1 00:00:00,709 --> 00:00:01,616 - [Voiceover] Oh, it's time. 2 00:00:01,616 --> 00:00:03,391 It's time for the super hot tension problem. 3 00:00:03,391 --> 00:00:04,837 We're about to do this right here. 4 00:00:04,837 --> 00:00:07,032 We've got our super hot can of red peppers 5 00:00:07,032 --> 00:00:08,821 hanging from these strings. 6 00:00:08,821 --> 00:00:11,761 We want to know what the tension is in these ropes. 7 00:00:11,761 --> 00:00:15,077 This is for real now, this is a real tension problem. 8 00:00:15,077 --> 00:00:16,309 And here's the deal. 9 00:00:16,309 --> 00:00:18,377 You might look at this, you might get frightened. 10 00:00:18,377 --> 00:00:20,232 You might think, I've gotta come up with 11 00:00:20,232 --> 00:00:22,551 a completely new strategy to tackle this. 12 00:00:22,551 --> 00:00:24,439 I've gotta throw away everything I've learned 13 00:00:24,439 --> 00:00:25,902 and just try something new. 14 00:00:25,902 --> 00:00:27,058 And that's a lie. 15 00:00:27,058 --> 00:00:28,499 You should not lie to yourself. 16 00:00:28,499 --> 00:00:30,439 Use the same process. 17 00:00:30,439 --> 00:00:32,290 We're gonna use the same process we used 18 00:00:32,290 --> 00:00:33,408 for the easy tension problems, 19 00:00:33,408 --> 00:00:35,134 because it's gonna lead us to the answer again. 20 00:00:35,134 --> 00:00:39,170 Be careful. Don't stray from the strategy here. 21 00:00:39,170 --> 00:00:40,109 The strategy works. 22 00:00:40,109 --> 00:00:41,650 So we're gonna draw our force diagram first. 23 00:00:41,650 --> 00:00:43,178 That's what we always do. 24 00:00:43,178 --> 00:00:45,031 We're gonna say that the forces are 25 00:00:45,031 --> 00:00:48,170 force of gravity on this can of red peppers, 26 00:00:48,170 --> 00:00:50,544 which is MG, and if it's 3 kilograms, 27 00:00:50,544 --> 00:00:52,962 we know 3 kilograms times about 10, 28 00:00:52,962 --> 00:00:55,708 we're gonna say, let's approximate G as 10 again 29 00:00:55,708 --> 00:00:57,417 to make the numbers come out nice. 30 00:00:57,417 --> 00:01:00,816 So instead of using 9.8, we'll say G is about 10, 31 00:01:00,816 --> 00:01:02,910 and so we'll say 3 kilograms 32 00:01:02,910 --> 00:01:07,077 times 10 meters per second squared is gonna be 30 Newtons. 33 00:01:09,947 --> 00:01:12,728 And so the force of gravity downward is 30 Newtons. 34 00:01:12,728 --> 00:01:14,078 What other forces do we have? 35 00:01:14,078 --> 00:01:17,330 We've got this T1, remember tension does not push. 36 00:01:17,330 --> 00:01:20,197 Ropes can't push, ropes can only pull, 37 00:01:20,197 --> 00:01:21,880 so T1's gonna pull that way. 38 00:01:21,880 --> 00:01:23,961 So I'm gonna draw T1 coming this way. 39 00:01:23,961 --> 00:01:25,378 So here's our T1. 40 00:01:26,219 --> 00:01:28,447 And then we're gonna have T2 pointing this way, 41 00:01:28,447 --> 00:01:29,630 so this is T2. 42 00:01:29,630 --> 00:01:31,731 Again, T2 pulls, just like all tension. 43 00:01:31,731 --> 00:01:34,018 Tension pulls, tension can't push. 44 00:01:34,018 --> 00:01:37,101 So I've got tension 2 going this way. 45 00:01:39,580 --> 00:01:41,656 That's it, that's our force diagram. 46 00:01:41,656 --> 00:01:42,584 There's no other forces. 47 00:01:42,584 --> 00:01:43,849 I don't draw a normal force, 48 00:01:43,849 --> 00:01:46,276 'cause this can isn't in contact with another surface. 49 00:01:46,276 --> 00:01:48,956 So there's no normal force, you've got these two tensions, 50 00:01:48,956 --> 00:01:50,436 the force of gravity. 51 00:01:50,436 --> 00:01:52,155 And now we do the same thing we always do. 52 00:01:52,155 --> 00:01:54,742 After our force diagram, we use Newton's Second Law 53 00:01:54,742 --> 00:01:56,809 in one direction or another. 54 00:01:56,809 --> 00:01:57,863 So let's do it. 55 00:01:57,863 --> 00:02:00,687 Let's say that acceleration is the net force 56 00:02:00,687 --> 00:02:03,779 in a given direction, divided by the mass. 57 00:02:03,779 --> 00:02:05,827 Which direction did we pick again? 58 00:02:05,827 --> 00:02:08,786 It's hard to say, we've got forces vertical, 59 00:02:08,786 --> 00:02:10,137 we've got forces horizontal. 60 00:02:10,137 --> 00:02:13,258 There's only two directions to pick, X or Y in this problem. 61 00:02:13,258 --> 00:02:15,222 We're gonna pick the vertical direction, 62 00:02:15,222 --> 00:02:17,094 even though it doesn't really matter too much. 63 00:02:17,094 --> 00:02:19,416 But because we know one of the forces in the vertical 64 00:02:19,416 --> 00:02:20,909 direction, we know the force of gravity. 65 00:02:20,909 --> 00:02:23,089 Force of gravity is 30 Newtons. 66 00:02:23,089 --> 00:02:25,267 Usually that's a guod strategy, pick the direction 67 00:02:25,267 --> 00:02:27,770 that you know something about at least. 68 00:02:27,770 --> 00:02:29,526 So we're gonna do that here. 69 00:02:29,526 --> 00:02:31,392 We're gonna say that the acceleration vertically 70 00:02:31,392 --> 00:02:33,896 equals to the net force vertically over the mass. 71 00:02:33,896 --> 00:02:34,968 And so now we plug in. 72 00:02:34,968 --> 00:02:36,784 If this can is just sitting here, 73 00:02:36,784 --> 00:02:39,473 if there's no acceleration, if this is in not an elevator 74 00:02:39,473 --> 00:02:42,557 transporting these peppers up or down, 75 00:02:42,557 --> 00:02:44,447 and it's not in a rocket, if it's just sitting here 76 00:02:44,447 --> 00:02:47,466 with no acceleration, our acceleration will be zero. 77 00:02:47,466 --> 00:02:50,995 That's gonna equal the net force and the vertical direction. 78 00:02:50,995 --> 00:02:52,377 So what are we gonna have? 79 00:02:52,377 --> 00:02:54,197 So what are the forces in the vertical direction here? 80 00:02:54,197 --> 00:02:57,222 One force is this 30 Newton force of gravity. 81 00:02:57,222 --> 00:03:00,571 This points down, we're gonna assume upward is positive, 82 00:03:00,571 --> 00:03:02,194 that means down in a negative. 83 00:03:02,194 --> 00:03:03,608 So I'll just put -30 Newtons. 84 00:03:03,608 --> 00:03:05,916 I could have written -MG, 85 00:03:05,916 --> 00:03:07,462 but we already knew it was 30 Newtons, 86 00:03:07,462 --> 00:03:08,921 so I'll write -30 Newtons. 87 00:03:08,921 --> 00:03:10,583 Then we've got T1 and T2. 88 00:03:10,583 --> 00:03:12,255 Both of those point up. 89 00:03:12,255 --> 00:03:13,648 But they don't completely point up, 90 00:03:13,648 --> 00:03:15,004 they partially point up. 91 00:03:15,004 --> 00:03:17,326 So part of them points to the right, 92 00:03:17,326 --> 00:03:19,584 part of them points upward. 93 00:03:19,584 --> 00:03:22,656 Only this vertical component, we'll call it T1Y, 94 00:03:22,656 --> 00:03:25,179 is gonna get included into this calculation, 95 00:03:25,179 --> 00:03:28,212 'cause this calculation only uses Y directed forces. 96 00:03:28,212 --> 00:03:30,868 And the reason is only Y directed forces, 97 00:03:30,868 --> 00:03:34,325 vertical forces, affect the vertical acceleration. 98 00:03:34,325 --> 00:03:36,427 So this T1Y points upward, 99 00:03:36,427 --> 00:03:39,785 I'll do plus T1 in the Y direction. 100 00:03:39,785 --> 00:03:41,675 And similarly, this T2. 101 00:03:41,675 --> 00:03:44,113 It doesn't all point vertically, 102 00:03:44,113 --> 00:03:45,594 only part of it points vertically. 103 00:03:45,594 --> 00:03:49,138 So I'll write this as T2 in the Y direction. 104 00:03:49,138 --> 00:03:50,437 And that's also upward, 105 00:03:50,437 --> 00:03:52,597 so since that's up, I'll count it 106 00:03:52,597 --> 00:03:55,314 as plus T2 in the Y direction. 107 00:03:55,314 --> 00:03:57,101 And that's it, that's all our forces. 108 00:03:57,101 --> 00:04:00,483 Notice we can't plug in the total amount T2 in this 109 00:04:00,483 --> 00:04:02,870 formula, 'cause only part of it points up. 110 00:04:02,870 --> 00:04:05,340 Similarly, we have to plug in only the vertical component 111 00:04:05,340 --> 00:04:09,031 of the T1 force because only part of it points vertically. 112 00:04:09,031 --> 00:04:13,175 And then we divide by the mass, the mass is 3 kilograms. 113 00:04:13,175 --> 00:04:15,875 But we're gonna multiply both sides by 3 kilograms, 114 00:04:15,875 --> 00:04:19,668 and we're gonna get zero equals all of this right here, 115 00:04:19,668 --> 00:04:21,065 so I'll just copy this right here. 116 00:04:21,065 --> 00:04:25,233 We use this over again, that comes down right there. 117 00:04:26,489 --> 00:04:29,331 But now there's nothing on the bottom here. 118 00:04:29,331 --> 00:04:30,227 So what do we do at this point? 119 00:04:30,227 --> 00:04:31,556 Now you might think we're stuck. 120 00:04:31,556 --> 00:04:33,525 I mean, we've got two unknowns in here. 121 00:04:33,525 --> 00:04:35,605 I can't solve for either one, 122 00:04:35,605 --> 00:04:36,938 I don't know either one of these. 123 00:04:36,938 --> 00:04:38,624 I know they have to add up to 30, 124 00:04:38,624 --> 00:04:40,920 so I'd do fine, if I added 30 to both sides, 125 00:04:40,920 --> 00:04:43,129 I'd realize that these two vertical components 126 00:04:43,129 --> 00:04:45,097 of these tension forces added up 127 00:04:45,097 --> 00:04:47,308 have to add up to 30, and that makes sense. 128 00:04:47,308 --> 00:04:49,319 They have to balance the force downward. 129 00:04:49,319 --> 00:04:52,664 But I don't know either of them, so how do I solve here? 130 00:04:52,664 --> 00:04:54,193 Well, let's do this. 131 00:04:54,193 --> 00:04:57,380 If you ever get stuck on one of the force equations 132 00:04:57,380 --> 00:05:00,090 for a single direction, just go to the next equation. 133 00:05:00,090 --> 00:05:01,889 Let's try A in the X direction. 134 00:05:01,889 --> 00:05:04,053 So for A in the X direction, we have the net force 135 00:05:04,053 --> 00:05:06,198 in the X direction, over the mass, 136 00:05:06,198 --> 00:05:09,231 again, the acceleration is gonna be zero 137 00:05:09,231 --> 00:05:11,887 if these peppers are not accelerating horizontally. 138 00:05:11,887 --> 00:05:14,270 So unless this thing's in a train car or something, 139 00:05:14,270 --> 00:05:16,435 and the whole thing's accelerating, 140 00:05:16,435 --> 00:05:18,076 then you might have horizontal acceleration. 141 00:05:18,076 --> 00:05:19,946 And if it did, it's not that big of a deal, 142 00:05:19,946 --> 00:05:21,639 you just plug it in there. 143 00:05:21,639 --> 00:05:23,341 But assuming it's acceleration zero, 144 00:05:23,341 --> 00:05:24,725 because the peppers are just sitting there, 145 00:05:24,725 --> 00:05:25,768 not changing their velocity, we'll plug in zero. 146 00:05:25,768 --> 00:05:28,206 We'll plug in the forces in the X direction. 147 00:05:28,206 --> 00:05:31,193 These are gonna be T1 in the X. 148 00:05:31,193 --> 00:05:33,927 So part of this T1 points in the X direction. 149 00:05:33,927 --> 00:05:37,566 Similarly, part of T2 points in the X direction. 150 00:05:37,566 --> 00:05:38,706 We'll call this T2X. 151 00:05:38,706 --> 00:05:40,833 We use these as the magnitude. 152 00:05:40,833 --> 00:05:42,994 Let's say T2X is the magnitude of the force 153 00:05:42,994 --> 00:05:45,140 that T2 pulls with to the left, 154 00:05:45,140 --> 00:05:49,307 and T1 is the magnitude that T1 pulls with to the right. 155 00:05:50,723 --> 00:05:53,079 So to plug these in, we've got to decide 156 00:05:53,079 --> 00:05:55,585 whether they should be positive or negative. 157 00:05:55,585 --> 00:05:57,988 So this T1X, since it pulls to the right, 158 00:05:57,988 --> 00:05:59,696 T1X will be positive. 159 00:05:59,696 --> 00:06:02,835 We're gonna consider rightward to be the positive direction, 160 00:06:02,835 --> 00:06:05,252 'cause that's the typical convention that we're gonna adopt. 161 00:06:05,252 --> 00:06:08,229 And T2X pulls to the left. 162 00:06:08,229 --> 00:06:10,033 That's gonna be a negative contribution, 163 00:06:10,033 --> 00:06:12,587 so minus T2 in the X direction. 164 00:06:12,587 --> 00:06:14,115 'Cause leftward would be negative. 165 00:06:14,115 --> 00:06:16,597 We divided by the mass, the mass was 3 kilograms, 166 00:06:16,597 --> 00:06:19,314 but again, we'll multiply both sides by 3, 167 00:06:19,314 --> 00:06:23,313 we'll get zero equals, and then we just get T, 168 00:06:23,313 --> 00:06:24,502 the same thing up here, 169 00:06:24,502 --> 00:06:28,669 so we'll just copy this thing here, put it down here. 170 00:06:30,614 --> 00:06:31,673 And again, you might be concerned. 171 00:06:31,673 --> 00:06:33,526 I can't solve this either. 172 00:06:33,526 --> 00:06:37,132 I mean, I can solve for T1X, but look at what I get. 173 00:06:37,132 --> 00:06:40,870 If I just multi, or if I added T2X to both sides, 174 00:06:40,870 --> 00:06:42,432 I'm just gonna get T1 in the X direction 175 00:06:42,432 --> 00:06:45,349 has to equal T2 in the X direction. 176 00:06:46,224 --> 00:06:47,684 And that makes sense. 177 00:06:47,684 --> 00:06:49,472 These two forces have to be equal and opposite, 178 00:06:49,472 --> 00:06:52,137 because they have to cancel so that you have no acceleration 179 00:06:52,137 --> 00:06:53,144 in the X direction. 180 00:06:53,144 --> 00:06:55,373 And this was not drawn proportionately, sorry, 181 00:06:55,373 --> 00:06:57,452 this should be the exact same size as this force 182 00:06:57,452 --> 00:06:59,423 because they have to cancel, 183 00:06:59,423 --> 00:07:01,517 since there's no horizontal acceleration. 184 00:07:01,517 --> 00:07:02,817 But what do we do? 185 00:07:02,817 --> 00:07:05,187 We can't solve this equation we got from X direction. 186 00:07:05,187 --> 00:07:08,635 We can't solve this equation we got from the Y direction. 187 00:07:08,635 --> 00:07:11,223 Whenever this happens, when you get two equations, 188 00:07:11,223 --> 00:07:12,761 and you can't solve either 189 00:07:12,761 --> 00:07:13,995 because there's too many unknowns, 190 00:07:13,995 --> 00:07:16,666 you're gonna have to end up plugging one into the other. 191 00:07:16,666 --> 00:07:18,407 But I can't even do that yet. 192 00:07:18,407 --> 00:07:20,358 I've got four different variables here. 193 00:07:20,358 --> 00:07:22,275 T1X, T2X, T1Y, and T2Y, 194 00:07:24,305 --> 00:07:25,720 these are all four different variables, 195 00:07:25,720 --> 00:07:28,387 I've only got two equations, I can't solve this. 196 00:07:28,387 --> 00:07:30,383 So the trick, the trick we're gonna use 197 00:07:30,383 --> 00:07:31,717 that a lot of people don't like doing 198 00:07:31,717 --> 00:07:33,147 because it's a little more sophisticated, 199 00:07:33,147 --> 00:07:37,144 now we've gotta put these all in terms of T1 and T2 200 00:07:37,144 --> 00:07:38,595 so that we can solve. 201 00:07:38,595 --> 00:07:41,859 If I put T1Y in terms of the total T1, 202 00:07:41,859 --> 00:07:44,706 and then sines of angles, and cosines of angles, 203 00:07:44,706 --> 00:07:48,039 and I put T2Y in terms of T2 and angles, 204 00:07:49,582 --> 00:07:51,240 and I do the same thing for 1X and 2X, 205 00:07:51,240 --> 00:07:53,042 I'll have two equations, and the only two unknowns 206 00:07:53,042 --> 00:07:57,501 will be T1 and T2, then we can finally solve. 207 00:07:57,501 --> 00:07:59,435 If that didn't make any sense, here's what I'm saying. 208 00:07:59,435 --> 00:08:02,751 I'm saying figure out what T1Y is in terms of T1. 209 00:08:02,751 --> 00:08:05,629 So I know this angle here, let's figure out these angles. 210 00:08:05,629 --> 00:08:08,666 So these angles here are, if this is 30, 211 00:08:08,666 --> 00:08:11,202 this angle down here has to be 30 because these 212 00:08:11,202 --> 00:08:13,312 are alternate interior angles. 213 00:08:13,312 --> 00:08:14,696 And if you don't believe me, 214 00:08:14,696 --> 00:08:16,811 imagine this big triangle over here, 215 00:08:16,811 --> 00:08:18,692 where this is a right angle. 216 00:08:18,692 --> 00:08:20,401 So this triangle from here to there, down to here, 217 00:08:20,401 --> 00:08:24,725 up to here, if this is 30, that's 90, this has gotta be 60, 218 00:08:24,725 --> 00:08:27,212 'cause it all adds up to 180 for a triangle. 219 00:08:27,212 --> 00:08:29,132 And if this right angle is 90, and this side's 60, 220 00:08:29,132 --> 00:08:31,291 this side's gotta be 30. 221 00:08:31,291 --> 00:08:34,102 Similarly, this side's a right angle. 222 00:08:34,102 --> 00:08:36,131 Look at this triangle, 60, 90, 223 00:08:36,131 --> 00:08:37,840 that means this would have to be 30. 224 00:08:37,841 --> 00:08:40,751 And so if I come down here, this angle would have to be 60. 225 00:08:40,751 --> 00:08:42,334 Just like this one, 226 00:08:43,188 --> 00:08:45,919 'cause it's an alternate interior angle, so that's 60. 227 00:08:45,919 --> 00:08:49,788 So this angle here is 60, this angle here is 30, 228 00:08:49,788 --> 00:08:51,934 we can figure out what these components are 229 00:08:51,934 --> 00:08:53,462 in terms of the total vectors. 230 00:08:53,462 --> 00:08:54,957 Once we find those, 231 00:08:54,957 --> 00:08:56,502 we're gonna plug those expressions into here, 232 00:08:56,502 --> 00:08:57,753 and that will let us solve. 233 00:08:57,753 --> 00:08:59,362 In other words, T1Y is gonna be, 234 00:08:59,362 --> 00:09:02,108 once you do this for awhile you realize, 235 00:09:02,108 --> 00:09:03,604 this is the opposite side. 236 00:09:03,604 --> 00:09:05,440 So this component here is going to be 237 00:09:05,440 --> 00:09:07,607 total T1 times sine of 30. 238 00:09:09,082 --> 00:09:11,132 Because it's the opposite side. 239 00:09:11,132 --> 00:09:13,779 And if that didn't make sense, we'll derive it right here. 240 00:09:13,779 --> 00:09:15,957 So what we're saying is that sine of 30, 241 00:09:15,957 --> 00:09:19,207 sine of 30 is opposite over hypotenuse, 242 00:09:21,588 --> 00:09:25,150 and in this case, the opposite side is T1Y. 243 00:09:25,150 --> 00:09:29,150 So T1Y over the total T1 is equal to sine of 30. 244 00:09:30,918 --> 00:09:33,275 And we can solve this for T1Y now, 245 00:09:33,275 --> 00:09:36,106 we can get the T1Y if I multiply both sides by T1. 246 00:09:36,106 --> 00:09:39,273 I get that that's T1 times sine of 30. 247 00:09:40,764 --> 00:09:42,199 So that's what I said down here. 248 00:09:42,199 --> 00:09:45,074 T1 is just T, oh sorry, forgot the one. 249 00:09:45,074 --> 00:09:46,741 T1 times sine of 30. 250 00:09:48,048 --> 00:09:50,506 Similarly, if you do the same thing with cosine 30, 251 00:09:50,506 --> 00:09:53,506 you'll get that T1X is T1 cosine 30, 252 00:09:55,852 --> 00:09:57,702 by the exact same process. 253 00:09:57,702 --> 00:10:01,147 Similarly over here, T2 is going to be, 254 00:10:01,147 --> 00:10:03,564 I'm sorry, T2X is gonna be 2. 255 00:10:05,004 --> 00:10:09,171 So T2 cosine 60, because this is the adjacent side. 256 00:10:10,133 --> 00:10:12,966 And T2Y is gonna be T2 sine of 60. 257 00:10:15,068 --> 00:10:16,531 And if any of that doesn't make sense, 258 00:10:16,531 --> 00:10:19,712 just go back to the definition of sine and cosine, 259 00:10:19,712 --> 00:10:22,152 write what the opposite side is, 260 00:10:22,152 --> 00:10:25,600 the total hypotenuse side, solve for your expression, 261 00:10:25,600 --> 00:10:26,719 you'll get these. 262 00:10:26,719 --> 00:10:29,191 If you don't believe me on those, try those out yourselves. 263 00:10:29,191 --> 00:10:31,494 But those are what these components are, 264 00:10:31,494 --> 00:10:35,966 in terms of T2 and the angles T2, T1 and the angles. 265 00:10:35,966 --> 00:10:37,574 And why are we doing this? 266 00:10:37,574 --> 00:10:39,839 We're doing this so that when plug in over here, 267 00:10:39,839 --> 00:10:41,251 we'll only have two variables. 268 00:10:41,251 --> 00:10:45,008 In other words, if I plug T1Y, this expression here, 269 00:10:45,008 --> 00:10:48,758 T1 sine 30 in for T1Y, similarly if I plug in 270 00:10:49,703 --> 00:10:53,870 T2Y is T2 sine 60 into this expression right there 271 00:10:55,357 --> 00:10:57,408 for T2Y, look at what I'll get. 272 00:10:57,408 --> 00:10:58,987 I'll get zero equals. 273 00:10:58,987 --> 00:11:01,654 So I'll get negative 30 Newtons, 274 00:11:02,712 --> 00:11:06,795 and then I'll get plus T1Y was T1 sine 30, so T1, 275 00:11:08,495 --> 00:11:11,145 and then sine 30, we can clean this up a little bit. 276 00:11:11,145 --> 00:11:12,334 Sine 30 is just a half. 277 00:11:12,334 --> 00:11:15,938 So I'll just write T1 over 2, and then 278 00:11:15,938 --> 00:11:17,874 'cause sine 30 is just one half. 279 00:11:17,874 --> 00:11:20,874 And then T2Y is gonna be T2 sine 60, 280 00:11:23,726 --> 00:11:26,651 and sine 60 is just root 3 over 2. 281 00:11:26,651 --> 00:11:30,818 So I'll write this as plus T2 over 2, and then times root 3. 282 00:11:33,088 --> 00:11:35,524 And you might think this is no better. 283 00:11:35,524 --> 00:11:37,477 I mean this is still a horrible mess right here. 284 00:11:37,477 --> 00:11:40,986 But, look at. This is in terms of T1 and T2. 285 00:11:40,986 --> 00:11:42,266 That's what I'm gonna do over here. 286 00:11:42,266 --> 00:11:44,237 I'm gonna put these in terms of T1 and T2, 287 00:11:44,237 --> 00:11:45,505 and then we can solve. 288 00:11:45,505 --> 00:11:48,557 So T1X is T1 over cosine 30, 289 00:11:48,557 --> 00:11:52,798 so I'm gonna write this as T1 times cosine 30, 290 00:11:52,798 --> 00:11:55,381 and cosine 30 is root 3 over 2, 291 00:11:56,568 --> 00:11:59,576 so this is T1 over 2 times root 3. 292 00:11:59,576 --> 00:12:02,909 And that should equal T2X is right here, 293 00:12:03,937 --> 00:12:07,354 That's T2 cosine 60, cosine 60 is a half. 294 00:12:08,224 --> 00:12:10,641 So T2X is gonna be T2 over 2. 295 00:12:12,348 --> 00:12:13,946 So T2 over 2. 296 00:12:13,946 --> 00:12:16,139 So what I'm doing is, if this doesn't make sense, 297 00:12:16,139 --> 00:12:18,873 I'm just substituting what these components are 298 00:12:18,873 --> 00:12:22,750 in terms of the total magnitude in the angle. 299 00:12:22,750 --> 00:12:25,269 And I do this, because look at what I have now, 300 00:12:25,269 --> 00:12:26,683 I have got one equation with T1 and T2. 301 00:12:26,683 --> 00:12:28,663 I've got another equation with T1 and T2. 302 00:12:28,663 --> 00:12:30,927 So what I'm gonna do to solve these, 303 00:12:30,927 --> 00:12:32,843 when we have two equations and two unknowns, 304 00:12:32,843 --> 00:12:34,925 you have to solve for one of these variables, 305 00:12:34,925 --> 00:12:37,198 and then substitute it into the other equation. 306 00:12:37,198 --> 00:12:40,127 That way you'll get one equation with one unknown. 307 00:12:40,127 --> 00:12:42,220 And you try to get the math right, 308 00:12:42,220 --> 00:12:43,197 and you'll get the problem. 309 00:12:43,197 --> 00:12:44,920 So I'm gonna solve this one is easier, 310 00:12:44,920 --> 00:12:47,572 so I'm gonna solve this one for, let's just say T2. 311 00:12:47,572 --> 00:12:48,982 So if we solve this for T2, 312 00:12:48,982 --> 00:12:52,235 I get that T2 equals, well, I can multiply 313 00:12:52,235 --> 00:12:56,577 both sides by 2, and I'll get T1 times root 3. 314 00:12:56,577 --> 00:13:00,947 So T1 times root 3, because the 2 here cancels with this 2, 315 00:13:00,947 --> 00:13:04,182 or when I multiply both sides by 2 it cancels out. 316 00:13:04,182 --> 00:13:07,936 So we get that T2 equals T1 root 3. This is great. 317 00:13:07,936 --> 00:13:12,103 I can substitute T2 as T1 root 3 into here for T2. 318 00:13:13,998 --> 00:13:15,431 And the reason I do that, 319 00:13:15,431 --> 00:13:17,655 is I'll get one equation with one unknown. 320 00:13:17,655 --> 00:13:19,541 I'll only have T1 in that equation now. 321 00:13:19,541 --> 00:13:23,168 So if I do this, I'll get zero equals negative, 322 00:13:23,168 --> 00:13:25,118 you know what, let's just move the -30 over. 323 00:13:25,118 --> 00:13:26,690 This is kind of annoying here. 324 00:13:26,690 --> 00:13:28,316 Just add 30 to both sides, 325 00:13:28,316 --> 00:13:30,804 then take this calculation here. 326 00:13:30,804 --> 00:13:35,326 We get plus 30 equals, and then we're gonna have 327 00:13:35,326 --> 00:13:38,409 T1 over 2, from this T1, so T1 over 2 328 00:13:39,487 --> 00:13:42,570 plus, I've got plus, T2 is T1 root 3. 329 00:13:44,894 --> 00:13:48,375 So when I plug T1 root 3 in for T2, 330 00:13:48,375 --> 00:13:52,292 what I'm gonna get is, I'm gonna get T1 root 3, 331 00:13:55,086 --> 00:13:56,893 and then times another route 3, 332 00:13:56,893 --> 00:13:59,560 because T2 itself was T1 root 3. 333 00:14:00,938 --> 00:14:02,452 So I'm taking this expression here, 334 00:14:02,452 --> 00:14:05,518 plugging it in for T2, but I still have to multiply 335 00:14:05,518 --> 00:14:08,577 that T2 by a root 3 and divide by 2. 336 00:14:08,577 --> 00:14:10,430 And so, what do we get? 337 00:14:10,430 --> 00:14:13,226 Root 3 times root 3 is just 3. 338 00:14:13,226 --> 00:14:16,976 So we have T2 times 3 halves, plus T1 over 2. 339 00:14:18,133 --> 00:14:19,941 So I'll get 30 equals, 340 00:14:19,941 --> 00:14:24,232 and then I get T1 over 2, we're almost there, I promise. 341 00:14:24,232 --> 00:14:29,122 T1 over 2, plus, and this is gonna be T1 times 3 over 2, 342 00:14:29,122 --> 00:14:33,768 so it's gonna be 3 T1 over 2, or what does that equal? 343 00:14:33,768 --> 00:14:37,435 T1 over 2 plus 3 T1 over 2 is just 4 halves. 344 00:14:39,037 --> 00:14:42,372 So that's just 2 T1. So this cleaned up beautifully. 345 00:14:42,372 --> 00:14:45,929 So this is just 2 times T1, and now we can solve for T1. 346 00:14:45,929 --> 00:14:49,346 We get that T1 is simply 30 divided by 2. 347 00:14:51,128 --> 00:14:54,497 If I divide both sides, this left hand side by 2, 348 00:14:54,497 --> 00:14:56,512 and this side here, this right side by 2, 349 00:14:56,512 --> 00:14:59,012 I get T1 is 30 over 2 Newtons, 350 00:15:00,037 --> 00:15:02,396 which is just, these should be Newtons, 351 00:15:02,396 --> 00:15:06,563 I should have units on these, which is just 15 Newtons. 352 00:15:08,227 --> 00:15:10,910 Whoo, I did it, 15 Newtons. 353 00:15:10,910 --> 00:15:12,554 T1 is 15 Newtons. 354 00:15:12,554 --> 00:15:14,436 We got T1. That's one of them. 355 00:15:14,436 --> 00:15:15,881 How do we get the other? 356 00:15:15,881 --> 00:15:19,200 You start back over at the very beginning. 357 00:15:19,200 --> 00:15:21,559 No, not really, that would be terrible. 358 00:15:21,559 --> 00:15:23,130 You actually just take this T1, 359 00:15:23,130 --> 00:15:25,946 and you plug it right into here, boop, there it goes. 360 00:15:25,946 --> 00:15:27,977 So T2, we already got it. 361 00:15:27,977 --> 00:15:29,426 T2 is just T1 root 3. 362 00:15:29,426 --> 00:15:32,348 So all I have to do is multiply root 3 by my T1, 363 00:15:32,348 --> 00:15:33,181 which I know now. 364 00:15:33,181 --> 00:15:37,521 And I get that T2 is just 15 times root 3 Newton. 365 00:15:37,521 --> 00:15:39,908 So once you get one of the forces, 366 00:15:39,908 --> 00:15:41,386 the next one is really easy. 367 00:15:41,386 --> 00:15:42,884 This is just T2. 368 00:15:42,884 --> 00:15:46,051 So T2 is 15 root 3, and T1 is just 15. 369 00:15:47,500 --> 00:15:49,544 So in case you got lost in the details, 370 00:15:49,544 --> 00:15:52,015 the big picture recap is this. 371 00:15:52,015 --> 00:15:55,349 We drew a force diagram, we used Newton's Second Law 372 00:15:55,349 --> 00:15:57,643 in the vertical direction we couldn't solve, 373 00:15:57,643 --> 00:15:59,152 because there were too many unknowns. 374 00:15:59,152 --> 00:16:02,501 We used Newton's Second Law in the horizontal direction, 375 00:16:02,501 --> 00:16:04,158 we couldn't solve because there were two unknowns. 376 00:16:04,158 --> 00:16:06,225 We put all four of these unknowns 377 00:16:06,225 --> 00:16:09,376 in terms of only two unknowns, T1 and T2, 378 00:16:09,376 --> 00:16:12,560 by writing how those components depended 379 00:16:12,560 --> 00:16:13,994 on those total vectors. 380 00:16:13,994 --> 00:16:17,324 We substituted these expressions in for each component. 381 00:16:17,324 --> 00:16:19,256 Once we did that, we had two equations, 382 00:16:19,256 --> 00:16:23,190 with only T1, T2, and T1 and T2 in them. 383 00:16:23,190 --> 00:16:27,680 We solved one of these equations for T2 in terms of T1, 384 00:16:27,680 --> 00:16:29,923 substituted that into the other equation. 385 00:16:29,923 --> 00:16:32,794 We got a single equation with only one unknown. 386 00:16:32,794 --> 00:16:34,929 We were able to solve for that unknown. 387 00:16:34,929 --> 00:16:37,508 Once we got that, which is our T1, 388 00:16:37,508 --> 00:16:39,137 once we have that variable, 389 00:16:39,137 --> 00:16:40,618 we plug it back into that first equation 390 00:16:40,618 --> 00:16:42,290 that we had solved for T2. 391 00:16:42,290 --> 00:16:45,560 We plug this 15 in, we get what the second tension is. 392 00:16:45,560 --> 00:16:48,060 So even when it seems like Newton's Second Law 393 00:16:48,060 --> 00:16:49,896 won't get you there, if you have faith, 394 00:16:49,896 --> 00:16:52,725 and you persevere, you will make it. 395 00:16:52,725 --> 00:00:00,000 Good job.