55,170 --> 01:01:58,550
and there is a woman singing loud voice, 755
01:01:58,550 --> 01:02:04,920
uniform frequency. Bingo! Glass breaks. 756
01:02:04,920 --> 01:02:08,270
He comes home and he tells his wife this story. 757
01:02:08,270 --> 01:02:11,630
Well, his wife was smart enough of course not to believe it. 758
01:02:11,630 --> 01:02:19,100
And he just wanna and just who happens that I recorded it on my Memorax tape. 759
01:02:19,100 --> 01:02:22,490
Well, so he plays the tape at home, 760
01:02:22,490 --> 01:02:26,930
and that moment that the woman's voice goes, what happens, 761
01:02:26,930 --> 01:02:31,970
the glasses break at home in his cabinet. 762
01:02:31,970 --> 01:02:33,820
So much for the physics of Memorax 763
01:02:33,820 --> 01:02:37,450
because you can imagine that the resonance frequency of the glasses at home 764
01:02:37,450 --> 01:02:40,600
were very different than the glasses in the orchestra. 765
01:02:40,600 --> 01:02:44,640
So it is all the swindle but that's, of course, what commercials are all about. 766
01:02:44,640 --> 01:02:48,420
What's now the bottom line? 767
01:02:48,420 --> 01:02:54,180
The bottom line was that if you buy Memorax, these tapes, 768
01:02:54,180 --> 01:02:59,170
then the reproduction is so perfect that 769
01:02:59,170 --> 01:03:07,260
you can even take it home and you see the glasses break. 770
01:03:07,260 --> 01:03:12,980
This brings up now the 64 million dollar question 771
01:03:12,980 --> 01:03:19,040
and that is: can it be done or can it not be done? 772
01:03:19,040 --> 01:03:23,100
So any women in my audience who want to give it a try? 773
01:03:23,100 --> 01:03:26,270
That would be wonderful. 774
01:03:26,270 --> 01:03:28,400
We will ask ourselves that question: 775
01:03:28,400 --> 01:03:31,180
Can this be done or can this not be done by a person? 776
01:03:31,180 --> 01:03:33,560
And I think we came to the conclusion that 777
01:03:33,560 --> 01:03:38,380
the person alone without all kinds of electronic equipments could not do that. 778
01:03:38,380 --> 01:03:40,850
And I still believe that today. 779
01:03:40,850 --> 01:03:44,920
But Prof. Felts, Michael Felts here in MIT, 780
01:03:44,920 --> 01:03:48,400
with one of his graduate students many years ago, 781
01:03:48,400 --> 01:03:52,760
developed some powerful equipment which you see here, 782
01:03:52,760 --> 01:03:58,890
which was designed to make an attempt to break a wine glass. 783
01:03:58,890 --> 01:04:02,340
It doesn't always work, but it works often. 784
01:04:02,340 --> 01:04:05,120
The idea being then that here's the wine glass. 785
01:04:05,120 --> 01:04:08,680
It almost a carbon copy of this one. 786
01:04:08,680 --> 01:04:14,890
When they finally make this do work which near 440Hz. 787
01:04:14,890 --> 01:04:16,070
They went to create a barrow 788
01:04:16,070 --> 01:04:19,660
and they asked how many of these wine glasses do you have? 789
01:04:19,660 --> 01:04:27,040
They said we have 5, 000 and they bought them all 5, 000. 790
01:04:27,040 --> 01:04:29,570
Because it's not obvious if you have to check the glass 791
01:04:29,570 --> 01:04:31,560
that it will ever work again. 792
01:04:31,560 --> 01:04:35,230
And to here with one of those glasses, 793
01:04:35,230 --> 01:04:40,620
here is a loud speaker, the sound comes from this side. 794
01:04:40,620 --> 01:04:43,400
And what is nice about this arrangement? 795
01:04:43,400 --> 01:04:51,820
That we can make you see the distortion of the glass in this resonance mode. 796
01:04:51,820 --> 01:04:54,570
The glass oscillates like this 797
01:04:54,570 --> 01:04:57,670
coz from oval to circular to oval to circular. 798
01:04:57,670 --> 01:04:59,220
And the way we can make you see that 799
01:04:59,220 --> 01:05:04,180
is by strobing the glass at the frequency 800
01:05:04,180 --> 01:05:08,780
which is a little bit different from the frequency of the sound. 801
01:05:08,780 --> 01:05:10,570
Think about it. 802
01:05:10,570 --> 01:05:14,460
Two frequencies almost the same give you a beat phenomenon. 803
01:05:14,460 --> 01:05:16,130
So what that comes down to it? 804
01:05:16,130 --> 01:05:19,670
You see the motion of the glass very slowly. 805
01:05:19,670 --> 01:05:22,170
And then if we hit that resonance frequency, 806
01:05:22,170 --> 01:05:28,540
we will increase the volume, and then maybe it will break. 807
01:05:28,540 --> 01:05:35,560
Now I have to warn you that the sound will be unbearably high. 808
01:05:35,560 --> 01:05:38,680
And so for some of you here in the front row, 809
01:05:38,680 --> 01:05:40,540
you may even want to cover your ears 810
01:05:40,540 --> 01:05:42,040
or you may want to move back. 811
01:05:42,040 --> 01:05:43,420
It is up to you. 812
01:05:43,420 --> 01:05:44,700
But be careful. 813
01:05:44,700 --> 01:05:48,200
This is really an enormously strong signal that they are going to here. 814
01:05:48,200 --> 01:05:50,620
So all be careful. You can move back if you want to. 815
01:05:50,620 --> 01:05:54,960
I have a hearing aids as you have noticed 816
01:05:54,960 --> 01:06:00,310
and I have the option that I can turn them off. 817
01:06:00,310 --> 01:06:03,620
But in spite of that, I am not deaf without them. 818
01:06:03,620 --> 01:06:06,710
I will still cover my ears. 819
01:06:06,710 --> 01:06:12,920
So let me first give you the light setting that we want. 820
01:06:12,920 --> 01:06:22,780
So we make it a little dark and then I will show you the glass. 821
01:06:22,780 --> 01:06:28,410
Oh, I have to change the setting here. 822
01:06:28,410 --> 01:06:32,470
There is the glass. There is no sound. Not oscillating. 823
01:06:32,470 --> 01:06:39,900
Now I turn on the speaker. 824
01:06:39,900 --> 01:06:48,950
So I am now going to turn my hearing aids off and put this on. 825
01:06:48,950 --> 01:06:52,650
And I am going to increase the volume. 826
01:06:52,650 --> 01:06:55,060
And if you doesn't want to break, 827
01:06:55,060 --> 01:06:58,010
I will trend the sound frequency a little bit 828
01:06:58,010 --> 01:07:03,000
to sweep over that resonance curve so that I get onto the maximum. 829
01:07:03,000 --> 01:07:07,250
Because it is only of the Q is so high of this system. 830
01:07:07,250 --> 01:07:09,920
That if I am a little bit off the frequency, 831
01:07:09,920 --> 01:07:14,250
then the amplitude will be low. 832
01:07:14,250 --> 01:07:25,390
For that, let's first see what happens when I increase the volume. 833
01:07:26,350 --> 01:07:29,030
Did I do something wrong? 834
01:07:29,030 --> 01:07:30,400
Yes, I changed the frequency. 835
01:09:03,660 --> 01:09:11,580
I think I have to convince you that a woman cannot do this. 836
01:09:11,580 --> 01:09:16,980
I have some last words of wisdom for you and that is 837
01:09:16,980 --> 01:09:19,930
falling in love is also a form of resonance 838
01:09:19,930 --> 01:09:25,770
and it too can be destructive because it can break you heart. 839
01:09:25,770 --> 01:09:29,360
So try to remember that next time. 840
01:09:29,360 --> 01:09:31,830
Have a good weekend.
| Name | Version | Size | Date | User |
| 03_c.srt | 1 | 76065 | 2/17/06 4:16 AM | OOPSSJTU |
Last Modified 2/21/06 8:35 AM
|
