1 00:00:00,349 --> 00:00:01,624 - So, check out this problem. 2 00:00:01,624 --> 00:00:04,415 You've got a 12 kilogram mass sitting on a table, 3 00:00:04,415 --> 00:00:06,723 and on the left hand side it's tied to a rope 4 00:00:06,723 --> 00:00:08,306 that passes over a pulley 5 00:00:08,306 --> 00:00:11,413 and that rope gets tied to a three kilogram mass. 6 00:00:11,413 --> 00:00:13,942 And then on the right side of this 12 kilogram box, 7 00:00:13,942 --> 00:00:15,433 you've got another rope 8 00:00:15,433 --> 00:00:18,174 and that rope passes over another pulley on the right 9 00:00:18,174 --> 00:00:20,932 and is tied to the five kilogram box over here. 10 00:00:20,932 --> 00:00:22,680 The question is, what's the acceleration 11 00:00:22,680 --> 00:00:24,454 of the 12 kilogram box? 12 00:00:24,454 --> 00:00:26,777 Let's make it even harder. 13 00:00:26,777 --> 00:00:28,728 Let's say there's a coefficient of kinetic friction 14 00:00:28,728 --> 00:00:32,895 between this 12 kilogram box and the table of 0.1. 15 00:00:34,446 --> 00:00:36,433 Now you're looking at a really hard problem 16 00:00:36,433 --> 00:00:38,820 if you try to solve this the hard way. 17 00:00:38,820 --> 00:00:42,159 And by the hard way, I mean using Newton's second law 18 00:00:42,159 --> 00:00:45,160 for each box individually and then trying to solve 19 00:00:45,160 --> 00:00:46,778 what'd you end up with is at least 20 00:00:46,778 --> 00:00:48,777 three equations and three unknowns 21 00:00:48,777 --> 00:00:52,072 because you're gonna have three different accelerations. 22 00:00:52,072 --> 00:00:54,506 For each of these, you'll have two different tensions 23 00:00:54,506 --> 00:00:56,411 cause this left rope is under a different 24 00:00:56,411 --> 00:00:58,594 tension from the right rope now. 25 00:00:58,594 --> 00:01:00,426 It's only when you have a single rope can you say 26 00:01:00,426 --> 00:01:01,841 that it's the same tension. 27 00:01:01,841 --> 00:01:02,947 This problem's gonna be hard. 28 00:01:02,947 --> 00:01:06,528 There's gonna be tons of Algebra mistakes potentially. 29 00:01:06,528 --> 00:01:09,328 And so to avoid that, we can solve this the easy way. 30 00:01:09,328 --> 00:01:11,960 And if you remember, the easy way is just by saying, 31 00:01:11,960 --> 00:01:14,033 well, let's treat all of these boxes 32 00:01:14,033 --> 00:01:16,317 as if they're a single object. 33 00:01:16,317 --> 00:01:18,421 And we can do that cause they're all gonna have 34 00:01:18,421 --> 00:01:21,096 the same magnitude of acceleration 35 00:01:21,096 --> 00:01:23,716 that I'm just calling a system. 36 00:01:23,716 --> 00:01:26,407 That's gonna be the magnitude of acceleration of our system. 37 00:01:26,407 --> 00:01:29,292 All these boxes will accelerate with the same magnitude. 38 00:01:29,292 --> 00:01:32,095 Some may have negative accelerations, 39 00:01:32,095 --> 00:01:35,060 some may have positive accelerations like these, 40 00:01:35,060 --> 00:01:36,875 but they're gonna all have the same 41 00:01:36,875 --> 00:01:38,745 magnitude of acceleration cause we're gonna assume 42 00:01:38,745 --> 00:01:40,576 that these ropes don't break. 43 00:01:40,576 --> 00:01:43,400 And if they broke, then they'll be different magnitudes 44 00:01:43,400 --> 00:01:45,859 or if they stretch, but we're assuming that doesn't happen. 45 00:01:45,859 --> 00:01:47,372 And the way we can find this is by just saying, 46 00:01:47,372 --> 00:01:49,345 well, if this is just a single object, 47 00:01:49,345 --> 00:01:52,035 I don't have to worry about any internal forces, 48 00:01:52,035 --> 00:01:54,539 now these tensions become internal forces. 49 00:01:54,539 --> 00:01:57,272 And those don't make a system accelerate, 50 00:01:57,272 --> 00:01:59,940 only external forces are gonna make a system accelerate. 51 00:01:59,940 --> 00:02:01,276 So, all I have to do is find out 52 00:02:01,276 --> 00:02:02,782 what are all the external forces 53 00:02:02,782 --> 00:02:04,879 that try to make this system go, 54 00:02:04,879 --> 00:02:06,178 try to accelerate it, 55 00:02:06,178 --> 00:02:08,875 and ones that try to prevent acceleration. 56 00:02:08,875 --> 00:02:10,828 I'll call this F external 57 00:02:10,828 --> 00:02:12,557 and then I divide by the total mass 58 00:02:12,557 --> 00:02:16,260 because this is just simply Newton's second law 59 00:02:16,260 --> 00:02:18,729 as if this were one big object. 60 00:02:18,729 --> 00:02:20,841 So, what are my external forces? 61 00:02:20,841 --> 00:02:22,721 Well, the force that makes it go 62 00:02:22,721 --> 00:02:26,678 is gonna be this five kilogram's force of gravity 63 00:02:26,678 --> 00:02:28,492 so I'm gonna have a force of gravity over here. 64 00:02:28,492 --> 00:02:30,546 That tries to propel the system forward. 65 00:02:30,546 --> 00:02:33,520 This is the one that's gonna be driving the system. 66 00:02:33,520 --> 00:02:35,304 If I let go of these boxes, 67 00:02:35,304 --> 00:02:38,563 it's gonna start shifting in this direction 68 00:02:38,563 --> 00:02:40,657 because this five kilogram mass 69 00:02:40,657 --> 00:02:43,582 has a larger force of gravity than this three kilogram mass. 70 00:02:43,582 --> 00:02:45,027 So, I'm gonna include that as a positive. 71 00:02:45,027 --> 00:02:46,518 I'm just gonna define direction of motion 72 00:02:46,518 --> 00:02:48,525 as positive, 'cause it's easy. 73 00:02:48,525 --> 00:02:50,450 You could do it differently if you wanted to. 74 00:02:50,450 --> 00:02:51,907 You could find the other way as positive. 75 00:02:51,907 --> 00:02:54,168 So five times 9.8 meters per second squared 76 00:02:54,168 --> 00:02:55,824 is how you find this force of gravity. 77 00:02:55,824 --> 00:02:59,129 Are there any other forces that are propelling this forward? 78 00:02:59,129 --> 00:03:01,296 No, no external forces are. 79 00:03:01,296 --> 00:03:03,608 So, are there any forces that are trying 80 00:03:03,608 --> 00:03:05,438 to reduce the acceleration? 81 00:03:05,438 --> 00:03:07,413 Yeah, there's this force of gravity over here. 82 00:03:07,413 --> 00:03:10,073 This force of gravity on the three kilogram mass 83 00:03:10,073 --> 00:03:12,447 is trying to prevent the acceleration because 84 00:03:12,447 --> 00:03:16,078 it's pointing opposite the direction of motion. 85 00:03:16,078 --> 00:03:19,124 The motion of this system is upright and down 86 00:03:19,124 --> 00:03:21,560 across this direction but this force 87 00:03:21,560 --> 00:03:24,088 is pointing opposite that direction. 88 00:03:24,088 --> 00:03:26,251 This force of gravity right here. 89 00:03:26,251 --> 00:03:29,529 So, I'm gonna have to subtract three kilograms 90 00:03:29,529 --> 00:03:32,529 times 9.8 meters per second squared. 91 00:03:35,653 --> 00:03:38,319 Am I gonna have any other forces that 92 00:03:38,319 --> 00:03:40,554 try to prevent the system from moving? 93 00:03:40,554 --> 00:03:41,998 You might think the force of gravity 94 00:03:41,998 --> 00:03:43,745 on this 12 kilogram box, 95 00:03:43,745 --> 00:03:46,675 but look, that doesn't really, in and of itself, 96 00:03:46,675 --> 00:03:49,013 prevent the system from moving or not moving. 97 00:03:49,013 --> 00:03:50,800 That's perpendicular to this direction. 98 00:03:50,800 --> 00:03:52,946 I've called the direction of motion, 99 00:03:52,946 --> 00:03:54,611 this positive direction. 100 00:03:54,611 --> 00:03:56,979 If it were a force this way, 101 00:03:56,979 --> 00:04:00,195 if it were a force this way or a force that way 102 00:04:00,195 --> 00:04:02,359 it'd try to cause acceleration of the system. 103 00:04:02,359 --> 00:04:04,443 This force of gravity just gets negated 104 00:04:04,443 --> 00:04:06,743 by the normal force, so I don't even have 105 00:04:06,743 --> 00:04:08,534 to worry about that force. 106 00:04:08,534 --> 00:04:10,572 So, are there any forces associated 107 00:04:10,572 --> 00:04:13,219 with the 12 kilogram box that try to prevent motion? 108 00:04:13,219 --> 00:04:14,771 It turns out there is. 109 00:04:14,771 --> 00:04:16,584 There is going to be a force of friction 110 00:04:16,584 --> 00:04:18,413 between the table because there's this 111 00:04:18,413 --> 00:04:20,356 coefficient of kinetic friction. 112 00:04:20,356 --> 00:04:22,237 So, I've got a force this way, 113 00:04:22,237 --> 00:04:24,672 this kinetic frictional force, 114 00:04:24,672 --> 00:04:26,714 that's gonna be, have a size of 115 00:04:26,714 --> 00:04:27,726 Mu K times f n. 116 00:04:27,726 --> 00:04:29,859 That's how you find the normal force 117 00:04:29,859 --> 00:04:33,692 and so this is gonna be minus, the Mu K is 0.1 118 00:04:35,259 --> 00:04:37,740 and the normal force will be the normal force 119 00:04:37,740 --> 00:04:39,499 for this 12 kilogram mass. 120 00:04:39,499 --> 00:04:41,549 So, I'll use 12 kilograms 121 00:04:41,549 --> 00:04:44,549 times 9.8 meters per second squared. 122 00:04:46,480 --> 00:04:47,750 You might object, you might say, 123 00:04:47,750 --> 00:04:51,147 "Hey, hold on, 12 times 9.8, that's the force of gravity. 124 00:04:51,147 --> 00:04:53,312 "Why are you using this force? 125 00:04:53,312 --> 00:04:55,165 "I thought you said we didn't use it?" 126 00:04:55,165 --> 00:04:57,188 Well, we don't use this force by itself, 127 00:04:57,188 --> 00:04:59,195 but it turns out this force of friction 128 00:04:59,195 --> 00:05:00,770 depends on this force. 129 00:05:00,770 --> 00:05:03,505 So, we're really using a horizontal force, 130 00:05:03,505 --> 00:05:05,408 a force that tries to prevent motion, 131 00:05:05,408 --> 00:05:07,598 which is why we've got this negative sign here, 132 00:05:07,598 --> 00:05:09,160 but it's a horizontal force. 133 00:05:09,160 --> 00:05:11,670 It just so happens that this horizontal force 134 00:05:11,670 --> 00:05:15,098 depends on a vertical force, which is the normal force. 135 00:05:15,098 --> 00:05:17,327 And so that's why we're multiplying by this .1 136 00:05:17,327 --> 00:05:19,358 that turns this vertical force, 137 00:05:19,358 --> 00:05:21,193 which is not propelling the system, 138 00:05:21,193 --> 00:05:23,747 or trying to stop it, into a horizontal force 139 00:05:23,747 --> 00:05:27,520 which is trying to reduce the acceleration of the system. 140 00:05:27,520 --> 00:05:30,068 That's why I subtracted and then I divide by the total mass 141 00:05:30,068 --> 00:05:32,742 and my total mass is gonna be three plus 12 142 00:05:32,742 --> 00:05:35,659 plus five is gonna be 20 kilograms. 143 00:05:36,813 --> 00:05:38,037 Now, I can just solve. 144 00:05:38,037 --> 00:05:40,387 If I solve this, I'll get that the acceleration 145 00:05:40,387 --> 00:05:44,554 of this system is gonna be 0.392 meters per second squared. 146 00:05:47,628 --> 00:05:49,412 So, this is a very fast way. 147 00:05:49,412 --> 00:05:51,504 Look it, this is basically a one-liner. 148 00:05:51,504 --> 00:05:53,391 If you could put this together right, 149 00:05:53,391 --> 00:05:54,616 it's a one-liner. 150 00:05:54,616 --> 00:05:57,767 There's much less chance for error than when you're trying 151 00:05:57,767 --> 00:06:00,475 to solve three equations with three unknowns. 152 00:06:00,475 --> 00:06:01,701 This is beautiful. 153 00:06:01,701 --> 00:06:03,824 When you apply this though, be careful. 154 00:06:03,824 --> 00:06:06,259 The acceleration of the five kilogram mass would be 155 00:06:06,259 --> 00:06:10,426 negative 0.392 because it's accelerating downward. 156 00:06:12,549 --> 00:06:16,343 The acceleration of the 12, we'd call positive 0.392 157 00:06:16,343 --> 00:06:18,568 because it's accelerating to the right 158 00:06:18,568 --> 00:06:21,929 and we typically call rightward accelerations positive. 159 00:06:21,929 --> 00:06:24,187 And then the three kilogram mass also would have 160 00:06:24,187 --> 00:00:00,000 positive .392 because it's accelerating upward.