1 00:00:00,020 --> 00:00:02,640 - [Voiceover] You bought a huge canister, 2 00:00:02,640 --> 00:00:05,210 aluminum can, of super hot red peppers, 3 00:00:05,210 --> 00:00:08,770 three kilograms' worth, and you hung them from two strings 4 00:00:08,770 --> 00:00:10,910 from the ceiling 'cause you don't want anyone 5 00:00:10,910 --> 00:00:13,010 to get your super hot red peppers, 6 00:00:13,010 --> 00:00:14,500 and you wanted to know, 7 00:00:14,500 --> 00:00:16,350 what's the tension in both of these strings? 8 00:00:16,350 --> 00:00:17,830 If this is the angle 9 00:00:17,830 --> 00:00:20,960 that the strings make with the ceiling, 10 00:00:20,960 --> 00:00:23,280 what are these two tensions? 11 00:00:23,280 --> 00:00:24,590 This problem's hard. 12 00:00:24,590 --> 00:00:27,910 This problem's spicy, this is a spicy tension problem. 13 00:00:27,910 --> 00:00:29,950 Let's start with something a little more mild 14 00:00:29,950 --> 00:00:31,370 and we'll work our way up to this 15 00:00:31,370 --> 00:00:32,770 to see how this works. 16 00:00:32,770 --> 00:00:35,760 We're gonna do that because even though this one 17 00:00:35,760 --> 00:00:38,320 seems very difficult, and easier problems 18 00:00:38,320 --> 00:00:39,680 seem much different from it, 19 00:00:39,680 --> 00:00:42,510 the process that you use to figure out the answer 20 00:00:42,510 --> 00:00:43,940 is always the same. 21 00:00:43,940 --> 00:00:45,610 So I'm gonna try to show you 22 00:00:45,610 --> 00:00:46,740 that you shouldn't get distracted. 23 00:00:46,740 --> 00:00:49,510 Even though the details are different, the process, 24 00:00:49,510 --> 00:00:52,190 the overall strategy is the same. 25 00:00:52,190 --> 00:00:54,050 Let's start with something a little easier. 26 00:00:54,050 --> 00:00:56,970 Let's start with this, a nice, red apple. 27 00:00:56,970 --> 00:00:59,160 Let's say this apple is three kilograms 28 00:00:59,160 --> 00:01:01,220 so we can keep the same number. 29 00:01:01,220 --> 00:01:03,610 Three kilogram apple, hanging from a string. 30 00:01:03,610 --> 00:01:06,200 We wanna know what's the tension in the rope. 31 00:01:06,200 --> 00:01:07,600 Well, this one's easy. 32 00:01:07,600 --> 00:01:09,320 The way you solve it is the way you solve 33 00:01:09,320 --> 00:01:10,320 all of these problems. 34 00:01:10,320 --> 00:01:12,170 So even though it's easy, we're gonna go through 35 00:01:12,170 --> 00:01:13,890 the entire process so you can see how it works, 36 00:01:13,890 --> 00:01:15,230 and it won't take long. 37 00:01:15,230 --> 00:01:16,270 We draw the forces. 38 00:01:16,270 --> 00:01:19,760 The force of gravity is exerted on this apple 39 00:01:19,760 --> 00:01:21,950 because it's always exerted on everything near the Earth, 40 00:01:21,950 --> 00:01:23,180 and it's mg. 41 00:01:23,180 --> 00:01:24,520 Now, we've got a tension force. 42 00:01:24,520 --> 00:01:26,280 And the tension does not push. 43 00:01:26,280 --> 00:01:27,520 You can't push with a rope. 44 00:01:27,520 --> 00:01:30,000 You can pull with a rope, so this tension force 45 00:01:30,000 --> 00:01:32,800 points upward, I'll call it T. 46 00:01:32,800 --> 00:01:33,910 That's always the first step. 47 00:01:33,910 --> 00:01:35,390 You draw your force diagram, 48 00:01:35,390 --> 00:01:37,690 and now you use Newton's Second Law 49 00:01:37,690 --> 00:01:39,660 for either the horizontal direction 50 00:01:39,660 --> 00:01:42,460 or the vertical direction, of both if you have to. 51 00:01:42,460 --> 00:01:44,280 So we're gonna use Newton's Second Law, 52 00:01:44,280 --> 00:01:45,600 which looks like this. 53 00:01:45,600 --> 00:01:48,100 And the acceleration in a certain direction 54 00:01:48,100 --> 00:01:50,100 equals the net force in that direction 55 00:01:50,100 --> 00:01:51,250 divided by the mass. 56 00:01:51,250 --> 00:01:52,490 Which direction do we pick? 57 00:01:52,490 --> 00:01:53,560 It's pretty obvious here. 58 00:01:53,560 --> 00:01:54,720 We'll pick the vertical direction 59 00:01:54,720 --> 00:01:58,040 because there are no forces in the horizontal direction. 60 00:01:58,040 --> 00:01:59,340 And I'll plug in the acceleration. 61 00:01:59,340 --> 00:02:02,040 If this apple's just hanging here and not moving, 62 00:02:02,040 --> 00:02:04,060 then it's not accelerating, it's just sitting here. 63 00:02:04,060 --> 00:02:06,720 Unless this was like an elevator or a rocket ship 64 00:02:06,720 --> 00:02:08,300 that had acceleration upward. 65 00:02:08,300 --> 00:02:10,580 You could do that, you'd plug in that acceleration. 66 00:02:10,580 --> 00:02:13,070 But if it's just hanging from the ceiling at rest, 67 00:02:13,070 --> 00:02:15,140 this acceleration's gonna be zero. 68 00:02:15,140 --> 00:02:17,220 That's gonna equal the net force. 69 00:02:17,220 --> 00:02:18,310 We've got tension upward. 70 00:02:18,310 --> 00:02:20,310 To figure out what goes here, 71 00:02:20,310 --> 00:02:22,250 like we draw this force diagram for a reason. 72 00:02:22,250 --> 00:02:23,820 This isn't just like 73 00:02:23,820 --> 00:02:26,060 happy fun painting time over here. 74 00:02:26,060 --> 00:02:27,320 This is a strategy. 75 00:02:27,320 --> 00:02:30,250 We draw these forces 'cause this lets us know 76 00:02:30,250 --> 00:02:33,350 what we plug into Newton's Second Law for the net force. 77 00:02:33,350 --> 00:02:35,100 If it's not on here, I don't put it up here. 78 00:02:35,100 --> 00:02:36,950 And if it is on here, I have to put it in here 79 00:02:36,950 --> 00:02:38,050 if it's in that direction. 80 00:02:38,050 --> 00:02:41,050 So tension is vertical, and this is the vertical direction 81 00:02:41,050 --> 00:02:42,130 for the net force. 82 00:02:42,130 --> 00:02:43,320 So I put tension in here. 83 00:02:43,320 --> 00:02:44,850 I'm gonna call upward positive. 84 00:02:44,850 --> 00:02:46,890 That means I make this tension positive. 85 00:02:46,890 --> 00:02:48,650 And how about mg over here? 86 00:02:48,650 --> 00:02:51,067 mg points down so I minus mg. 87 00:02:52,130 --> 00:02:53,970 Then I divide by the mass. 88 00:02:53,970 --> 00:02:56,490 Well, now I just solve for the tension. 89 00:02:56,490 --> 00:02:57,680 So this is the process. 90 00:02:57,680 --> 00:03:00,210 You draw your force diagram, you use Newton's Second Law, 91 00:03:00,210 --> 00:03:02,110 and then you try to solve for the force that you want. 92 00:03:02,110 --> 00:03:03,690 In this case, it's the tension. 93 00:03:03,690 --> 00:03:05,760 So I multiply both sides by m. 94 00:03:05,760 --> 00:03:07,660 It's still zero on the left-hand side, 95 00:03:07,660 --> 00:03:11,460 so zero equals t minus mg. 96 00:03:11,460 --> 00:03:13,340 And if I solve this for t, 97 00:03:13,340 --> 00:03:16,920 I get something that might not be all that surprising. 98 00:03:16,920 --> 00:03:19,820 I just get that the tension is equal to mg. 99 00:03:19,820 --> 00:03:21,800 And you might be like, "Well, duh. 100 00:03:22,740 --> 00:03:24,980 "That was like way more trouble than it had to be." 101 00:03:24,980 --> 00:03:26,340 I mean, you just knew it, it was mg. 102 00:03:26,340 --> 00:03:28,220 This tension just has to balance gravity, 103 00:03:28,220 --> 00:03:31,340 so it's just mg, why did we go through all this trouble? 104 00:03:31,340 --> 00:03:34,460 And the reason is, I mean, it is equal to mg, 105 00:03:34,460 --> 00:03:37,320 but it won't always be equal to mg. 106 00:03:37,320 --> 00:03:40,670 So if you wanna know what to do when it's not equal to mg, 107 00:03:40,670 --> 00:03:43,060 you have to know how to use this process well. 108 00:03:43,060 --> 00:03:45,200 Why would it not be equal to mg? 109 00:03:45,200 --> 00:03:46,200 Well, imagine this. 110 00:03:46,200 --> 00:03:48,400 Imagine I had, say, two tensions up here. 111 00:03:48,400 --> 00:03:50,120 Or let's start with this, let's say I just pull down 112 00:03:50,120 --> 00:03:51,310 on the rope. 113 00:03:51,310 --> 00:03:54,630 I just come over here and someone pulls down on the apple. 114 00:03:54,630 --> 00:03:56,140 Let's say someone just pulls down the apple 115 00:03:56,140 --> 00:03:57,890 with an extra five Newtons. 116 00:03:57,890 --> 00:04:00,470 Right, pulling this apple, which pulls this string, 117 00:04:00,470 --> 00:04:01,590 which makes it tighter. 118 00:04:01,590 --> 00:04:02,950 What would I do in that case? 119 00:04:02,950 --> 00:04:05,330 Well, I've got one more force here in my force diagram. 120 00:04:05,330 --> 00:04:06,890 I just add that. 121 00:04:06,890 --> 00:04:08,050 So that's five Newtons. 122 00:04:08,050 --> 00:04:10,220 It's pointing downward, so when I come over to here 123 00:04:10,220 --> 00:04:12,720 to my net force, I've got to subtract five Newtons 124 00:04:12,720 --> 00:04:14,730 'cause that's pointing downward. 125 00:04:14,730 --> 00:04:16,760 And now I do my algebra just as before. 126 00:04:16,760 --> 00:04:18,709 I multiply both sides by m. 127 00:04:18,709 --> 00:04:22,530 I'll have another minus five Newtons here. 128 00:04:22,530 --> 00:04:25,090 And then when I solve for t I'll add mg to both sides, 129 00:04:25,090 --> 00:04:27,880 and I'll add five Newtons to both sides. 130 00:04:27,880 --> 00:04:29,410 And so this would be my force. 131 00:04:29,410 --> 00:04:31,170 So with numbers, what are we gonna get? 132 00:04:31,170 --> 00:04:34,460 The mass was three, so we'll have three kilograms 133 00:04:35,520 --> 00:04:37,390 times the acceleration due to gravity 134 00:04:37,390 --> 00:04:40,050 is 9.8, but to make the numbers nice, 135 00:04:40,050 --> 00:04:41,850 let's just round to 10 for now, 136 00:04:41,850 --> 00:04:43,650 10 meters per second squared. 137 00:04:43,650 --> 00:04:46,900 That way we don't get lost in details and decimals. 138 00:04:46,900 --> 00:04:49,030 So plus five Newtons. 139 00:04:49,030 --> 00:04:51,510 It's mg when it's just hanging there, 140 00:04:51,510 --> 00:04:53,800 but if there's extra forces, it won't be mg. 141 00:04:53,800 --> 00:04:57,040 In this case, with a five Newton force downward, 142 00:04:57,040 --> 00:04:58,980 it's gonna be 30 plus five, 143 00:04:58,980 --> 00:05:01,320 so it's gonna be 35 Newtons. 144 00:05:01,320 --> 00:05:03,820 So it's equal to mg in the easiest possible case, 145 00:05:03,820 --> 00:05:07,270 but if there's extra forces, it won't be equal to that. 146 00:05:07,270 --> 00:05:10,100 Also, if there was another rope pulling up 147 00:05:10,100 --> 00:05:11,590 in the same spot as the first rope, 148 00:05:11,590 --> 00:05:13,800 now I'd have two tensions pulling up. 149 00:05:13,800 --> 00:05:17,590 Now I've got another tension force pulling up this way. 150 00:05:17,590 --> 00:05:18,630 So what would I do now? 151 00:05:18,630 --> 00:05:19,860 I'd have to add another T. 152 00:05:19,860 --> 00:05:21,490 I'd have plus T. 153 00:05:21,490 --> 00:05:24,470 Well T plus T is just 2T. 154 00:05:24,470 --> 00:05:27,850 So I'd have a 2T here, I'd have a 2T here. 155 00:05:27,850 --> 00:05:30,820 To solve for T, I'd just have to do 35 Newtons 156 00:05:30,820 --> 00:05:31,820 divided by two. 157 00:05:31,820 --> 00:05:34,810 You could start making these harder, and harder, and harder, 158 00:05:34,810 --> 00:05:36,960 and you could do that by adding extra ropes, 159 00:05:36,960 --> 00:05:38,610 or adding forces down. 160 00:05:38,610 --> 00:05:40,860 And you might convince yourself erroneously 161 00:05:40,860 --> 00:05:42,820 you have to do something new, but you don't. 162 00:05:42,820 --> 00:05:45,000 You still just draw your force diagram, 163 00:05:45,000 --> 00:05:46,730 then you go to Newton's Second Law, 164 00:05:46,730 --> 00:05:48,410 you put those forces in carefully, 165 00:05:48,410 --> 00:05:50,350 you solve for what you wanna know. 166 00:05:50,350 --> 00:05:52,950 And if this is confusing, why is it 35 over two? 167 00:05:52,950 --> 00:05:53,950 It kind of makes sense. 168 00:05:53,950 --> 00:05:56,300 The total downward force is 35 Newtons 169 00:05:56,300 --> 00:05:59,990 'cause this is 30 Newtons of weight here. 170 00:05:59,990 --> 00:06:01,920 We've got 35 Newtons in downward. 171 00:06:01,920 --> 00:06:04,320 The total upward force has to be 35. 172 00:06:04,320 --> 00:06:06,470 And if these strings are attached at the same point, 173 00:06:06,470 --> 00:06:08,270 they're both gonna bear the same amount of weight. 174 00:06:08,270 --> 00:06:10,450 They have to total up to 35, 175 00:06:10,450 --> 00:06:13,950 so you just get 35 over two amongst each of them 176 00:06:13,950 --> 00:06:16,310 'cause they each bear half the weight. 177 00:06:16,310 --> 00:06:18,000 So that was one of the easiest problems. 178 00:06:18,000 --> 00:06:20,090 We turned it into a little bit of a harder problem. 179 00:06:20,090 --> 00:06:22,450 It gets even harder if we add an angle. 180 00:06:22,450 --> 00:06:24,020 Remember, those jalapenos were at an angle. 181 00:06:24,020 --> 00:06:25,220 We gotta do this. 182 00:06:25,220 --> 00:06:26,630 We gotta do one at an angle. 183 00:06:26,630 --> 00:06:29,200 Say you got a chalkboard hanging from two strings. 184 00:06:29,200 --> 00:06:32,200 One string's horizontal, one string is up here at an angle. 185 00:06:32,200 --> 00:06:33,610 What do you do now? 186 00:06:33,610 --> 00:06:36,450 It's easy to convince yourself you gotta try something new, 187 00:06:36,450 --> 00:06:38,560 or go for a new strategy, but you don't. 188 00:06:38,560 --> 00:06:39,950 You solve this the same way. 189 00:06:39,950 --> 00:06:41,510 We're gonna draw our forces. 190 00:06:41,510 --> 00:06:43,540 We've still got a force of gravity down. 191 00:06:43,540 --> 00:06:46,850 So this force of gravity is just mg. 192 00:06:46,850 --> 00:06:50,450 And for consistency's sake, let's say the mass 193 00:06:50,450 --> 00:06:53,610 of this chalkboard is also three kilograms. 194 00:06:54,690 --> 00:06:55,860 All right, what else do I have? 195 00:06:55,860 --> 00:06:59,960 I'm gonna have a tension that points up and to the right. 196 00:07:00,970 --> 00:07:02,710 It doesn't point this way. 197 00:07:02,710 --> 00:07:04,830 Tension doesn't push, tension pulls. 198 00:07:04,830 --> 00:07:06,890 So this tension's gotta pull this way. 199 00:07:06,890 --> 00:07:08,990 We'll call this T one. 200 00:07:08,990 --> 00:07:10,950 This is T one. 201 00:07:10,950 --> 00:07:13,290 And over on my force diagram, it would look like this. 202 00:07:13,290 --> 00:07:17,310 I'd have a T one that points something like that. 203 00:07:17,310 --> 00:07:19,020 So here's my T one. 204 00:07:19,930 --> 00:07:22,280 I'll put it here, T one. 205 00:07:22,280 --> 00:07:23,900 And I've got one more force now. 206 00:07:23,900 --> 00:07:25,830 I've got this horizontal force here. 207 00:07:25,830 --> 00:07:27,450 Again, it does not push. 208 00:07:27,450 --> 00:07:29,360 This is a rope, it can only pull. 209 00:07:29,360 --> 00:07:31,040 So it pulls to the left. 210 00:07:31,040 --> 00:07:32,630 I'll call this T two. 211 00:07:32,630 --> 00:07:35,180 So on my force diagram, I would have 212 00:07:35,180 --> 00:07:37,700 T two, and that's it. 213 00:07:37,700 --> 00:07:38,860 Those are all my forces. 214 00:07:38,860 --> 00:07:40,040 I don't have a normal force. 215 00:07:40,040 --> 00:07:41,740 Sometimes people wanna draw, 216 00:07:41,740 --> 00:07:44,070 people are so used to there always being normal force. 217 00:07:44,070 --> 00:07:45,470 They're like, "Normal force, right?" 218 00:07:45,470 --> 00:07:46,640 And it's like, "No." 219 00:07:46,640 --> 00:07:49,400 There's no surface touching this chalkboard. 220 00:07:49,400 --> 00:07:50,940 It's just hanging by strings. 221 00:07:50,940 --> 00:07:53,260 The only force keeping it up would be the vertical component 222 00:07:53,260 --> 00:07:54,940 of this T one, so this is it. 223 00:07:54,940 --> 00:07:56,780 These are the only forces we've got. 224 00:07:56,780 --> 00:07:57,830 What do you do after that? 225 00:07:57,830 --> 00:07:58,970 Same as the apple problem. 226 00:07:58,970 --> 00:08:00,620 We go to Newton's Second Law, 227 00:08:00,620 --> 00:08:04,250 and we say that the acceleration will be the net force 228 00:08:04,250 --> 00:08:05,820 divided by the mass. 229 00:08:05,820 --> 00:08:07,240 Which direction do we pick? 230 00:08:07,240 --> 00:08:08,780 It's not quite as obvious here. 231 00:08:08,780 --> 00:08:11,590 We've got forces vertically and horizontally. 232 00:08:11,590 --> 00:08:13,030 So here's my advice. 233 00:08:13,030 --> 00:08:14,520 Look for something that you know. 234 00:08:14,520 --> 00:08:17,150 in this case, I know the mass is three kilograms, 235 00:08:17,150 --> 00:08:18,890 and I know the acceleration due to gravity, 236 00:08:18,890 --> 00:08:20,960 or the magnitude of it, 9.8, 237 00:08:20,960 --> 00:08:22,090 but in this case, 10. 238 00:08:22,090 --> 00:08:23,400 Since I know this force, remember, 239 00:08:23,400 --> 00:08:25,680 this force is just 30 Newtons 240 00:08:25,680 --> 00:08:28,250 'cause three times 9.8, 241 00:08:28,250 --> 00:08:30,550 or three times 10 meters per second squared 242 00:08:30,550 --> 00:08:31,650 is just 30 Newtons. 243 00:08:31,650 --> 00:08:34,220 Since I know this force, it's a vertical force, 244 00:08:34,220 --> 00:08:35,840 I know something about that direction. 245 00:08:35,840 --> 00:08:37,580 I'm just gonna start with that direction 246 00:08:37,580 --> 00:08:39,820 'cause I already know something about it. 247 00:08:39,820 --> 00:08:41,640 So I'm gonna do a in the y direction, 248 00:08:41,640 --> 00:08:42,659 F in the y direction. 249 00:08:42,659 --> 00:08:44,139 We'll start with the vertical direction. 250 00:08:44,140 --> 00:08:46,430 If you make a mistake and you pick the wrong direction, 251 00:08:46,430 --> 00:08:47,870 it's not the end of the world. 252 00:08:47,870 --> 00:08:48,970 Just pick the other direction. 253 00:08:48,970 --> 00:08:50,650 There's only two to worry about, 254 00:08:50,650 --> 00:08:52,810 so if you screw one up, just go on to the next one. 255 00:08:52,810 --> 00:08:54,170 It's not that big of a deal. 256 00:08:54,170 --> 00:08:55,960 All right, acceleration vertically again. 257 00:08:55,960 --> 00:08:57,780 Let's say this is at rest, 258 00:08:57,780 --> 00:08:59,980 just hanging from these strings. 259 00:08:59,980 --> 00:09:01,890 So we don't have to worry about any acceleration. 260 00:09:01,890 --> 00:09:03,200 Although, if there was, you would just 261 00:09:03,200 --> 00:09:04,600 plug that acceleration in here. 262 00:09:04,600 --> 00:09:07,400 It's not that much harder of a problem. 263 00:09:07,400 --> 00:09:10,060 Zero equals, all right, what do we got? 264 00:09:10,060 --> 00:09:12,770 We need to put our vertical forces up top. 265 00:09:12,770 --> 00:09:13,780 I know this one. 266 00:09:13,780 --> 00:09:16,490 30 Newtons downward. 267 00:09:16,490 --> 00:09:18,480 So I'm gonna have negative 30 Newtons 268 00:09:18,480 --> 00:09:20,060 'cause those 30 Newtons point down, 269 00:09:20,060 --> 00:09:22,720 and I'm gonna consider downward as a negative, 270 00:09:22,720 --> 00:09:24,400 upward as positive. 271 00:09:24,400 --> 00:09:25,740 This is really just mg. 272 00:09:25,740 --> 00:09:28,120 I could've wrote it as negative mg. 273 00:09:28,120 --> 00:09:29,150 What else do I have? 274 00:09:29,150 --> 00:09:31,820 I've got T one; T one points up, 275 00:09:31,820 --> 00:09:34,100 but I can't add all of T one here 276 00:09:34,100 --> 00:09:36,060 'cause it doesn't all point up. 277 00:09:36,060 --> 00:09:37,500 I can only add all of T one 278 00:09:37,500 --> 00:09:40,040 if T one pointed straight upward. 279 00:09:40,040 --> 00:09:41,040 But it doesn't. 280 00:09:41,040 --> 00:09:42,370 Part of it points up. 281 00:09:42,370 --> 00:09:45,320 So this part of T one points to the right, 282 00:09:45,320 --> 00:09:46,330 pulls to the right. 283 00:09:46,330 --> 00:09:47,800 This part of T one pulls up. 284 00:09:47,800 --> 00:09:50,530 It's this component right here that's gonna be 285 00:09:50,530 --> 00:09:52,970 the component that actually causes this chalkboard 286 00:09:52,970 --> 00:09:56,160 to stay up, that keeps it from falling down 287 00:09:56,160 --> 00:09:58,170 because that's the part that's fighting gravity. 288 00:09:58,170 --> 00:10:00,750 We'll call this T one in the y direction. 289 00:10:00,750 --> 00:10:04,120 We'll call this T one in the x direction. 290 00:10:04,120 --> 00:10:08,500 So we need to add plus T one in the y direction. 291 00:10:08,500 --> 00:10:10,640 And that's it, those are the only two forces 292 00:10:10,640 --> 00:10:11,780 that are vertical. 293 00:10:11,780 --> 00:10:13,410 T one x is not vertical. 294 00:10:13,410 --> 00:10:15,790 That's horizontal, and this T two is horizontal. 295 00:10:15,790 --> 00:10:17,600 So I've included T one y and mg. 296 00:10:17,600 --> 00:10:19,440 Those are the only two forces that are vertical. 297 00:10:19,440 --> 00:10:21,190 So now I divide by them my mass. 298 00:10:22,080 --> 00:10:24,190 I can multiply both sides by mass. 299 00:10:24,190 --> 00:10:27,000 m times zero is still zero. 300 00:10:27,000 --> 00:10:31,270 I get zero equals negative 30 Newtons 301 00:10:31,270 --> 00:10:34,070 plus T one y. 302 00:10:34,070 --> 00:10:35,720 And now we have to figure out, okay, 303 00:10:35,720 --> 00:10:37,090 T one y, 304 00:10:37,090 --> 00:10:38,940 T one y has to equal what? 305 00:10:38,940 --> 00:10:41,020 I can solve this for T one y. 306 00:10:41,020 --> 00:10:43,310 I add 30 to both sides. 307 00:10:43,310 --> 00:10:44,900 I'm gonna get T one y 308 00:10:44,900 --> 00:10:48,040 equals positive 30 Newtons. 309 00:10:48,040 --> 00:10:49,490 And that makes sense. 310 00:10:49,490 --> 00:10:52,410 I mean, this T one y is the only component 311 00:10:52,410 --> 00:10:53,760 that's balancing out gravity. 312 00:10:53,760 --> 00:10:54,760 We know it has to balance 313 00:10:54,760 --> 00:10:57,460 because there's no acceleration vertically. 314 00:10:57,460 --> 00:10:59,780 So this T one y has to be the exact same size 315 00:10:59,780 --> 00:11:01,080 as the force of gravity. 316 00:11:01,080 --> 00:11:02,740 I drew it, it's not proportional here. 317 00:11:02,740 --> 00:11:03,770 Sorry about that, I should've drawn it 318 00:11:03,770 --> 00:11:05,850 with this component exactly the same length 319 00:11:05,850 --> 00:11:07,720 as this component 'cause they to be the same, 320 00:11:07,720 --> 00:11:09,310 they have to cancel. 321 00:11:09,310 --> 00:11:10,790 But that just tells me T one y. 322 00:11:10,790 --> 00:11:12,810 I wanna know what T one is. 323 00:11:12,810 --> 00:11:15,240 How do I solve for what T one is, 324 00:11:15,240 --> 00:11:17,640 and what T two is? 325 00:11:17,640 --> 00:11:18,830 These are what I wanna figure out. 326 00:11:18,830 --> 00:11:19,840 What are the tensions? 327 00:11:19,840 --> 00:11:22,440 I don't just want the component, I want the tension. 328 00:11:22,440 --> 00:11:26,500 And so now I say that this component, T one y, 329 00:11:26,500 --> 00:11:28,840 is gonna be related to the total T one, 330 00:11:28,840 --> 00:11:31,500 and it's related through this angle here. 331 00:11:31,500 --> 00:11:33,170 So I can say that T one y, 332 00:11:33,170 --> 00:11:35,820 whatever this angle is right here. 333 00:11:35,820 --> 00:11:37,710 Remember, we can use trigonometry 334 00:11:37,710 --> 00:11:41,360 and we can say that sine theta 335 00:11:41,360 --> 00:11:45,330 is gonna be the opposite side over the hypotenuse. 336 00:11:45,330 --> 00:11:47,830 And in this case the opposite side to this angle, 337 00:11:47,830 --> 00:11:50,700 opposite is T one y. 338 00:11:50,700 --> 00:11:52,880 It's gonna be T one y divided by, 339 00:11:52,880 --> 00:11:55,140 the hypotenuse side is the total tension. 340 00:11:55,140 --> 00:11:57,270 So that's always the total magnitude of the force. 341 00:11:57,270 --> 00:12:00,460 In this case, we're calling that T one. 342 00:12:00,460 --> 00:12:02,130 So I wanna solve for T one. 343 00:12:02,130 --> 00:12:04,710 So if I multiply both sides by T one, 344 00:12:04,710 --> 00:12:06,550 I'll get T one times sine theta, 345 00:12:06,550 --> 00:12:08,890 and then I divide both sides by sine theta. 346 00:12:08,890 --> 00:12:11,900 I'll end up with T one equals 347 00:12:11,900 --> 00:12:16,330 T one in the y direction divided by sine theta. 348 00:12:16,330 --> 00:12:17,960 I know T one in the y direction. 349 00:12:17,960 --> 00:12:20,490 That was 30 degree, or sorry, not 30 degrees. 350 00:12:20,490 --> 00:12:22,160 That was 30 Newtons. 351 00:12:22,160 --> 00:12:25,900 So I've got 30 Newtons, that's my force, upward. 352 00:12:25,900 --> 00:12:28,320 This vertical component right here had to be 30 Newtons 353 00:12:28,320 --> 00:12:29,830 'cause it had to balance gravity, 354 00:12:29,830 --> 00:12:32,640 divided by sine of the angle. 355 00:12:32,640 --> 00:12:34,230 But what is this angle? 356 00:12:34,230 --> 00:12:36,260 We know this angle's 30. 357 00:12:36,260 --> 00:12:37,570 And you could probably convince yourself, 358 00:12:37,570 --> 00:12:39,400 if I draw a triangle this way, 359 00:12:39,400 --> 00:12:41,110 let's try to figure out, 360 00:12:41,110 --> 00:12:43,960 we wanna figure out what this angle is right here 361 00:12:43,960 --> 00:12:45,460 'cause that's what this angle is here. 362 00:12:45,460 --> 00:12:47,490 So if this is 30 and that's 90, 363 00:12:47,490 --> 00:12:49,520 then this has to be 60. 364 00:12:49,520 --> 00:12:52,840 And if that's 60, and this is 90, 365 00:12:52,840 --> 00:12:54,220 then this has to be 30. 366 00:12:54,220 --> 00:12:57,220 So this angle is 30 degrees right here. 367 00:12:57,220 --> 00:12:58,930 So that's 30 degrees. 368 00:12:58,930 --> 00:13:01,400 So this angle right here, 369 00:13:01,400 --> 00:13:04,100 which is this angle right here, has to be 30. 370 00:13:04,100 --> 00:13:05,210 So when I'm taking my sine, 371 00:13:05,210 --> 00:13:07,500 I'm taking my sine of 30 degrees. 372 00:13:07,500 --> 00:13:10,880 And I get 30 Newtons divided by sine of 30, 373 00:13:10,880 --> 00:13:12,980 and sine of 30 is 1/2. 374 00:13:12,980 --> 00:13:17,080 So .5, so I get that this is 60 Newtons. 375 00:13:19,190 --> 00:13:20,610 And that might seem crazy. 376 00:13:20,610 --> 00:13:22,640 You might be like, "Wait a minute. 377 00:13:22,640 --> 00:13:25,180 "T one is 60 Newtons? 378 00:13:25,180 --> 00:13:26,340 "60 Newtons? 379 00:13:26,340 --> 00:13:29,270 "The weight of this chalkboard is only 30 Newtons. 380 00:13:29,270 --> 00:13:32,340 "How in the world can the tension in this rope 381 00:13:32,340 --> 00:13:34,020 "be 60 Newtons?" 382 00:13:34,880 --> 00:13:38,110 I mean, if we just hung it by a single string, 383 00:13:38,110 --> 00:13:40,540 if we just hung this chalkboard by a single string 384 00:13:40,540 --> 00:13:41,740 over the center of mass, 385 00:13:41,740 --> 00:13:43,600 you'd just get a tension of 30 Newtons. 386 00:13:43,600 --> 00:13:44,960 How can this be 60 Newtons? 387 00:13:44,960 --> 00:13:48,170 And the reason is, this part's gotta be 30 Newtons. 388 00:13:48,170 --> 00:13:50,450 We know that 'cause it has to balance gravity. 389 00:13:50,450 --> 00:13:53,120 But that's only part of the total tension. 390 00:13:53,120 --> 00:13:56,700 So if the total tension, if part of the total tension is 30, 391 00:13:56,700 --> 00:13:58,870 all of the tension's gotta be more than 30. 392 00:13:58,870 --> 00:14:01,290 And in this case, it's 60 Newtons. 393 00:14:01,290 --> 00:14:03,130 So that's why it's larger in this case, 394 00:14:03,130 --> 00:14:04,310 'cause it's at an angle. 395 00:14:04,310 --> 00:14:07,080 So this component has to equal gravity, 396 00:14:07,080 --> 00:14:08,930 and this total amount has to be bigger than that 397 00:14:08,930 --> 00:14:11,750 so that its component is equal to gravity. 398 00:14:11,750 --> 00:14:13,640 Right, how do we figure out T two? 399 00:14:13,640 --> 00:14:16,070 Well, you don't invent a new strategy. 400 00:14:16,070 --> 00:14:18,470 We keep going, we're just gonna say that the acceleration 401 00:14:18,470 --> 00:14:20,290 in the horizontal direction 402 00:14:20,290 --> 00:14:22,470 is the net force in the horizontal direction 403 00:14:22,470 --> 00:14:24,170 divided by the mass, so we still 404 00:14:24,170 --> 00:14:25,820 stick with Newton's Second Law 405 00:14:25,820 --> 00:14:28,160 even when we wanna find this other force. 406 00:14:28,160 --> 00:14:30,040 This force is horizontal, so it makes sense 407 00:14:30,040 --> 00:14:31,830 that we're gonna use Newton's Second Law 408 00:14:31,830 --> 00:14:33,240 for the horizontal direction. 409 00:14:33,240 --> 00:14:35,580 Again, if this chalkboard is not accelerating, 410 00:14:35,580 --> 00:14:37,620 the acceleration is zero, so I'll draw a line here 411 00:14:37,620 --> 00:14:40,300 to keep my calculations separate. 412 00:14:41,920 --> 00:14:44,270 Equals net force in the x direction. 413 00:14:44,270 --> 00:14:46,470 Okay, now I'm gonna have T one in the x direction. 414 00:14:46,470 --> 00:14:48,880 So this is gonna be T one in the x. 415 00:14:48,880 --> 00:14:51,810 So I'll have T one in the x direction. 416 00:14:51,810 --> 00:14:53,490 That's positive 'cause it points right, 417 00:14:53,490 --> 00:14:55,550 and I'm gonna consider rightward positive. 418 00:14:55,550 --> 00:14:58,120 Minus T two, all of T two. 419 00:14:58,120 --> 00:14:59,470 I don't have to break T two up. 420 00:14:59,470 --> 00:15:02,760 T two points completely in the horizontal direction. 421 00:15:02,760 --> 00:15:04,710 And I divide that by the mass. 422 00:15:04,710 --> 00:15:06,930 And well, I can multiply both sides by mass. 423 00:15:06,930 --> 00:15:09,450 I'd get zero equals 424 00:15:09,450 --> 00:15:11,540 T one in the x direction 425 00:15:12,750 --> 00:15:16,130 puh, minus, excuse me, minus T two. 426 00:15:16,130 --> 00:15:19,180 So if I solve this for T two, 427 00:15:19,180 --> 00:15:22,070 I'm gonna get that T two, if I add T two to both sides, 428 00:15:22,070 --> 00:15:24,470 I get that T two just equals 429 00:15:24,470 --> 00:15:26,550 T one in the x direction. 430 00:15:26,550 --> 00:15:28,760 But how big is T one in the x direction? 431 00:15:28,760 --> 00:15:31,430 We know T one is 60 Newtons. 432 00:15:31,430 --> 00:15:33,560 We know T one in the y direction 433 00:15:33,560 --> 00:15:35,670 of this piece here was 30 Newtons. 434 00:15:35,670 --> 00:15:37,550 How big is this piece? 435 00:15:37,550 --> 00:15:39,250 Well we can use, instead of sine now, 436 00:15:39,250 --> 00:15:41,320 we can use cosine. 437 00:15:41,320 --> 00:15:45,040 So if I use cosine, I can get that cosine theta, 438 00:15:45,040 --> 00:15:48,220 which is 30 degrees here, 'cause this angle here's 30. 439 00:15:48,220 --> 00:15:51,190 Cosine of 30 would be the adjacent. 440 00:15:51,190 --> 00:15:54,100 That's this T one x, so it's adjacent 441 00:15:54,100 --> 00:15:55,500 over the hypotenuse. 442 00:15:55,500 --> 00:15:58,290 The hypotenuse is T one, and we know T one. 443 00:15:58,290 --> 00:16:00,190 T one was 60. 444 00:16:00,190 --> 00:16:01,640 So I can solve for T one x, 445 00:16:01,640 --> 00:16:04,210 and I get that T one in the x direction, 446 00:16:04,210 --> 00:16:06,940 if I multiply both sides by 60 Newtons, 447 00:16:06,940 --> 00:16:09,420 this is 60 Newtons, I get that T one in the x direction 448 00:16:09,420 --> 00:16:11,540 would be 60 Newtons 449 00:16:11,540 --> 00:16:14,210 times cosine of 30. 450 00:16:14,210 --> 00:16:17,080 And the cosine of 30 is, 451 00:16:17,080 --> 00:16:19,130 cosine of 30 is root three over two. 452 00:16:19,130 --> 00:16:21,030 So I get 60 Newtons 453 00:16:21,030 --> 00:16:23,970 times root three over two, 454 00:16:23,970 --> 00:16:27,130 which means that T one in the x direction is, 455 00:16:27,130 --> 00:16:28,830 60 over two would be 30, 456 00:16:28,830 --> 00:16:32,350 so this is 30 root three Newtons. 457 00:16:32,350 --> 00:16:33,870 And that's what I can bring up here. 458 00:16:33,870 --> 00:16:35,460 This is T one x. 459 00:16:35,460 --> 00:16:39,110 So since that's T one x, I can say that T one x right here 460 00:16:39,110 --> 00:16:42,780 is 30 root three Newtons. 461 00:16:42,780 --> 00:16:45,770 And by Newton's Second Law in the horizontal direction, 462 00:16:45,770 --> 00:16:48,430 that's what T two had to equal. 463 00:16:48,430 --> 00:16:51,720 So T two equals 30 root three Newtons. 464 00:16:51,720 --> 00:16:52,720 So we figured it out. 465 00:16:52,720 --> 00:16:55,640 T two equals 30 root three Newtons, 466 00:16:55,640 --> 00:16:57,280 and that should be surprising. 467 00:16:57,280 --> 00:16:59,080 This force here, in order to make it 468 00:16:59,080 --> 00:17:01,450 so that there's no acceleration horizontally, 469 00:17:01,450 --> 00:17:04,030 just has to equal this force here. 470 00:17:04,030 --> 00:17:06,220 Those are the only two horizontal forces. 471 00:17:06,220 --> 00:17:09,210 We knew T one x was 30 root three. 472 00:17:09,210 --> 00:17:10,349 That's what we found. 473 00:17:10,349 --> 00:17:13,619 So that means T two also has to be 30 root three 474 00:17:13,619 --> 00:17:15,519 to make it so that these forces are balanced 475 00:17:15,520 --> 00:17:17,180 in the horizontal direction. 476 00:17:17,180 --> 00:17:18,180 All right, so we did it. 477 00:17:18,180 --> 00:17:20,050 We figured out T one, 60 Newtons. 478 00:17:20,050 --> 00:17:23,260 We figured out T two, 30 root three Newtons. 479 00:17:23,260 --> 00:17:25,450 Now we're ready, now we could figure out 480 00:17:25,450 --> 00:17:27,579 the super hot jalapeno problem. 481 00:17:27,579 --> 00:00:00,000 We'll do that in the next video.