1 00:00:00,900 --> 00:00:05,200 So what we ended up is with a vector ... 2 00:00:05,650 --> 00:00:08,650 ... representing geometric property of ... 3 00:00:08,750 --> 00:00:12,700 ... a point of the trajectory. And ... 4 00:00:12,900 --> 00:00:17,000 ... a number positive or negative representing ... 5 00:00:18,100 --> 00:00:21,100 ... the same or different -- well some property ... 6 00:00:21,200 --> 00:00:23,800 ... of the same curve at the same point. 7 00:00:24,500 --> 00:00:27,600 And well the natural curiosity ... 8 00:00:27,850 --> 00:00:32,100 ... suggests that we should relate those things. 9 00:00:32,500 --> 00:00:36,600 And we especially should relate those things to the ... 10 00:00:37,450 --> 00:00:41,500 ... very simple case of the trajectory being a circle. 11 00:00:42,950 --> 00:00:47,500 Because what we ultimately want is the relation between ... 12 00:00:47,900 --> 00:00:49,600 ... this Physics ... 13 00:00:50,300 --> 00:00:56,000 ... idea of orthogonal component of acceleration and the idea of the radius of a circle. 14 00:00:56,900 --> 00:00:59,400 So to do that I will ... 15 00:01:00,700 --> 00:01:03,000 ... look at a circle. 16 00:01:07,300 --> 00:01:09,500 Let's look at a circle ... 17 00:01:10,700 --> 00:01:12,500 ... of radius ... 18 00:01:15,100 --> 00:01:16,100 ... R. 19 00:01:18,800 --> 00:01:20,900 And er ... 20 00:01:21,000 --> 00:01:24,000 I will think about a particle moving along the circle. 21 00:01:24,550 --> 00:01:29,600 And the simplest way to let a particle move along a circle would be to ... 22 00:01:33,000 --> 00:01:39,000 ... let it move [...] be parametrized using cosine and sine. 23 00:01:39,100 --> 00:01:45,700 R times cosine of t and R times sine of t ... 24 00:01:46,300 --> 00:01:49,500 ... is the simplest way to parameterize a circle. 25 00:01:49,800 --> 00:01:54,200 My particle is going to be moving around a circle. 26 00:01:54,300 --> 00:01:59,650 So that at some time t its position is going to be ... 27 00:01:59,750 --> 00:02:02,800 ... radius times cosine of t ... 28 00:02:03,300 --> 00:02:08,400 ... being x-coordinate and radius times sine of t being the y-coordinate. 29 00:02:09,100 --> 00:02:11,000 So that's my position. 30 00:02:11,800 --> 00:02:13,900 Let's find the ... 31 00:02:16,200 --> 00:02:17,700 ... velocity. 32 00:02:18,000 --> 00:02:22,500 Well now as I have a formula for the position I can simply differentiate. 33 00:02:22,650 --> 00:02:24,700 And I know how to differentiate cosine. 34 00:02:24,800 --> 00:02:28,200 So it's R times negative ... 35 00:02:28,300 --> 00:02:30,100 ... sine t. 36 00:02:30,300 --> 00:02:34,700 And the derivative of sine is cosine. So it's R times cosine t. 37 00:02:34,950 --> 00:02:37,900 And I also find the second derivative. 38 00:02:40,500 --> 00:02:42,800 That is negative ... 39 00:02:43,600 --> 00:02:46,200 ... R cosine t ... 40 00:02:46,300 --> 00:02:49,600 ... and negative again ... 41 00:02:49,700 --> 00:02:51,700 ... R sine t. 42 00:02:53,600 --> 00:02:57,800 And then I would like to evaluate those two formulas. 43 00:02:59,500 --> 00:03:03,000 So first I will evaluate the ... 44 00:03:05,000 --> 00:03:06,150 ... Physics invariant ... 45 00:03:06,250 --> 00:03:11,700 ... orthogonal component of the second derivative of f with respect to the first ... 46 00:03:11,800 --> 00:03:16,800 ... divided by the magnitude of the first derivative squared. 47 00:03:21,450 --> 00:03:23,300 So what is that? 48 00:03:28,950 --> 00:03:31,800 Well let's find the magnitude of the first derivative. 49 00:03:31,900 --> 00:03:35,800 Magnitude of the velocity, that's the speed. 50 00:03:35,950 --> 00:03:38,400 So what's the magnitude of ... 51 00:03:38,700 --> 00:03:40,300 ... f prime? 52 00:03:40,900 --> 00:03:43,800 It's a square root of sum of these squares. 53 00:03:44,400 --> 00:03:48,500 Right? Square root of R squared sine squared ... 54 00:03:49,550 --> 00:03:51,000 ... plus ... 55 00:03:51,350 --> 00:03:54,000 ... R squared cosine squared. 56 00:03:55,800 --> 00:03:58,700 And since R squared factors ... 57 00:04:00,250 --> 00:04:04,600 ... the remaining part is sine squared plus cosine squared that gives you one. 58 00:04:04,700 --> 00:04:09,200 And square root of R squared is going to be simply R. 59 00:04:09,600 --> 00:04:12,900 So the number in denominator is going to be ... 60 00:04:13,700 --> 00:04:16,100 ... radius of a circle squared. 61 00:04:17,350 --> 00:04:19,600 All right. What about the numerator? 62 00:04:21,300 --> 00:04:23,100 For the numerator ... 63 00:04:23,800 --> 00:04:27,900 ... I'll have to take the second derivative ... 64 00:04:28,800 --> 00:04:33,200 ... and subtract the projection of that second derivative on the first. 65 00:04:33,500 --> 00:04:38,800 The projection is computed as the second derivative dot the first derivative ... 66 00:04:39,400 --> 00:04:41,800 ... divided by the ... 67 00:04:42,550 --> 00:04:47,600 ... square of the magnitude of the first derivative times the first derivative. 68 00:04:47,700 --> 00:04:50,900 That's the formula for orthogonal component. 69 00:04:51,700 --> 00:04:53,900 And [...] that. 70 00:04:55,500 --> 00:04:57,100 What's the f double prime? 71 00:04:57,200 --> 00:05:06,200 It is minus R cosine t, minus R sine t. 72 00:05:08,100 --> 00:05:09,100 And then ... 73 00:05:10,100 --> 00:05:12,200 The dot product here ... 74 00:05:12,750 --> 00:05:14,700 ... is going to be ... 75 00:05:15,000 --> 00:05:17,300 ... minus minus gives plus ... 76 00:05:17,600 --> 00:05:19,500 ... R squared ... 77 00:05:19,900 --> 00:05:24,100 ... times sine t times cosine t ... 78 00:05:24,300 --> 00:05:27,150 ... and then plus minus gives minus ... 79 00:05:27,250 --> 00:05:34,200 ... R squared times cosine t times sine t. 80 00:05:35,200 --> 00:05:38,400 And what we notice here is that it is zero. 81 00:05:39,400 --> 00:05:41,400 The whole thing is zero. 82 00:05:41,500 --> 00:05:44,600 So no matter what we divide by ... 83 00:05:46,450 --> 00:05:48,600 ... or multiply by ... 84 00:05:49,200 --> 00:05:51,200 ... the whole thing is zero. 85 00:05:51,600 --> 00:05:57,300 So the whole formula is going to be equal to that over radius squared. 86 00:05:59,100 --> 00:06:00,100 And ... 87 00:06:01,700 --> 00:06:04,100 Well that's a vector. 88 00:06:04,550 --> 00:06:08,400 One radius cancels. So have one over R ... 89 00:06:09,150 --> 00:06:12,600 ... multiplied by the vector ... 90 00:06:12,950 --> 00:06:15,300 ... minus cosine t ... 91 00:06:15,900 --> 00:06:17,700 ... minus sine t. 92 00:06:25,600 --> 00:06:29,800 So what is the relation of this vector to the radius of the circle? 93 00:06:31,600 --> 00:06:34,450 Well if you look at the magnitude of the vector ... 94 00:06:34,550 --> 00:06:37,600 ... the magnitude of that part ... 95 00:06:38,650 --> 00:06:40,900 ... magnitude of this is going to be one. 96 00:06:41,000 --> 00:06:44,400 Cosine squared plus sine squared is equal to one. 97 00:06:45,000 --> 00:06:51,250 So what we see here is that the magnitude of orthogonal component ... 98 00:06:51,350 --> 00:06:55,500 ... of f double prime with respect to f prime ... 99 00:06:55,600 --> 00:06:57,900 ... divided by the ... 100 00:06:59,000 --> 00:07:00,650 ... speed squared ... 101 00:07:00,750 --> 00:07:02,700 ... magnitude of that vector ... 102 00:07:03,000 --> 00:07:06,200 ... is equal to one over R. 103 00:07:09,750 --> 00:07:12,100 That's the relation ... 104 00:07:12,300 --> 00:07:14,900 ... the exact relation between ... 105 00:07:15,000 --> 00:07:19,300 ... the idea of normal acceleration that we experience in Physics ... 106 00:07:20,300 --> 00:07:24,600 ... related to the radius of the circle that we see geometrically. 107 00:07:28,450 --> 00:07:31,150 All right. Now er ... 108 00:07:31,250 --> 00:07:33,800 Well basically what it say is that ... 109 00:07:33,900 --> 00:07:38,100 ... from Physics point of view if we look at that vector ... 110 00:07:38,200 --> 00:07:40,400 ... suggested as geometric invariant ... 111 00:07:40,500 --> 00:07:44,000 ... the magnitude of that vector is one over the radius. 112 00:07:44,650 --> 00:07:48,150 Now let's look at that other formula. 113 00:07:48,250 --> 00:07:51,400 The algebraic formula involving area. 114 00:07:53,850 --> 00:07:56,700 How different is that invariant? 115 00:07:57,500 --> 00:08:04,700 So I need to compute the area on f prime and f double prime ... 116 00:08:04,800 --> 00:08:09,400 ... divided by the magnitude of f prime cubed. 117 00:08:13,400 --> 00:08:17,700 And I have both vectors computed explicitly. 118 00:08:17,800 --> 00:08:23,000 So the area is going to be determinant of this matrix. 119 00:08:23,100 --> 00:08:26,800 So I have to multiply these two quantities. 120 00:08:26,900 --> 00:08:29,100 Minus times minus gives plus. 121 00:08:29,200 --> 00:08:31,800 R squared sine times sine. 122 00:08:31,900 --> 00:08:36,800 So R squared times sine squared t ... 123 00:08:37,300 --> 00:08:39,500 ... minus that product. 124 00:08:39,600 --> 00:08:41,900 And minus minus gives plus. 125 00:08:42,000 --> 00:08:44,500 R squared cosine squared ... 126 00:08:45,000 --> 00:08:49,000 ... plus R squared cosine squared. 127 00:08:49,700 --> 00:08:51,500 That's the area. 128 00:08:52,000 --> 00:08:58,100 And it should be divided by the magnitude of f prime which is R cubed. 129 00:09:00,900 --> 00:09:03,500 And of course sine squared plus cosine squared is one ... 130 00:09:03,600 --> 00:09:06,200 ... after we factor R squared. 131 00:09:06,700 --> 00:09:10,000 And this is immediately one over the radius. 132 00:09:16,950 --> 00:09:23,100 So that is also related to the radius in exactly the same way. 133 00:09:24,700 --> 00:09:26,800 That is also one over radius. 134 00:09:29,000 --> 00:09:33,700 So somehow that formula coming from Physics ... 135 00:09:34,050 --> 00:09:38,850 ... is [...] related to that formula coming from Algebra. 136 00:09:38,950 --> 00:09:40,800 But they look very different. 137 00:09:42,500 --> 00:09:46,800 Now let's figure out the relation geometrically. 138 00:09:47,400 --> 00:09:49,100 If I have ... 139 00:09:50,250 --> 00:09:53,500 ... vector f double prime ... 140 00:09:53,700 --> 00:09:57,500 ... and ... Well let's say f prime here. 141 00:09:58,200 --> 00:10:01,400 And f double prime there. 142 00:10:04,400 --> 00:10:08,550 Then what's the area on these two vectors? 143 00:10:08,650 --> 00:10:12,200 Well that is going to be the area of this parallelogram. 144 00:10:13,600 --> 00:10:16,200 That's the area we are talking about. 145 00:10:17,400 --> 00:10:22,000 So area on f prime and f double prime. 146 00:10:22,500 --> 00:10:24,500 Now I want to relate it to what? 147 00:10:24,600 --> 00:10:28,550 I want to relate it to the magnitude of orthogonal component. 148 00:10:28,650 --> 00:10:33,000 Now how can I see orthogonal component of f double prime with respect to f prime? 149 00:10:33,200 --> 00:10:35,300 It is going to be ... 150 00:10:37,200 --> 00:10:39,400 ... this vector, right? 151 00:10:41,400 --> 00:10:43,700 So this is going to be ... 152 00:10:43,800 --> 00:10:48,400 ... orthogonal component of f double prime with respect to f prime. 153 00:10:50,100 --> 00:10:52,500 Now the magnitude of that ... 154 00:10:54,850 --> 00:10:56,300 ... of that vector ... 155 00:10:56,600 --> 00:10:59,500 ... is going to be the height of the parallelogram. 156 00:11:00,150 --> 00:11:03,300 So this is the same as the height ... 157 00:11:04,400 --> 00:11:07,200 ... of that parallelogram we ara looking at. 158 00:11:07,300 --> 00:11:11,400 And we know that the area is equal to the height ... 159 00:11:12,400 --> 00:11:15,400 ... times the magnitude of the base. 160 00:11:18,150 --> 00:11:22,800 And now we see that there is no mystery in seeing that ... 161 00:11:23,100 --> 00:11:28,500 ... the area divided by magnitude of f prime ... 162 00:11:29,000 --> 00:11:31,400 ... is the same thing as ... 163 00:11:31,700 --> 00:11:34,400 ... the magnitude of orthogonal component. 164 00:11:35,150 --> 00:11:41,700 So that's exactly why we see extra f prime in the denominator of the algebraic formula. 165 00:11:41,800 --> 00:11:45,200 So these two formulas are exactly about the same thing. 166 00:11:46,400 --> 00:11:49,400 And that same thing is one over radius. 167 00:11:49,700 --> 00:11:53,700 It is now obviously related to the radius of the circle. |

Video Lectures > Calculus III (2012 Spring) > Lecture 2012.02.21 Curvature Formula > Part 09 Movement along a Circle >