WEBVTT 00:00:00.000 --> 00:00:01.543 - [Narrator] Imagine that in an effort 00:00:01.543 --> 00:00:03.588 to make bowling more exciting, 00:00:03.588 --> 00:00:05.735 bowling alleys put a big loop-the-loop 00:00:05.735 --> 00:00:08.004 in the middle of the lane, so you had to bowl the ball 00:00:08.004 --> 00:00:11.229 really fast to get the ball up and around the loop 00:00:11.229 --> 00:00:14.151 and then only afterward, it would go hit the bowling pins 00:00:14.151 --> 00:00:16.932 kinda like mini golf bowling or something like that. 00:00:16.932 --> 00:00:17.765 Well if you were gonna build this, 00:00:17.765 --> 00:00:19.469 you'd have to know at the top of the loop, 00:00:19.469 --> 00:00:20.991 this structure's gonna have to withstand 00:00:20.991 --> 00:00:22.927 a certain minimum amount of force. 00:00:22.927 --> 00:00:25.256 You might wanna know how strong do you have to make this. 00:00:25.256 --> 00:00:26.507 You can't have this thing breaking 00:00:26.507 --> 00:00:29.265 because it can't withstand the force of the bowling ball. 00:00:29.265 --> 00:00:30.646 So let's ask ourselves that question. 00:00:30.646 --> 00:00:34.320 How much force is this loop structure gonna have to be able 00:00:34.320 --> 00:00:37.569 to exert while this bowling ball is going around in a circle 00:00:37.569 --> 00:00:40.102 and let's pick this point at the top to analyze. 00:00:40.102 --> 00:00:41.461 We'll put some numbers in here. 00:00:41.461 --> 00:00:44.074 Let's say the ball was going eight meters per second 00:00:44.074 --> 00:00:45.064 at the top of the loop. 00:00:45.064 --> 00:00:48.214 That's pretty darn fast so someone really hurled 00:00:48.214 --> 00:00:49.450 this thing through here. 00:00:49.450 --> 00:00:51.889 Now let's say the loop has a radius of two meters 00:00:51.889 --> 00:00:54.755 and the bowling ball has a mass of four kilograms, 00:00:54.755 --> 00:00:56.687 which is around eight or nine pounds. 00:00:56.687 --> 00:00:58.555 Now that we have these numbers, we can ask the question: 00:00:58.555 --> 00:01:00.963 How much normal force is there gonna be 00:01:00.963 --> 00:01:03.010 between the loop and the ball? 00:01:03.010 --> 00:01:06.022 So in other words, what is the size of that normal force, 00:01:06.022 --> 00:01:08.129 the force between the two surfaces? 00:01:08.129 --> 00:01:09.975 This is what we'd have to know in order to figure out 00:01:09.975 --> 00:01:13.015 if our structure is strong enough to contain 00:01:13.015 --> 00:01:15.356 this bowling ball as it goes around in a circle. 00:01:15.356 --> 00:01:18.097 And it's also a classic centripetal force problem, 00:01:18.097 --> 00:01:19.069 so let's do this. 00:01:19.069 --> 00:01:20.280 What do we do first? 00:01:20.280 --> 00:01:22.457 We should always draw a force diagram. 00:01:22.457 --> 00:01:25.108 If we're looking for a force, you draw a force diagram. 00:01:25.108 --> 00:01:26.623 So what are the forces on this ball? 00:01:26.623 --> 00:01:28.712 You're gonna have a force of gravity downward, 00:01:28.712 --> 00:01:31.073 and the magnitude of the force of gravity is always 00:01:31.073 --> 00:01:34.960 given by M times G, where G represents the magnitude 00:01:34.960 --> 00:01:36.915 of the acceleration due to gravity. 00:01:36.915 --> 00:01:38.386 And we're gonna have a normal force as well. 00:01:38.386 --> 00:01:40.721 Now which way does this normal force point? 00:01:40.721 --> 00:01:42.851 A common misconception, people wanna say that 00:01:42.851 --> 00:01:44.891 that normal force points up because 00:01:44.891 --> 00:01:48.090 in a lot of other situations, the normal force points up. 00:01:48.090 --> 00:01:49.802 If you're just standing on the ground over here, 00:01:49.802 --> 00:01:52.190 the normal force on you is upward 00:01:52.190 --> 00:01:54.293 because it keeps you from falling through the ground, 00:01:54.293 --> 00:01:56.504 but that's not what this loop structure's doing up here. 00:01:56.504 --> 00:01:58.773 The loop structure isn't keeping you up. 00:01:58.773 --> 00:02:01.659 The loop structure's keeping you from flying out of the loop 00:02:01.659 --> 00:02:03.184 and that means this normal force is gonna 00:02:03.184 --> 00:02:04.365 have to point downward. 00:02:04.365 --> 00:02:06.076 So this is weird for a lot of people to think about, 00:02:06.076 --> 00:02:09.227 but because the surface is above this ball, 00:02:09.227 --> 00:02:10.973 the surface pushes down. 00:02:10.973 --> 00:02:12.650 Surfaces can only push. 00:02:12.650 --> 00:02:15.724 If the surface is below you, the surface has to push up. 00:02:15.724 --> 00:02:17.731 If the surface was to the side of you, 00:02:17.731 --> 00:02:19.242 the surface would have to push right. 00:02:19.242 --> 00:02:21.121 And if the surface was to the right of you, 00:02:21.121 --> 00:02:22.605 the surface would have to push left. 00:02:22.605 --> 00:02:25.425 Normal forces in other words, always push. 00:02:25.425 --> 00:02:27.881 So the force on the ball from the track 00:02:27.881 --> 00:02:30.361 is gonna be downward but vice versa. 00:02:30.361 --> 00:02:34.066 The force on the track from the ball is gonna be upward. 00:02:34.066 --> 00:02:35.918 So if this ball were going a little too fast 00:02:35.918 --> 00:02:37.141 and this were made out of wood, 00:02:37.141 --> 00:02:38.900 you might see this thing splinter 00:02:38.900 --> 00:02:40.670 because there's too much force pushing 00:02:40.670 --> 00:02:41.814 on the track this way. 00:02:41.814 --> 00:02:44.674 But if we're analyzing the ball, the force on the ball 00:02:44.674 --> 00:02:46.490 from the track is downward. 00:02:46.490 --> 00:02:48.073 And after you draw a force diagram, 00:02:48.073 --> 00:02:50.825 the next step is usually, if you wanna find a force, 00:02:50.825 --> 00:02:52.736 to use Newton's Second Law. 00:02:52.736 --> 00:02:54.415 And to keep the calculation simple, 00:02:54.415 --> 00:02:58.038 we typically use Newton's Second Law for a single dimension 00:02:58.038 --> 00:03:02.282 at at time, i.e. vertical, horizontal, centripetal. 00:03:02.282 --> 00:03:04.001 And that's what we're gonna use in this case 00:03:04.001 --> 00:03:07.051 because the normal force is pointing toward 00:03:07.051 --> 00:03:10.224 the center of the circular path and the normal force 00:03:10.224 --> 00:03:11.788 is the force we wanna find, 00:03:11.788 --> 00:03:13.956 we're gonna use Newton's Second Law 00:03:13.956 --> 00:03:15.947 for the centripetal direction and remember 00:03:15.947 --> 00:03:17.915 centripetal is just a fancy word 00:03:17.915 --> 00:03:20.314 for pointing toward the center of the circle. 00:03:20.314 --> 00:03:21.147 So, let's do it. 00:03:21.147 --> 00:03:23.233 Let's write down that the centripetal acceleration 00:03:23.233 --> 00:03:26.051 should equal the net centripetal force 00:03:26.051 --> 00:03:28.685 divided by the mass that's going in the circle. 00:03:28.685 --> 00:03:30.221 So if we choose this, we know that 00:03:30.221 --> 00:03:33.072 the centripetal acceleration can always be re-written 00:03:33.072 --> 00:03:36.790 as the speed squared divided by the radius 00:03:36.790 --> 00:03:39.636 of the circular path that the object is taking, 00:03:39.636 --> 00:03:41.752 and this should equal the net centripetal force 00:03:41.752 --> 00:03:45.100 divided by the mass of the object that's going in the circle 00:03:45.100 --> 00:03:47.319 and you gotta remember how we deal with signs here 00:03:47.319 --> 00:03:49.974 because we put a positive sign over here 00:03:49.974 --> 00:03:51.400 because we have a positive sign 00:03:51.400 --> 00:03:52.894 for our centripetal acceleration 00:03:52.894 --> 00:03:55.551 and our centripetal acceleration points toward 00:03:55.551 --> 00:03:57.648 the center of the circle always. 00:03:57.648 --> 00:04:00.967 Then in toward the center of the circle is going to be 00:04:00.967 --> 00:04:02.489 our positive direction, 00:04:02.489 --> 00:04:03.711 and that means for these forces, 00:04:03.711 --> 00:04:05.482 we're gonna plug in forces toward 00:04:05.482 --> 00:04:07.154 the center of the circle as positive. 00:04:07.154 --> 00:04:07.987 So let's do that. 00:04:07.987 --> 00:04:10.484 This is the part where most of the problem is happening. 00:04:10.484 --> 00:04:11.429 You gotta be careful here. 00:04:11.429 --> 00:04:12.408 I'm just gonna plug in. 00:04:12.408 --> 00:04:13.839 What are the centripetal forces? 00:04:13.839 --> 00:04:16.382 To figure that out, we just look at our force diagram. 00:04:16.382 --> 00:04:18.108 What forces do we have in our diagram. 00:04:18.108 --> 00:04:20.682 We've got the normal force and the force of gravity. 00:04:20.682 --> 00:04:22.083 Let's start with gravity. 00:04:22.083 --> 00:04:24.766 Is the gravitational force going to be a centripetal force. 00:04:24.766 --> 00:04:26.607 First of all, that's the question you have to ask. 00:04:26.607 --> 00:04:28.595 Does it even get included in here at all? 00:04:28.595 --> 00:04:30.039 And to figure that out you ask: 00:04:30.039 --> 00:04:31.505 Does it point centripetally? 00:04:31.505 --> 00:04:33.767 I.e. does it point toward the center of the circle? 00:04:33.767 --> 00:04:36.379 And it does so we're gonna include the force of gravity 00:04:36.379 --> 00:04:39.501 moreover because it points toward the center of the circle 00:04:39.501 --> 00:04:42.731 as opposed to radially away from the center of the circle. 00:04:42.731 --> 00:04:45.272 We're gonna include this as a positive centripetal force. 00:04:45.272 --> 00:04:48.182 Similarly, for the normal force, it also points 00:04:48.182 --> 00:04:50.551 toward the center of the circle, 00:04:50.551 --> 00:04:52.765 so we include it in this calculation 00:04:52.765 --> 00:04:55.973 and it as well will be a positive centripetal force. 00:04:55.973 --> 00:04:57.602 And now we can solve for the normal force. 00:04:57.602 --> 00:05:00.263 If I solve algebraically, I can multiply both sides 00:05:00.263 --> 00:05:03.598 by the mass and then I'd subtract MG from both sides. 00:05:03.598 --> 00:05:06.897 And that would give me the mass times V squared over R 00:05:06.897 --> 00:05:09.837 minus the magnitude of the force of gravity, 00:05:09.837 --> 00:05:13.306 which if we plug in numbers, gives us four kilograms 00:05:13.306 --> 00:05:16.339 times eight meters per second squared, 00:05:16.339 --> 00:05:17.813 you can't forget the square, 00:05:17.813 --> 00:05:21.381 divided by a two meter radius minus the magnitude 00:05:21.381 --> 00:05:24.840 of the force of gravity which is four kilograms times G 00:05:24.840 --> 00:05:29.588 which if you multiply that out gives you 88.8 newtons. 00:05:29.588 --> 00:05:32.987 This is how much downward force is exerted on the ball 00:05:32.987 --> 00:05:35.431 from the track but from Newton's Third Law, 00:05:35.431 --> 00:05:38.063 we know that that is also how much force the ball 00:05:38.063 --> 00:05:39.926 exerts upward on the track. 00:05:39.926 --> 00:05:41.324 So whatever you make this loop out of, 00:05:41.324 --> 00:05:44.694 it better be able to withstand 88.8 newtons 00:05:44.694 --> 00:05:47.566 if people are gonna be rolling this ball around the loop 00:05:47.566 --> 00:05:48.804 with eight meters per second. 00:05:48.804 --> 00:05:49.749 Now let me ask you this. 00:05:49.749 --> 00:05:51.866 What if the ball makes it over to here, right? 00:05:51.866 --> 00:05:54.572 So the ball rolls around and now it's over at this point. 00:05:54.572 --> 00:05:57.253 Now how much normal force is there at this point? 00:05:57.253 --> 00:05:59.421 Is it gonna be greater than, less than, 00:05:59.421 --> 00:06:02.130 or equal to 88.8 newtons. 00:06:02.130 --> 00:06:04.485 Well to figure it out, we should draw a force diagram. 00:06:04.485 --> 00:06:06.173 So there's gonna be a force of gravity. 00:06:06.173 --> 00:06:08.342 Again, it's gonna point straight down, 00:06:08.342 --> 00:06:09.819 and again, it's gonna be equal to 00:06:09.819 --> 00:06:11.815 at least the magnitude of it will be equal to 00:06:11.815 --> 00:06:15.199 the mass times the magnitude of acceleration due to gravity. 00:06:15.199 --> 00:06:16.856 And then we also have a normal force, 00:06:16.856 --> 00:06:19.655 but this time, the normal force does not push down. 00:06:19.655 --> 00:06:21.773 Remember, surfaces push outward 00:06:21.773 --> 00:06:24.202 and if this surface is to the left of the ball, 00:06:24.202 --> 00:06:26.070 the surface pushes to the right. 00:06:26.070 --> 00:06:28.181 This time our normal force points to the right. 00:06:28.181 --> 00:06:30.537 And let's assume this a well oiled track 00:06:30.537 --> 00:06:32.304 so there's really no friction to worry about. 00:06:32.304 --> 00:06:35.100 In that case, these would again be the only two forces. 00:06:35.100 --> 00:06:36.751 So what about the answer to our question. 00:06:36.751 --> 00:06:40.072 Will this normal force now be bigger, less than, 00:06:40.072 --> 00:06:43.283 or equal to what the normal force was at the top. 00:06:43.283 --> 00:06:45.378 Well I'm gonna argue it's gotta be bigger, 00:06:45.378 --> 00:06:47.310 and I'm gonna argue it's gonna have to be much bigger 00:06:47.310 --> 00:06:49.352 because when you plug in over here, 00:06:49.352 --> 00:06:50.690 into the centripetal forces, 00:06:50.690 --> 00:06:54.162 you only plug in forces that point radially. 00:06:54.162 --> 00:06:55.709 That is to say centripetally, 00:06:55.709 --> 00:06:57.994 either into the circle, which would be positive, 00:06:57.994 --> 00:07:00.922 or radially out of the circle, which would be negative. 00:07:00.922 --> 00:07:04.085 If they neither point into nor out of the circle, 00:07:04.085 --> 00:07:06.776 you don't include them in this calculation at all 00:07:06.776 --> 00:07:08.908 because they aren't pointing in the direction 00:07:08.908 --> 00:07:10.469 of the centripetal acceleration. 00:07:10.469 --> 00:07:12.087 In other words, they're not causing 00:07:12.087 --> 00:07:13.705 the centripetal acceleration. 00:07:13.705 --> 00:07:16.772 So for this case over here, gravity is no longer 00:07:16.772 --> 00:07:19.355 a centripetal force because the force of gravity 00:07:19.355 --> 00:07:22.630 no longer points toward the center of the circle. 00:07:22.630 --> 00:07:25.325 This force of gravity is tangential to the circle. 00:07:25.325 --> 00:07:27.977 It's neither pointing into nor out of, 00:07:27.977 --> 00:07:29.918 which means it doesn't factor into 00:07:29.918 --> 00:07:31.362 the centripetal motion at all. 00:07:31.362 --> 00:07:34.280 It merely tries to speed the ball up at this point. 00:07:34.280 --> 00:07:36.442 It does not change the ball's direction, 00:07:36.442 --> 00:07:39.096 which means it doesn't contribute to making this ball 00:07:39.096 --> 00:07:42.211 go in a circle, so we don't include it in this calculation. 00:07:42.211 --> 00:07:43.874 So when we solved for the normal force, 00:07:43.874 --> 00:07:45.650 we'd multiply both sides by M, 00:07:45.650 --> 00:07:47.326 we would not have an MG anymore. 00:07:47.326 --> 00:07:49.817 So we wouldn't be subtracting this term 00:07:49.817 --> 00:07:51.890 and that's gonna make our normal force bigger. 00:07:51.890 --> 00:07:55.468 Moreover, the speed of this ball's gonna increase 00:07:55.468 --> 00:07:56.810 compared to what it was up here. 00:07:56.810 --> 00:07:59.617 So as the ball falls down, gravity's going to 00:07:59.617 --> 00:08:03.256 speed this ball up and now that it's speed is larger, 00:08:03.256 --> 00:08:05.559 and we're not subtracting anything from it, 00:08:05.559 --> 00:08:08.404 The normal force will be much greater at this point 00:08:08.404 --> 00:08:10.603 compared to what it was at the top of the loop. 00:08:10.603 --> 00:08:12.136 So recapping, when you wanna solve 00:08:12.136 --> 00:08:13.405 the centripetal force problem, 00:08:13.405 --> 00:08:15.411 always draw your force diagram first. 00:08:15.411 --> 00:08:17.342 If you choose to analyze the forces 00:08:17.342 --> 00:08:19.574 in the centripetal direction, in other words, 00:08:19.574 --> 00:08:22.360 for the direction in toward the center of the circle, 00:08:22.360 --> 00:08:25.544 make sure you only plug in forces that are into, 00:08:25.544 --> 00:08:28.820 radially into the circle or radially out of the circle. 00:08:28.820 --> 00:08:31.586 If they're radially into the circle, you make them positive. 00:08:31.586 --> 00:08:33.520 If they were radially out of the circle, 00:08:33.520 --> 00:08:35.030 you would make them negative. 00:08:35.030 --> 00:08:37.591 And if they neither point radially inward, 00:08:37.591 --> 00:08:38.741 toward the center of the circle 00:08:38.741 --> 00:08:41.907 or radially outward, away from the center of the circle, 00:08:41.907 --> 00:08:44.251 you just do not include those forces at all 00:08:44.251 --> 00:08:47.418 when using this centripetal direction.