1 00:00:03,320 --> 00:00:06,640 We are now in position to find the accelerations a1, a2, 2 00:00:06,640 --> 00:00:10,580 and the tension, because we have Newton's second law 3 00:00:10,580 --> 00:00:12,970 and our constraint condition for the acceleration. 4 00:00:12,970 --> 00:00:15,340 Let's recall the equations that we found. 5 00:00:15,340 --> 00:00:20,530 We had m1g minus t was m1a1. 6 00:00:20,530 --> 00:00:26,660 And we had m2g minus 2t was equal to m2a2. 7 00:00:26,660 --> 00:00:29,160 And we also had the condition-- constraint condition 8 00:00:29,160 --> 00:00:34,910 between the accelerations that a1 was minus 2a2. 9 00:00:34,910 --> 00:00:38,310 So now we have a system of three equations. 10 00:00:38,310 --> 00:00:41,620 And we can find-- we can solve for any of these quantities 11 00:00:41,620 --> 00:00:42,790 that we want. 12 00:00:42,790 --> 00:00:46,240 So one way to do it is to identify an equation 13 00:00:46,240 --> 00:00:49,610 and to identify a quantity that we would like to solve for. 14 00:00:49,610 --> 00:00:51,800 So for instance, let's identify that we 15 00:00:51,800 --> 00:00:55,630 want to solve for a1 first. 16 00:00:55,630 --> 00:00:58,110 So how are we going to develop a strategy for that? 17 00:00:58,110 --> 00:01:02,390 Well, I'll choose equation 1 as my backbone equation. 18 00:01:02,390 --> 00:01:04,550 And I have an unknown t. 19 00:01:04,550 --> 00:01:08,580 But in equation 2, that unknown t is appearing there, 20 00:01:08,580 --> 00:01:11,170 but it's expressed in terms of a2. 21 00:01:11,170 --> 00:01:15,920 And a2 , though, from equation 3 is expressed in terms of a1. 22 00:01:15,920 --> 00:01:20,810 So my first step is to write equation-- rewrite equation 23 00:01:20,810 --> 00:01:25,500 2 as m2g minus 2t equals m2. 24 00:01:25,500 --> 00:01:28,080 Now here I'm going to make the substitution, which 25 00:01:28,080 --> 00:01:33,990 is minus a1 over 2, so we'll call that 2a. 26 00:01:33,990 --> 00:01:38,670 And now I have equation 1 and equation 2a, 27 00:01:38,670 --> 00:01:41,300 two unknowns t, and a1. 28 00:01:41,300 --> 00:01:46,150 And what I can do is I can solve for either equation. 29 00:01:46,150 --> 00:01:48,020 And I can make a choice what's easiest. 30 00:01:48,020 --> 00:01:49,539 When I look at these equations, it's 31 00:01:49,539 --> 00:01:54,259 easiest for me to identify what t is in terms of a1. 32 00:01:54,259 --> 00:02:00,240 So I'll write m1g minus m1a1 is equal to t. 33 00:02:00,240 --> 00:02:01,943 I'll call that equation 1a. 34 00:02:01,943 --> 00:02:06,795 And now I can substitute that value of t into equation 2a. 35 00:02:06,795 --> 00:02:16,920 And I get m2g minus 2 times m1g minus m1a1. 36 00:02:16,920 --> 00:02:24,170 And that's equal to minus m2 over 2 times a1. 37 00:02:27,260 --> 00:02:31,550 And now I'd like to collect my a1 terms. 38 00:02:31,550 --> 00:02:34,600 And what I have over here-- let's 39 00:02:34,600 --> 00:02:37,390 bring all the a1 terms to this side. 40 00:02:37,390 --> 00:02:42,900 And I get m2g minus 2m1g. 41 00:02:42,900 --> 00:02:47,430 I have a plus, so I'm going to bring that over to this side. 42 00:02:47,430 --> 00:02:51,504 And I get equal to-- that's plus plus, 43 00:02:51,504 --> 00:02:54,260 so I'll have two minus signs, and I'll 44 00:02:54,260 --> 00:03:03,230 have 2m1 plus m2 over 2 a1. 45 00:03:03,230 --> 00:03:04,780 And now I can solve for a1. 46 00:03:07,290 --> 00:03:15,650 And what I get is I get 2m1g minus m2g. 47 00:03:15,650 --> 00:03:21,240 And I want to divide through by this denominator. 48 00:03:21,240 --> 00:03:29,579 And what I have downstairs is 2m1 plus m2 divided by 2. 49 00:03:29,579 --> 00:03:31,930 And that's my expression for a1. 50 00:03:31,930 --> 00:03:34,360 Once I have that expression for a1, 51 00:03:34,360 --> 00:03:38,420 I can easily come back and find out what a2 is. 52 00:03:38,420 --> 00:03:42,450 Or I can substitute it into the equation here for a1 53 00:03:42,450 --> 00:03:44,540 and find out what the tension is. 54 00:03:44,540 --> 00:03:55,970 And so I can now easily find my expressions for a2 and t. 55 00:03:55,970 --> 00:03:59,607 But I'll leave that as an exercise.