1 00:00:03,640 --> 00:00:06,730 When we did our analytic analysis of the constraint 2 00:00:06,730 --> 00:00:09,010 conditions between the accelerations of objects 1 3 00:00:09,010 --> 00:00:11,200 and 2, we came up with the condition 4 00:00:11,200 --> 00:00:13,980 that a1 was equal to minus 2 a2. 5 00:00:13,980 --> 00:00:16,880 Now let's do something which we call a virtual displacement 6 00:00:16,880 --> 00:00:17,610 argument. 7 00:00:17,610 --> 00:00:22,410 Suppose that b and 2 move down a certain amount. 8 00:00:22,410 --> 00:00:25,580 Let's imagine-- and I'm going to draw in a different color, 9 00:00:25,580 --> 00:00:31,280 too-- so we have object 2, and pulley 2 has moved down. 10 00:00:31,280 --> 00:00:35,350 Now, this object has displaced by distance delta 11 00:00:35,350 --> 00:00:38,900 y 2, which is also equal to delta y 12 00:00:38,900 --> 00:00:40,980 b because they're connected. 13 00:00:40,980 --> 00:00:45,020 Now, what happens when the system does that, 14 00:00:45,020 --> 00:00:52,640 is our rope has to extend downwards around this pulley 15 00:00:52,640 --> 00:00:54,140 and come back up. 16 00:00:54,140 --> 00:00:56,860 And that means that the rope that object 1 17 00:00:56,860 --> 00:01:00,830 has been foreshortened by not just delta y2, 18 00:01:00,830 --> 00:01:06,530 but on both sides, delta y2 and delta y2, 19 00:01:06,530 --> 00:01:13,289 object 1 has displaced up by that amount. 20 00:01:13,289 --> 00:01:18,635 So we'll just make this so we can draw it 21 00:01:18,635 --> 00:01:20,640 in a reasonable way. 22 00:01:20,640 --> 00:01:25,560 So what we see here is that delta y1 is 23 00:01:25,560 --> 00:01:32,660 equal to-- now notice, if 2 goes down, by delta y2, 24 00:01:32,660 --> 00:01:35,600 then delta y1-- which is this whole distance-- 25 00:01:35,600 --> 00:01:37,430 is a negative quantity. 26 00:01:37,430 --> 00:01:39,090 And it's going upwards. 27 00:01:39,090 --> 00:01:44,150 And so we see that that's minus 2 delta y2. 28 00:01:44,150 --> 00:01:47,850 And if we took two derivatives-- or displacement 29 00:01:47,850 --> 00:01:50,370 and then look at the change in displacement-- 30 00:01:50,370 --> 00:01:54,100 we would see that this implies that the acceleration of 1 31 00:01:54,100 --> 00:01:57,039 is minus 2 a2. 32 00:01:57,039 --> 00:02:00,430 But let's come back to our two conditions for length 33 00:02:00,430 --> 00:02:02,400 and see the same thing here. 34 00:02:02,400 --> 00:02:06,540 Because delta l2 is 0, this tells us 35 00:02:06,540 --> 00:02:10,470 that delta y2 is equal to delta y d. 36 00:02:10,470 --> 00:02:13,100 So we'll write minus equals 0. 37 00:02:13,100 --> 00:02:16,140 And that was our condition that the block and 2 38 00:02:16,140 --> 00:02:17,660 were moving together. 39 00:02:17,660 --> 00:02:24,770 And up here, we see that, because delta l1 is also 0, 40 00:02:24,770 --> 00:02:30,220 this implies-- and now I'll make that substitution that delta 41 00:02:30,220 --> 00:02:37,600 y b is equal to delta y2-- that 2 delta y2 here 42 00:02:37,600 --> 00:02:44,579 and here plus delta y1 has to be 0 coming from that piece. 43 00:02:44,579 --> 00:02:50,260 And so we see that delta y1 is minus 2 delta y2. 44 00:02:50,260 --> 00:02:53,310 Which is what our virtual displacement 45 00:02:53,310 --> 00:02:54,740 argument showed us. 46 00:02:54,740 --> 00:03:00,590 And again, if you take two derivatives here, 47 00:03:00,590 --> 00:03:04,770 we have that recall that, in the simplest way, 48 00:03:04,770 --> 00:03:10,840 that the velocity is dy1 dt. 49 00:03:10,840 --> 00:03:17,579 And the acceleration, a1, is d squared y1 dt squared. 50 00:03:17,579 --> 00:03:21,550 Then, this same proportionality is maintained 51 00:03:21,550 --> 00:03:23,290 under the two derivatives. 52 00:03:23,290 --> 00:03:25,680 And that's another way of thinking 53 00:03:25,680 --> 00:03:27,420 about how to get the relationship 54 00:03:27,420 --> 00:03:28,870 between the accelerations. 55 00:03:28,870 --> 00:03:32,900 But you have to be extremely careful about that sign 56 00:03:32,900 --> 00:03:35,579 because this 2 goes in the positive direction, 57 00:03:35,579 --> 00:03:38,520 1 will go in the negative direction.