1 00:00:00,538 --> 00:00:04,054 What I want to do in this video is a calculus proof of the famous 2 00:00:04,054 --> 00:00:07,862 centripetal acceleration formula that tells us the magnitude 3 00:00:07,862 --> 00:00:10,824 of centripetal acceleration, the actual direction will change 4 00:00:10,824 --> 00:00:14,901 it's always going to be pointing inwards, but the magnitude of centripetal acceleration is equal to the 5 00:00:14,901 --> 00:00:19,235 magnitude of the velocity-squared divided by the radius 6 00:00:19,358 --> 00:00:23,082 I want to be very clear, this is a scalar formula right over here; we're talking about the magnitude 7 00:00:23,082 --> 00:00:26,410 of the acceleration and the magnitude of the velocity. If these were vectors, 8 00:00:26,410 --> 00:00:30,045 we would have arrows drawn over it. So this really, I don't want people to get confused 9 00:00:30,045 --> 00:00:34,313 because this is a v, this is really referring to the speed-squared. 10 00:00:34,313 --> 00:00:36,859 and this is the magnitude, and these are all 11 00:00:36,859 --> 00:00:41,094 scalar quantities. So to do that, let's imagine some 12 00:00:41,094 --> 00:00:45,366 object, maybe it's some object in orbit around a planet or something, 13 00:00:45,366 --> 00:00:48,830 so let's say that's the planet, and you have some object 14 00:00:48,830 --> 00:00:52,087 that is in orbit around the planet, and it is going in a counter-clockwise 15 00:00:52,087 --> 00:00:56,273 direction, and so let's specify its position vector as 16 00:00:56,273 --> 00:00:58,657 a function as time. 17 00:01:01,519 --> 00:01:04,199 So that is its position vector and it is going to 18 00:01:04,199 --> 00:01:07,999 change as a function of time as this thing spins around 19 00:01:07,999 --> 00:01:11,569 We're going to assume, for the purposes of this 20 00:01:11,569 --> 00:01:13,430 proof. 21 00:01:18,630 --> 00:01:19,893 So that is our 22 00:01:19,893 --> 00:01:23,307 y-axis, and this is our x-axis. 23 00:01:23,645 --> 00:01:27,532 We're going to define theta as the angle between the positive 24 00:01:27,532 --> 00:01:30,051 x-axis and our vector. 25 00:01:31,189 --> 00:01:35,199 And we're going to assume this thing is in orbit with the radius of r. So the 26 00:01:35,199 --> 00:01:39,284 magnitude of our position vector, even though the direction is going to change 27 00:01:39,284 --> 00:01:43,400 the magnitude of our position vector is not going to change. It's always going to have 28 00:01:43,400 --> 00:01:47,068 length r. So this is going in a circle with radius r. 29 00:01:47,345 --> 00:01:50,345 The magnitude of our position vector, 30 00:01:50,345 --> 00:01:54,635 which is changing as a function of time, is going to be 31 00:01:54,635 --> 00:01:58,628 r. So how can we write 32 00:01:58,628 --> 00:02:03,348 the position vector in terms of its components at any given time? 33 00:02:03,348 --> 00:02:06,881 We can write the position vector, and I'll do it in engineering notation, 34 00:02:06,881 --> 00:02:09,562 and so you might want to review those videos if some of this looks foreign and I'll 35 00:02:09,562 --> 00:02:14,026 do a bit of basic trigonometry in breaking down the vector into its components 36 00:02:14,026 --> 00:02:17,557 and I encourage you to review some of those videos if some of that looks a bit 37 00:02:17,557 --> 00:02:23,006 daunting. If you take the position vector at any time, 38 00:02:23,006 --> 00:02:25,263 the magnitude is r, 39 00:02:25,263 --> 00:02:28,475 this angle is theta, 40 00:02:28,475 --> 00:02:33,274 its x-component, in blue, 41 00:02:33,274 --> 00:02:37,905 this vector right over here, the magnitude of the vector, I should say, 42 00:02:37,905 --> 00:02:41,380 is going to be r cosine of theta 43 00:02:41,380 --> 00:02:44,512 We learned that this came from basic trigonometry when we started 44 00:02:44,512 --> 00:02:48,443 two-dimensional projectile motion, we saw how to break these 45 00:02:48,443 --> 00:02:55,263 vectors down into its components, and the y-component of this vector 46 00:02:55,294 --> 00:02:58,457 is going to be r sine of theta 47 00:03:00,704 --> 00:03:03,989 So this is going to be r sine of 48 00:03:03,989 --> 00:03:09,113 theta. So the position vector at any time 49 00:03:09,113 --> 00:03:11,972 can be written as a sum of its x- and y-components. 50 00:03:11,972 --> 00:03:15,083 So it's the magnitude of its x-component, it's going to be 51 00:03:15,083 --> 00:03:20,174 r cosine of theta, and I could write theta as a function of 52 00:03:20,174 --> 00:03:23,886 time if I'd like, but I'm just going to write r cosine of theta 53 00:03:23,886 --> 00:03:27,035 actually, let me write it that way, so it shows theta is a function of time 54 00:03:27,035 --> 00:03:30,503 This thing is moving, and there's going to be that times the 55 00:03:30,503 --> 00:03:34,870 i-unit vector, we're in engineering notation over here. 56 00:03:34,870 --> 00:03:38,353 So that's the i-unit vector, it tells us that the x-component 57 00:03:38,353 --> 00:03:42,507 is going in the positive x direction. Plus 58 00:03:42,507 --> 00:03:45,479 the magnitude of the y-component, which is 59 00:03:45,479 --> 00:03:50,443 r sine of theta, which is going to be a function of time. 60 00:03:50,443 --> 00:03:54,715 So to be clear, the function of time applies to the theta. 61 00:03:57,484 --> 00:03:58,371 And that is going 62 00:03:58,371 --> 00:04:00,603 in the j-direction. 63 00:04:02,818 --> 00:04:05,284 So that is our j-unit vector. 64 00:04:07,853 --> 00:04:10,774 So now we have position as a function of theta 65 00:04:10,774 --> 00:04:14,122 which is actually a function of time. So let's take the derivative of 66 00:04:14,122 --> 00:04:19,627 this thing right here. So what is 67 00:04:19,627 --> 00:04:26,818 the derivative of our position vector with respect to time 68 00:04:26,818 --> 00:04:30,754 Well that's just going to be our velocity vector, 69 00:04:30,754 --> 00:04:34,625 as a function of time, and it's going to be equal to 70 00:04:34,625 --> 00:04:38,375 we just have to take the derivative of each of these parts with respect to 71 00:04:38,375 --> 00:04:42,222 time. And you just do the chain rule. So you're going to have 72 00:04:42,222 --> 00:04:45,646 the r sit outside cause that's just a constant. So you're going to have r 73 00:04:45,646 --> 00:04:50,277 the derivative of cosine of theta t with respect to theta t 74 00:04:50,277 --> 00:04:53,523 So I'm just doing the chain rule right over here. That's going to be 75 00:04:53,523 --> 00:04:58,361 negative sine of theta t 76 00:04:58,361 --> 00:05:01,746 and then as the chain rule, we also have to multiply that 77 00:05:01,746 --> 00:05:04,705 times the derivative of the theta of t 78 00:05:04,705 --> 00:05:10,122 with respect to t. So times d-theta, dt 79 00:05:10,122 --> 00:05:12,929 so this is just the chain rule right over here. 80 00:05:12,929 --> 00:05:16,882 So that's going to be how it's changing in the x-direction, and in the y-direction 81 00:05:16,882 --> 00:05:19,999 we do something very similar. In the y-direction 82 00:05:19,999 --> 00:05:23,305 we take the same derivative. We have the r scalar out front 83 00:05:23,305 --> 00:05:26,964 r, and then the derivative of sine of theta 84 00:05:26,964 --> 00:05:31,030 with respect to theta is going to be cosine of theta 85 00:05:31,030 --> 00:05:34,947 and I'll write it as a function of time, and then do the chain rule 86 00:05:34,947 --> 00:05:39,513 you'll also have to multiply that by the rate at which theta is changing with respect to t, 87 00:05:39,513 --> 00:05:42,885 times d-theta, dt, 88 00:05:42,885 --> 00:05:50,287 and this is all going to be times our j-unit vector. Now, 89 00:05:50,287 --> 00:05:54,418 there's something you might already realize, and you should rewatch the video 90 00:05:54,418 --> 00:05:57,691 on angular velocity if this is foreign to you, but 91 00:05:57,691 --> 00:06:01,759 d-theta, dt, this is our angular velocity. 92 00:06:01,759 --> 00:06:05,741 That's why I said to rewatch that video. This right over here, the rate at which 93 00:06:05,741 --> 00:06:09,333 the angle changes with respect to time, that is angular velocity. 94 00:06:09,333 --> 00:06:13,314 So this right over here is 95 00:06:13,314 --> 00:06:17,311 angular velocity. And for the sake of this 96 00:06:17,311 --> 00:06:21,362 video, this is an assumption we'll have to make for this formula right over here 97 00:06:21,362 --> 00:06:25,432 we're going to assume, that omega, which is the rate of change 98 00:06:25,432 --> 00:06:28,315 of our angle with respect to time, we're going to assume 99 00:06:28,315 --> 00:06:31,931 that this is constant. 100 00:06:32,754 --> 00:06:36,911 So this is an assumption we're making for this proof. This is we are going 101 00:06:36,911 --> 00:06:40,357 to assume that omega is constant. And if omega is 102 00:06:40,357 --> 00:06:44,227 constant, then we can treat it as a constant and we can factor 103 00:06:44,227 --> 00:06:47,413 it out of this expression. So let's factor out 104 00:06:47,413 --> 00:06:52,948 a negative omega-r from this expression over here. So we can rewrite 105 00:06:53,533 --> 00:06:56,420 our velocity as a function of time is equal to 106 00:06:56,420 --> 00:07:00,285 I'm going to factor out a negative omega 107 00:07:00,285 --> 00:07:04,420 times r, and if you factor out a negative omega-r, 108 00:07:04,420 --> 00:07:08,681 what you're left with is 109 00:07:08,681 --> 00:07:10,896 this first term, 110 00:07:14,215 --> 00:07:20,069 sine of theta-t 111 00:07:21,438 --> 00:07:25,932 And I didn't have to make it explicit that theta is a function of t, but this makes it explicit 112 00:07:25,932 --> 00:07:30,920 that theta is a function of t, and then times 113 00:07:30,920 --> 00:07:35,772 our i-unit vector, 114 00:07:35,772 --> 00:07:39,531 plus, so if we're factoring out 115 00:07:39,531 --> 00:07:44,312 a negative omega-r, this becomes negative 116 00:07:44,312 --> 00:07:46,817 cosine of theta, 117 00:07:46,817 --> 00:07:51,164 which is a function of t, and that 118 00:07:51,164 --> 00:07:55,544 is times our j-unit vector. 119 00:07:58,177 --> 00:07:59,389 So we factored out 120 00:07:59,420 --> 00:08:02,114 a negative omega-r. Now let's take the derivative of 121 00:08:02,114 --> 00:08:05,784 this with respect to time. So if we take 122 00:08:07,184 --> 00:08:09,999 the derivative of velocity 123 00:08:09,999 --> 00:08:13,964 with respect to time, this is clearly just what the acceleration is 124 00:08:13,964 --> 00:08:17,515 as a function of time, and we're going to assume that the magnitude of this thing 125 00:08:17,515 --> 00:08:21,048 is constant, but the actual direction is changing, so this is the acceleration 126 00:08:21,048 --> 00:08:26,564 as a function of time, is going to be equal to 127 00:08:28,795 --> 00:08:29,620 this negative 128 00:08:29,620 --> 00:08:32,655 omega-r, so what is the derivative of 129 00:08:32,655 --> 00:08:36,934 this thing right over here? So the derivative of sine with respect to theta, 130 00:08:36,985 --> 00:08:38,581 we're just doing the chain rule here, 131 00:08:38,581 --> 00:08:40,635 the derivative of sine with respect to theta 132 00:08:40,635 --> 00:08:46,828 is going to be cosine of theta 133 00:08:46,878 --> 00:08:49,049 as a function of t. 134 00:08:49,049 --> 00:08:52,464 And then chain rule, we also have to take that and multiply it with the derivative of theta 135 00:08:52,514 --> 00:08:54,701 with respect to t. 136 00:08:56,498 --> 00:08:59,788 I could write d-theta, dt here 137 00:08:59,788 --> 00:09:03,759 But that once again, is just omega. 138 00:09:03,759 --> 00:09:08,043 So that is just omega, and that of course 139 00:09:08,043 --> 00:09:12,774 is in the i-direction. And from that, 140 00:09:12,774 --> 00:09:14,808 and next to that, we take the derivative of 141 00:09:14,808 --> 00:09:19,515 cosine of theta of t with respect to theta, so that's going to be 142 00:09:19,515 --> 00:09:23,115 that would be negative sine of theta, so we would have a negative out front 143 00:09:23,115 --> 00:09:27,543 so it becomes positive sine of 144 00:09:27,543 --> 00:09:30,942 theta as a function of t. 145 00:09:30,942 --> 00:09:35,396 And then we have to do the chain rule, the derivative of theta with respect to t. 146 00:09:35,396 --> 00:09:38,125 We have to multiply by this, and for that we could write 147 00:09:38,125 --> 00:09:42,806 d-theta, dt right there, but that again is the same thing 148 00:09:42,806 --> 00:09:45,824 as omega. And all that is being multiplied by the 149 00:09:45,824 --> 00:09:48,728 j-unit vector. 150 00:09:51,120 --> 00:09:54,998 So now let's factor out this other omega, and we get something interesting, 151 00:09:54,998 --> 00:09:58,795 we get the acceleration vector as a function of 152 00:09:58,795 --> 00:10:02,513 time is equal to, and if we factor out another omega, we get 153 00:10:02,513 --> 00:10:06,455 negative omega-squared r, 154 00:10:06,455 --> 00:10:11,224 I'm just factoring out another negative omega, 155 00:10:11,224 --> 00:10:16,550 times, and I'll write it in parentheses here, 156 00:10:16,550 --> 00:10:21,040 cosine of theta 157 00:10:21,040 --> 00:10:23,025 as a function of t 158 00:10:23,025 --> 00:10:26,668 times our i-unit vector 159 00:10:26,668 --> 00:10:29,707 plus 160 00:10:29,707 --> 00:10:34,025 sine of theta, which is a function of t 161 00:10:34,025 --> 00:10:37,659 times our j-unit vector. 162 00:10:37,659 --> 00:10:45,467 Now what is all of this business right over here? 163 00:10:45,467 --> 00:10:49,401 Just look at this part right over here, well r times this, 164 00:10:49,401 --> 00:10:53,813 especially if you distributed the r, that is exactly this thing right over here 165 00:10:53,813 --> 00:10:57,462 If you distribute the r, you get exactly r cosine theta as a function 166 00:10:57,462 --> 00:11:01,241 of t times our i-unit vector plus our sine theta as a 167 00:11:01,241 --> 00:11:06,582 function of t times the j-unit vector. So everything that I squared-off in orange right over here, 168 00:11:06,582 --> 00:11:13,135 this is our position vector as a function of time. 169 00:11:13,135 --> 00:11:14,975 So all that work we did, we just got a 170 00:11:14,975 --> 00:11:19,135 very interesting result. We got that our acceleration vector 171 00:11:19,135 --> 00:11:23,384 as a function of time is equal to the negative 172 00:11:23,384 --> 00:11:32,767 of our constant angular velocity-squared 173 00:11:32,767 --> 00:11:34,829 times our position vector 174 00:11:34,829 --> 00:11:37,928 And just to be clear, angular velocity is kind of the pseudo vector, 175 00:11:37,928 --> 00:11:42,248 it tends to be treated like a scalar, especially when you're dealing with two-dimensionals 176 00:11:42,248 --> 00:11:47,311 like this, it's really a pseudo scalar, but let's just go with this. 177 00:11:47,311 --> 00:11:51,294 We're assuming this right over here is a constant scalar quantity. 178 00:11:51,909 --> 00:11:55,721 Now, we're very very very very close here. 179 00:11:55,721 --> 00:11:58,978 Now what we want to do is to relate this 180 00:11:58,978 --> 00:12:02,591 this is essentially the scalar version of it, so if we wanted 181 00:12:02,591 --> 00:12:05,774 to take the magnitudes of both sides, 182 00:12:05,774 --> 00:12:10,021 so we're saying the acceleration vector is equal to this constant times 183 00:12:10,021 --> 00:12:19,116 the position vector, so let's take the magnitude of both sides of this thing 184 00:12:19,116 --> 00:12:22,615 So then we get the magnitude of the acceleration vector, 185 00:12:22,615 --> 00:12:24,462 which I'm just going to call a sub c, 186 00:12:24,462 --> 00:12:26,880 is going to be equal to 187 00:12:26,880 --> 00:12:29,790 you could say the magnitude of this negative omega-squared 188 00:12:29,790 --> 00:12:32,645 but when you take the magnitude, it's like taking the absolute value 189 00:12:32,645 --> 00:12:35,464 in fact, absolute value is just the one-dimensional version of magnitude, 190 00:12:35,464 --> 00:12:38,107 that's just going to be positive omega-squared 191 00:12:38,107 --> 00:12:41,488 we don't care about the direction, sign gives us the direction, we just care about 192 00:12:41,488 --> 00:12:48,076 the actual size. So this is going to be 193 00:12:48,076 --> 00:12:50,255 the magnitude of negative omega-squared 194 00:12:50,255 --> 00:12:54,642 times the magnitude of our position vector 195 00:12:54,642 --> 00:12:58,517 the magnitude of omega-squared is just going to be omega-squared 196 00:12:58,517 --> 00:13:01,797 you can get rid of the sign, and the magnitude of our position vector 197 00:13:01,797 --> 00:13:05,167 we saw at the beginning of this video, is just r, 198 00:13:05,167 --> 00:13:08,563 our radius 199 00:13:09,317 --> 00:13:13,277 so this right over here is just going to be equal to 200 00:13:13,277 --> 00:13:16,051 the radius of the circle that we're going around. 201 00:13:16,189 --> 00:13:19,884 Now, we also know the angular velocity, or 202 00:13:19,884 --> 00:13:24,367 the magnitude of the angular velocity, is equal to the 203 00:13:24,367 --> 00:13:28,433 magnitude of our velocity, or the speed of our 204 00:13:28,433 --> 00:13:32,101 object, divided by the radius of the circle 205 00:13:32,101 --> 00:13:35,913 that it is going around. So we could substitute that right over here. 206 00:13:35,913 --> 00:13:42,648 So if we square it, this is going to be (v over r)-squared, now we saw that in the video on angular velocity, 207 00:13:42,648 --> 00:13:44,687 times r 208 00:13:44,687 --> 00:13:47,424 and this is all going to be the magnitude of our acceleration, 209 00:13:47,424 --> 00:13:51,490 which is really our centripetal acceleration, our inward directed acceleration. 210 00:13:51,798 --> 00:13:56,021 So this is going to be equal to, and I think you see where this is going, 211 00:13:56,021 --> 00:13:59,869 This is equal to v-squared over r-squared 212 00:13:59,869 --> 00:14:02,970 times r, but this r cancels out 213 00:14:02,970 --> 00:14:06,688 with the r-squared, so you're just left with v-squared 214 00:14:06,688 --> 00:14:09,004 over r, and you're done! 215 00:14:09,004 --> 00:14:13,654 The magnitude of the centripetal acceleration is equal to your speed, 216 00:14:13,654 --> 00:14:17,161 the magnitude of your velocity, squared, divided by 217 00:14:17,161 --> 00:00:00,000 your radius. And, we are done.