WEBVTT 00:00:03.640 --> 00:00:06.730 When we did our analytic analysis of the constraint 00:00:06.730 --> 00:00:09.010 conditions between the accelerations of objects 1 00:00:09.010 --> 00:00:11.200 and 2, we came up with the condition 00:00:11.200 --> 00:00:13.980 that a1 was equal to minus 2 a2. 00:00:13.980 --> 00:00:16.880 Now let's do something which we call a virtual displacement 00:00:16.880 --> 00:00:17.610 argument. 00:00:17.610 --> 00:00:22.410 Suppose that b and 2 move down a certain amount. 00:00:22.410 --> 00:00:25.580 Let's imagine-- and I'm going to draw in a different color, 00:00:25.580 --> 00:00:31.280 too-- so we have object 2, and pulley 2 has moved down. 00:00:31.280 --> 00:00:35.350 Now, this object has displaced by distance delta 00:00:35.350 --> 00:00:38.900 y 2, which is also equal to delta y 00:00:38.900 --> 00:00:40.980 b because they're connected. 00:00:40.980 --> 00:00:45.020 Now, what happens when the system does that, 00:00:45.020 --> 00:00:52.640 is our rope has to extend downwards around this pulley 00:00:52.640 --> 00:00:54.140 and come back up. 00:00:54.140 --> 00:00:56.860 And that means that the rope that object 1 00:00:56.860 --> 00:01:00.830 has been foreshortened by not just delta y2, 00:01:00.830 --> 00:01:06.530 but on both sides, delta y2 and delta y2, 00:01:06.530 --> 00:01:13.289 object 1 has displaced up by that amount. 00:01:13.289 --> 00:01:18.635 So we'll just make this so we can draw it 00:01:18.635 --> 00:01:20.640 in a reasonable way. 00:01:20.640 --> 00:01:25.560 So what we see here is that delta y1 is 00:01:25.560 --> 00:01:32.660 equal to-- now notice, if 2 goes down, by delta y2, 00:01:32.660 --> 00:01:35.600 then delta y1-- which is this whole distance-- 00:01:35.600 --> 00:01:37.430 is a negative quantity. 00:01:37.430 --> 00:01:39.090 And it's going upwards. 00:01:39.090 --> 00:01:44.150 And so we see that that's minus 2 delta y2. 00:01:44.150 --> 00:01:47.850 And if we took two derivatives-- or displacement 00:01:47.850 --> 00:01:50.370 and then look at the change in displacement-- 00:01:50.370 --> 00:01:54.100 we would see that this implies that the acceleration of 1 00:01:54.100 --> 00:01:57.039 is minus 2 a2. 00:01:57.039 --> 00:02:00.430 But let's come back to our two conditions for length 00:02:00.430 --> 00:02:02.400 and see the same thing here. 00:02:02.400 --> 00:02:06.540 Because delta l2 is 0, this tells us 00:02:06.540 --> 00:02:10.470 that delta y2 is equal to delta y d. 00:02:10.470 --> 00:02:13.100 So we'll write minus equals 0. 00:02:13.100 --> 00:02:16.140 And that was our condition that the block and 2 00:02:16.140 --> 00:02:17.660 were moving together. 00:02:17.660 --> 00:02:24.770 And up here, we see that, because delta l1 is also 0, 00:02:24.770 --> 00:02:30.220 this implies-- and now I'll make that substitution that delta 00:02:30.220 --> 00:02:37.600 y b is equal to delta y2-- that 2 delta y2 here 00:02:37.600 --> 00:02:44.579 and here plus delta y1 has to be 0 coming from that piece. 00:02:44.579 --> 00:02:50.260 And so we see that delta y1 is minus 2 delta y2. 00:02:50.260 --> 00:02:53.310 Which is what our virtual displacement 00:02:53.310 --> 00:02:54.740 argument showed us. 00:02:54.740 --> 00:03:00.590 And again, if you take two derivatives here, 00:03:00.590 --> 00:03:04.770 we have that recall that, in the simplest way, 00:03:04.770 --> 00:03:10.840 that the velocity is dy1 dt. 00:03:10.840 --> 00:03:17.579 And the acceleration, a1, is d squared y1 dt squared. 00:03:17.579 --> 00:03:21.550 Then, this same proportionality is maintained 00:03:21.550 --> 00:03:23.290 under the two derivatives. 00:03:23.290 --> 00:03:25.680 And that's another way of thinking 00:03:25.680 --> 00:03:27.420 about how to get the relationship 00:03:27.420 --> 00:03:28.870 between the accelerations. 00:03:28.870 --> 00:03:32.900 But you have to be extremely careful about that sign 00:03:32.900 --> 00:03:35.579 because this 2 goes in the positive direction, 00:03:35.579 --> 00:03:38.520 1 will go in the negative direction.