1 00:00:00,631 --> 00:00:01,664 - [Voiceover] So in the previous video 2 00:00:01,664 --> 00:00:04,175 we solved this problem the hard way. 3 00:00:04,175 --> 00:00:05,850 Maybe you watched it, maybe you didn't, 4 00:00:05,850 --> 00:00:07,604 maybe you just skipped right to here and you're like, 5 00:00:07,604 --> 00:00:08,855 "I don't even wanna know the hard way. 6 00:00:08,855 --> 00:00:11,037 "Just show me the easy way please." 7 00:00:11,037 --> 00:00:12,769 Well, that's what we're gonna talk about now. 8 00:00:12,769 --> 00:00:14,015 Turns out there's a trick 9 00:00:14,015 --> 00:00:17,837 and the trick is after you solve this problem the hard way 10 00:00:17,837 --> 00:00:20,576 with a five kilogram mass and a three kilogram mass, 11 00:00:20,576 --> 00:00:22,030 when you find the acceleration, 12 00:00:22,030 --> 00:00:23,103 what you get is this. 13 00:00:23,103 --> 00:00:25,300 That the acceleration of the five kilogram mass 14 00:00:25,300 --> 00:00:29,320 is just 29.4 divided by eight kilograms. 15 00:00:29,320 --> 00:00:31,908 But when you do enough of these, you might start realizing, 16 00:00:31,908 --> 00:00:34,580 "Wait a minute. 29.4 Newtons. 17 00:00:34,580 --> 00:00:37,122 "That was just the force of gravity 18 00:00:37,122 --> 00:00:39,180 "pulling on this three kilogram mass." 19 00:00:39,180 --> 00:00:40,074 In other words, 20 00:00:40,074 --> 00:00:42,012 the only force that was really propelling 21 00:00:42,012 --> 00:00:45,272 this whole entire system forward. 22 00:00:45,272 --> 00:00:48,248 Or at least the only external force propelling it forward. 23 00:00:48,248 --> 00:00:50,090 And then eight kilograms down here. 24 00:00:50,090 --> 00:00:51,980 You're gonna be like, "Wait. Eight kilograms? 25 00:00:51,980 --> 00:00:55,402 "That's just five kilograms plus three kilograms. 26 00:00:55,402 --> 00:00:57,442 "Is that just a coincidence 27 00:00:57,442 --> 00:01:00,137 "or is this telling us something deep and fundamental?" 28 00:01:00,137 --> 00:01:02,080 And it's not a coincidence. 29 00:01:02,080 --> 00:01:03,893 Turns out you'll always get this. 30 00:01:03,893 --> 00:01:06,905 That what you'll end up with after solving this hard way, 31 00:01:06,905 --> 00:01:08,775 you'll get in the very end, 32 00:01:08,775 --> 00:01:12,485 you'll get all the external forces added up here 33 00:01:12,485 --> 00:01:14,629 where forces that make it go 34 00:01:14,629 --> 00:01:17,487 like this force of gravity end up being positive 35 00:01:17,487 --> 00:01:20,275 and forces that try to resist the motion. 36 00:01:20,275 --> 00:01:23,274 So if there was friction, that would be an external force 37 00:01:23,274 --> 00:01:26,343 that tries to resist motion, would be up top 38 00:01:26,343 --> 00:01:29,531 and then you get the total mass on the bottom. 39 00:01:29,531 --> 00:01:30,661 And this makes sense. 40 00:01:30,661 --> 00:01:33,208 The acceleration of this entire system, 41 00:01:33,208 --> 00:01:36,016 if we think about it as a single object--- 42 00:01:36,016 --> 00:01:39,437 So if you imagine this was just one single object 43 00:01:39,437 --> 00:01:40,668 and you asked yourself, 44 00:01:40,668 --> 00:01:44,098 "What's the total acceleration of this entire system?" 45 00:01:44,098 --> 00:01:46,885 Well, it's only gonna depend on the external forces 46 00:01:46,885 --> 00:01:49,928 and in this case, the only external force making it go 47 00:01:49,928 --> 00:01:52,469 was this force of gravity right here. 48 00:01:52,469 --> 00:01:53,302 You might object. 49 00:01:53,302 --> 00:01:54,135 You might be like, "Wait. 50 00:01:54,135 --> 00:01:55,783 "What about this tension right here? 51 00:01:55,783 --> 00:01:59,191 "Isn't the tension pulling on this five kilogram mass 52 00:01:59,191 --> 00:02:00,986 "making this system go?" 53 00:02:00,986 --> 00:02:03,976 It is but since it's an internal force now, 54 00:02:03,976 --> 00:02:07,461 if we're treating this entire system as our one object, 55 00:02:07,461 --> 00:02:09,894 since this tension is trying to make it go, 56 00:02:09,894 --> 00:02:11,126 you've got another tension over here 57 00:02:11,126 --> 00:02:12,980 resisting the motion on this mass, 58 00:02:12,980 --> 00:02:14,280 trying to make it stop. 59 00:02:14,280 --> 00:02:16,236 That's what internal forces do. 60 00:02:16,236 --> 00:02:17,538 There's always equal and opposite 61 00:02:17,538 --> 00:02:19,377 on one part of the object than the other 62 00:02:19,377 --> 00:02:20,859 so you can't propel yourself forward 63 00:02:20,859 --> 00:02:22,388 with an internal force. 64 00:02:22,388 --> 00:02:24,736 So these end up cancelling out essentially. 65 00:02:24,736 --> 00:02:26,030 The only force you have in this case 66 00:02:26,030 --> 00:02:27,805 was the force of gravity on top, 67 00:02:27,805 --> 00:02:29,442 only external forces, 68 00:02:29,442 --> 00:02:31,894 and the total mass on the bottom. 69 00:02:31,894 --> 00:02:33,113 And that's trick. 70 00:02:33,113 --> 00:02:35,233 That's the trick to quickly find 71 00:02:35,233 --> 00:02:37,417 the acceleration of some system 72 00:02:37,417 --> 00:02:38,890 that might be complicated 73 00:02:38,890 --> 00:02:40,454 if you had to do it in multiple equations 74 00:02:40,454 --> 00:02:41,852 and multiple unknowns 75 00:02:41,852 --> 00:02:44,729 but much, much easier once you realize this. 76 00:02:44,729 --> 00:02:46,417 So the trick, sometimes it's called 77 00:02:46,417 --> 00:02:50,649 just "Treating systems as a single object". 78 00:02:50,649 --> 00:02:51,847 Let me just show you really quick. 79 00:02:51,847 --> 00:02:54,858 If that made no sense, let me just show you what this means. 80 00:02:54,858 --> 00:02:56,652 If we just get rid of this. 81 00:02:56,652 --> 00:02:58,767 So what I'm claiming is this. 82 00:02:58,767 --> 00:03:00,245 If you ever have a system 83 00:03:00,245 --> 00:03:02,692 where multiple objects are required to move 84 00:03:02,692 --> 00:03:06,052 with the exact same magnitude of acceleration, right? 85 00:03:06,052 --> 00:03:08,338 Because maybe they're tied together by rope 86 00:03:08,338 --> 00:03:10,326 or maybe they're pushing on each other. 87 00:03:10,326 --> 00:03:12,660 Maybe there's many boxes in a row 88 00:03:12,660 --> 00:03:14,954 and these boxes all have to be pushed 89 00:03:14,954 --> 00:03:16,252 at the same acceleration 90 00:03:16,252 --> 00:03:19,427 because they can't get pushed through each other. 91 00:03:19,427 --> 00:03:22,884 Right, if there's some condition where multiple objects 92 00:03:22,884 --> 00:03:26,191 must have the same magnitude of acceleration, 93 00:03:26,191 --> 00:03:30,540 then you can simply find the acceleration of that system 94 00:03:30,540 --> 00:03:32,045 as if it were a single object. 95 00:03:32,045 --> 00:03:34,281 I'm writing, "SYS" for system. 96 00:03:34,281 --> 00:03:36,223 By just using Newton's second law, 97 00:03:36,223 --> 00:03:38,204 but instead of looking at an individual object 98 00:03:38,204 --> 00:03:40,034 for a given direction, 99 00:03:40,034 --> 00:03:43,731 we're just gonna do all of the external forces, 100 00:03:43,731 --> 00:03:46,999 all of the external forces on our system, 101 00:03:46,999 --> 00:03:49,043 treat it as if it were a single object, 102 00:03:49,043 --> 00:03:52,233 divide it by the total mass of our system. 103 00:03:52,233 --> 00:03:54,843 And so when you plug in these external forces--- 104 00:03:54,843 --> 00:03:56,025 These are forces that are external 105 00:03:56,025 --> 00:03:58,755 so external means not internal to the system. 106 00:03:58,755 --> 00:04:02,524 So if I think of this five kilogram box 107 00:04:02,524 --> 00:04:06,555 and this three kilogram box as a single mass, 108 00:04:06,555 --> 00:04:08,509 tension would be an internal force 109 00:04:08,509 --> 00:04:13,440 because it's applied internally between these two objects, 110 00:04:13,440 --> 00:04:15,973 between objects inside of our system. 111 00:04:15,973 --> 00:04:18,485 But the force of gravity on the three kilogram mass, 112 00:04:18,485 --> 00:04:21,021 that's an external force 'cause that's the Earth 113 00:04:21,021 --> 00:04:23,275 pulling down on the three kilogram mass 114 00:04:23,275 --> 00:04:26,498 and the Earth is not part of our system. 115 00:04:26,498 --> 00:04:29,607 Similarly, the normal force is an external force 116 00:04:29,607 --> 00:04:33,203 but it's exactly cancelled by the gravitational force. 117 00:04:33,203 --> 00:04:35,114 So even though those are external, 118 00:04:35,114 --> 00:04:36,212 they're not gonna make it in here. 119 00:04:36,212 --> 00:04:37,072 I mean, you can put 'em in there 120 00:04:37,072 --> 00:04:38,509 but they're just gonna cancel anyway. 121 00:04:38,509 --> 00:04:41,515 We only look at forces in the direction of motion 122 00:04:41,515 --> 00:04:44,249 and if it's a force that causes motion, 123 00:04:44,249 --> 00:04:46,155 we're gonna make that a positive force. 124 00:04:46,155 --> 00:04:48,282 If it's a force in the direction of motion 125 00:04:48,282 --> 00:04:49,936 like this force of gravity is, 126 00:04:49,936 --> 00:04:51,884 we make those positive forces. 127 00:04:51,884 --> 00:04:55,708 So forces will be plugged in positive into here 128 00:04:55,708 --> 00:04:58,649 if they make the system go. 129 00:04:58,649 --> 00:05:00,608 And that might seem weird. 130 00:05:00,608 --> 00:05:01,441 You might be like, "Wait. 131 00:05:01,441 --> 00:05:03,717 "How do I decide if it makes the system go?" 132 00:05:03,717 --> 00:05:06,081 Well, just ask yourself, "Is that force directed 133 00:05:06,081 --> 00:05:09,357 "in the same direction as the motion of the system?" 134 00:05:09,357 --> 00:05:12,167 So, we're just saying the system is gonna accelerate 135 00:05:12,167 --> 00:05:14,251 if there's forces that make it go 136 00:05:14,251 --> 00:05:17,695 and we're gonna plug in negative forces, 137 00:05:17,695 --> 00:05:20,321 the forces that make the system stop 138 00:05:20,321 --> 00:05:22,976 or resist the motion of system. 139 00:05:22,976 --> 00:05:27,037 So maybe I should say, "Resist motion of the system." 140 00:05:27,037 --> 00:05:28,611 In this case, for this one down here, 141 00:05:28,611 --> 00:05:30,610 I don't have any of those. 142 00:05:30,610 --> 00:05:33,193 So resist motion of the system. 143 00:05:35,100 --> 00:05:35,933 I don't have any of those. 144 00:05:35,933 --> 00:05:39,215 I could have if I had a force of friction. 145 00:05:39,215 --> 00:05:42,994 Then there'd be a external force that resist the motion. 146 00:05:42,994 --> 00:05:45,835 I would plug in that external force as a negative 147 00:05:45,835 --> 00:05:47,814 'cause it resist the motion. 148 00:05:47,814 --> 00:05:49,719 So even though this might sound weird, 149 00:05:49,719 --> 00:05:51,929 it makes sense if you think about it. 150 00:05:51,929 --> 00:05:54,805 The acceleration of our system treated as a single object 151 00:05:54,805 --> 00:05:56,671 is only gonna depend on the forces 152 00:05:56,671 --> 00:05:58,547 that try to make the system go 153 00:05:58,547 --> 00:06:00,846 and the forces that try to make the system stop 154 00:06:00,846 --> 00:06:02,269 or resist the motion. 155 00:06:02,269 --> 00:06:05,254 So if we add those accordingly with positives and negatives, 156 00:06:05,254 --> 00:06:06,614 we divide it by the total mass 157 00:06:06,614 --> 00:06:09,395 which gives the total measure of the inertia of our system, 158 00:06:09,395 --> 00:06:11,387 we'll get the acceleration of our system. 159 00:06:11,387 --> 00:06:13,190 It makes sense and it works. 160 00:06:13,190 --> 00:06:14,237 Turns out it always works 161 00:06:14,237 --> 00:06:16,653 and it saves a ridiculous amount of time. 162 00:06:16,653 --> 00:06:19,110 For instance, if we wanted to do this problem, 163 00:06:19,110 --> 00:06:21,124 if you just gave me this problem straight away 164 00:06:21,124 --> 00:06:22,928 and you were told, "Do this however you want.", 165 00:06:22,928 --> 00:06:24,006 I would use this trick. 166 00:06:24,006 --> 00:06:27,302 And I would say that the acceleration of this system 167 00:06:27,302 --> 00:06:29,497 which is composed of this five kilogram mass 168 00:06:29,497 --> 00:06:31,377 and our three kilogram mass 169 00:06:31,377 --> 00:06:32,513 is just gonna be equal to--- 170 00:06:32,513 --> 00:06:35,443 I'd ask myself, "What force makes this system go? 171 00:06:35,443 --> 00:06:37,587 "What force drives this system?" 172 00:06:37,587 --> 00:06:40,705 And it's this force of gravity on the three kilogram mass 173 00:06:40,705 --> 00:06:42,569 that's driving this system, right? 174 00:06:42,569 --> 00:06:44,510 If you took this force away, 175 00:06:44,510 --> 00:06:46,206 if you eliminated that force, 176 00:06:46,206 --> 00:06:47,657 nothing's gonna happen here. 177 00:06:47,657 --> 00:06:49,869 This is the force making the system go 178 00:06:49,869 --> 00:06:54,036 so I'd put it in my three kilograms times nine point eight. 179 00:06:55,643 --> 00:06:56,476 And at this point, 180 00:06:56,476 --> 00:06:58,654 you might be like, "Well, okay, that gravity made it go. 181 00:06:58,654 --> 00:07:00,698 "Should I include this gravity, too?" 182 00:07:00,698 --> 00:07:04,347 But no, that gravity is perpendicular to the motion for one 183 00:07:04,347 --> 00:07:06,890 so this gravity isn't making the system, 184 00:07:06,890 --> 00:07:09,420 that's just causing this mass to sit on the table 185 00:07:09,420 --> 00:07:12,308 and for two, it's cancelled by that normal force. 186 00:07:12,308 --> 00:07:13,756 So those cancel anyway, 187 00:07:13,756 --> 00:07:16,010 even though they're external forces. 188 00:07:16,010 --> 00:07:18,391 This is it, this is the only one that drives the system. 189 00:07:18,391 --> 00:07:21,386 So I put that in here and I divide by my total mass 190 00:07:21,386 --> 00:07:23,428 'cause that tells me how much my system resists 191 00:07:23,428 --> 00:07:25,245 through inertia, 192 00:07:25,245 --> 00:07:27,483 changes in velocity, and this is what I get. 193 00:07:27,483 --> 00:07:28,534 I get the same thing I got before, 194 00:07:28,534 --> 00:07:32,085 I get back my three point six eight 195 00:07:32,085 --> 00:07:35,037 meters per second squared, and I get in one line. 196 00:07:35,037 --> 00:07:37,646 I mean, this trick is amazing and it works, 197 00:07:37,646 --> 00:07:38,958 and it works in every example 198 00:07:38,958 --> 00:07:40,825 where two masses or more masses 199 00:07:40,825 --> 00:07:44,209 are forced to move with the same acceleration. 200 00:07:44,209 --> 00:07:46,864 So this is great. This'll save you a ton of time. 201 00:07:46,864 --> 00:07:48,661 This is supposed to be a three here. 202 00:07:48,661 --> 00:07:50,238 And to show you how useful it is, 203 00:07:50,238 --> 00:07:51,704 let say there was friction, 204 00:07:51,704 --> 00:07:53,867 let's say there was a coefficient of friction 205 00:07:53,867 --> 00:07:56,124 of zero point three. 206 00:07:56,124 --> 00:07:57,787 Well, now I'd have a frictional force 207 00:07:57,787 --> 00:08:00,509 so there'd be an external frictional force here. 208 00:08:00,509 --> 00:08:02,640 It'd be applied this five kilogram mass. 209 00:08:02,640 --> 00:08:04,508 I'd have to subtract it up here. 210 00:08:04,508 --> 00:08:06,203 So if I get rid of this--- 211 00:08:06,203 --> 00:08:08,535 So it's not gonna be three point six eight anymore. 212 00:08:08,535 --> 00:08:12,699 I'm gonna have a force of friction that I have to subtract. 213 00:08:12,699 --> 00:08:16,630 So minus mu K so the force of friction--- 214 00:08:16,630 --> 00:08:18,309 I'll just put force of friction. 215 00:08:18,309 --> 00:08:21,232 And so to solve for the force of friction, 216 00:08:21,232 --> 00:08:23,403 the force of friction is gonna be equal to--- 217 00:08:23,403 --> 00:08:25,610 Well, I know three times nine point eight is--- 218 00:08:25,610 --> 00:08:26,599 Let me just write this in here, 219 00:08:26,599 --> 00:08:30,766 29.4 Newtons minus the force of friction it's given by. 220 00:08:32,011 --> 00:08:34,850 So there's a formula for force of friction. 221 00:08:34,851 --> 00:08:38,184 The force of friction is always mu K FN. 222 00:08:39,193 --> 00:08:41,410 So the force of friction on this five kilogram mass 223 00:08:41,410 --> 00:08:44,370 is gonna be mu K which is point three. 224 00:08:44,370 --> 00:08:48,067 So it's gonna be zero point three times the normal force, 225 00:08:48,067 --> 00:08:50,492 not the normal force on our entire system. 226 00:08:50,492 --> 00:08:52,451 I don't include this three kilogram mass. 227 00:08:52,451 --> 00:08:55,318 It's only the normal force on this five kilogram mass 228 00:08:55,318 --> 00:08:58,332 that's contributing to this force of friction here. 229 00:08:58,332 --> 00:09:00,445 So even though we're treating this system as a whole, 230 00:09:00,445 --> 00:09:03,102 we still have to find individual forces 231 00:09:03,102 --> 00:09:06,434 on this individual boxes correctly. 232 00:09:06,434 --> 00:09:08,819 So it won't be the entire mass that goes here. 233 00:09:08,819 --> 00:09:10,491 The normal force on the five kilogram mass 234 00:09:10,491 --> 00:09:14,658 is just gonna be five kilograms times nine point eight 235 00:09:17,510 --> 00:09:19,399 meters per second squared. 236 00:09:19,399 --> 00:09:21,535 I divide by my total mass down here 237 00:09:21,535 --> 00:09:26,257 because the entire mass is resisting motion through inertia. 238 00:09:26,257 --> 00:09:28,880 And if I solve this from my acceleration of the system, 239 00:09:28,880 --> 00:09:33,110 I get one point eight four meters per seconds squared. 240 00:09:33,110 --> 00:09:35,871 So this is less, less than our three point six eight 241 00:09:35,871 --> 00:09:36,849 and that makes sense. 242 00:09:36,849 --> 00:09:38,642 Now, there's a resistive force, 243 00:09:38,642 --> 00:09:40,324 a resistive external force, 244 00:09:40,324 --> 00:09:42,508 tryna prevent the system from moving. 245 00:09:42,508 --> 00:09:43,824 But you have to be careful. 246 00:09:43,824 --> 00:09:45,583 What I'm really finding here, 247 00:09:45,583 --> 00:09:49,100 I'm really finding the magnitude of the acceleration. 248 00:09:49,100 --> 00:09:50,508 This is just giving me the magnitude. 249 00:09:50,508 --> 00:09:51,350 If I'm playing this game 250 00:09:51,350 --> 00:09:54,007 where positive forces are ones that make it go 251 00:09:54,007 --> 00:09:56,802 and negative forces are ones that resist motion, 252 00:09:56,802 --> 00:09:58,389 external forces that is, 253 00:09:58,389 --> 00:10:00,643 I'm just getting the magnitude of the acceleration. 254 00:10:00,643 --> 00:10:03,489 Individual boxes will have that magnitude 255 00:10:03,489 --> 00:10:04,529 of the acceleration 256 00:10:04,529 --> 00:10:07,104 but they may have positive or negative accelerations. 257 00:10:07,104 --> 00:10:08,039 In other words, 258 00:10:08,039 --> 00:10:11,289 this five kilogram mass accelerating to the right, 259 00:10:11,289 --> 00:10:13,339 gonna have a positive acceleration. 260 00:10:13,339 --> 00:10:15,877 In other words, the acceleration of the five kilogram mass 261 00:10:15,877 --> 00:10:18,197 will be positive one point eight four 262 00:10:18,197 --> 00:10:20,181 and the acceleration of the three kilogram mass 263 00:10:20,181 --> 00:10:21,677 since it's accelerating downward 264 00:10:21,677 --> 00:10:24,291 will be negative one point eight four 265 00:10:24,291 --> 00:00:00,000 meters per seconds squared.