WEBVTT 00:00:00.709 --> 00:00:01.616 - [Voiceover] Oh, it's time. 00:00:01.616 --> 00:00:03.391 It's time for the super hot tension problem. 00:00:03.391 --> 00:00:04.837 We're about to do this right here. 00:00:04.837 --> 00:00:07.032 We've got our super hot can of red peppers 00:00:07.032 --> 00:00:08.821 hanging from these strings. 00:00:08.821 --> 00:00:11.761 We want to know what the tension is in these ropes. 00:00:11.761 --> 00:00:15.077 This is for real now, this is a real tension problem. 00:00:15.077 --> 00:00:16.309 And here's the deal. 00:00:16.309 --> 00:00:18.377 You might look at this, you might get frightened. 00:00:18.377 --> 00:00:20.232 You might think, I've gotta come up with 00:00:20.232 --> 00:00:22.551 a completely new strategy to tackle this. 00:00:22.551 --> 00:00:24.439 I've gotta throw away everything I've learned 00:00:24.439 --> 00:00:25.902 and just try something new. 00:00:25.902 --> 00:00:27.058 And that's a lie. 00:00:27.058 --> 00:00:28.499 You should not lie to yourself. 00:00:28.499 --> 00:00:30.439 Use the same process. 00:00:30.439 --> 00:00:32.290 We're gonna use the same process we used 00:00:32.290 --> 00:00:33.408 for the easy tension problems, 00:00:33.408 --> 00:00:35.134 because it's gonna lead us to the answer again. 00:00:35.134 --> 00:00:39.170 Be careful. Don't stray from the strategy here. 00:00:39.170 --> 00:00:40.109 The strategy works. 00:00:40.109 --> 00:00:41.650 So we're gonna draw our force diagram first. 00:00:41.650 --> 00:00:43.178 That's what we always do. 00:00:43.178 --> 00:00:45.031 We're gonna say that the forces are 00:00:45.031 --> 00:00:48.170 force of gravity on this can of red peppers, 00:00:48.170 --> 00:00:50.544 which is MG, and if it's 3 kilograms, 00:00:50.544 --> 00:00:52.962 we know 3 kilograms times about 10, 00:00:52.962 --> 00:00:55.708 we're gonna say, let's approximate G as 10 again 00:00:55.708 --> 00:00:57.417 to make the numbers come out nice. 00:00:57.417 --> 00:01:00.816 So instead of using 9.8, we'll say G is about 10, 00:01:00.816 --> 00:01:02.910 and so we'll say 3 kilograms 00:01:02.910 --> 00:01:07.077 times 10 meters per second squared is gonna be 30 Newtons. 00:01:09.947 --> 00:01:12.728 And so the force of gravity downward is 30 Newtons. 00:01:12.728 --> 00:01:14.078 What other forces do we have? 00:01:14.078 --> 00:01:17.330 We've got this T1, remember tension does not push. 00:01:17.330 --> 00:01:20.197 Ropes can't push, ropes can only pull, 00:01:20.197 --> 00:01:21.880 so T1's gonna pull that way. 00:01:21.880 --> 00:01:23.961 So I'm gonna draw T1 coming this way. 00:01:23.961 --> 00:01:25.378 So here's our T1. 00:01:26.219 --> 00:01:28.447 And then we're gonna have T2 pointing this way, 00:01:28.447 --> 00:01:29.630 so this is T2. 00:01:29.630 --> 00:01:31.731 Again, T2 pulls, just like all tension. 00:01:31.731 --> 00:01:34.018 Tension pulls, tension can't push. 00:01:34.018 --> 00:01:37.101 So I've got tension 2 going this way. 00:01:39.580 --> 00:01:41.656 That's it, that's our force diagram. 00:01:41.656 --> 00:01:42.584 There's no other forces. 00:01:42.584 --> 00:01:43.849 I don't draw a normal force, 00:01:43.849 --> 00:01:46.276 'cause this can isn't in contact with another surface. 00:01:46.276 --> 00:01:48.956 So there's no normal force, you've got these two tensions, 00:01:48.956 --> 00:01:50.436 the force of gravity. 00:01:50.436 --> 00:01:52.155 And now we do the same thing we always do. 00:01:52.155 --> 00:01:54.742 After our force diagram, we use Newton's Second Law 00:01:54.742 --> 00:01:56.809 in one direction or another. 00:01:56.809 --> 00:01:57.863 So let's do it. 00:01:57.863 --> 00:02:00.687 Let's say that acceleration is the net force 00:02:00.687 --> 00:02:03.779 in a given direction, divided by the mass. 00:02:03.779 --> 00:02:05.827 Which direction did we pick again? 00:02:05.827 --> 00:02:08.786 It's hard to say, we've got forces vertical, 00:02:08.786 --> 00:02:10.137 we've got forces horizontal. 00:02:10.137 --> 00:02:13.258 There's only two directions to pick, X or Y in this problem. 00:02:13.258 --> 00:02:15.222 We're gonna pick the vertical direction, 00:02:15.222 --> 00:02:17.094 even though it doesn't really matter too much. 00:02:17.094 --> 00:02:19.416 But because we know one of the forces in the vertical 00:02:19.416 --> 00:02:20.909 direction, we know the force of gravity. 00:02:20.909 --> 00:02:23.089 Force of gravity is 30 Newtons. 00:02:23.089 --> 00:02:25.267 Usually that's a guod strategy, pick the direction 00:02:25.267 --> 00:02:27.770 that you know something about at least. 00:02:27.770 --> 00:02:29.526 So we're gonna do that here. 00:02:29.526 --> 00:02:31.392 We're gonna say that the acceleration vertically 00:02:31.392 --> 00:02:33.896 equals to the net force vertically over the mass. 00:02:33.896 --> 00:02:34.968 And so now we plug in. 00:02:34.968 --> 00:02:36.784 If this can is just sitting here, 00:02:36.784 --> 00:02:39.473 if there's no acceleration, if this is in not an elevator 00:02:39.473 --> 00:02:42.557 transporting these peppers up or down, 00:02:42.557 --> 00:02:44.447 and it's not in a rocket, if it's just sitting here 00:02:44.447 --> 00:02:47.466 with no acceleration, our acceleration will be zero. 00:02:47.466 --> 00:02:50.995 That's gonna equal the net force and the vertical direction. 00:02:50.995 --> 00:02:52.377 So what are we gonna have? 00:02:52.377 --> 00:02:54.197 So what are the forces in the vertical direction here? 00:02:54.197 --> 00:02:57.222 One force is this 30 Newton force of gravity. 00:02:57.222 --> 00:03:00.571 This points down, we're gonna assume upward is positive, 00:03:00.571 --> 00:03:02.194 that means down in a negative. 00:03:02.194 --> 00:03:03.608 So I'll just put -30 Newtons. 00:03:03.608 --> 00:03:05.916 I could have written -MG, 00:03:05.916 --> 00:03:07.462 but we already knew it was 30 Newtons, 00:03:07.462 --> 00:03:08.921 so I'll write -30 Newtons. 00:03:08.921 --> 00:03:10.583 Then we've got T1 and T2. 00:03:10.583 --> 00:03:12.255 Both of those point up. 00:03:12.255 --> 00:03:13.648 But they don't completely point up, 00:03:13.648 --> 00:03:15.004 they partially point up. 00:03:15.004 --> 00:03:17.326 So part of them points to the right, 00:03:17.326 --> 00:03:19.584 part of them points upward. 00:03:19.584 --> 00:03:22.656 Only this vertical component, we'll call it T1Y, 00:03:22.656 --> 00:03:25.179 is gonna get included into this calculation, 00:03:25.179 --> 00:03:28.212 'cause this calculation only uses Y directed forces. 00:03:28.212 --> 00:03:30.868 And the reason is only Y directed forces, 00:03:30.868 --> 00:03:34.325 vertical forces, affect the vertical acceleration. 00:03:34.325 --> 00:03:36.427 So this T1Y points upward, 00:03:36.427 --> 00:03:39.785 I'll do plus T1 in the Y direction. 00:03:39.785 --> 00:03:41.675 And similarly, this T2. 00:03:41.675 --> 00:03:44.113 It doesn't all point vertically, 00:03:44.113 --> 00:03:45.594 only part of it points vertically. 00:03:45.594 --> 00:03:49.138 So I'll write this as T2 in the Y direction. 00:03:49.138 --> 00:03:50.437 And that's also upward, 00:03:50.437 --> 00:03:52.597 so since that's up, I'll count it 00:03:52.597 --> 00:03:55.314 as plus T2 in the Y direction. 00:03:55.314 --> 00:03:57.101 And that's it, that's all our forces. 00:03:57.101 --> 00:04:00.483 Notice we can't plug in the total amount T2 in this 00:04:00.483 --> 00:04:02.870 formula, 'cause only part of it points up. 00:04:02.870 --> 00:04:05.340 Similarly, we have to plug in only the vertical component 00:04:05.340 --> 00:04:09.031 of the T1 force because only part of it points vertically. 00:04:09.031 --> 00:04:13.175 And then we divide by the mass, the mass is 3 kilograms. 00:04:13.175 --> 00:04:15.875 But we're gonna multiply both sides by 3 kilograms, 00:04:15.875 --> 00:04:19.668 and we're gonna get zero equals all of this right here, 00:04:19.668 --> 00:04:21.065 so I'll just copy this right here. 00:04:21.065 --> 00:04:25.233 We use this over again, that comes down right there. 00:04:26.489 --> 00:04:29.331 But now there's nothing on the bottom here. 00:04:29.331 --> 00:04:30.227 So what do we do at this point? 00:04:30.227 --> 00:04:31.556 Now you might think we're stuck. 00:04:31.556 --> 00:04:33.525 I mean, we've got two unknowns in here. 00:04:33.525 --> 00:04:35.605 I can't solve for either one, 00:04:35.605 --> 00:04:36.938 I don't know either one of these. 00:04:36.938 --> 00:04:38.624 I know they have to add up to 30, 00:04:38.624 --> 00:04:40.920 so I'd do fine, if I added 30 to both sides, 00:04:40.920 --> 00:04:43.129 I'd realize that these two vertical components 00:04:43.129 --> 00:04:45.097 of these tension forces added up 00:04:45.097 --> 00:04:47.308 have to add up to 30, and that makes sense. 00:04:47.308 --> 00:04:49.319 They have to balance the force downward. 00:04:49.319 --> 00:04:52.664 But I don't know either of them, so how do I solve here? 00:04:52.664 --> 00:04:54.193 Well, let's do this. 00:04:54.193 --> 00:04:57.380 If you ever get stuck on one of the force equations 00:04:57.380 --> 00:05:00.090 for a single direction, just go to the next equation. 00:05:00.090 --> 00:05:01.889 Let's try A in the X direction. 00:05:01.889 --> 00:05:04.053 So for A in the X direction, we have the net force 00:05:04.053 --> 00:05:06.198 in the X direction, over the mass, 00:05:06.198 --> 00:05:09.231 again, the acceleration is gonna be zero 00:05:09.231 --> 00:05:11.887 if these peppers are not accelerating horizontally. 00:05:11.887 --> 00:05:14.270 So unless this thing's in a train car or something, 00:05:14.270 --> 00:05:16.435 and the whole thing's accelerating, 00:05:16.435 --> 00:05:18.076 then you might have horizontal acceleration. 00:05:18.076 --> 00:05:19.946 And if it did, it's not that big of a deal, 00:05:19.946 --> 00:05:21.639 you just plug it in there. 00:05:21.639 --> 00:05:23.341 But assuming it's acceleration zero, 00:05:23.341 --> 00:05:24.725 because the peppers are just sitting there, 00:05:24.725 --> 00:05:25.768 not changing their velocity, we'll plug in zero. 00:05:25.768 --> 00:05:28.206 We'll plug in the forces in the X direction. 00:05:28.206 --> 00:05:31.193 These are gonna be T1 in the X. 00:05:31.193 --> 00:05:33.927 So part of this T1 points in the X direction. 00:05:33.927 --> 00:05:37.566 Similarly, part of T2 points in the X direction. 00:05:37.566 --> 00:05:38.706 We'll call this T2X. 00:05:38.706 --> 00:05:40.833 We use these as the magnitude. 00:05:40.833 --> 00:05:42.994 Let's say T2X is the magnitude of the force 00:05:42.994 --> 00:05:45.140 that T2 pulls with to the left, 00:05:45.140 --> 00:05:49.307 and T1 is the magnitude that T1 pulls with to the right. 00:05:50.723 --> 00:05:53.079 So to plug these in, we've got to decide 00:05:53.079 --> 00:05:55.585 whether they should be positive or negative. 00:05:55.585 --> 00:05:57.988 So this T1X, since it pulls to the right, 00:05:57.988 --> 00:05:59.696 T1X will be positive. 00:05:59.696 --> 00:06:02.835 We're gonna consider rightward to be the positive direction, 00:06:02.835 --> 00:06:05.252 'cause that's the typical convention that we're gonna adopt. 00:06:05.252 --> 00:06:08.229 And T2X pulls to the left. 00:06:08.229 --> 00:06:10.033 That's gonna be a negative contribution, 00:06:10.033 --> 00:06:12.587 so minus T2 in the X direction. 00:06:12.587 --> 00:06:14.115 'Cause leftward would be negative. 00:06:14.115 --> 00:06:16.597 We divided by the mass, the mass was 3 kilograms, 00:06:16.597 --> 00:06:19.314 but again, we'll multiply both sides by 3, 00:06:19.314 --> 00:06:23.313 we'll get zero equals, and then we just get T, 00:06:23.313 --> 00:06:24.502 the same thing up here, 00:06:24.502 --> 00:06:28.669 so we'll just copy this thing here, put it down here. 00:06:30.614 --> 00:06:31.673 And again, you might be concerned. 00:06:31.673 --> 00:06:33.526 I can't solve this either. 00:06:33.526 --> 00:06:37.132 I mean, I can solve for T1X, but look at what I get. 00:06:37.132 --> 00:06:40.870 If I just multi, or if I added T2X to both sides, 00:06:40.870 --> 00:06:42.432 I'm just gonna get T1 in the X direction 00:06:42.432 --> 00:06:45.349 has to equal T2 in the X direction. 00:06:46.224 --> 00:06:47.684 And that makes sense. 00:06:47.684 --> 00:06:49.472 These two forces have to be equal and opposite, 00:06:49.472 --> 00:06:52.137 because they have to cancel so that you have no acceleration 00:06:52.137 --> 00:06:53.144 in the X direction. 00:06:53.144 --> 00:06:55.373 And this was not drawn proportionately, sorry, 00:06:55.373 --> 00:06:57.452 this should be the exact same size as this force 00:06:57.452 --> 00:06:59.423 because they have to cancel, 00:06:59.423 --> 00:07:01.517 since there's no horizontal acceleration. 00:07:01.517 --> 00:07:02.817 But what do we do? 00:07:02.817 --> 00:07:05.187 We can't solve this equation we got from X direction. 00:07:05.187 --> 00:07:08.635 We can't solve this equation we got from the Y direction. 00:07:08.635 --> 00:07:11.223 Whenever this happens, when you get two equations, 00:07:11.223 --> 00:07:12.761 and you can't solve either 00:07:12.761 --> 00:07:13.995 because there's too many unknowns, 00:07:13.995 --> 00:07:16.666 you're gonna have to end up plugging one into the other. 00:07:16.666 --> 00:07:18.407 But I can't even do that yet. 00:07:18.407 --> 00:07:20.358 I've got four different variables here. 00:07:20.358 --> 00:07:22.275 T1X, T2X, T1Y, and T2Y, 00:07:24.305 --> 00:07:25.720 these are all four different variables, 00:07:25.720 --> 00:07:28.387 I've only got two equations, I can't solve this. 00:07:28.387 --> 00:07:30.383 So the trick, the trick we're gonna use 00:07:30.383 --> 00:07:31.717 that a lot of people don't like doing 00:07:31.717 --> 00:07:33.147 because it's a little more sophisticated, 00:07:33.147 --> 00:07:37.144 now we've gotta put these all in terms of T1 and T2 00:07:37.144 --> 00:07:38.595 so that we can solve. 00:07:38.595 --> 00:07:41.859 If I put T1Y in terms of the total T1, 00:07:41.859 --> 00:07:44.706 and then sines of angles, and cosines of angles, 00:07:44.706 --> 00:07:48.039 and I put T2Y in terms of T2 and angles, 00:07:49.582 --> 00:07:51.240 and I do the same thing for 1X and 2X, 00:07:51.240 --> 00:07:53.042 I'll have two equations, and the only two unknowns 00:07:53.042 --> 00:07:57.501 will be T1 and T2, then we can finally solve. 00:07:57.501 --> 00:07:59.435 If that didn't make any sense, here's what I'm saying. 00:07:59.435 --> 00:08:02.751 I'm saying figure out what T1Y is in terms of T1. 00:08:02.751 --> 00:08:05.629 So I know this angle here, let's figure out these angles. 00:08:05.629 --> 00:08:08.666 So these angles here are, if this is 30, 00:08:08.666 --> 00:08:11.202 this angle down here has to be 30 because these 00:08:11.202 --> 00:08:13.312 are alternate interior angles. 00:08:13.312 --> 00:08:14.696 And if you don't believe me, 00:08:14.696 --> 00:08:16.811 imagine this big triangle over here, 00:08:16.811 --> 00:08:18.692 where this is a right angle. 00:08:18.692 --> 00:08:20.401 So this triangle from here to there, down to here, 00:08:20.401 --> 00:08:24.725 up to here, if this is 30, that's 90, this has gotta be 60, 00:08:24.725 --> 00:08:27.212 'cause it all adds up to 180 for a triangle. 00:08:27.212 --> 00:08:29.132 And if this right angle is 90, and this side's 60, 00:08:29.132 --> 00:08:31.291 this side's gotta be 30. 00:08:31.291 --> 00:08:34.102 Similarly, this side's a right angle. 00:08:34.102 --> 00:08:36.131 Look at this triangle, 60, 90, 00:08:36.131 --> 00:08:37.841 that means this would have to be 30. 00:08:37.841 --> 00:08:40.751 And so if I come down here, this angle would have to be 60. 00:08:40.751 --> 00:08:42.333 Just like this one, 00:08:43.188 --> 00:08:45.919 'cause it's an alternate interior angle, so that's 60. 00:08:45.919 --> 00:08:49.788 So this angle here is 60, this angle here is 30, 00:08:49.788 --> 00:08:51.934 we can figure out what these components are 00:08:51.934 --> 00:08:53.462 in terms of the total vectors. 00:08:53.462 --> 00:08:54.957 Once we find those, 00:08:54.957 --> 00:08:56.502 we're gonna plug those expressions into here, 00:08:56.502 --> 00:08:57.753 and that will let us solve. 00:08:57.753 --> 00:08:59.362 In other words, T1Y is gonna be, 00:08:59.362 --> 00:09:02.108 once you do this for awhile you realize, 00:09:02.108 --> 00:09:03.604 this is the opposite side. 00:09:03.604 --> 00:09:05.440 So this component here is going to be 00:09:05.440 --> 00:09:07.607 total T1 times sine of 30. 00:09:09.082 --> 00:09:11.132 Because it's the opposite side. 00:09:11.132 --> 00:09:13.779 And if that didn't make sense, we'll derive it right here. 00:09:13.779 --> 00:09:15.957 So what we're saying is that sine of 30, 00:09:15.957 --> 00:09:19.207 sine of 30 is opposite over hypotenuse, 00:09:21.588 --> 00:09:25.150 and in this case, the opposite side is T1Y. 00:09:25.150 --> 00:09:29.150 So T1Y over the total T1 is equal to sine of 30. 00:09:30.918 --> 00:09:33.275 And we can solve this for T1Y now, 00:09:33.275 --> 00:09:36.106 we can get the T1Y if I multiply both sides by T1. 00:09:36.106 --> 00:09:39.273 I get that that's T1 times sine of 30. 00:09:40.764 --> 00:09:42.199 So that's what I said down here. 00:09:42.199 --> 00:09:45.074 T1 is just T, oh sorry, forgot the one. 00:09:45.074 --> 00:09:46.741 T1 times sine of 30. 00:09:48.048 --> 00:09:50.506 Similarly, if you do the same thing with cosine 30, 00:09:50.506 --> 00:09:53.506 you'll get that T1X is T1 cosine 30, 00:09:55.852 --> 00:09:57.702 by the exact same process. 00:09:57.702 --> 00:10:01.147 Similarly over here, T2 is going to be, 00:10:01.147 --> 00:10:03.564 I'm sorry, T2X is gonna be 2. 00:10:05.004 --> 00:10:09.171 So T2 cosine 60, because this is the adjacent side. 00:10:10.133 --> 00:10:12.966 And T2Y is gonna be T2 sine of 60. 00:10:15.068 --> 00:10:16.531 And if any of that doesn't make sense, 00:10:16.531 --> 00:10:19.712 just go back to the definition of sine and cosine, 00:10:19.712 --> 00:10:22.152 write what the opposite side is, 00:10:22.152 --> 00:10:25.600 the total hypotenuse side, solve for your expression, 00:10:25.600 --> 00:10:26.719 you'll get these. 00:10:26.719 --> 00:10:29.191 If you don't believe me on those, try those out yourselves. 00:10:29.191 --> 00:10:31.494 But those are what these components are, 00:10:31.494 --> 00:10:35.966 in terms of T2 and the angles T2, T1 and the angles. 00:10:35.966 --> 00:10:37.574 And why are we doing this? 00:10:37.574 --> 00:10:39.839 We're doing this so that when plug in over here, 00:10:39.839 --> 00:10:41.251 we'll only have two variables. 00:10:41.251 --> 00:10:45.008 In other words, if I plug T1Y, this expression here, 00:10:45.008 --> 00:10:48.758 T1 sine 30 in for T1Y, similarly if I plug in 00:10:49.703 --> 00:10:53.870 T2Y is T2 sine 60 into this expression right there 00:10:55.357 --> 00:10:57.408 for T2Y, look at what I'll get. 00:10:57.408 --> 00:10:58.987 I'll get zero equals. 00:10:58.987 --> 00:11:01.654 So I'll get negative 30 Newtons, 00:11:02.712 --> 00:11:06.795 and then I'll get plus T1Y was T1 sine 30, so T1, 00:11:08.495 --> 00:11:11.145 and then sine 30, we can clean this up a little bit. 00:11:11.145 --> 00:11:12.334 Sine 30 is just a half. 00:11:12.334 --> 00:11:15.938 So I'll just write T1 over 2, and then 00:11:15.938 --> 00:11:17.874 'cause sine 30 is just one half. 00:11:17.874 --> 00:11:20.874 And then T2Y is gonna be T2 sine 60, 00:11:23.726 --> 00:11:26.651 and sine 60 is just root 3 over 2. 00:11:26.651 --> 00:11:30.818 So I'll write this as plus T2 over 2, and then times root 3. 00:11:33.088 --> 00:11:35.524 And you might think this is no better. 00:11:35.524 --> 00:11:37.477 I mean this is still a horrible mess right here. 00:11:37.477 --> 00:11:40.986 But, look at. This is in terms of T1 and T2. 00:11:40.986 --> 00:11:42.266 That's what I'm gonna do over here. 00:11:42.266 --> 00:11:44.237 I'm gonna put these in terms of T1 and T2, 00:11:44.237 --> 00:11:45.505 and then we can solve. 00:11:45.505 --> 00:11:48.557 So T1X is T1 over cosine 30, 00:11:48.557 --> 00:11:52.798 so I'm gonna write this as T1 times cosine 30, 00:11:52.798 --> 00:11:55.381 and cosine 30 is root 3 over 2, 00:11:56.568 --> 00:11:59.576 so this is T1 over 2 times root 3. 00:11:59.576 --> 00:12:02.909 And that should equal T2X is right here, 00:12:03.937 --> 00:12:07.354 That's T2 cosine 60, cosine 60 is a half. 00:12:08.224 --> 00:12:10.641 So T2X is gonna be T2 over 2. 00:12:12.348 --> 00:12:13.946 So T2 over 2. 00:12:13.946 --> 00:12:16.139 So what I'm doing is, if this doesn't make sense, 00:12:16.139 --> 00:12:18.873 I'm just substituting what these components are 00:12:18.873 --> 00:12:22.750 in terms of the total magnitude in the angle. 00:12:22.750 --> 00:12:25.269 And I do this, because look at what I have now, 00:12:25.269 --> 00:12:26.683 I have got one equation with T1 and T2. 00:12:26.683 --> 00:12:28.663 I've got another equation with T1 and T2. 00:12:28.663 --> 00:12:30.927 So what I'm gonna do to solve these, 00:12:30.927 --> 00:12:32.843 when we have two equations and two unknowns, 00:12:32.843 --> 00:12:34.925 you have to solve for one of these variables, 00:12:34.925 --> 00:12:37.198 and then substitute it into the other equation. 00:12:37.198 --> 00:12:40.127 That way you'll get one equation with one unknown. 00:12:40.127 --> 00:12:42.220 And you try to get the math right, 00:12:42.220 --> 00:12:43.197 and you'll get the problem. 00:12:43.197 --> 00:12:44.920 So I'm gonna solve this one is easier, 00:12:44.920 --> 00:12:47.572 so I'm gonna solve this one for, let's just say T2. 00:12:47.572 --> 00:12:48.982 So if we solve this for T2, 00:12:48.982 --> 00:12:52.235 I get that T2 equals, well, I can multiply 00:12:52.235 --> 00:12:56.577 both sides by 2, and I'll get T1 times root 3. 00:12:56.577 --> 00:13:00.947 So T1 times root 3, because the 2 here cancels with this 2, 00:13:00.947 --> 00:13:04.182 or when I multiply both sides by 2 it cancels out. 00:13:04.182 --> 00:13:07.936 So we get that T2 equals T1 root 3. This is great. 00:13:07.936 --> 00:13:12.103 I can substitute T2 as T1 root 3 into here for T2. 00:13:13.998 --> 00:13:15.431 And the reason I do that, 00:13:15.431 --> 00:13:17.655 is I'll get one equation with one unknown. 00:13:17.655 --> 00:13:19.541 I'll only have T1 in that equation now. 00:13:19.541 --> 00:13:23.168 So if I do this, I'll get zero equals negative, 00:13:23.168 --> 00:13:25.118 you know what, let's just move the -30 over. 00:13:25.118 --> 00:13:26.690 This is kind of annoying here. 00:13:26.690 --> 00:13:28.316 Just add 30 to both sides, 00:13:28.316 --> 00:13:30.804 then take this calculation here. 00:13:30.804 --> 00:13:35.326 We get plus 30 equals, and then we're gonna have 00:13:35.326 --> 00:13:38.409 T1 over 2, from this T1, so T1 over 2 00:13:39.487 --> 00:13:42.570 plus, I've got plus, T2 is T1 root 3. 00:13:44.894 --> 00:13:48.375 So when I plug T1 root 3 in for T2, 00:13:48.375 --> 00:13:52.292 what I'm gonna get is, I'm gonna get T1 root 3, 00:13:55.086 --> 00:13:56.893 and then times another route 3, 00:13:56.893 --> 00:13:59.560 because T2 itself was T1 root 3. 00:14:00.938 --> 00:14:02.452 So I'm taking this expression here, 00:14:02.452 --> 00:14:05.518 plugging it in for T2, but I still have to multiply 00:14:05.518 --> 00:14:08.577 that T2 by a root 3 and divide by 2. 00:14:08.577 --> 00:14:10.430 And so, what do we get? 00:14:10.430 --> 00:14:13.226 Root 3 times root 3 is just 3. 00:14:13.226 --> 00:14:16.976 So we have T2 times 3 halves, plus T1 over 2. 00:14:18.133 --> 00:14:19.941 So I'll get 30 equals, 00:14:19.941 --> 00:14:24.232 and then I get T1 over 2, we're almost there, I promise. 00:14:24.232 --> 00:14:29.122 T1 over 2, plus, and this is gonna be T1 times 3 over 2, 00:14:29.122 --> 00:14:33.768 so it's gonna be 3 T1 over 2, or what does that equal? 00:14:33.768 --> 00:14:37.435 T1 over 2 plus 3 T1 over 2 is just 4 halves. 00:14:39.037 --> 00:14:42.372 So that's just 2 T1. So this cleaned up beautifully. 00:14:42.372 --> 00:14:45.929 So this is just 2 times T1, and now we can solve for T1. 00:14:45.929 --> 00:14:49.346 We get that T1 is simply 30 divided by 2. 00:14:51.128 --> 00:14:54.497 If I divide both sides, this left hand side by 2, 00:14:54.497 --> 00:14:56.512 and this side here, this right side by 2, 00:14:56.512 --> 00:14:59.012 I get T1 is 30 over 2 Newtons, 00:15:00.037 --> 00:15:02.396 which is just, these should be Newtons, 00:15:02.396 --> 00:15:06.563 I should have units on these, which is just 15 Newtons. 00:15:08.227 --> 00:15:10.910 Whoo, I did it, 15 Newtons. 00:15:10.910 --> 00:15:12.554 T1 is 15 Newtons. 00:15:12.554 --> 00:15:14.436 We got T1. That's one of them. 00:15:14.436 --> 00:15:15.881 How do we get the other? 00:15:15.881 --> 00:15:19.200 You start back over at the very beginning. 00:15:19.200 --> 00:15:21.559 No, not really, that would be terrible. 00:15:21.559 --> 00:15:23.130 You actually just take this T1, 00:15:23.130 --> 00:15:25.946 and you plug it right into here, boop, there it goes. 00:15:25.946 --> 00:15:27.977 So T2, we already got it. 00:15:27.977 --> 00:15:29.426 T2 is just T1 root 3. 00:15:29.426 --> 00:15:32.348 So all I have to do is multiply root 3 by my T1, 00:15:32.348 --> 00:15:33.181 which I know now. 00:15:33.181 --> 00:15:37.521 And I get that T2 is just 15 times root 3 Newton. 00:15:37.521 --> 00:15:39.908 So once you get one of the forces, 00:15:39.908 --> 00:15:41.386 the next one is really easy. 00:15:41.386 --> 00:15:42.884 This is just T2. 00:15:42.884 --> 00:15:46.051 So T2 is 15 root 3, and T1 is just 15. 00:15:47.500 --> 00:15:49.544 So in case you got lost in the details, 00:15:49.544 --> 00:15:52.015 the big picture recap is this. 00:15:52.015 --> 00:15:55.349 We drew a force diagram, we used Newton's Second Law 00:15:55.349 --> 00:15:57.643 in the vertical direction we couldn't solve, 00:15:57.643 --> 00:15:59.152 because there were too many unknowns. 00:15:59.152 --> 00:16:02.501 We used Newton's Second Law in the horizontal direction, 00:16:02.501 --> 00:16:04.158 we couldn't solve because there were two unknowns. 00:16:04.158 --> 00:16:06.225 We put all four of these unknowns 00:16:06.225 --> 00:16:09.376 in terms of only two unknowns, T1 and T2, 00:16:09.376 --> 00:16:12.560 by writing how those components depended 00:16:12.560 --> 00:16:13.994 on those total vectors. 00:16:13.994 --> 00:16:17.324 We substituted these expressions in for each component. 00:16:17.324 --> 00:16:19.256 Once we did that, we had two equations, 00:16:19.256 --> 00:16:23.190 with only T1, T2, and T1 and T2 in them. 00:16:23.190 --> 00:16:27.680 We solved one of these equations for T2 in terms of T1, 00:16:27.680 --> 00:16:29.923 substituted that into the other equation. 00:16:29.923 --> 00:16:32.794 We got a single equation with only one unknown. 00:16:32.794 --> 00:16:34.929 We were able to solve for that unknown. 00:16:34.929 --> 00:16:37.508 Once we got that, which is our T1, 00:16:37.508 --> 00:16:39.137 once we have that variable, 00:16:39.137 --> 00:16:40.618 we plug it back into that first equation 00:16:40.618 --> 00:16:42.290 that we had solved for T2. 00:16:42.290 --> 00:16:45.560 We plug this 15 in, we get what the second tension is. 00:16:45.560 --> 00:16:48.060 So even when it seems like Newton's Second Law 00:16:48.060 --> 00:16:49.896 won't get you there, if you have faith, 00:16:49.896 --> 00:16:52.725 and you persevere, you will make it. 00:16:52.725 --> 00:16:53.558 Good job.