1 00:00:00,000 --> 00:00:01,155 - [Instructor] Let's solve some more 2 00:00:01,155 --> 00:00:03,006 of these systems problems. 3 00:00:03,006 --> 00:00:04,981 If you remember, there's a hard way to do this, 4 00:00:04,981 --> 00:00:06,540 and an easy way to do this. 5 00:00:06,540 --> 00:00:08,812 The hard way is to solve Newton's second law 6 00:00:08,812 --> 00:00:11,936 for each box individually, and then combine them, 7 00:00:11,936 --> 00:00:14,419 and you get two equations with two unknowns, 8 00:00:14,419 --> 00:00:16,451 you try your best to solve the algebra 9 00:00:16,451 --> 00:00:18,980 without losing any sins, but let's be honest, 10 00:00:18,980 --> 00:00:20,612 it usually goes wrong. 11 00:00:20,612 --> 00:00:22,435 So, the easy way to do this, 12 00:00:22,435 --> 00:00:25,814 the way to get the magnitude of the acceleration 13 00:00:25,814 --> 00:00:27,986 of the objects in your system, 14 00:00:27,986 --> 00:00:30,200 that is to say, if I wanna know the magnitude 15 00:00:30,200 --> 00:00:32,548 at which this five kilogram box accelerates, 16 00:00:32,548 --> 00:00:35,060 or that this three kilogram box accelerates, 17 00:00:35,060 --> 00:00:38,545 all I need to do is take the net external force 18 00:00:38,545 --> 00:00:41,745 that tries to make my system go, 19 00:00:41,745 --> 00:00:44,785 and then I divide by my total mass of my system. 20 00:00:44,785 --> 00:00:47,205 This is a quick way to get what the magnitude 21 00:00:47,205 --> 00:00:50,355 of the acceleration is of the objects in my system, 22 00:00:50,355 --> 00:00:52,417 but it's good to note, it'll only work 23 00:00:52,417 --> 00:00:55,409 if the objects in your system are required to move 24 00:00:55,409 --> 00:00:57,831 with the same magnitude of acceleration. 25 00:00:57,831 --> 00:00:58,848 And in this case they are, 26 00:00:58,848 --> 00:01:00,895 what I have here is a five kilogram mass 27 00:01:00,895 --> 00:01:03,469 tied to a rope, and that rope passes over a pulley, 28 00:01:03,469 --> 00:01:05,875 pulls over and connects to this three kilogram mass 29 00:01:05,875 --> 00:01:07,788 so that if this five kilogram mass 30 00:01:07,788 --> 00:01:11,094 has some acceleration downward, this three kilogram mass 31 00:01:11,094 --> 00:01:13,866 has to be accelerating upward at the same rate, 32 00:01:13,866 --> 00:01:17,147 otherwise this rope would break or snap or stretch, 33 00:01:17,147 --> 00:01:18,697 and we're assuming that that doesn't happen. 34 00:01:18,697 --> 00:01:21,381 So this rope is the condition that requires 35 00:01:21,381 --> 00:01:23,083 the fact that this rope doesn't break 36 00:01:23,083 --> 00:01:25,396 is what allows us to say that the system 37 00:01:25,396 --> 00:01:28,143 is just a single, big total mass 38 00:01:28,143 --> 00:01:30,325 with external forces exerted on it. 39 00:01:30,325 --> 00:01:31,988 So how would we solve this? 40 00:01:31,988 --> 00:01:34,671 I'd just say that, well, what are the external forces? 41 00:01:34,671 --> 00:01:37,548 Keep in mind, external forces are forces that are exerted 42 00:01:37,548 --> 00:01:40,556 on the objects in our system from objects 43 00:01:40,556 --> 00:01:42,080 outside of our system. 44 00:01:42,080 --> 00:01:44,730 So one external force would just be the force of gravity 45 00:01:44,730 --> 00:01:46,695 on this five kilogram mass. 46 00:01:46,695 --> 00:01:49,416 So I'm gonna have a force of gravity this way, 47 00:01:49,416 --> 00:01:51,725 and that force of gravity is just going to be equal 48 00:01:51,725 --> 00:01:55,892 to five kilograms times 9.8 meters per second squared, 49 00:01:58,763 --> 00:02:00,884 because that's how we find the force of gravity. 50 00:02:00,884 --> 00:02:02,488 Should I make it positive or negative? 51 00:02:02,488 --> 00:02:04,198 Well, this five kilogram is gonna be the one 52 00:02:04,198 --> 00:02:06,893 that's pulling downward, so if the question is, 53 00:02:06,893 --> 00:02:10,271 I hold these masses and I let go, what's the acceleration? 54 00:02:10,271 --> 00:02:13,931 This five kilogram mass is gonna accelerate downward, 55 00:02:13,931 --> 00:02:15,952 it's gonna drive the system forward. 56 00:02:15,952 --> 00:02:18,700 That's the force making the system go, 57 00:02:18,700 --> 00:02:21,112 so I'm gonna make that a positive force. 58 00:02:21,112 --> 00:02:22,784 And then I figure out, are there any other forces 59 00:02:22,784 --> 00:02:24,948 making this system go? 60 00:02:24,948 --> 00:02:26,171 No, there are not. 61 00:02:26,171 --> 00:02:28,858 You might say, well what about this tension over here? 62 00:02:28,858 --> 00:02:32,095 Isn't the tension on this three kilogram mass? 63 00:02:32,095 --> 00:02:34,758 Isn't that tension making this system go? 64 00:02:34,758 --> 00:02:36,945 Not really, because that's an internal force 65 00:02:36,945 --> 00:02:39,085 exerted between the objects in our system 66 00:02:39,085 --> 00:02:42,709 and internal forces are always opposed 67 00:02:42,709 --> 00:02:44,127 by another internal force. 68 00:02:44,127 --> 00:02:46,573 This tension will be pulling the three kilogram, 69 00:02:46,573 --> 00:02:48,674 trying to make it move, but it opposes the motion 70 00:02:48,674 --> 00:02:50,253 of the five kilogram mass, 71 00:02:50,253 --> 00:02:53,246 and if we think of this three plus five kilogram mass 72 00:02:53,246 --> 00:02:56,843 as a single object, these end up just canceling 73 00:02:56,843 --> 00:02:58,674 on our single object that we're viewing 74 00:02:58,674 --> 00:03:01,036 as one big eight kilogram mass. 75 00:03:01,036 --> 00:03:02,578 So those are internal forces. 76 00:03:02,578 --> 00:03:05,059 We don't include them, they're not part of this trick. 77 00:03:05,059 --> 00:03:06,335 We have to figure out what other forces 78 00:03:06,335 --> 00:03:07,894 would try to make this system go 79 00:03:07,894 --> 00:03:09,115 or try to prevent it from moving. 80 00:03:09,115 --> 00:03:10,659 Another force that tries to prevent it from moving 81 00:03:10,659 --> 00:03:14,188 is the force of gravity on the three kilogram mass. 82 00:03:14,188 --> 00:03:16,375 Or, one force that tries to prevent the system 83 00:03:16,375 --> 00:03:18,467 from moving would be this force of gravity. 84 00:03:18,467 --> 00:03:19,300 How big is that? 85 00:03:19,300 --> 00:03:23,467 That's three kilograms times 9.8 meters per second squared. 86 00:03:25,797 --> 00:03:27,514 And that's trying to prevent the system from moving. 87 00:03:27,514 --> 00:03:30,713 This five kilogram mass is accelerating downward, 88 00:03:30,713 --> 00:03:33,261 and this force is in the opposite direction of motion. 89 00:03:33,261 --> 00:03:34,853 That trips people out sometimes. 90 00:03:34,853 --> 00:03:35,889 They're like, I don't understand, 91 00:03:35,889 --> 00:03:37,960 they're both pointing down. 92 00:03:37,960 --> 00:03:39,915 Shouldn't they have the same sin? 93 00:03:39,915 --> 00:03:42,733 They would when we're using Newton's second law 94 00:03:42,733 --> 00:03:43,950 the way we usually use it, 95 00:03:43,950 --> 00:03:46,287 but when we're using this trick, what we're concerned with 96 00:03:46,287 --> 00:03:48,816 are forces in the direction of motion, 97 00:03:48,816 --> 00:03:50,075 this is an easy way to figure it out, 98 00:03:50,075 --> 00:03:52,821 forces in the direction of motion we're gonna call positive. 99 00:03:52,821 --> 00:03:55,272 And any forces opposite the direction of motion 100 00:03:55,272 --> 00:03:57,201 we're gonna call negative. 101 00:03:57,201 --> 00:03:59,578 So, forces that propel the system forward 102 00:03:59,578 --> 00:04:01,542 we'll just call that positive direction. 103 00:04:01,542 --> 00:04:03,902 Forces that resist the motion, 104 00:04:03,902 --> 00:04:06,046 we're just gonna call that the negative direction. 105 00:04:06,046 --> 00:04:09,808 And since this is on this side of the motion of the system, 106 00:04:09,808 --> 00:04:13,389 this system is, everything in this system is going this way. 107 00:04:13,389 --> 00:04:15,365 The three kilogram mass goes up. 108 00:04:15,365 --> 00:04:17,115 The string over here goes up. 109 00:04:17,115 --> 00:04:18,646 The string up here goes to the right. 110 00:04:18,646 --> 00:04:20,043 The string right here goes down. 111 00:04:20,043 --> 00:04:22,041 The five kilogram mass goes down. 112 00:04:22,041 --> 00:04:24,672 Because all the motion in the system is this way, 113 00:04:24,672 --> 00:04:26,291 we'd find that way's positive, 114 00:04:26,291 --> 00:04:28,441 but this force of gravity on the three kilogram mass 115 00:04:28,441 --> 00:04:30,127 is the opposite direction. 116 00:04:30,127 --> 00:04:32,435 It's opposing the motion of the system. 117 00:04:32,435 --> 00:04:34,324 It's preventing the system from accelerating 118 00:04:34,324 --> 00:04:35,977 as fast as it would have. 119 00:04:35,977 --> 00:04:37,519 That's why we subtract it. 120 00:04:37,519 --> 00:04:39,413 And now we just divide by the total mass. 121 00:04:39,413 --> 00:04:41,322 And the total mass is just five plus three, 122 00:04:41,322 --> 00:04:44,114 is gonna be eight kilograms, 123 00:04:44,114 --> 00:04:45,604 and I get the acceleration of my system. 124 00:04:45,604 --> 00:04:47,594 So if I just add this up, 125 00:04:47,594 --> 00:04:50,677 I get 2.45 meters per second squared. 126 00:04:52,722 --> 00:04:54,986 So this is a really fast way to get what the acceleration 127 00:04:54,986 --> 00:04:56,999 of our system is, but you have to be careful. 128 00:04:56,999 --> 00:04:58,480 If the question is, what's the acceleration 129 00:04:58,480 --> 00:05:00,310 of a five kilogram box? 130 00:05:00,310 --> 00:05:01,830 Well, technically, that acceleration 131 00:05:01,830 --> 00:05:05,345 of the five kilogram box would be negative 2.45. 132 00:05:05,345 --> 00:05:06,756 What we really found here, 133 00:05:06,756 --> 00:05:08,307 since we were just finding the magnitude, 134 00:05:08,307 --> 00:05:10,350 was the size of the acceleration, 135 00:05:10,350 --> 00:05:12,817 since this five kilogram box is accelerating down, 136 00:05:12,817 --> 00:05:14,808 and we usually treat down as negative. 137 00:05:14,808 --> 00:05:16,412 You won't wanna forget that negative 138 00:05:16,412 --> 00:05:18,146 in putting in that answer the acceleration 139 00:05:18,146 --> 00:05:20,672 of the three kilogram box, however, 140 00:05:20,672 --> 00:05:24,755 would be positive 2.45 meters per second squared. 141 00:05:25,761 --> 00:05:28,008 So when you're applying this to an individual box, 142 00:05:28,008 --> 00:05:29,065 you have to be very careful 143 00:05:29,065 --> 00:05:31,679 and make sure that you apply that acceleration 144 00:05:31,679 --> 00:05:35,323 with the correct sin for that particular box. 145 00:05:35,323 --> 00:05:37,140 And if you wanted to find the tension now, 146 00:05:37,140 --> 00:05:38,260 now it's easy to find the tension. 147 00:05:38,260 --> 00:05:40,320 I could find this tension right here if I wanted to. 148 00:05:40,320 --> 00:05:43,542 If the next step was find the tension in the string 149 00:05:43,542 --> 00:05:44,939 connected to the boxes, 150 00:05:44,939 --> 00:05:47,477 now I can just use Newton's second law, 151 00:05:47,477 --> 00:05:49,081 but the way we always use it. 152 00:05:49,081 --> 00:05:49,946 I'm done with the trick. 153 00:05:49,946 --> 00:05:50,863 The trick is just the way to get 154 00:05:50,863 --> 00:05:52,400 the magnitude of the acceleration. 155 00:05:52,400 --> 00:05:55,845 Now that I have that, I'm done treating it as a system 156 00:05:55,845 --> 00:05:56,852 or a single object. 157 00:05:56,852 --> 00:06:00,326 I'll look at this single five kilogram mass all alone, 158 00:06:00,326 --> 00:06:03,571 and I'll say that the acceleration 159 00:06:03,571 --> 00:06:07,099 of the five kilogram mass, which is Newton's second law, 160 00:06:07,099 --> 00:06:10,322 is gonna equal the net force on the five kilogram mass 161 00:06:10,322 --> 00:06:13,235 divided by the mass of the five kilogram mass. 162 00:06:13,235 --> 00:06:15,701 I know the acceleration of the five kilogram mass, 163 00:06:15,701 --> 00:06:17,435 but if I'm gonna treat up as positive now, 164 00:06:17,435 --> 00:06:20,410 I gotta plug this acceleration in with a negative sign. 165 00:06:20,410 --> 00:06:24,615 So negative 2.45 meters per second squared 166 00:06:24,615 --> 00:06:27,895 is gonna equal the net force on the five kilogram mass. 167 00:06:27,895 --> 00:06:29,390 I've got tension up, 168 00:06:29,390 --> 00:06:31,915 you might be like, wait, we said that was an internal force. 169 00:06:31,915 --> 00:06:34,492 It was an internal force, and we didn't include it up here, 170 00:06:34,492 --> 00:06:36,020 but we're doing the old rules now. 171 00:06:36,020 --> 00:06:39,288 Normal second law in the vertical direction. 172 00:06:39,288 --> 00:06:41,278 So I use vertical forces, and if they're upward 173 00:06:41,278 --> 00:06:42,658 I'm gonna treat them as positive, 174 00:06:42,658 --> 00:06:46,108 and if they're downward like this five times 9.8, 175 00:06:46,108 --> 00:06:47,888 I'm gonna treat it as a negative, 176 00:06:47,888 --> 00:06:49,648 because it points down. 177 00:06:49,648 --> 00:06:52,193 Five times 9.8 meters per second squared, 178 00:06:52,193 --> 00:06:53,848 and I divide by the five kilogram mass, 179 00:06:53,848 --> 00:06:55,263 'cause that's the box I'm analyzing. 180 00:06:55,263 --> 00:06:57,094 I'm not analyzing the whole system. 181 00:06:57,094 --> 00:06:59,896 I'm just analyzing the five kilogram box now. 182 00:06:59,896 --> 00:07:01,926 And I can solve and I can get my tension. 183 00:07:01,926 --> 00:07:03,338 The alternate way to do this would be 184 00:07:03,338 --> 00:07:05,487 to say, all right, let's just treat down as positive 185 00:07:05,487 --> 00:07:07,422 for this five kilogram mass. 186 00:07:07,422 --> 00:07:09,715 I'd then plug my acceleration in as positive, 187 00:07:09,715 --> 00:07:12,246 and I'd plug my force of gravity in positive, 188 00:07:12,246 --> 00:07:14,166 then my tension would be negative. 189 00:07:14,166 --> 00:07:15,430 I'd get the same value. 190 00:07:15,430 --> 00:07:16,954 Here I'm just solving for the magnitude 191 00:07:16,954 --> 00:07:18,328 of the tension anyway. 192 00:07:18,328 --> 00:07:20,118 So if I solve this, if I plug this into the calculator 193 00:07:20,118 --> 00:07:24,285 and solve for tension, I'm gonna get 36.75 Newtons, 194 00:07:25,987 --> 00:07:28,441 which is less than the force of gravity, 195 00:07:28,441 --> 00:07:30,940 which it has to be, 'cause if the tension was greater than 196 00:07:30,940 --> 00:07:33,070 the force of gravity, this five kilogram mass 197 00:07:33,070 --> 00:07:34,281 would accelerate up. 198 00:07:34,281 --> 00:07:35,554 We know that doesn't happen. 199 00:07:35,554 --> 00:07:37,642 The tension's gotta be less than the force of gravity, 200 00:07:37,642 --> 00:07:40,541 so that this five kilogram mass can accelerate downward. 201 00:07:40,541 --> 00:07:42,785 So that's a quick way to solve for the magnitude 202 00:07:42,785 --> 00:07:45,506 of the acceleration of the system by treating it 203 00:07:45,506 --> 00:07:46,881 as a single object. 204 00:07:46,881 --> 00:07:48,958 We're saying that if it's a single object, 205 00:07:48,958 --> 00:07:50,646 or thought of as a single object, 206 00:07:50,646 --> 00:07:52,448 which we can do, 'cause these are required 207 00:07:52,448 --> 00:07:53,980 to have the same acceleration, 208 00:07:53,980 --> 00:07:56,008 or same magnitude of the acceleration, 209 00:07:56,008 --> 00:07:57,920 that if we're treating it like a single object, 210 00:07:57,920 --> 00:07:59,933 only external forces matter, 211 00:07:59,933 --> 00:08:02,109 and those external forces that make the system go 212 00:08:02,109 --> 00:08:03,882 are going to accelerate the system. 213 00:08:03,882 --> 00:08:06,539 And those external forces that resist the motion 214 00:08:06,539 --> 00:08:09,092 are trying to reduce the acceleration, 215 00:08:09,092 --> 00:08:11,615 and we divide by the total mass of the system 216 00:08:11,615 --> 00:08:14,426 that we're treating as one object, we get the acceleration. 217 00:08:14,426 --> 00:08:16,793 If that still seems like mathematical witchcraft, 218 00:08:16,793 --> 00:08:19,684 or if you're not sure about this whole idea, 219 00:08:19,684 --> 00:08:21,836 I encourage you to go back and watch the video. 220 00:08:21,836 --> 00:08:24,624 We solved one of these types of problems the hard way. 221 00:08:24,624 --> 00:08:26,538 And you see, you really do end up with 222 00:08:26,538 --> 00:08:28,916 the force that tries to make the system go externally, 223 00:08:28,916 --> 00:08:30,814 and the external force that tries to stop it 224 00:08:30,814 --> 00:08:33,573 divided by the total mass gives you the acceleration. 225 00:08:33,573 --> 00:08:34,784 Essentially, what we're saying 226 00:08:34,784 --> 00:08:36,886 is that these internal forces cancel 227 00:08:36,886 --> 00:08:41,382 if you're thinking of this system as one single object, 228 00:08:41,383 --> 00:08:43,270 'cause these are applied internally, 229 00:08:43,270 --> 00:08:45,404 and they're opposed to each other. 230 00:08:45,404 --> 00:08:46,849 One tries to make the system go, 231 00:08:46,849 --> 00:00:00,000 one tries to make the system stop.