1 00:00:00,440 --> 00:00:01,498 - [Instructor] I found that when students 2 00:00:01,498 --> 00:00:04,466 have to do problems involving the force of tension, 3 00:00:04,466 --> 00:00:07,795 they get annoyed maybe more than any kind of force problem, 4 00:00:07,795 --> 00:00:10,068 and we'll talk about why in a minute. 5 00:00:10,068 --> 00:00:12,407 We'll go over some examples involving tension 6 00:00:12,407 --> 00:00:14,706 to try to demystify this force, 7 00:00:14,706 --> 00:00:16,354 make you more comfortable with finding it 8 00:00:16,354 --> 00:00:18,645 so it's not so annoying when you get one of these problems. 9 00:00:18,645 --> 00:00:20,052 And at the same time we'll talk about 10 00:00:20,052 --> 00:00:22,737 the misconceptions that a lot of people have about tension 11 00:00:22,737 --> 00:00:25,480 so that you don't make those mistakes as well. 12 00:00:25,480 --> 00:00:27,500 All right, so to make this a little more clear, 13 00:00:27,500 --> 00:00:28,437 what is tension? 14 00:00:28,437 --> 00:00:31,313 That's the first good question is what is this tension? 15 00:00:31,313 --> 00:00:33,615 Well, tension is the force exerted by a rope 16 00:00:33,615 --> 00:00:36,907 or a string or a cable or any rope-like object. 17 00:00:36,907 --> 00:00:40,513 If you had a box of cheese snacks and we tied a rope to it. 18 00:00:40,513 --> 00:00:41,830 We tie a rope over to here 19 00:00:41,830 --> 00:00:44,879 and we figure out how much force do I have to pull with 20 00:00:44,879 --> 00:00:47,558 since the force is being transmitted through a rope, 21 00:00:47,558 --> 00:00:49,476 we'd call that tension. 22 00:00:49,476 --> 00:00:51,788 I mean, it's a force just like any other force. 23 00:00:51,788 --> 00:00:52,989 I'm gonna call this T one 24 00:00:52,989 --> 00:00:55,460 because we're gonna add more ropes in a minute. 25 00:00:55,460 --> 00:00:57,356 It's a force exerted just like any other force. 26 00:00:57,356 --> 00:00:59,180 You treat it just like any other force. 27 00:00:59,180 --> 00:01:03,331 It just is a force that happens to be transmitted by a rope. 28 00:01:03,331 --> 00:01:04,858 What's happening here in this rope? 29 00:01:04,858 --> 00:01:07,135 Ropes are typically composed of fibers 30 00:01:07,135 --> 00:01:09,010 that have been braided together 31 00:01:09,010 --> 00:01:11,445 or wound around each other so that 32 00:01:11,445 --> 00:01:13,792 when I exert a force at this end, right, 33 00:01:13,792 --> 00:01:15,492 when I exert a force down here in this end, 34 00:01:15,492 --> 00:01:16,965 I pull on this end of the rope, 35 00:01:16,965 --> 00:01:19,705 that force gets transmitted through the rope 36 00:01:19,705 --> 00:01:20,855 all the way to this other end 37 00:01:20,855 --> 00:01:22,909 and it'll exert a force on this box. 38 00:01:22,909 --> 00:01:25,352 And the way that works is I pull on the fibers here. 39 00:01:25,352 --> 00:01:26,563 They're braided around each other 40 00:01:26,563 --> 00:01:29,303 so these fibers will now pull on this fibers here 41 00:01:29,303 --> 00:01:30,802 which pull on the fibers here, 42 00:01:30,802 --> 00:01:33,154 and this could be parts of the same fiber. 43 00:01:33,154 --> 00:01:34,710 But the same idea holds. 44 00:01:34,710 --> 00:01:36,509 Once this force is exerted here 45 00:01:36,509 --> 00:01:38,677 it pulls on the ones behind them. 46 00:01:38,677 --> 00:01:39,923 Keeps pulling on the ones behind them, 47 00:01:39,923 --> 00:01:41,337 eventually that force gets transmitted 48 00:01:41,337 --> 00:01:42,870 all the way to this other end. 49 00:01:42,870 --> 00:01:45,479 And so, if I pull on this end with a force, 50 00:01:45,479 --> 00:01:47,476 this end gets pulled with a force. 51 00:01:47,476 --> 00:01:49,212 Tension is useful, ropes are useful 52 00:01:49,212 --> 00:01:51,483 because they allow us to transmit a force 53 00:01:51,483 --> 00:01:53,282 over some large distance. 54 00:01:53,282 --> 00:01:54,709 And so, what you might hear, 55 00:01:54,709 --> 00:01:56,255 a typical problem might say this. 56 00:01:56,255 --> 00:01:57,730 Typical problem might say all right, 57 00:01:57,730 --> 00:02:00,890 so let's say there's a force of tension on this box 58 00:02:00,890 --> 00:02:02,924 and it causes the box to accelerate. 59 00:02:02,924 --> 00:02:05,752 With some acceleration a zero, 60 00:02:05,752 --> 00:02:06,785 and the question might say, 61 00:02:06,785 --> 00:02:08,210 how much tension is required 62 00:02:08,210 --> 00:02:12,408 in order to accelerate this box of mass m 63 00:02:12,408 --> 00:02:13,904 with this acceleration a zero? 64 00:02:13,904 --> 00:02:15,457 What tension is required for that. 65 00:02:15,457 --> 00:02:17,730 Now, a lot of the problems will say 66 00:02:17,730 --> 00:02:20,550 assuming the rope is massless 67 00:02:20,550 --> 00:02:22,957 and you might be like what? 68 00:02:22,957 --> 00:02:25,282 First of all, how can you have a massless rope? 69 00:02:25,282 --> 00:02:27,830 Second of all, why would you ever want one? 70 00:02:27,830 --> 00:02:30,138 Well, the reason physics problem say that a lot of times 71 00:02:30,138 --> 00:02:33,196 is because imagine if the rope was not massless. 72 00:02:33,196 --> 00:02:35,428 Imagine the rope is heavy, very massive. 73 00:02:35,428 --> 00:02:37,425 One of those thick massive ropes. 74 00:02:37,425 --> 00:02:40,454 Well then, these fibers here at this end 75 00:02:40,454 --> 00:02:42,695 would not only have to be pulling the box, 76 00:02:42,695 --> 00:02:45,474 it also be pulling all of the rope in between them 77 00:02:45,474 --> 00:02:46,307 so they don't have to be pulling 78 00:02:46,307 --> 00:02:48,410 all this heavy rope in between them. 79 00:02:48,410 --> 00:02:51,014 Whereas, the fibers right here would have to be pulling 80 00:02:51,014 --> 00:02:53,419 this amount of rope, half of it. 81 00:02:53,419 --> 00:02:54,815 All right, the fibers here would have to pulling 82 00:02:54,815 --> 00:02:56,276 this amount of rope which is heavy. 83 00:02:56,276 --> 00:02:59,736 Not as heavy as all of the rope and the box. 84 00:02:59,736 --> 00:03:01,687 And then, so the tension here would be a little less 85 00:03:01,687 --> 00:03:02,710 than the tension at this end. 86 00:03:02,710 --> 00:03:04,475 And then the tension over at this end, 87 00:03:04,475 --> 00:03:06,505 well, these fibers would only be pulling the box. 88 00:03:06,505 --> 00:03:09,197 They don't have to pull any heavy rope behind them. 89 00:03:09,197 --> 00:03:11,630 Since they're not dragging any heavy rope behind them, 90 00:03:11,630 --> 00:03:13,034 the tension over here would be less 91 00:03:13,034 --> 00:03:14,087 than the tension over here. 92 00:03:14,087 --> 00:03:17,263 You'd have a tension gradient or a tension, 93 00:03:17,263 --> 00:03:18,298 a varying amount of tension 94 00:03:18,298 --> 00:03:19,971 where the tension is big at this end, 95 00:03:19,971 --> 00:03:22,019 smaller, smaller, smaller, smaller. 96 00:03:22,019 --> 00:03:22,893 That's complicated. 97 00:03:22,893 --> 00:03:24,567 We don't have to deal with that. 98 00:03:24,567 --> 00:03:25,716 Most of these physics problems 99 00:03:25,716 --> 00:03:26,827 don't wanna have to deal with that 100 00:03:26,827 --> 00:03:29,578 so what we say is that the rope is massless 101 00:03:29,578 --> 00:03:31,521 but we don't really mean the rope is massless. 102 00:03:31,521 --> 00:03:33,147 The rope's got to be made out of something. 103 00:03:33,147 --> 00:03:36,072 What we mean is that the rope's mass is negligible. 104 00:03:36,072 --> 00:03:38,331 It's small compared to any mass here 105 00:03:38,331 --> 00:03:39,551 so that even though there is 106 00:03:39,551 --> 00:03:42,874 some small variation within this rope of the tension, 107 00:03:42,874 --> 00:03:44,362 it really doesn't matter much. 108 00:03:44,362 --> 00:03:47,442 In other words, maybe the tension at this end is 50 newtons 109 00:03:47,442 --> 00:03:51,109 and the tension at this end is like 49.9998. 110 00:03:52,432 --> 00:03:54,066 Okay, yeah they're a little different. 111 00:03:54,066 --> 00:03:55,924 The tension here is a little greater 112 00:03:55,924 --> 00:03:58,611 but it's insignificant, that's what we're saying. 113 00:03:58,611 --> 00:03:59,574 So let's try to do this problem. 114 00:03:59,574 --> 00:04:00,875 Let's get rid of that. 115 00:04:00,875 --> 00:04:02,904 Let me get rid of all of these 116 00:04:02,904 --> 00:04:06,304 and let's ask, what is this required tension right here 117 00:04:06,304 --> 00:04:10,271 in order to pull this box with an acceleration of a zero? 118 00:04:10,271 --> 00:04:12,724 To do these problems, we're gonna draw a force diagram 119 00:04:12,724 --> 00:04:14,651 the way you draw any force diagram. 120 00:04:14,651 --> 00:04:16,414 You draw the forces on the object. 121 00:04:16,414 --> 00:04:18,593 So we're gonna say that the force of gravity 122 00:04:18,593 --> 00:04:19,611 is equal to mg. 123 00:04:19,611 --> 00:04:21,281 Force of gravity is wonderful. 124 00:04:21,281 --> 00:04:23,665 Force of gravity has its own formula, mg. 125 00:04:23,665 --> 00:04:26,473 I just plug right in and I get the force of gravity 126 00:04:26,473 --> 00:04:27,306 and if you're near the earth 127 00:04:27,306 --> 00:04:29,508 there's always a force of gravity pulling down. 128 00:04:29,508 --> 00:04:32,508 Since this box is in contact with the floor 129 00:04:32,508 --> 00:04:33,646 there's gonna be a normal force 130 00:04:33,646 --> 00:04:35,182 because the floor is a surface, 131 00:04:35,182 --> 00:04:36,555 the box is a surface. 132 00:04:36,555 --> 00:04:38,084 When two surfaces are in contact 133 00:04:38,084 --> 00:04:40,093 you'll have this normal force. 134 00:04:40,093 --> 00:04:41,744 And then you're gonna have this tension. 135 00:04:41,744 --> 00:04:43,281 Here's the first big misconception. 136 00:04:43,281 --> 00:04:45,783 People look at this line of this rope 137 00:04:45,783 --> 00:04:48,636 and they say, well, it looks like an arrow pushing that way. 138 00:04:48,636 --> 00:04:51,124 So they think that this rope's pushing on this box 139 00:04:51,124 --> 00:04:53,231 and that doesn't make any sense. 140 00:04:53,231 --> 00:04:54,757 You can't push with the rope. 141 00:04:54,757 --> 00:04:58,109 If you don't believe me, go right now, go. 142 00:04:58,109 --> 00:04:59,740 Pause this video, tie a rope to something 143 00:04:59,740 --> 00:05:01,296 and try to push on it and you'll realize, 144 00:05:01,296 --> 00:05:04,712 oh, yeah, if I try that the rope just goes slack 145 00:05:04,712 --> 00:05:06,249 and I can't really push. 146 00:05:06,249 --> 00:05:08,219 But you can pull on things with a rope. 147 00:05:08,219 --> 00:05:11,424 Ropes cause this tension which is a pulling force. 148 00:05:11,424 --> 00:05:12,638 Tension's a pulling force 149 00:05:12,638 --> 00:05:14,429 because this rope gets taut, it gets tight 150 00:05:14,429 --> 00:05:16,396 and now I can pull on things. 151 00:05:16,396 --> 00:05:18,940 Normally force is a pushing force, right? 152 00:05:18,940 --> 00:05:21,662 Force is the two surfaces push on each other, 153 00:05:21,662 --> 00:05:25,084 the ground pushes out to keep the box out of the ground. 154 00:05:25,084 --> 00:05:27,128 But ropes, tension is a pulling force 155 00:05:27,128 --> 00:05:29,471 so I have to draw this force this way. 156 00:05:29,471 --> 00:05:31,928 This tension T one, I'm in my force diagram 157 00:05:31,928 --> 00:05:33,423 would point to the right. 158 00:05:33,423 --> 00:05:35,280 I'd call this T one. 159 00:05:35,280 --> 00:05:36,324 Those are all my forces. 160 00:05:36,324 --> 00:05:38,457 I could have friction here but let's say 161 00:05:38,457 --> 00:05:41,789 this cheese snack manufacturing plant has made 162 00:05:41,789 --> 00:05:45,610 transporting their cheese snacks as efficient as possible. 163 00:05:45,610 --> 00:05:47,494 They've made a frictionless ground 164 00:05:47,494 --> 00:05:48,984 and if that sounds unbelievable 165 00:05:48,984 --> 00:05:51,207 maybe there's ball bearings under here 166 00:05:51,207 --> 00:05:54,668 to prevent there from being basically any friction. 167 00:05:54,668 --> 00:05:56,207 We'll just keep it simple to start. 168 00:05:56,207 --> 00:05:57,714 We'll make it more complicated here in a minute 169 00:05:57,714 --> 00:05:59,936 but let's just keep this simple to start. 170 00:05:59,936 --> 00:06:02,296 Now, how do I solve for tension? 171 00:06:02,296 --> 00:06:05,102 The reason people don't like solving for tension I think 172 00:06:05,102 --> 00:06:07,800 is because tension doesn't have a nice formula like gravity. 173 00:06:07,800 --> 00:06:09,345 Look at gravity's formula. 174 00:06:09,345 --> 00:06:10,791 Force of gravity's just mg. 175 00:06:10,791 --> 00:06:13,146 You could just find it right away, it's so nice. 176 00:06:13,146 --> 00:06:14,288 But to find the force of tension, 177 00:06:14,288 --> 00:06:16,370 there's no corresponding formula that's like 178 00:06:16,370 --> 00:06:19,990 T equals and then something analogous to mg. 179 00:06:19,990 --> 00:06:22,678 The way you find tension in almost all problems 180 00:06:22,678 --> 00:06:25,126 is by using Newton's Second Law. 181 00:06:25,126 --> 00:06:27,066 Newton's Second Law says that the acceleration 182 00:06:27,066 --> 00:06:29,879 equals the net force over the mass. 183 00:06:29,879 --> 00:06:31,891 Now if you don't like Newton's Second Law 184 00:06:31,891 --> 00:06:34,199 that's probably why you don't like solving for tension 185 00:06:34,199 --> 00:06:36,426 because this is what you have to do to find the tension. 186 00:06:36,426 --> 00:06:38,078 Since there's no formula dedicated 187 00:06:38,078 --> 00:06:40,298 to just tension itself. 188 00:06:40,298 --> 00:06:41,564 What's the acceleration? 189 00:06:41,564 --> 00:06:43,726 We'll have to pick a direction first. 190 00:06:43,726 --> 00:06:45,389 Do I wanna treat the vertical direction 191 00:06:45,389 --> 00:06:47,007 or the horizontal direction? 192 00:06:47,007 --> 00:06:48,273 I'm gonna treat the horizontal direction 193 00:06:48,273 --> 00:06:49,940 because my tension that I wanna find 194 00:06:49,940 --> 00:06:51,979 is in the horizontal direction. 195 00:06:51,979 --> 00:06:53,919 My acceleration in this horizontal direction 196 00:06:53,919 --> 00:06:55,373 is a zero. 197 00:06:55,373 --> 00:06:57,759 That's gonna equal my net force 198 00:06:57,759 --> 00:07:00,414 and in the x direction, I only have one force. 199 00:07:00,414 --> 00:07:02,149 I've got this tension force. 200 00:07:02,149 --> 00:07:05,508 Only force I have is T one in the x direction. 201 00:07:05,508 --> 00:07:07,249 And since that points right 202 00:07:07,249 --> 00:07:10,048 and I'm gonna consider rightward as positive. 203 00:07:10,048 --> 00:07:12,031 I'm gonna call this positive T one 204 00:07:12,031 --> 00:07:13,595 even though that's pretty much implied. 205 00:07:13,595 --> 00:07:15,496 But positive because it points to the right 206 00:07:15,496 --> 00:07:18,106 and I'm gonna assume rightward is positive. 207 00:07:18,106 --> 00:07:21,028 You could call leftward positive if you really wanted to. 208 00:07:21,028 --> 00:07:22,339 It'd be kind of weird in this case. 209 00:07:22,339 --> 00:07:23,531 Now I divide by the mass, 210 00:07:23,531 --> 00:07:25,690 I can solve for T one now. 211 00:07:25,690 --> 00:07:27,002 I just do a little algebra. 212 00:07:27,002 --> 00:07:31,169 I get the T one, the tension in this first rope right here. 213 00:07:32,076 --> 00:07:35,109 It's gonna equal the mass times whatever the acceleration 214 00:07:35,109 --> 00:07:38,673 of this box is that we're causing with this rope. 215 00:07:38,673 --> 00:07:40,364 Don't draw acceleration as a force. 216 00:07:40,364 --> 00:07:41,197 This is a no-no. 217 00:07:41,197 --> 00:07:43,120 People try to draw this sometimes. 218 00:07:43,120 --> 00:07:44,784 Acceleration is not a force. 219 00:07:44,784 --> 00:07:47,027 Acceleration is caused by a force. 220 00:07:47,027 --> 00:07:50,597 Acceleration itself is not a force so don't ever draw that. 221 00:07:50,597 --> 00:07:52,677 Okay, so we found tension, not too bad 222 00:07:52,677 --> 00:07:55,451 but this is probably the easiest imaginable 223 00:07:55,451 --> 00:07:57,821 tension problem you could ever come up with. 224 00:07:57,821 --> 00:07:58,654 Let's step it up. 225 00:07:58,654 --> 00:07:59,601 You're probably gonna face problems 226 00:07:59,601 --> 00:08:01,307 that are more difficult than this. 227 00:08:01,307 --> 00:08:02,325 Let's say we made it harder, 228 00:08:02,325 --> 00:08:04,558 let's say over here someone's pulling on this side 229 00:08:04,558 --> 00:08:05,925 with another rope. 230 00:08:05,925 --> 00:08:07,232 Let's say there's another T two 231 00:08:07,232 --> 00:08:09,700 so people are fighting over these cheese snacks. 232 00:08:09,700 --> 00:08:11,022 People are hungry. 233 00:08:11,022 --> 00:08:13,001 And someone's pulling on this end. 234 00:08:13,001 --> 00:08:14,100 What would that change? 235 00:08:14,100 --> 00:08:16,268 Let's say I steal this person over here. 236 00:08:16,268 --> 00:08:18,435 He's like, uh-uh, you're not gonna get my cheese snacks. 237 00:08:18,435 --> 00:08:19,935 Say they pull with a force 238 00:08:19,935 --> 00:08:22,772 to maintain this acceleration to be the same. 239 00:08:22,772 --> 00:08:24,655 All right, they're gonna pull whatever they need to 240 00:08:24,655 --> 00:08:25,882 even with this new force here 241 00:08:25,882 --> 00:08:30,098 so that the acceleration just remains a zero to the right. 242 00:08:30,098 --> 00:08:32,623 What would that change appear in my calculation? 243 00:08:32,623 --> 00:08:34,950 Well, my force diagram I've got another force 244 00:08:34,950 --> 00:08:37,871 but I can't, I don't draw this force pushing on the box. 245 00:08:37,871 --> 00:08:39,947 Again, you can't push with tension, 246 00:08:39,947 --> 00:08:41,717 you can only pull with tension. 247 00:08:41,717 --> 00:08:44,286 This rope can pull to the left, 248 00:08:44,287 --> 00:08:47,062 so I'm gonna draw that as T two. 249 00:08:47,062 --> 00:08:48,309 And how do I include that here. 250 00:08:48,309 --> 00:08:50,747 Well, that's the force to the left so I'd subtract it 251 00:08:50,747 --> 00:08:54,195 because leftward forces we're gonna consider negative. 252 00:08:54,195 --> 00:08:55,368 And now I do my algebra, 253 00:08:55,368 --> 00:08:57,642 I multiply both sides by m to get ma 254 00:08:57,642 --> 00:08:59,961 but then I have to add T two to both sides 255 00:08:59,961 --> 00:09:01,120 and this makes sense. 256 00:09:01,120 --> 00:09:02,278 If I'm gonna pull over here, 257 00:09:02,278 --> 00:09:04,538 if I want my T one to compensate 258 00:09:04,538 --> 00:09:07,902 and make it so that this box still accelerates with a zero, 259 00:09:07,902 --> 00:09:10,048 even though this people over here are pulling to the left, 260 00:09:10,048 --> 00:09:11,920 this tension has to increase 261 00:09:11,920 --> 00:09:14,954 in order to maintain the same acceleration to the right. 262 00:09:14,954 --> 00:09:15,994 And I'll step it up even more. 263 00:09:15,994 --> 00:09:18,595 Let's say it's about to, war's about to breakout 264 00:09:18,595 --> 00:09:20,185 over these cheese snacks right here. 265 00:09:20,185 --> 00:09:22,462 Let's say someone pulls this way. 266 00:09:22,462 --> 00:09:26,424 Someone pulls that way with a force T three. 267 00:09:26,424 --> 00:09:27,740 We'll call this T three, 268 00:09:27,740 --> 00:09:30,273 someone's pulling at an angle this time. 269 00:09:30,273 --> 00:09:33,773 Let's say this force is at an angle theta. 270 00:09:34,663 --> 00:09:36,067 Now what does that change? 271 00:09:36,067 --> 00:09:39,150 Again, let's say this T one has to be such that 272 00:09:39,150 --> 00:09:41,133 you get the same acceleration. 273 00:09:41,133 --> 00:09:42,755 But with that change up here you're gonna have a 274 00:09:42,755 --> 00:09:46,042 tension force up into the right in my force diagram. 275 00:09:46,042 --> 00:09:48,294 So you get a tension force this way. 276 00:09:48,294 --> 00:09:51,127 This is T three at an angle theta. 277 00:09:53,809 --> 00:09:56,559 Now I can't plug all of T three into this formula. 278 00:09:56,559 --> 00:09:59,532 This formula was just for the horizontal direction 279 00:09:59,532 --> 00:10:02,639 so I have to plug only the horizontal component 280 00:10:02,639 --> 00:10:04,599 of this T three force. 281 00:10:04,599 --> 00:10:07,192 This component right here. 282 00:10:07,192 --> 00:10:09,691 Only that component of T three do I plug in, 283 00:10:09,691 --> 00:10:11,695 I'm gonna call that T three x. 284 00:10:11,695 --> 00:10:14,005 T three x is what I plug in up here. 285 00:10:14,005 --> 00:10:16,552 T three y, this isn't gonna get plugged 286 00:10:16,552 --> 00:10:17,682 into this formula at all. 287 00:10:17,682 --> 00:10:21,371 The T three y does not affect the horizontal acceleration. 288 00:10:21,371 --> 00:10:23,377 It will only affect the vertical acceleration 289 00:10:23,377 --> 00:10:26,591 and maybe any forces that are exerted vertically. 290 00:10:26,591 --> 00:10:28,292 I'll call this T three y. 291 00:10:28,292 --> 00:10:29,814 How do I find T three x? 292 00:10:29,814 --> 00:10:31,019 I have to use trigonometry, 293 00:10:31,019 --> 00:10:33,220 the way you find components of these vectors 294 00:10:33,220 --> 00:10:35,547 is always trigonometry so I'm gonna say 295 00:10:35,547 --> 00:10:38,464 cosine theta and I'm gonna use cosine because 296 00:10:38,464 --> 00:10:41,707 I know this side, T three x is adjacent 297 00:10:41,707 --> 00:10:43,604 to this angle that I'm given. 298 00:10:43,604 --> 00:10:45,796 Since this is adjacent to the angle I'm given 299 00:10:45,796 --> 00:10:48,161 I use cosine because the definition of cosine 300 00:10:48,161 --> 00:10:51,192 is adjacent over hypotenuse 301 00:10:51,192 --> 00:10:54,826 and my adjacent side is T three x. 302 00:10:54,826 --> 00:10:58,414 My hypotenuse is this side here which is the entire 303 00:10:58,414 --> 00:10:59,747 tension T three. 304 00:11:00,847 --> 00:11:02,634 If this tension was like 50 newtons 305 00:11:02,634 --> 00:11:04,698 at an angle of 30 degrees, 306 00:11:04,698 --> 00:11:06,947 I couldn't plug the whole 50 newtons in here. 307 00:11:06,947 --> 00:11:11,114 I'd say that 50 newtons times if I solve this for T three x. 308 00:11:12,338 --> 00:11:14,471 I get T three x equals. 309 00:11:14,471 --> 00:11:16,533 I'm gonna multiply both sides by T three 310 00:11:16,533 --> 00:11:18,985 so that would be like our 50 newtons. 311 00:11:18,985 --> 00:11:22,466 T three, the entire magnitude of the tension force 312 00:11:22,466 --> 00:11:25,820 times the cosine theta, whatever that theta is. 313 00:11:25,820 --> 00:11:28,760 If it was 30 degrees I'd plug in 30 degrees. 314 00:11:28,760 --> 00:11:30,487 This is what I can plug in up here. 315 00:11:30,487 --> 00:11:33,323 Now I can plug this into my Newton's Second Law. 316 00:11:33,323 --> 00:11:34,735 I couldn't plug the whole force in 317 00:11:34,735 --> 00:11:37,757 because the entire force was not in the x direction. 318 00:11:37,757 --> 00:11:39,852 The entire force was composed of this vertical 319 00:11:39,852 --> 00:11:41,523 and horizontal component 320 00:11:41,523 --> 00:11:43,648 and the vertical component does not affect 321 00:11:43,648 --> 00:11:45,369 the acceleration in the horizontal direction, 322 00:11:45,369 --> 00:11:48,593 only the horizontal component of this tension 323 00:11:48,593 --> 00:11:51,042 which is this amount. 324 00:11:51,042 --> 00:11:52,283 If I plug this in up here, 325 00:11:52,283 --> 00:11:55,273 I'll put a plus because this horizontal component 326 00:11:55,273 --> 00:11:56,509 points to the right, 327 00:11:56,509 --> 00:11:59,176 plus T three times cosine theta. 328 00:12:00,737 --> 00:12:02,943 And again, the way I'd solve for T one 329 00:12:02,943 --> 00:12:05,228 is I'd multiply both sides by m 330 00:12:05,228 --> 00:12:08,343 so I'd get ma knot and then I'd add T two to both sides 331 00:12:08,343 --> 00:12:11,508 and then I'd have to subtract T three cosine theta 332 00:12:11,508 --> 00:12:16,017 from both sides in order to solve for this algebraically. 333 00:12:16,017 --> 00:12:17,967 And this makes sense. 334 00:12:17,967 --> 00:12:20,639 My tension T one doesn't have to be as big anymore 335 00:12:20,639 --> 00:12:23,296 because it's got a force helping it pull to the right. 336 00:12:23,296 --> 00:12:25,951 There's someone on its side pulling to the right 337 00:12:25,951 --> 00:12:28,094 so it doesn't have to exert as much force 338 00:12:28,094 --> 00:12:30,346 that's why this ends up subtracting up here. 339 00:12:30,346 --> 00:12:33,322 T one decreases if you give it a helper force 340 00:12:33,322 --> 00:12:35,777 to pull on the same direction that it's pulling. 341 00:12:35,777 --> 00:12:37,217 Conceptually that's why this tension 342 00:12:37,217 --> 00:12:39,046 might increase or decrease. 343 00:12:39,046 --> 00:12:40,201 That's how you would deal with it 344 00:12:40,201 --> 00:12:41,956 if there were forces involved. 345 00:12:41,956 --> 00:12:44,907 You can keep adding forces here even friction 346 00:12:44,907 --> 00:12:47,058 if you had a frictional force to the left, 347 00:12:47,058 --> 00:12:49,470 you would just have to include that as a force up here. 348 00:12:49,470 --> 00:12:51,602 You'd keep doing it using Newton's Second Law 349 00:12:51,602 --> 00:12:53,367 and then solve algebraically 350 00:12:53,367 --> 00:12:55,511 for the tension that you wanted to find. 351 00:12:55,511 --> 00:12:58,146 To recap, remember the way you solve for tension 352 00:12:58,146 --> 00:13:00,105 is by using Newton's Second Law, 353 00:13:00,105 --> 00:13:02,184 carefully getting all the signs right 354 00:13:02,184 --> 00:13:03,847 and doing your algebra to solve 355 00:13:03,847 --> 00:13:05,905 for that tension that you wanna find. 356 00:13:05,905 --> 00:13:09,681 Also, remember the force of tension is not a pushing force. 357 00:13:09,681 --> 00:13:11,504 The force of tension is a pulling force. 358 00:13:11,504 --> 00:13:12,805 You can pull with the rope 359 00:13:12,805 --> 00:13:14,491 but you can't push with the rope. 360 00:13:14,491 --> 00:13:17,117 And in this problem, the tension throughout the rope 361 00:13:17,117 --> 00:13:18,962 was the same because we assumed 362 00:13:18,962 --> 00:13:21,491 that either the rope was massless 363 00:13:21,491 --> 00:13:23,711 or the mass of the rope was so insignificant 364 00:13:23,711 --> 00:13:25,511 compared to the mass of this box 365 00:13:25,511 --> 00:13:27,444 that any variation didn't matter. 366 00:13:27,444 --> 00:13:29,780 In other words, the tension at every point 367 00:13:29,780 --> 00:13:31,543 in this rope was the same 368 00:13:31,543 --> 00:13:32,841 and that could have made a difference 369 00:13:32,841 --> 00:13:34,409 because if we were asking the question 370 00:13:34,409 --> 00:13:36,695 how hard does this person over here 371 00:13:36,695 --> 00:13:40,365 have to pull on this rope to cause this acceleration? 372 00:13:40,365 --> 00:13:42,762 If the rope itself was massive 373 00:13:42,762 --> 00:13:44,500 this person would have to not only pull 374 00:13:44,500 --> 00:13:47,731 on this massive box but also on this massive rope 375 00:13:47,731 --> 00:13:49,631 and there'd be a variation in tension here 376 00:13:49,631 --> 00:13:52,511 that honestly, we often don't wanna have to deal with. 377 00:13:52,511 --> 00:13:53,929 So we assume the rope is massless 378 00:13:53,929 --> 00:13:54,841 and then we can just assume 379 00:13:54,841 --> 00:13:57,790 that whatever force this person pulls with 380 00:13:57,790 --> 00:13:59,410 because tension's a pulling force, 381 00:13:59,410 --> 00:14:02,003 is also transmitted here undiluted. 382 00:14:02,003 --> 00:14:04,158 If this person pulled with 50 newtons 383 00:14:04,158 --> 00:14:06,390 then this point of the rope would also pull 384 00:14:06,390 --> 00:00:00,000 on the box with 50 newtons.