1 00:00:05,420 --> 00:00:08,960 The little prince is sitting on his little planet, 2 00:00:08,960 --> 00:00:11,840 and he's watching the planets go by. 3 00:00:11,840 --> 00:00:14,690 And so suddenly, he's seeing three of them. 4 00:00:14,690 --> 00:00:17,160 One, two, three. 5 00:00:17,160 --> 00:00:20,330 And he wonders, hmm, what would the center of mass 6 00:00:20,330 --> 00:00:23,180 of these three planets be? 7 00:00:23,180 --> 00:00:26,060 So let's calculate it. 8 00:00:26,060 --> 00:00:28,890 We have one planet here that's three times 9 00:00:28,890 --> 00:00:31,030 the mass of this guy. 10 00:00:31,030 --> 00:00:35,390 And then this one has half of m1. 11 00:00:35,390 --> 00:00:38,030 And I've written this up there already. 12 00:00:38,030 --> 00:00:40,070 And the center of mass, when we want 13 00:00:40,070 --> 00:00:43,900 to determine its coordinate-- it is a coordinate 14 00:00:43,900 --> 00:00:46,610 that depends on this coordinate system, 15 00:00:46,610 --> 00:00:51,430 and this origin here-- then we need 16 00:00:51,430 --> 00:00:53,430 to have the total mass of the system, 17 00:00:53,430 --> 00:00:57,160 because it's a mass-weighted coordinate. 18 00:00:57,160 --> 00:01:00,680 And so the total mass is going to be 4.5 m1. 19 00:01:00,680 --> 00:01:06,860 And if we want to calculate this position 20 00:01:06,860 --> 00:01:10,150 function of the center of mass, Rcm, 21 00:01:10,150 --> 00:01:13,720 that is the mass weight here. 22 00:01:13,720 --> 00:01:19,810 And then we need the sum of all of our masses 23 00:01:19,810 --> 00:01:23,470 times our distances. 24 00:01:23,470 --> 00:01:28,390 We're going to sum here over j from 1 to n. 25 00:01:28,390 --> 00:01:30,520 And what that means is we need to now write out 26 00:01:30,520 --> 00:01:33,690 the sum for our three planets. 27 00:01:33,690 --> 00:01:35,800 And we need to give this a radius here, 28 00:01:35,800 --> 00:01:40,020 so the r1 would be going from here to here. 29 00:01:40,020 --> 00:01:42,020 r2 goes from here to there. 30 00:01:42,020 --> 00:01:44,270 And r3 from here to here. 31 00:01:44,270 --> 00:01:48,150 You need to write them out in the i hat direction 32 00:01:48,150 --> 00:01:49,335 and in the j hat direction. 33 00:01:52,810 --> 00:01:55,160 Let's add that here. 34 00:01:55,160 --> 00:01:59,620 And then sum it all up and calculate our r. 35 00:01:59,620 --> 00:02:02,040 So let's write this out. 36 00:02:02,040 --> 00:02:08,139 We have m1, and we're going to have 35 and 10. 37 00:02:08,139 --> 00:02:15,870 35 i hat plus 10 j hat plus m2. 38 00:02:15,870 --> 00:02:20,525 It's going to be 5 and 20. 39 00:02:20,525 --> 00:02:24,390 5 and 20. 40 00:02:24,390 --> 00:02:29,440 And then m3, we have 40 and 30. 41 00:02:35,670 --> 00:02:37,780 And what we can do-- oh, and of course that 42 00:02:37,780 --> 00:02:44,190 has to be divided by my system mass. 43 00:02:44,190 --> 00:02:46,579 And what we can do is we can concentrate first 44 00:02:46,579 --> 00:02:52,910 on the x component, and then on the y component. 45 00:02:52,910 --> 00:02:56,510 Maybe we'll just continue here. 46 00:02:56,510 --> 00:03:01,760 So we're going to have-- and we can plug it in all the m's. 47 00:03:01,760 --> 00:03:04,720 We'll do it for the x component first. 48 00:03:04,720 --> 00:03:16,290 We're going to have m1, and then we have 35 i hat plus here 49 00:03:16,290 --> 00:03:19,720 we're going to have 1/2 m2. 50 00:03:19,720 --> 00:03:35,440 That is 3m1 5 plus-- and for m3, we have 0.5 m1 40. 51 00:03:35,440 --> 00:03:38,829 that's in the i hat direction. 52 00:03:38,829 --> 00:03:44,510 And we'll have to divide that over our system mass. 53 00:03:44,510 --> 00:03:51,100 And then we do the same for the y component. 54 00:03:51,100 --> 00:04:10,600 So m1, 10 plus 3. m1, 20 plus 0.5 m1 30 j hat. 55 00:04:10,600 --> 00:04:16,839 And again, we have to divide this over our system mass. 56 00:04:16,839 --> 00:04:23,030 So this boils down to-- hang on. 57 00:04:23,030 --> 00:04:24,100 Let me redo this again. 58 00:04:24,100 --> 00:04:26,050 Let me actually look at the answer first. 59 00:04:28,807 --> 00:04:30,270 What do I have here? 60 00:04:30,270 --> 00:04:34,310 70 and 85, OK. 61 00:04:34,310 --> 00:04:42,150 So this boils to m1 over the system mass. 62 00:04:42,150 --> 00:04:50,120 And we have 70 in the i hat plus 85 in the j hat direction. 63 00:04:50,120 --> 00:04:53,320 So the 70 comes from this term, the 80 comes from this. 64 00:04:53,320 --> 00:04:56,050 And I put it back together, and now we need to plug in this one 65 00:04:56,050 --> 00:04:57,500 here. 66 00:04:57,500 --> 00:05:06,220 And so we will get in the end of that Rcm equals m over 4.5 m. 67 00:05:06,220 --> 00:05:10,970 So the m goes away, and we have a factor of 1/4.5 here. 68 00:05:10,970 --> 00:05:13,160 We'll divide this through, and we're 69 00:05:13,160 --> 00:05:30,580 going to have 15.5 in the i hat direction, and 18.9 70 00:05:30,580 --> 00:05:32,148 in the j hat direction. 71 00:05:34,960 --> 00:05:39,930 All right, so let's see where this fits on our graph here. 72 00:05:39,930 --> 00:05:43,740 So 15 in the i hat is somewhere here. 73 00:05:43,740 --> 00:05:47,380 And 19 is almost 20, so it's going 74 00:05:47,380 --> 00:05:50,680 to be here, so about there. 75 00:05:54,060 --> 00:05:57,830 So this is my Rcm, and this here is 76 00:05:57,830 --> 00:06:06,090 my center of mass of the system of these three little planets. 77 00:06:06,090 --> 00:06:09,750 Of course, we used approximate math here for all the planets. 78 00:06:09,750 --> 00:06:11,770 But if we look at the real numbers, 79 00:06:11,770 --> 00:06:13,740 imagine that this would be Earth, 80 00:06:13,740 --> 00:06:18,110 and it has one Earth mass. 81 00:06:18,110 --> 00:06:21,350 And if this were Saturn, it would have something 82 00:06:21,350 --> 00:06:25,180 like 318 Earth masses. 83 00:06:25,180 --> 00:06:34,600 And if this is Pluto, it would have 0.0025 Earth masses. 84 00:06:34,600 --> 00:06:40,220 You will see that Saturn really holds all the weight. 85 00:06:40,220 --> 00:06:42,880 And if we were to do this calculation with these numbers 86 00:06:42,880 --> 00:06:50,100 here, then our Rcm would-- and keeping this coordinate system 87 00:06:50,100 --> 00:06:52,350 in the arrangement of the planets, 88 00:06:52,350 --> 00:06:55,810 then it would go right into-- if here's the center, 89 00:06:55,810 --> 00:07:02,920 it would go right next to the center right over here, 90 00:07:02,920 --> 00:07:06,720 because Saturn just weighs so, so, so much more than Pluto 91 00:07:06,720 --> 00:07:08,930 and Earth together.