WEBVTT 00:00:00.000 --> 00:00:02.860 This is Giancoli Answers with Mr. Dychko. 00:00:04.251 --> 00:00:06.251 The distance from the center of the Earth 00:00:06.250 --> 00:00:09.531 to the spacecraft, we'll label it d subscript E 00:00:09.931 --> 00:00:11.805 and then we have the distance to the moon 00:00:11.800 --> 00:00:14.651 and we know that this position, where there is zero net force 00:00:14.650 --> 00:00:16.240 of gravity on the spacecraft, 00:00:16.240 --> 00:00:18.422 has to be closer to the moon than it is to the Earth 00:00:18.491 --> 00:00:20.457 because the moon is so much less massive; 00:00:20.450 --> 00:00:25.508 we have to get closer to it for its force to get up to the point where it can match 00:00:25.771 --> 00:00:28.674 the force exerted by this mass of Earth. 00:00:30.400 --> 00:00:32.491 Now, to reduce the number of variables, 00:00:32.490 --> 00:00:34.308 we can use this 00:00:34.457 --> 00:00:37.325 value for the distance between the Earth and moon center's, 00:00:37.320 --> 00:00:39.531 384 times 10 to the 6 meters 00:00:39.645 --> 00:00:44.525 to express this d subscript M, in terms of d E. 00:00:45.954 --> 00:00:47.611 So the distance to the moon 00:00:47.610 --> 00:00:50.982 is the total distance, which we know, minus d E. 00:00:51.108 --> 00:00:52.662 And then when we read our 00:00:52.800 --> 00:00:54.994 gravity due to the Earth equals 00:00:54.990 --> 00:00:56.594 gravity due to the moon formula, 00:00:56.800 --> 00:00:59.440 we can substitute for d M 00:00:59.440 --> 00:01:01.977 and write d minus d E in its place. 00:01:02.605 --> 00:01:06.342 And then we have an equation with only one unknown, which we can solve. 00:01:07.120 --> 00:01:09.702 So this is gravitational constant times 00:01:09.702 --> 00:01:11.700 mass of the Earth times mass of the spacecraft 00:01:11.908 --> 00:01:14.720 divided by distance to the spacecraft from the Earth squared 00:01:15.051 --> 00:01:18.262 equals the force of gravity exerted by the moon, 00:01:18.537 --> 00:01:21.417 G mass of the moon times mass of the spacecraft 00:01:21.410 --> 00:01:23.108 divided by the distance to the moon squared. 00:01:23.268 --> 00:01:27.062 And the G's cancel and so does the mass of the spacecraft 00:01:27.462 --> 00:01:30.788 and so this position is 00:01:30.780 --> 00:01:33.977 does not depend on the spacecraft's mass, as it turns out. 00:01:35.005 --> 00:01:39.074 And, we have also made a substitution here for that 00:01:39.070 --> 00:01:43.268 distance to the moon in terms of the distance from the Earth. 00:01:44.160 --> 00:01:47.405 So, we'll take the reciprocal of both sides by 00:01:47.400 --> 00:01:49.325 raising both sides to the exponent negative 1 00:01:49.320 --> 00:01:51.405 to get our unknown in the numerator which is a 00:01:51.400 --> 00:01:55.142 easier place to work on it, with unknowns in the numerator. 00:01:55.497 --> 00:01:57.508 And then we'll take the square root of both sides 00:01:57.794 --> 00:01:59.634 to get rid of that square there. 00:01:59.862 --> 00:02:03.314 And so we have d E over square root of mass of the Earth equals 00:02:03.310 --> 00:02:06.925 d minus d E divided by square root mass of the moon. 00:02:07.211 --> 00:02:13.851 And then, we get rid of the denominators here by multiplying 00:02:14.068 --> 00:02:16.994 by square root mass of the moon 00:02:17.154 --> 00:02:20.685 and also multiplying by square root mass of the Earth both sides 00:02:26.022 --> 00:02:29.634 and mass of the Earth cancels there leaving us with distance to the Earth 00:02:29.630 --> 00:02:33.234 times square root mass of the moon equals 00:02:33.348 --> 00:02:38.125 d times mass of the Earth minus d to the Earth 00:02:38.125 --> 00:02:40.120 times square root mass of the Earth. 00:02:40.342 --> 00:02:44.582 And we'll put both the d E terms on the same side 00:02:44.685 --> 00:02:47.382 and when we move this to the left, it becomes positive 00:02:47.520 --> 00:02:49.382 and so we can factor out this common factor, 00:02:49.380 --> 00:02:52.320 distance to the Earth multiplied by square root mass of the moon 00:02:52.320 --> 00:02:53.691 plus square root mass of the Earth 00:02:53.862 --> 00:02:56.948 equals total separation between the Earth and the moon 00:02:56.940 --> 00:02:58.491 times square root mass of the Earth 00:02:58.662 --> 00:03:01.097 and then divide both sides by this bracket here 00:03:01.257 --> 00:03:03.714 to get that the distance to the Earth 00:03:04.262 --> 00:03:07.245 from this position where there's zero force of gravity 00:03:07.531 --> 00:03:10.320 is separation between the Earth and the moon 00:03:10.320 --> 00:03:12.685 times square root mass of the Earth divided by 00:03:12.680 --> 00:03:15.222 the sum of the square roots of the masses of the Earth and the moon. 00:03:15.771 --> 00:03:18.171 And on the calculator, it looks like this. 00:03:19.142 --> 00:03:21.897 So, we have 384 times 10 to the 6 meters 00:03:22.022 --> 00:03:25.942 times square root 5.98 times 10 to the 24 kilograms—mass of the Earth— 00:03:26.274 --> 00:03:30.868 divided by square root of 7.35 times 10 to the 22 kilograms—mass of the moon— 00:03:30.860 --> 00:03:33.097 plus square root mass of the Earth 00:03:33.090 --> 00:03:36.834 and we get 3.46 times 10 to the 8 meters 00:03:36.830 --> 00:03:39.851 is the distance from the Earth, Earth's center I should say, 00:03:39.977 --> 00:03:44.194 to the point where there is zero net force due to gravity.