1 00:00:00,250 --> 00:00:01,978 - [Voiceover] Alright, this problem is a classic, 2 00:00:01,978 --> 00:00:03,648 you're gonna see this in basically 3 00:00:03,648 --> 00:00:05,617 every single physics text book. 4 00:00:05,617 --> 00:00:07,457 And the problem is this, if you've got two masses 5 00:00:07,457 --> 00:00:10,978 tied together by a rope, and that rope passes over a pulley, 6 00:00:10,978 --> 00:00:13,659 what's the acceleration of the masses? 7 00:00:13,659 --> 00:00:17,826 In other words, what's the acceleration of the 3 kg mass? 8 00:00:18,890 --> 00:00:21,961 And then what's the acceleration of the 5 kg mass? 9 00:00:21,961 --> 00:00:26,072 And if you're wondering what the heck is a pulley? 10 00:00:26,072 --> 00:00:28,027 The pulley is this part right here. 11 00:00:28,027 --> 00:00:30,027 This right here is the pulley. 12 00:00:30,027 --> 00:00:33,498 So what a pulley does, a pulley is a little piece of plastic 13 00:00:33,498 --> 00:00:35,788 or metal that can rotate. 14 00:00:35,788 --> 00:00:38,258 And it's usually got a groove in it so that a string 15 00:00:38,258 --> 00:00:40,017 or a rope can pass over it. 16 00:00:40,017 --> 00:00:44,149 What it does, is it rotates freely so that you can turn 17 00:00:44,149 --> 00:00:46,429 what's a horizontal tension on one side, 18 00:00:46,429 --> 00:00:48,380 into a vertical tension on the other. 19 00:00:48,380 --> 00:00:49,912 Or vice versa. 20 00:00:49,912 --> 00:00:52,461 It turns vertical forces into horizontal forces. 21 00:00:52,461 --> 00:00:55,342 It allows you to transfer a force from one direction 22 00:00:55,342 --> 00:00:56,741 to another direction. 23 00:00:56,741 --> 00:00:58,793 So that's what these pulleys are useful for. 24 00:00:58,793 --> 00:01:01,263 And if they can spin freely, and if this pulley 25 00:01:01,263 --> 00:01:03,954 has basically no mass, if there's no resistance 26 00:01:03,954 --> 00:01:07,161 to motion at all, then this tension on this side 27 00:01:07,161 --> 00:01:10,542 is gonna be equal to the tension on this side. 28 00:01:10,542 --> 00:01:14,092 This vertical tension gets transferred fully undiluted, 29 00:01:14,092 --> 00:01:17,459 into a horizontal tension and these tension values 30 00:01:17,459 --> 00:01:20,898 will just be the same if this pulley can spin freely. 31 00:01:20,898 --> 00:01:22,878 And if its mass is really small so that 32 00:01:22,878 --> 00:01:26,926 there's no inertial reason why it doesn't wanna spin. 33 00:01:26,926 --> 00:01:28,398 So that's the problem. 34 00:01:28,398 --> 00:01:30,459 Let's say you wanted to figure this out. 35 00:01:30,459 --> 00:01:33,899 What is the acceleration of the 3 kg mass 36 00:01:33,899 --> 00:01:36,528 with the acceleration of the 5 kg mass? 37 00:01:36,528 --> 00:01:38,949 Now, I've gotta warn you, there's an easy way to do this, 38 00:01:38,949 --> 00:01:40,616 and a hard way to do this. 39 00:01:40,616 --> 00:01:43,376 Now, I'm gonna show you the hard way first. 40 00:01:43,376 --> 00:01:46,034 Sorry, no one ever wants to hear that. 41 00:01:46,034 --> 00:01:50,495 But, the reason is that, the easy way won't make any sense 42 00:01:50,495 --> 00:01:51,975 unless I show you the hard way first. 43 00:01:51,975 --> 00:01:54,436 It won't make any sense why the easy way works unless 44 00:01:54,436 --> 00:01:56,074 I show you the hard way. 45 00:01:56,074 --> 00:01:59,045 And for two, the hard way isn't really all that hard. 46 00:01:59,045 --> 00:02:00,186 So I'm calling it the hard way, 47 00:02:00,186 --> 00:02:01,936 but it's not really that bad. 48 00:02:01,936 --> 00:02:04,185 And for three, sometimes teachers and professors just wanna 49 00:02:04,185 --> 00:02:05,944 see you do it the hard way, so you should know how 50 00:02:05,944 --> 00:02:07,186 to do this. 51 00:02:07,186 --> 00:02:08,019 So what do we do? 52 00:02:08,019 --> 00:02:09,946 We wanna find acceleration, well you know how 53 00:02:09,946 --> 00:02:11,145 to find acceleration. 54 00:02:11,145 --> 00:02:13,142 We're gonna use Newton's second law. 55 00:02:13,142 --> 00:02:15,966 So we'll say that the acceleration in a given direction 56 00:02:15,966 --> 00:02:18,166 is gonna equal the net force in that direction, 57 00:02:18,166 --> 00:02:19,653 divided by the mass. 58 00:02:19,653 --> 00:02:20,965 Now what do we do? 59 00:02:20,965 --> 00:02:21,964 What mass are we gonna choose? 60 00:02:21,964 --> 00:02:23,824 We've got a couple masses here. 61 00:02:23,824 --> 00:02:26,123 One thing we could do, let's just pick the 5 kg mass. 62 00:02:26,123 --> 00:02:27,761 Just pick one of them. 63 00:02:27,761 --> 00:02:30,703 So I'm gonna say that the acceleration of the 5 kg mass 64 00:02:30,703 --> 00:02:34,613 is the net force on the 5 kg mass, divided by the mass 65 00:02:34,613 --> 00:02:36,313 of the 5 kg mass. 66 00:02:36,313 --> 00:02:38,845 And remember, we should always pick a direction as well. 67 00:02:38,845 --> 00:02:40,694 So do we wanna pick the vertical direction 68 00:02:40,694 --> 00:02:42,403 or the horizontal direction? 69 00:02:42,403 --> 00:02:44,984 Well, since this box is gonna be accelerating horizontally, 70 00:02:44,984 --> 00:02:47,361 and that's what we're interested in, I'm gonna put one more 71 00:02:47,361 --> 00:02:49,843 sub-script up here, X, to let us know we're picking 72 00:02:49,843 --> 00:02:52,025 the horizontal direction. 73 00:02:52,025 --> 00:02:54,814 So I can fill this out now, I can plug stuff in. 74 00:02:54,814 --> 00:02:58,614 The acceleration of the 5 kg mass in the X direction 75 00:02:58,614 --> 00:03:00,474 is gonna be equal to. 76 00:03:00,474 --> 00:03:02,695 Alright what forces do we have to figure out what goes 77 00:03:02,695 --> 00:03:03,923 up here? 78 00:03:03,923 --> 00:03:06,091 You always draw a force diagram. 79 00:03:06,091 --> 00:03:08,242 So what forces do I have on the 5 kg mass? 80 00:03:08,242 --> 00:03:10,375 I'm gonna have a force of gravity, I'll draw 81 00:03:10,375 --> 00:03:11,933 that straight down. 82 00:03:11,933 --> 00:03:14,923 FG, and there's gonna be an equal force, 83 00:03:14,923 --> 00:03:16,821 normal force upward. 84 00:03:16,821 --> 00:03:19,532 So this normal force up should be equal to the force 85 00:03:19,532 --> 00:03:23,113 of gravity and magnitude because this box is probably not 86 00:03:23,113 --> 00:03:24,823 gonna be accelerating vertically. 87 00:03:24,823 --> 00:03:26,283 There's no real reason why it should be 88 00:03:26,283 --> 00:03:28,172 if this table is rigid. 89 00:03:28,172 --> 00:03:30,323 And there's one more force on this box. 90 00:03:30,323 --> 00:03:31,981 There's a force to the right. 91 00:03:31,981 --> 00:03:34,012 That's gonna be the force of tension. 92 00:03:34,012 --> 00:03:36,131 And if there's no friction on this table, 93 00:03:36,131 --> 00:03:38,259 then I have no left orb forces here. 94 00:03:38,259 --> 00:03:39,722 I'm ignoring air resistance since 95 00:03:39,722 --> 00:03:42,444 we usually ignore air resistance. 96 00:03:42,444 --> 00:03:43,695 So that's is, the only horizontal force 97 00:03:43,695 --> 00:03:45,675 I've got is T, tension. 98 00:03:45,675 --> 00:03:50,606 And I divide by the mass of the 5 kg box, which is 5 kg. 99 00:03:50,606 --> 00:03:51,944 But we got a problem. 100 00:03:51,944 --> 00:03:54,994 Look it, we don't know the acceleration of the 5 kg mass, 101 00:03:54,994 --> 00:03:56,115 and we don't know the tension. 102 00:03:56,115 --> 00:03:58,065 I can't solve this. 103 00:03:58,065 --> 00:03:59,865 Normally what you do in this case, is you go to 104 00:03:59,865 --> 00:04:02,933 the vertical direction, the other direction in other words. 105 00:04:02,933 --> 00:04:04,204 But that's not gonna help me either. 106 00:04:04,204 --> 00:04:06,214 That's just gonna tell me that the normal force is gonna 107 00:04:06,214 --> 00:04:08,546 be equal to the force of gravity. 108 00:04:08,546 --> 00:04:09,594 And we kind of already knew that. 109 00:04:09,594 --> 00:04:10,894 So that doesn't help. 110 00:04:10,894 --> 00:04:12,035 So what do we do? 111 00:04:12,035 --> 00:04:14,492 Well, you might note, this is only the equation for 112 00:04:14,492 --> 00:04:16,096 the 5 kg mass. 113 00:04:16,096 --> 00:04:18,613 And so now I have to do this for the 3 kg mass. 114 00:04:18,613 --> 00:04:21,505 So let's come over here, let's say that the acceleration 115 00:04:21,505 --> 00:04:25,575 of the 3 kg mass is gonna be equal to the net force 116 00:04:25,575 --> 00:04:29,742 on the 3 kg mass, divided by the mass of the 3 kg mass. 117 00:04:30,875 --> 00:04:33,104 And again, which direction should we pick? 118 00:04:33,104 --> 00:04:35,444 Well this acceleration over here is gonna be vertical. 119 00:04:35,444 --> 00:04:37,813 So let's solve this for the vertical direction. 120 00:04:37,813 --> 00:04:40,854 I'm gonna add one more sub-script, Y, to remind myself. 121 00:04:40,854 --> 00:04:43,211 And you should do this too so you remember which direction 122 00:04:43,211 --> 00:04:44,296 you're picking. 123 00:04:44,296 --> 00:04:46,894 So what forces do I plug in here? 124 00:04:46,894 --> 00:04:48,554 You figure that out with a force diagram. 125 00:04:48,554 --> 00:04:52,606 I'm gonna have a force of gravity on this 3 kg mass, 126 00:04:52,606 --> 00:04:55,363 and then I'm gonna have the same size of friction, 127 00:04:55,363 --> 00:04:57,906 or sorry, the same as tension, that I had over here. 128 00:04:57,906 --> 00:04:59,784 So the tension on this side of the rope, it's gonna 129 00:04:59,784 --> 00:05:01,554 be the same as the tension on this side. 130 00:05:01,554 --> 00:05:05,744 Assuming this pulley offers no resistance either by its mass 131 00:05:05,744 --> 00:05:06,852 or friction. 132 00:05:06,852 --> 00:05:09,816 So assuming that its mass is negligible, there's basically 133 00:05:09,816 --> 00:05:11,792 no friction, then I'm gonna have a tension. 134 00:05:11,792 --> 00:05:14,681 That tension is gonna be the same size. 135 00:05:14,681 --> 00:05:16,733 So I'll draw that coming upward. 136 00:05:16,733 --> 00:05:19,592 But it's not gonna be as big as the force of gravity is 137 00:05:19,592 --> 00:05:21,662 on this 3 kg mass. 138 00:05:21,662 --> 00:05:23,230 I've got the force of gravity here. 139 00:05:23,230 --> 00:05:26,163 This tension is gonna be smaller, and the reason is, 140 00:05:26,163 --> 00:05:28,793 this 3 kg mass is accelerating downwards. 141 00:05:28,793 --> 00:05:30,691 So these forces can't be balanced. 142 00:05:30,691 --> 00:05:33,063 The upward force of tension has gotta be smaller 143 00:05:33,063 --> 00:05:36,142 than the force of gravity on this 3 kg mass. 144 00:05:36,142 --> 00:05:37,942 But this tension here should be the same 145 00:05:37,942 --> 00:05:39,780 as this tension here. 146 00:05:39,780 --> 00:05:40,934 So I'll plug those in. 147 00:05:40,934 --> 00:05:41,863 So let's plug this in. 148 00:05:41,863 --> 00:05:46,582 A of the 3 kg mass, in the Y direction is gonna be equal to, 149 00:05:46,582 --> 00:05:48,434 I've got two vertical forces. 150 00:05:48,434 --> 00:05:50,324 I've got tension up, so I'll make that positive, 151 00:05:50,324 --> 00:05:52,103 'cause we usually treat up as positive. 152 00:05:52,103 --> 00:05:55,525 I've got gravity down, and so I'm gonna have negative, 153 00:05:55,525 --> 00:05:59,692 'cause it's downward force of 3 kg times the acceleration 154 00:06:01,200 --> 00:06:04,783 on Earth, is 9.8 meters per second squared. 155 00:06:07,025 --> 00:06:07,858 Now what do we do? 156 00:06:07,858 --> 00:06:10,112 We divide by 3 kg, 'cause that's the mass. 157 00:06:10,112 --> 00:06:11,724 But I've still got a problem. 158 00:06:11,724 --> 00:06:14,974 I don't know this acceleration or this tension. 159 00:06:14,974 --> 00:06:16,737 So what do I do? 160 00:06:16,737 --> 00:06:18,522 You might notice, if you're clever you'll say wait, 161 00:06:18,522 --> 00:06:21,093 I've got my unknown on this side is acceleration 162 00:06:21,093 --> 00:06:22,362 and tension. 163 00:06:22,362 --> 00:06:24,924 My unknown on this side is acceleration and tension. 164 00:06:24,924 --> 00:06:27,223 It seems like I've got two equations, two unknowns, 165 00:06:27,223 --> 00:06:28,841 maybe we should combine them. 166 00:06:28,841 --> 00:06:30,784 And that's exactly how you do these. 167 00:06:30,784 --> 00:06:34,244 So I've got tension in both of these equations. 168 00:06:34,244 --> 00:06:35,962 Let me solve for tension over here, 169 00:06:35,962 --> 00:06:37,483 where it's kind of simple. 170 00:06:37,483 --> 00:06:41,564 And I'll just get the tension equals 5 kg, 171 00:06:41,564 --> 00:06:45,731 times the acceleration of the 5 kg mass in the X direction. 172 00:06:47,334 --> 00:06:48,402 So now I know what tension is. 173 00:06:48,402 --> 00:06:49,894 Tension is equal to this. 174 00:06:49,894 --> 00:06:51,714 And that tension over on this side is the same as 175 00:06:51,714 --> 00:06:53,374 the tension on this side. 176 00:06:53,374 --> 00:06:56,405 So I can take this and I can plug it in for 177 00:06:56,405 --> 00:06:57,842 this tension right here. 178 00:06:57,842 --> 00:06:59,383 And let's see what we get. 179 00:06:59,383 --> 00:07:02,443 We get that the acceleration of the 3 kg mass vertically, 180 00:07:02,443 --> 00:07:06,313 is gonna equal, alright, I'm gonna have a big mess on top, 181 00:07:06,313 --> 00:07:07,522 what am I gonna get? 182 00:07:07,522 --> 00:07:11,039 I'm gonna get, so T is the same as 5AX. 183 00:07:11,039 --> 00:07:14,622 So I'll plug in 5 kg times the acceleration 184 00:07:16,340 --> 00:07:19,131 of the 5 kg mass in the X direction. 185 00:07:19,131 --> 00:07:21,253 And then I get all of this stuff over here. 186 00:07:21,253 --> 00:07:24,586 So I'll get the rest of this right here. 187 00:07:27,383 --> 00:07:29,971 I'll just bring that down right there. 188 00:07:29,971 --> 00:07:31,322 Alright, now what do I have? 189 00:07:31,322 --> 00:07:33,533 I've got 3 kg on the bottom still, so I have to put 190 00:07:33,533 --> 00:07:35,022 that here. 191 00:07:35,022 --> 00:07:36,511 Are we any better off? 192 00:07:36,511 --> 00:07:38,412 Yeah, we're better, because now my only unknowns 193 00:07:38,412 --> 00:07:39,913 are acceleration. 194 00:07:39,913 --> 00:07:41,973 But these are not the same acceleration. 195 00:07:41,973 --> 00:07:43,831 Look, this acceleration here is the acceleration of 196 00:07:43,831 --> 00:07:46,473 the 3 kg mass, vertically. 197 00:07:46,473 --> 00:07:48,733 This acceleration here is the acceleration 198 00:07:48,733 --> 00:07:51,622 of the 5kg mass horizontally. 199 00:07:51,622 --> 00:07:53,781 Now here's where I'm gonna have to make an argument, 200 00:07:53,781 --> 00:07:55,100 and some people don't like this. 201 00:07:55,100 --> 00:07:58,213 But, it's crucial to figuring out this problem. 202 00:07:58,213 --> 00:08:03,043 And the key idea is this, if this 3 kg mass moves down, 203 00:08:03,043 --> 00:08:06,233 let's say one meter, let's say it moves downward one meter. 204 00:08:06,233 --> 00:08:10,502 Well then this 5 kg mass had better move forward one meter. 205 00:08:10,502 --> 00:08:14,281 Because if it doesn't, then it didn't provide the one meter 206 00:08:14,281 --> 00:08:17,980 of rope that this 3 kg mass needed to go downward. 207 00:08:17,980 --> 00:08:21,780 Which means either the rope broke, or the rope stretched. 208 00:08:21,780 --> 00:08:23,654 And we're gonna assume that our rope does not break 209 00:08:23,654 --> 00:08:25,114 or stretch. 210 00:08:25,114 --> 00:08:25,993 That's kind of a lie. 211 00:08:25,993 --> 00:08:29,023 All ropes are gonna stretch a little bit under tension. 212 00:08:29,023 --> 00:08:31,911 We're gonna assume that stretch is negligible. 213 00:08:31,911 --> 00:08:35,251 So the argument is that if this 3 kg mass moves downward 214 00:08:35,251 --> 00:08:38,131 a certain amount, this 5 kg mass has to move forward 215 00:08:38,131 --> 00:08:41,010 by that same amount in order to feed that amount of rope 216 00:08:41,010 --> 00:08:44,731 for this 3 kg mass to go downward by that amount. 217 00:08:44,731 --> 00:08:45,741 Otherwise, think about it. 218 00:08:45,741 --> 00:08:49,630 If this 5 kg mass just sat here and the 3 kg moved, 219 00:08:49,630 --> 00:08:53,070 or the 3 kg moved farther than the 5 kg mass, 220 00:08:53,070 --> 00:08:54,880 then this rope is stretching or breaking. 221 00:08:54,880 --> 00:08:56,359 So if you believe that, if you don't believe it, 222 00:08:56,359 --> 00:08:57,839 pause it and think about it. 223 00:08:57,839 --> 00:08:59,101 'Cause you've gotta convince yourself of that. 224 00:08:59,101 --> 00:09:01,560 If you believe that then you can also convince yourself 225 00:09:01,560 --> 00:09:04,369 that, well if the 3 kg mass was moving downward 226 00:09:04,369 --> 00:09:07,260 at a certain speed, let's say two meters per second. 227 00:09:07,260 --> 00:09:10,140 Then the 5 kg mass had better also be moving forward 228 00:09:10,140 --> 00:09:13,300 two meters per second because otherwise it wouldn't 229 00:09:13,300 --> 00:09:16,972 be feeding rope at a rate that this 3 kg needs 230 00:09:16,972 --> 00:09:18,641 to move downward at that rate. 231 00:09:18,641 --> 00:09:20,412 And finally, if you believe all that, 232 00:09:20,412 --> 00:09:22,452 it's not too much harder to convince yourself that 233 00:09:22,452 --> 00:09:25,198 this 3 kg mass, no matter what 234 00:09:25,198 --> 00:09:27,880 its acceleration downward must be, 235 00:09:27,880 --> 00:09:30,692 this 5 kg mass had better have the same magnitude 236 00:09:30,692 --> 00:09:34,341 of acceleration forward so that it's again, 237 00:09:34,341 --> 00:09:36,420 feeding the rope so this rope doesn't break, 238 00:09:36,420 --> 00:09:38,357 or snap, or stretch. 239 00:09:38,357 --> 00:09:40,586 'Cause we're gonna assume the rope doesn't do that. 240 00:09:40,586 --> 00:09:44,915 So what I'm saying is that the acceleration of the 3 kg mass 241 00:09:44,915 --> 00:09:48,592 in the Y direction had better equal the magnitude. 242 00:09:48,592 --> 00:09:50,327 So these magnitudes have to be the same. 243 00:09:50,327 --> 00:09:53,337 The sign doesn't have to be the same. 244 00:09:53,337 --> 00:09:56,707 So this 3 kg mass has a negative acceleration 245 00:09:56,707 --> 00:09:58,797 just 'cause it points down, and we're assuming up 246 00:09:58,797 --> 00:10:00,666 is positive, down is negative. 247 00:10:00,666 --> 00:10:03,897 This 5 kg mass has a positive acceleration 'cause 248 00:10:03,897 --> 00:10:05,957 it's pointing to the right, and we're assuming rightward 249 00:10:05,957 --> 00:10:08,477 is the positive horizontal direction. 250 00:10:08,477 --> 00:10:11,068 So, they can have different signs, but the magnitudes 251 00:10:11,068 --> 00:10:13,167 had better be the same so that your feeding this rope at 252 00:10:13,167 --> 00:10:16,217 a rate that the other one needs in order to move. 253 00:10:16,217 --> 00:10:18,897 And so we can say that the magnitudes are the same. 254 00:10:18,897 --> 00:10:21,175 In this case, since one is negative of the other, 255 00:10:21,175 --> 00:10:23,755 I can say that the acceleration of the 3 kg mass 256 00:10:23,755 --> 00:10:28,355 vertically downward is gonna be equal to, let's say negative 257 00:10:28,355 --> 00:10:33,075 of the acceleration of the 5 kg mass in the X direction. 258 00:10:33,075 --> 00:10:34,596 I could have written it the other way. 259 00:10:34,596 --> 00:10:37,206 I could have wrote that A of the 5 kg mass 260 00:10:37,206 --> 00:10:40,477 in the X direction is a negative A of the 3 kg mass 261 00:10:40,477 --> 00:10:41,365 in the Y direction. 262 00:10:41,365 --> 00:10:43,583 They're just different by a negative sign is all 263 00:10:43,583 --> 00:10:44,647 that's important here. 264 00:10:44,647 --> 00:10:46,447 Okay, so this is the link we need. 265 00:10:46,447 --> 00:10:47,324 This is it. 266 00:10:47,324 --> 00:10:49,425 So this allows us to put this final equation here 267 00:10:49,425 --> 00:10:51,358 in terms of only one variable. 268 00:10:51,358 --> 00:10:54,235 'Cause I know I've got A3Y on this left hand side. 269 00:10:54,235 --> 00:10:56,985 I know A3Y should always be -A5X. 270 00:10:57,835 --> 00:11:02,002 If I take this and just plug it in for A3Y right here, 271 00:11:02,986 --> 00:11:07,153 I'm gonna get -A5X =, well all of this stuff, so I'll 272 00:11:10,014 --> 00:11:11,135 just copy this. 273 00:11:11,135 --> 00:11:12,766 Save some time. 274 00:11:12,766 --> 00:11:14,545 Copy, paste. 275 00:11:14,545 --> 00:11:15,663 Just equals all of that. 276 00:11:15,663 --> 00:11:19,055 All I did was plug in what I know A3Y has to be equal to. 277 00:11:19,055 --> 00:11:22,094 'Cause now look, I've got one equation with one unknown. 278 00:11:22,094 --> 00:11:24,077 I just need to solve for what A5X is. 279 00:11:24,077 --> 00:11:25,584 It's on both sides. 280 00:11:25,584 --> 00:11:28,097 So I'll need to combine these and then isolate 281 00:11:28,097 --> 00:11:29,815 it on one side. 282 00:11:29,815 --> 00:11:31,566 So there's gonna be a little bit of algebra here. 283 00:11:31,566 --> 00:11:35,107 Let's just take this, let's give ourselves some room. 284 00:11:35,107 --> 00:11:36,816 Move this up just a little bit. 285 00:11:36,816 --> 00:11:38,534 Okay, so what do we do? 286 00:11:38,534 --> 00:11:39,783 We're gonna solve for A5X. 287 00:11:39,783 --> 00:11:41,315 Let me just get rid of this denominator. 288 00:11:41,315 --> 00:11:44,232 Let me multiply both sides by 3 kg. 289 00:11:45,387 --> 00:11:49,470 So I'm gonna get -3kg x A5 in the X direction, 290 00:11:52,465 --> 00:11:55,385 if I multiply both sides by 3 kg, 291 00:11:55,385 --> 00:11:59,302 and then I get 5 kg x A5 in the X direction. 292 00:12:01,753 --> 00:12:05,920 And I've still got minus, alright 3 x 9.8 is 29.4 Newtons. 293 00:12:08,694 --> 00:12:10,655 So we'll just turn this into what it's supposed to be. 294 00:12:10,655 --> 00:12:12,412 29.4 Newtons. 295 00:12:12,412 --> 00:12:14,533 So let's combine our A terms now. 296 00:12:14,533 --> 00:12:17,484 Let's move this negative 3A to the right hand side 297 00:12:17,484 --> 00:12:19,435 by adding it to both sides. 298 00:12:19,435 --> 00:12:22,184 And let's add this 29.4 to both sides. 299 00:12:22,184 --> 00:12:26,254 So I'll get the 29.4 Newtons over here with a positive, 300 00:12:26,254 --> 00:12:27,812 if I add it to both sides. 301 00:12:27,812 --> 00:12:29,794 And it'll disappear on the right hand side. 302 00:12:29,794 --> 00:12:31,965 And then I'll add this term to both sides. 303 00:12:31,965 --> 00:12:36,075 Add a positive 3 kg x A to both sides. 304 00:12:36,075 --> 00:12:40,242 It'll disappear on the left, and I'll get 5 kg x A5 305 00:12:41,574 --> 00:12:45,741 in the X direction + 3 kg x A5 in the X direction. 306 00:12:47,444 --> 00:12:50,411 Now we're close, look on the right hand side I can combine 307 00:12:50,411 --> 00:12:54,578 these terms because 5A plus 3A is the same thing as 8A. 308 00:12:55,905 --> 00:13:00,072 So 29.4 Newtons, .4 = 8 kg x, I'll put the parenthesis here, 309 00:13:05,115 --> 00:13:06,453 times five. 310 00:13:06,453 --> 00:13:08,165 85 in the X direction. 311 00:13:08,165 --> 00:13:12,180 Now I can divide both sides by eight and usually we put 312 00:13:12,180 --> 00:13:14,034 the thing we're solving for on the left, so I'm just gonna 313 00:13:14,034 --> 00:13:15,064 put that over here. 314 00:13:15,064 --> 00:13:18,647 I'll get 29.4 Newtons over 8 kg is equal to 315 00:13:22,094 --> 00:13:24,474 the acceleration of mass five. 316 00:13:24,474 --> 00:13:27,225 The 5 kg mass in the X direction. 317 00:13:27,225 --> 00:13:29,435 And if we calculate that, 318 00:13:29,435 --> 00:13:30,964 I'll just put that into my calculator. 319 00:13:30,964 --> 00:13:32,797 29.4 divided by eight. 320 00:13:34,293 --> 00:13:35,293 I get 3.675. 321 00:13:36,464 --> 00:13:37,472 So we'll just round. 322 00:13:37,472 --> 00:13:39,722 We'll just say that's 3.68. 323 00:13:40,566 --> 00:13:44,566 3.6 whoops, 3.68 and it's positive, that's good. 324 00:13:45,709 --> 00:13:49,154 We should get a positive because the 5 kg mass has 325 00:13:49,154 --> 00:13:50,742 a positive acceleration. 326 00:13:50,742 --> 00:13:55,534 So we get positive 3.68 meters per second squared. 327 00:13:55,534 --> 00:13:57,992 But that's just of the 5 kg mass. 328 00:13:57,992 --> 00:14:00,082 How do we get the acceleration of the 3 kg mass? 329 00:14:00,082 --> 00:14:00,978 Well that's easy. 330 00:14:00,978 --> 00:14:04,959 It's gotta have the same magnitude of the 5 kg mass. 331 00:14:04,959 --> 00:14:06,658 All I have to do is take this number now. 332 00:14:06,658 --> 00:14:08,521 I know what A5X is. 333 00:14:08,521 --> 00:14:10,579 So if I just plug that in right here, 334 00:14:10,579 --> 00:14:14,579 well then I know that A3Y is just gonna be equal 335 00:14:15,629 --> 00:14:18,546 to -3.68 meters per second squared. 336 00:14:22,258 --> 00:14:23,750 And I'm done, I did it. 337 00:14:23,750 --> 00:14:27,619 We figured out the acceleration of the 3 kg mass, 338 00:14:27,619 --> 00:14:28,452 it's negative. 339 00:14:28,452 --> 00:14:30,760 No surprise, 'cause it's accelerating downward. 340 00:14:30,760 --> 00:14:33,508 We figured out the acceleration of the 5 kg mass, 341 00:14:33,508 --> 00:14:34,841 it's positive, not a surprise. 342 00:14:34,841 --> 00:14:36,600 It was accelerating to the right. 343 00:14:36,600 --> 00:14:38,861 The way we did it, recapping really quick, 344 00:14:38,861 --> 00:14:41,780 we did Newton's second law for the 5 kg mass. 345 00:14:41,780 --> 00:14:43,104 That didn't let us solve. 346 00:14:43,104 --> 00:14:46,061 We did Newton's second law for the 3 kg mass, 347 00:14:46,061 --> 00:14:47,722 that didn't let us solve. 348 00:14:47,722 --> 00:14:49,811 In fact it got really bleak, because it seemed like we 349 00:14:49,811 --> 00:14:53,229 had three unknowns and only two equations, 350 00:14:53,229 --> 00:14:56,150 but the link that allowed us to make it so that we only 351 00:14:56,150 --> 00:14:59,140 had one equation with one unknown, is that we plugged 352 00:14:59,140 --> 00:15:01,401 one equation to the other first. 353 00:15:01,401 --> 00:15:03,799 We had to then write the accelerations in terms 354 00:15:03,799 --> 00:15:07,180 of each other, that's because these accelerations 355 00:15:07,180 --> 00:15:08,410 are not independent. 356 00:15:08,410 --> 00:15:10,851 The accelerations have to have the same magnitude. 357 00:15:10,851 --> 00:15:13,578 And in this case one had the opposite sign. 358 00:15:13,578 --> 00:15:15,490 So when we plugged that in, we have one equation 359 00:15:15,490 --> 00:15:18,018 with one unknown, we solve, we get the amount 360 00:15:18,018 --> 00:15:19,321 of acceleration. 361 00:15:19,321 --> 00:15:21,951 So that's the hard way to do these problems. 362 00:15:21,951 --> 00:15:24,551 So in the next video I'll show you the easy way 363 00:15:24,551 --> 00:00:00,000 to do these problems.