1 00:00:00,401 --> 00:00:02,778 - [Instructor] There are unfortunately quite a few 2 00:00:02,778 --> 00:00:04,990 common misconceptions that many people have 3 00:00:04,990 --> 00:00:07,904 when they deal with centripetal force problems, 4 00:00:07,904 --> 00:00:09,767 so in this video, we're gonna go over some examples 5 00:00:09,767 --> 00:00:12,069 to give you some problem solving strategies 6 00:00:12,069 --> 00:00:14,271 that you can use as well as going over 7 00:00:14,271 --> 00:00:17,058 a lot of the common misconceptions that people have 8 00:00:17,058 --> 00:00:20,058 when they deal with these centripetal motion problems. 9 00:00:20,058 --> 00:00:21,948 So, to start with, imagine this example, 10 00:00:21,948 --> 00:00:25,639 let's say a string is causing a ball to rotate in a circle. 11 00:00:25,639 --> 00:00:27,573 And to make it simple, let's say 12 00:00:27,573 --> 00:00:29,590 this ball is tracing out a perfect circle, 13 00:00:29,590 --> 00:00:32,798 and let's say it's sitting on a perfectly frictionless table 14 00:00:32,798 --> 00:00:34,600 so this would be the bird's eye view. 15 00:00:34,600 --> 00:00:36,289 This is the view from above. 16 00:00:36,289 --> 00:00:37,769 What it would look like from the side 17 00:00:37,769 --> 00:00:39,137 would be something like this. 18 00:00:39,137 --> 00:00:40,704 You'd have the ball tied to the rope 19 00:00:40,704 --> 00:00:42,550 and then you nail some sort of stake 20 00:00:42,550 --> 00:00:43,743 in the middle of the table. 21 00:00:43,743 --> 00:00:45,093 You tie the rope to the stake, 22 00:00:45,093 --> 00:00:46,268 and then you give the ball a push. 23 00:00:46,268 --> 00:00:49,159 And the ball's gonna take this circular path on the table 24 00:00:49,159 --> 00:00:50,718 when we view it from the side. 25 00:00:50,718 --> 00:00:52,249 But when we view it from above, 26 00:00:52,249 --> 00:00:54,340 you see this path traced out. 27 00:00:54,340 --> 00:00:55,704 So this is a bird's eye view 28 00:00:55,704 --> 00:00:57,362 that you would see if you were looking down 29 00:00:57,362 --> 00:00:59,012 from above the table, 30 00:00:59,012 --> 00:01:00,608 and this would be the side view. 31 00:01:00,608 --> 00:01:01,968 So let me ask you this question. 32 00:01:01,968 --> 00:01:05,319 What force is causing this ball to go in a circle? 33 00:01:05,319 --> 00:01:07,120 Now, a lot of people want to answer that question with 34 00:01:07,120 --> 00:01:08,799 the centripetal force. 35 00:01:08,799 --> 00:01:10,395 They'd say that it's the centripetal force 36 00:01:10,395 --> 00:01:14,229 that points inward that causes this ball to go in a circle, 37 00:01:14,229 --> 00:01:16,726 and that's not wrong. 38 00:01:16,726 --> 00:01:17,746 It's the truth, 39 00:01:17,746 --> 00:01:19,414 but it's not the whole truth. 40 00:01:19,414 --> 00:01:22,148 And the reason is that when we say centripetal force, 41 00:01:22,148 --> 00:01:25,000 all we really mean is a force that's directed 42 00:01:25,000 --> 00:01:26,628 toward the center of the circle. 43 00:01:26,628 --> 00:01:29,385 So saying the force that causes this ball to go in a circle 44 00:01:29,385 --> 00:01:30,729 is the centripetal force 45 00:01:30,729 --> 00:01:32,758 is a little unsatisfying. 46 00:01:32,758 --> 00:01:34,403 It'd be like answering the question, 47 00:01:34,403 --> 00:01:36,877 what force balances the force of gravity 48 00:01:36,877 --> 00:01:38,505 while the ball's on the table 49 00:01:38,505 --> 00:01:41,570 with the answer, the upward force. 50 00:01:41,570 --> 00:01:44,740 I mean, yeah, we knew it had to be an upward force, 51 00:01:44,740 --> 00:01:47,298 but that really doesn't tell us what force it is. 52 00:01:47,298 --> 00:01:49,482 Similarly, just saying the centripetal force 53 00:01:49,482 --> 00:01:52,472 just tells us what direction the force points. 54 00:01:52,472 --> 00:01:55,572 It doesn't really tell us what type of force this is, 55 00:01:55,572 --> 00:01:58,060 so to answer this question over here in a better way, 56 00:01:58,060 --> 00:02:00,474 if someone asked you what force counteracts gravity 57 00:02:00,474 --> 00:02:02,500 that keeps the ball from falling through the table, 58 00:02:02,500 --> 00:02:04,737 instead of saying upward force, 59 00:02:04,737 --> 00:02:07,628 it'd be better to just say that's the normal force. 60 00:02:07,628 --> 00:02:09,764 And we can do better over here as well. 61 00:02:09,764 --> 00:02:11,530 Instead of just saying the centripetal force, 62 00:02:11,530 --> 00:02:13,649 we could say what kind of force this is. 63 00:02:13,649 --> 00:02:16,764 It's gotta be one of the forces that we already know about. 64 00:02:16,764 --> 00:02:19,317 I mean, it's gotta be either the force of friction 65 00:02:19,317 --> 00:02:22,348 or normal force or tension or the force of gravity. 66 00:02:22,348 --> 00:02:24,994 The centripetal force isn't a new type of force. 67 00:02:24,994 --> 00:02:27,577 It's just one of the forces we already know 68 00:02:27,577 --> 00:02:30,515 that happens to be pointing toward the center of the circle. 69 00:02:30,515 --> 00:02:32,211 And that's important because this is our first, 70 00:02:32,211 --> 00:02:33,674 big common misconception. 71 00:02:33,674 --> 00:02:36,741 People think the centripetal force is a new kind of force, 72 00:02:36,741 --> 00:02:37,653 but it's not. 73 00:02:37,653 --> 00:02:39,822 It's just one of the forces we already know 74 00:02:39,822 --> 00:02:43,123 that happen to be pointing toward the center of the circle 75 00:02:43,123 --> 00:02:46,614 and that happen to be causing an object to move in a circle. 76 00:02:46,614 --> 00:02:48,706 So in this case over here, what force is it? 77 00:02:48,706 --> 00:02:50,822 Well, there's a rope tied to this mass, 78 00:02:50,822 --> 00:02:52,139 and that rope's gonna pull on it. 79 00:02:52,139 --> 00:02:54,948 And when a rope pulls, we call that the force of tension, 80 00:02:54,948 --> 00:02:56,832 so I'm gonna call this the tension. 81 00:02:56,832 --> 00:02:57,999 So that's a little better. 82 00:02:57,999 --> 00:02:59,547 Now we know what kind of force 83 00:02:59,547 --> 00:03:01,430 is acting as the centripetal force. 84 00:03:01,430 --> 00:03:02,727 Now, be careful out there. 85 00:03:02,727 --> 00:03:04,207 Sometimes, people want to do this, 86 00:03:04,207 --> 00:03:05,766 they're like, oh yeah, there's a force of tension, 87 00:03:05,766 --> 00:03:08,013 and there's also a centripetal force. 88 00:03:08,013 --> 00:03:11,034 But that's just crazy because this tension 89 00:03:11,034 --> 00:03:12,730 is the centripetal force. 90 00:03:12,730 --> 00:03:13,751 I wouldn't draw it twice 91 00:03:13,751 --> 00:03:16,833 anymore than I'd come over here and say, 92 00:03:16,833 --> 00:03:17,948 yeah, there's a normal force, 93 00:03:17,948 --> 00:03:20,133 there's also upward force. 94 00:03:20,133 --> 00:03:22,093 The upward force is the normal force. 95 00:03:22,093 --> 00:03:23,259 I wouldn't draw it again. 96 00:03:23,259 --> 00:03:25,592 Similarly, over here, I'm not gonna draw 97 00:03:25,592 --> 00:03:27,174 the centripetal force twice. 98 00:03:27,174 --> 00:03:29,416 The tension was the centripetal force. 99 00:03:29,416 --> 00:03:31,696 I mean, it's possible you could have two forces inward. 100 00:03:31,696 --> 00:03:33,012 Maybe there's two ropes 101 00:03:33,012 --> 00:03:35,180 and you had a second tension over here pulling inward, 102 00:03:35,180 --> 00:03:37,939 but you'd better be able to identify what force it is 103 00:03:37,939 --> 00:03:39,410 before you draw it. 104 00:03:39,410 --> 00:03:41,136 Don't just call it F centripetal, 105 00:03:41,136 --> 00:03:43,337 so you might be like, yeah, yeah, I get it. 106 00:03:43,337 --> 00:03:46,098 The centripetal force is just an extra title we give 107 00:03:46,098 --> 00:03:47,716 to a force that happens to point 108 00:03:47,716 --> 00:03:48,977 toward the center of the circle, 109 00:03:48,977 --> 00:03:51,038 but how would I ever solve a problem like this? 110 00:03:51,038 --> 00:03:52,622 What strategy do I use? 111 00:03:52,622 --> 00:03:55,418 I've got forces that are up, that are down, that are in. 112 00:03:55,418 --> 00:03:56,693 So let me show you how to solve some problems 113 00:03:56,693 --> 00:03:58,100 and some things to keep in mind. 114 00:03:58,100 --> 00:03:59,505 So let me add some numbers in here. 115 00:03:59,505 --> 00:04:00,871 So let's say I told you this. 116 00:04:00,871 --> 00:04:02,551 Let's say the mass of the ball was two kilograms, 117 00:04:02,551 --> 00:04:04,991 the rope's length was 0.5 meters, 118 00:04:04,991 --> 00:04:06,888 and the ball is traveling around the circle 119 00:04:06,888 --> 00:04:09,500 at a constant speed of five meters per second. 120 00:04:09,500 --> 00:04:11,615 So what kind of question might you be asked 121 00:04:11,615 --> 00:04:13,167 if given a problem like this? 122 00:04:13,167 --> 00:04:14,471 A possible question would be, 123 00:04:14,471 --> 00:04:17,135 well, what's the force of tension in the rope? 124 00:04:17,135 --> 00:04:18,357 And so, now's a good time for me 125 00:04:18,357 --> 00:04:19,916 to let you in on a little secret. 126 00:04:19,916 --> 00:04:22,432 The secret to solving centripetal force problems 127 00:04:22,432 --> 00:04:24,868 is that you solve them the same way you solve 128 00:04:24,868 --> 00:04:26,513 any force problem. 129 00:04:26,513 --> 00:04:30,260 In other words, first, you draw a quality force diagram. 130 00:04:30,260 --> 00:04:32,802 And then you use Newton's second law 131 00:04:32,802 --> 00:04:34,897 for one of the directions at a time. 132 00:04:34,897 --> 00:04:36,207 And if the direction you chose 133 00:04:36,207 --> 00:04:38,201 to analyze Newton's second law for 134 00:04:38,201 --> 00:04:40,316 didn't get you to where you needed to be, 135 00:04:40,316 --> 00:04:41,545 just do it again. 136 00:04:41,545 --> 00:04:43,650 Use Newton's second law again for another direction, 137 00:04:43,650 --> 00:04:45,458 and that'll get you to where you need to be. 138 00:04:45,458 --> 00:04:47,991 So in other words, let's draw a quality force diagram. 139 00:04:47,991 --> 00:04:49,758 We've got forces, but they're kind of all over here. 140 00:04:49,758 --> 00:04:52,267 This side view's gonna better illustrate 141 00:04:52,267 --> 00:04:53,589 all the forces involved. 142 00:04:53,589 --> 00:04:55,201 So we've already got the normal force upward 143 00:04:55,201 --> 00:04:57,037 and the force of gravity downward. 144 00:04:57,037 --> 00:04:59,710 Now, I'm gonna draw this tension pointing inward, 145 00:04:59,710 --> 00:05:02,550 that's the force that's acting as the centripetal force. 146 00:05:02,550 --> 00:05:04,483 Now, we're gonna use Newton's second law 147 00:05:04,483 --> 00:05:05,607 for one of the directions. 148 00:05:05,607 --> 00:05:07,268 Which direction should we pick? 149 00:05:07,268 --> 00:05:08,976 Well, which force do we want to find? 150 00:05:08,976 --> 00:05:10,383 We want to find this force of tension, 151 00:05:10,383 --> 00:05:12,564 so even though I could if I wanted to 152 00:05:12,564 --> 00:05:15,375 use Newton's second law for this vertical direction, 153 00:05:15,375 --> 00:05:17,553 the tension doesn't even point that way, 154 00:05:17,553 --> 00:05:19,756 so I'm not gonna bother with that direction first. 155 00:05:19,756 --> 00:05:22,035 I'm gonna see if I can get by doing this in one step, 156 00:05:22,035 --> 00:05:24,083 so I'm gonna use this horizontal direction 157 00:05:24,083 --> 00:05:26,260 and that's gonna be the centripetal direction, 158 00:05:26,260 --> 00:05:27,811 i.e., into the circle. 159 00:05:27,811 --> 00:05:29,731 And when we're dealing with the centripetal force, 160 00:05:29,731 --> 00:05:32,521 we're gonna be dealing with the centripetal acceleration, 161 00:05:32,521 --> 00:05:35,479 so over here, when I use a and set that equal to 162 00:05:35,479 --> 00:05:37,124 the net force over mass, 163 00:05:37,124 --> 00:05:39,278 if I'm gonna use the centripetal force, 164 00:05:39,278 --> 00:05:42,004 I'm gonna have to use the centripetal acceleration. 165 00:05:42,004 --> 00:05:43,770 In other words, I'm gonna only plug 166 00:05:43,770 --> 00:05:47,306 forces that go into, radially into the circle here, 167 00:05:47,306 --> 00:05:50,592 and I'm gonna have the radial centripetal acceleration 168 00:05:50,592 --> 00:05:51,425 right here. 169 00:05:51,425 --> 00:05:53,084 And we know the formula for centripetal acceleration, 170 00:05:53,084 --> 00:05:55,598 that's v squared over r, 171 00:05:55,598 --> 00:05:57,191 so I'm gonna plug v squared over r 172 00:05:57,191 --> 00:05:58,822 into the left hand side. 173 00:05:58,822 --> 00:06:00,581 That's the thing that's new. 174 00:06:00,581 --> 00:06:03,829 When we used Newton's second law for just regular forces, 175 00:06:03,829 --> 00:06:05,935 we just left it as a over here, 176 00:06:05,935 --> 00:06:07,544 but now, when you're using this law 177 00:06:07,544 --> 00:06:08,871 for the particular direction 178 00:06:08,871 --> 00:06:10,505 that is the centripetal direction, 179 00:06:10,505 --> 00:06:13,329 you're gonna replace a with v squared over r 180 00:06:13,329 --> 00:06:15,923 and then I set it equal to the net force 181 00:06:15,923 --> 00:06:18,814 in the centripetal direction over the mass. 182 00:06:18,814 --> 00:06:20,539 So what am I gonna plug in up here? 183 00:06:20,539 --> 00:06:22,295 What forces do I put up here? 184 00:06:22,295 --> 00:06:25,755 I mean, I've got normal force, tension, gravity. 185 00:06:25,755 --> 00:06:28,289 A common misconception is that people try to put them all 186 00:06:28,289 --> 00:06:29,122 into here. 187 00:06:29,122 --> 00:06:31,403 People put the gravitational force, the normal force, 188 00:06:31,403 --> 00:06:32,674 the tension, why not? 189 00:06:32,674 --> 00:06:35,999 But remember, if we've selected the centripetal direction, 190 00:06:35,999 --> 00:06:37,591 centripetal just means 191 00:06:37,591 --> 00:06:39,777 pointing toward the center of the circle, 192 00:06:39,777 --> 00:06:42,183 so I'm only going to plug in forces 193 00:06:42,183 --> 00:06:45,281 that are directed in toward the center of the circle, 194 00:06:45,281 --> 00:06:48,424 and that's not the normal force or the gravitational force. 195 00:06:48,424 --> 00:06:50,708 These forces do not point inward 196 00:06:50,708 --> 00:06:52,144 toward the center of the circle. 197 00:06:52,144 --> 00:06:53,999 The only force in this case 198 00:06:53,999 --> 00:06:55,510 that points toward the center of the circle 199 00:06:55,510 --> 00:06:56,698 is the tension force, 200 00:06:56,698 --> 00:06:57,935 and like we already said, 201 00:06:57,935 --> 00:06:59,903 that is the centripetal force. 202 00:06:59,903 --> 00:07:02,210 So over here, I'd have v squared over r, 203 00:07:02,210 --> 00:07:03,142 and that would equal 204 00:07:03,142 --> 00:07:05,397 the only force acting as the centripetal force 205 00:07:05,397 --> 00:07:07,163 is the tension. 206 00:07:07,163 --> 00:07:08,775 Now, should that be positive or negative? 207 00:07:08,775 --> 00:07:11,640 Well, we're gonna treat inward as positive, 208 00:07:11,640 --> 00:07:14,354 so any forces that point inward are gonna be positive. 209 00:07:14,354 --> 00:07:17,688 Is it possible for a centripetal force to be negative? 210 00:07:17,688 --> 00:07:18,718 It is. 211 00:07:18,718 --> 00:07:21,138 If there was some force that pointed outward, 212 00:07:21,138 --> 00:07:24,082 if for some reason there was another string 213 00:07:24,082 --> 00:07:26,111 pulling on the ball outward, 214 00:07:26,111 --> 00:07:28,618 we would include that force in this calculation, 215 00:07:28,618 --> 00:07:30,479 and we would include it with a negative sign, 216 00:07:30,479 --> 00:07:33,131 so forces that are directed out of the circle, 217 00:07:33,131 --> 00:07:34,628 we're gonna count as negative 218 00:07:34,628 --> 00:07:36,681 and forces that are directed into the circle, 219 00:07:36,681 --> 00:07:38,625 we're gonna count as positive in here. 220 00:07:38,625 --> 00:07:40,888 And if they're not directed into or out, 221 00:07:40,888 --> 00:07:43,837 we're not gonna include them in this calculation at all. 222 00:07:43,837 --> 00:07:44,670 Now, you might object. 223 00:07:44,670 --> 00:07:45,521 You might say, wait a minute. 224 00:07:45,521 --> 00:07:47,487 There is a force out of the circle. 225 00:07:47,487 --> 00:07:49,501 This ball wants to go out of the circle. 226 00:07:49,501 --> 00:07:51,624 There should be a force this way. 227 00:07:51,624 --> 00:07:54,960 This is often referred to as the centrifugal force, 228 00:07:54,960 --> 00:07:56,691 and that doesn't really exist. 229 00:07:56,691 --> 00:07:59,049 So when people say that there's an outward force 230 00:07:59,049 --> 00:08:02,048 trying to direct this ball out of the circle, 231 00:08:02,048 --> 00:08:05,183 they're usually referring to this centrifugal force, 232 00:08:05,183 --> 00:08:06,686 but this doesn't exist. 233 00:08:06,686 --> 00:08:08,714 It turns out this is not a real thing 234 00:08:08,714 --> 00:08:10,486 if you're using a good reference frame. 235 00:08:10,486 --> 00:08:13,220 There is no natural outward force 236 00:08:13,220 --> 00:08:14,996 for something going in a circle. 237 00:08:14,996 --> 00:08:15,829 You might object. 238 00:08:15,829 --> 00:08:16,806 You might be like, wait a minute. 239 00:08:16,806 --> 00:08:18,208 If I let go of this ball, 240 00:08:18,208 --> 00:08:19,967 it flies out of the circle. 241 00:08:19,967 --> 00:08:21,636 Won't it go flying off this way? 242 00:08:21,636 --> 00:08:23,442 And no, it won't. 243 00:08:23,442 --> 00:08:24,860 If you let go of the string right now, 244 00:08:24,860 --> 00:08:26,325 for some reason the string broke, 245 00:08:26,325 --> 00:08:29,799 at this moment this ball would not veer off that way. 246 00:08:29,799 --> 00:08:31,749 There's no force pushing it to the right. 247 00:08:31,749 --> 00:08:33,227 The ball, if the string broke, 248 00:08:33,227 --> 00:08:35,027 would just follow Newton's first law. 249 00:08:35,028 --> 00:08:37,374 It says it would just travel in a straight line 250 00:08:37,374 --> 00:08:39,785 with constant velocity, and it would roll off the table. 251 00:08:39,785 --> 00:08:41,683 So the reason you have to pull on the rope 252 00:08:41,683 --> 00:08:43,120 to get the ball to go in a circle 253 00:08:43,121 --> 00:08:45,800 is not because there's an outward force 254 00:08:45,800 --> 00:08:49,182 but because this ball wants to maintain its velocity. 255 00:08:49,182 --> 00:08:51,384 It has inertia, it wants to keep moving in a straight line, 256 00:08:51,384 --> 00:08:53,440 but you have to keep pulling on it 257 00:08:53,440 --> 00:08:56,775 to keep changing the direction of this velocity. 258 00:08:56,775 --> 00:08:58,255 So even though many people think 259 00:08:58,255 --> 00:09:00,404 there's an outward centrifugal force 260 00:09:00,404 --> 00:09:02,408 that's just naturally occurring 261 00:09:02,408 --> 00:09:03,421 on an object going in a circle, 262 00:09:03,421 --> 00:09:04,254 there is not. 263 00:09:04,254 --> 00:09:06,137 So finally, we can come back over to here. 264 00:09:06,137 --> 00:09:07,558 I can put my mass here. 265 00:09:07,558 --> 00:09:09,639 I can finally solve for my force of tension. 266 00:09:09,639 --> 00:09:12,161 If I do this, I'll multiply both sides by mass, 267 00:09:12,161 --> 00:09:14,054 and I just get that the force of tension is 268 00:09:14,054 --> 00:09:17,511 mass times the speed squared over the radius of the circle, 269 00:09:17,511 --> 00:09:18,739 and if I plug in my values, 270 00:09:18,739 --> 00:09:21,822 the mass was two, the speed was five, 271 00:09:22,690 --> 00:09:24,109 and you can't forget to square it. 272 00:09:24,109 --> 00:09:26,338 You divide by the radius which was 0.5, 273 00:09:26,338 --> 00:09:29,780 and you get that the force of tension had to be 100 Newtons. 274 00:09:29,780 --> 00:09:31,328 So in this case, the force of tension, 275 00:09:31,328 --> 00:09:35,788 which is the centripetal force, is equal to 100 Newtons. 276 00:09:35,788 --> 00:09:37,134 Now, some of you might be thinking, 277 00:09:37,134 --> 00:09:39,084 hey, this was way too much work 278 00:09:39,084 --> 00:09:41,525 for what ended up being a really simple problem. 279 00:09:41,525 --> 00:09:43,371 Why did we have to go through all the trouble 280 00:09:43,371 --> 00:09:46,594 of stating all of this problem solving strategy? 281 00:09:46,594 --> 00:09:47,450 And I agree. 282 00:09:47,450 --> 00:09:48,813 This one was easy, 283 00:09:48,813 --> 00:09:51,186 but other problems won't be easy. 284 00:09:51,186 --> 00:09:52,481 And if you don't have some sort of 285 00:09:52,481 --> 00:09:55,113 problem solving framework to fall back on, 286 00:09:55,113 --> 00:09:56,401 you'll be shooting blind 287 00:09:56,401 --> 00:09:59,388 and that's a lonely, lonely place to be. 288 00:09:59,388 --> 00:10:01,269 So let's use this same procedure, 289 00:10:01,269 --> 00:10:03,254 but let's look at a new problem. 290 00:10:03,254 --> 00:10:04,638 Let's say, you have this. 291 00:10:04,638 --> 00:10:05,779 Let's say you were riding your bike 292 00:10:05,779 --> 00:10:07,625 over a circular hill. 293 00:10:07,625 --> 00:10:09,862 So this gray line represents the pavement, 294 00:10:09,862 --> 00:10:11,178 and it starts off flat. 295 00:10:11,178 --> 00:10:12,814 But then the pavement veers upward 296 00:10:12,814 --> 00:10:14,870 and it creates this concrete hill 297 00:10:14,870 --> 00:10:17,186 that you ride over and then down 298 00:10:17,186 --> 00:10:18,395 and you ride over to this side. 299 00:10:18,395 --> 00:10:20,565 And all this purple circle is representing 300 00:10:20,565 --> 00:10:22,463 is the fact that if you were to continue 301 00:10:22,463 --> 00:10:24,916 this crest of the hill around into a circle, 302 00:10:24,916 --> 00:10:26,421 it would form this shape, 303 00:10:26,421 --> 00:10:29,217 so that gives us a way to define what the radius is 304 00:10:29,217 --> 00:10:30,777 of this top part of the hill. 305 00:10:30,777 --> 00:10:32,050 So, let's put some numbers in here. 306 00:10:32,050 --> 00:10:34,600 Let's say the radius of this hill was eight meters. 307 00:10:34,600 --> 00:10:37,013 Let's say the mass of you and your bike together 308 00:10:37,013 --> 00:10:39,257 are about 100 kilograms. 309 00:10:39,257 --> 00:10:40,740 And let's say you're riding over this hill 310 00:10:40,740 --> 00:10:42,499 at six meters per second. 311 00:10:42,499 --> 00:10:43,687 And let's say I asked you, 312 00:10:43,687 --> 00:10:46,016 what's the size of the normal force 313 00:10:46,016 --> 00:10:47,999 exerted on you and your bike 314 00:10:47,999 --> 00:10:51,014 as you ride over the crest of this hill 315 00:10:51,014 --> 00:10:53,087 at six meters per second? 316 00:10:53,087 --> 00:10:54,828 Now, let me show you what you can't do 317 00:10:54,828 --> 00:10:56,917 because most people would try to do this. 318 00:10:56,917 --> 00:10:58,830 They really want to say that the normal force 319 00:10:58,830 --> 00:11:01,625 is just gonna be equal to the force of gravity. 320 00:11:01,625 --> 00:11:04,692 Therefore, since the force of gravity is mg, 321 00:11:04,692 --> 00:11:06,784 the normal force should just be mg, 322 00:11:06,784 --> 00:11:07,947 but that can't be right. 323 00:11:07,947 --> 00:11:10,638 If the forces on an object are balanced and they cancel, 324 00:11:10,638 --> 00:11:13,393 the object is just gonna maintain its velocity, 325 00:11:13,393 --> 00:11:15,141 size, and direction, 326 00:11:15,141 --> 00:11:17,169 so this object, since it's going to the right, 327 00:11:17,169 --> 00:11:19,323 this bike would just continue going to the right 328 00:11:19,323 --> 00:11:21,613 and it would just hover straight off this hill. 329 00:11:21,613 --> 00:11:23,903 That'd be awesome, but that doesn't happen. 330 00:11:23,903 --> 00:11:25,697 This bike moves downward. 331 00:11:25,697 --> 00:11:27,776 It accelerates downward after this moment 332 00:11:27,776 --> 00:11:29,698 since it rides down the hill, 333 00:11:29,698 --> 00:11:32,036 so the downward force has got to be bigger. 334 00:11:32,036 --> 00:11:34,326 The force of gravity's gonna be bigger than the normal force 335 00:11:34,326 --> 00:11:36,624 'cause if it wasn't, this bike would just 336 00:11:36,624 --> 00:11:37,809 hover off into space. 337 00:11:37,809 --> 00:11:39,688 So how do you solve this problem? 338 00:11:39,688 --> 00:11:41,969 We use the same strategy we used before. 339 00:11:41,969 --> 00:11:43,466 We're gonna draw a force diagram, 340 00:11:43,466 --> 00:11:44,574 but we already did that. 341 00:11:44,574 --> 00:11:46,431 We're gonna use Newton's second law 342 00:11:46,431 --> 00:11:47,371 for one of the directions, 343 00:11:47,371 --> 00:11:48,544 and the direction we're gonna pick 344 00:11:48,544 --> 00:11:49,831 is the vertical direction. 345 00:11:49,831 --> 00:11:52,861 Now, is that vertical direction the centripetal direction? 346 00:11:52,861 --> 00:11:54,920 Yeah, it is because look at 347 00:11:54,920 --> 00:11:56,363 into the circle is downward. 348 00:11:56,363 --> 00:11:58,717 Because this bike is at the crest of the hill, 349 00:11:58,717 --> 00:12:02,771 down corresponds to pointing toward the center of the circle 350 00:12:02,771 --> 00:12:05,892 and upward corresponds to pointing away, 351 00:12:05,892 --> 00:12:08,582 radially away from the center of the circle. 352 00:12:08,582 --> 00:12:10,862 So, since I'm dealing with the centripetal direction, 353 00:12:10,862 --> 00:12:13,762 we plug in the formula for the centripetal acceleration, 354 00:12:13,762 --> 00:12:15,931 and the part where you have to be most careful 355 00:12:15,931 --> 00:12:19,371 is what you plug into the centripetal forces. 356 00:12:19,371 --> 00:12:22,599 Remember that into the circle counts as positive 357 00:12:22,599 --> 00:12:25,118 and out of the circle counts as negative. 358 00:12:25,118 --> 00:12:27,654 So both of these forces, normal and gravity, 359 00:12:27,654 --> 00:12:30,182 are gonna be included, but only one of them 360 00:12:30,182 --> 00:12:32,046 are gonna be included with a positive sign. 361 00:12:32,046 --> 00:12:33,499 Think about which one. 362 00:12:33,499 --> 00:12:34,989 Can you figure out which force 363 00:12:34,989 --> 00:12:38,089 would be included in here with a positive sign? 364 00:12:38,089 --> 00:12:39,827 If you said the force of gravity, 365 00:12:39,827 --> 00:12:41,424 you're right, which is weird. 366 00:12:41,424 --> 00:12:43,694 Usually, we treat the force of gravity as negative 367 00:12:43,694 --> 00:12:44,742 because it points down, 368 00:12:44,742 --> 00:12:47,745 but for centripetal forces, what we care about is into 369 00:12:47,745 --> 00:12:48,921 or out of the circle. 370 00:12:48,921 --> 00:12:51,740 So, I'm gonna treat gravity as a positive centripetal force. 371 00:12:51,740 --> 00:12:53,088 Gravity is the force 372 00:12:53,088 --> 00:12:55,051 pointing toward the center of the circle, 373 00:12:55,051 --> 00:12:56,540 and the normal force in this case 374 00:12:56,540 --> 00:12:58,708 is gonna be a negative centripetal force 375 00:12:58,708 --> 00:13:01,266 since it's directed out of the center of the circle. 376 00:13:01,266 --> 00:13:02,566 And then, we divide by our mass. 377 00:13:02,566 --> 00:13:04,934 And so, if we solve this for the normal force, 378 00:13:04,934 --> 00:13:05,953 if you do some algebra, 379 00:13:05,953 --> 00:13:07,825 we'll multiply both sides by m, 380 00:13:07,825 --> 00:13:10,733 we move over the F N and then move the m v squared 381 00:13:10,733 --> 00:13:13,044 to the other side and what we end up getting is that 382 00:13:13,044 --> 00:13:15,794 mg minus m times v squared over r 383 00:13:17,091 --> 00:13:18,867 is equal to the normal force, 384 00:13:18,867 --> 00:13:20,376 which if we plug in numbers, 385 00:13:20,376 --> 00:13:23,043 gives us 100 kilograms times 9.8 386 00:13:24,216 --> 00:13:28,186 minus 100 kilograms times the speed squared, 387 00:13:28,186 --> 00:13:31,319 that's gonna be six meters per second squared, 388 00:13:31,319 --> 00:13:33,550 divided by the radius of the circle we're traveling in 389 00:13:33,550 --> 00:13:34,717 which is eight meters, 390 00:13:34,717 --> 00:13:37,657 and you end up getting 530 Newtons. 391 00:13:37,657 --> 00:13:40,183 So the normal force on you and your bike 392 00:13:40,183 --> 00:13:41,342 as you ride over this hill 393 00:13:41,342 --> 00:13:43,067 is 530 Newtons. 394 00:13:43,067 --> 00:13:45,209 That is not equal to your weight. 395 00:13:45,209 --> 00:13:46,767 This is less than your weight. 396 00:13:46,767 --> 00:13:48,418 The force of gravity on you 397 00:13:48,418 --> 00:13:49,781 is gonna be m times g, 398 00:13:49,781 --> 00:13:52,280 that would be about 980 Newtons. 399 00:13:52,280 --> 00:13:54,318 So you experience less normal force, 400 00:13:54,318 --> 00:13:55,658 and this is natural. 401 00:13:55,658 --> 00:13:57,800 This is what happens when you ride over a hill fast. 402 00:13:57,800 --> 00:14:01,105 You feel slightly weightless as you go over that hill. 403 00:14:01,105 --> 00:14:02,321 If you've ever gone with a car 404 00:14:02,321 --> 00:14:03,513 a little too fast over a hill, 405 00:14:03,513 --> 00:14:06,067 you feel that whoa in your stomach, 406 00:14:06,067 --> 00:14:07,469 and you're like, hey, that was cool. 407 00:14:07,469 --> 00:14:09,827 That was the weightlessness you felt for a moment. 408 00:14:09,827 --> 00:14:12,855 If you go too fast, if you go too fast, 409 00:14:12,855 --> 00:14:14,651 this normal force will become zero. 410 00:14:14,651 --> 00:14:17,811 You'll subtract so much m v squared over r here, 411 00:14:17,811 --> 00:14:19,703 the normal force becomes zero. 412 00:14:19,703 --> 00:14:22,541 When that happens, you do become airborne, 413 00:14:22,541 --> 00:14:24,675 so be careful driving over those hills. 414 00:14:24,675 --> 00:14:27,269 If you drive too fast, you'll become airborne 415 00:14:27,269 --> 00:14:30,125 since your normal force is gonna become zero. 416 00:14:30,125 --> 00:14:32,534 So, recapping, when you solve centripetal force problems, 417 00:14:32,534 --> 00:14:34,652 be sure to draw a quality force diagram. 418 00:14:34,652 --> 00:14:36,534 Then use Newton's second law 419 00:14:36,534 --> 00:14:38,151 for one of the directions at a time. 420 00:14:38,151 --> 00:14:39,956 If you use the centripetal direction, 421 00:14:39,956 --> 00:14:42,610 the direction pointed radially into the circle, 422 00:14:42,610 --> 00:14:45,624 you can say that the acceleration in that direction 423 00:14:45,624 --> 00:14:46,818 is v squared over r, 424 00:14:46,818 --> 00:14:48,907 but be sure to only plug in forces 425 00:14:48,907 --> 00:14:50,489 that are directed radially, 426 00:14:50,489 --> 00:14:52,840 that is to say, forces that are pointed into 427 00:14:52,840 --> 00:14:54,106 or out of the circle. 428 00:14:54,106 --> 00:14:55,679 If they point into the circle, 429 00:14:55,679 --> 00:14:57,048 they're gonna be positive forces, 430 00:14:57,048 --> 00:14:58,399 and if they point out of the circle, 431 00:14:58,399 --> 00:00:00,000 they're gonna be negative forces.