1 00:00:00,060 --> 00:00:01,680 so let's say you wanted to know where 2 00:00:01,680 --> 00:00:03,840 the center of mass was between this two 3 00:00:03,840 --> 00:00:06,270 kilogram mass and the six kilogram mass 4 00:00:06,270 --> 00:00:08,580 now they're separated by 10 centimeters 5 00:00:08,580 --> 00:00:10,950 so it's somewhere in between them and we 6 00:00:10,950 --> 00:00:13,170 know it's going to be closer to the 7 00:00:13,170 --> 00:00:14,910 larger mass because the center of mass 8 00:00:14,910 --> 00:00:17,580 is always closer to the larger mass but 9 00:00:17,580 --> 00:00:20,189 exactly where is it going to be we need 10 00:00:20,189 --> 00:00:22,680 a formula to figure this out and the 11 00:00:22,680 --> 00:00:24,300 formula for the center of mass looks 12 00:00:24,300 --> 00:00:27,449 like this it says the location of the 13 00:00:27,449 --> 00:00:29,849 center of mass that's what this is this 14 00:00:29,849 --> 00:00:33,300 X cm is just the location of the center 15 00:00:33,300 --> 00:00:35,670 of mass is the position of the center of 16 00:00:35,670 --> 00:00:38,670 mass is going to equal you take all the 17 00:00:38,670 --> 00:00:40,290 masses that you're trying to find the 18 00:00:40,290 --> 00:00:42,390 center of mass between you take all 19 00:00:42,390 --> 00:00:44,640 those masses times their positions and 20 00:00:44,640 --> 00:00:47,280 you add up all of these M times X's 21 00:00:47,280 --> 00:00:49,500 until you've accounted for every single 22 00:00:49,500 --> 00:00:51,449 M times X there is in your system and 23 00:00:51,449 --> 00:00:53,610 then you just divide by all of the 24 00:00:53,610 --> 00:00:55,860 masses added together and what you get 25 00:00:55,860 --> 00:00:58,050 out of this is the location of the 26 00:00:58,050 --> 00:01:00,390 center of mass so let's use this let's 27 00:01:00,390 --> 00:01:02,129 use this for this example problem right 28 00:01:02,129 --> 00:01:03,719 here let's see what we get we'll have 29 00:01:03,719 --> 00:01:05,880 that the center of mass the position of 30 00:01:05,880 --> 00:01:08,100 the center of mass is going to be equal 31 00:01:08,100 --> 00:01:11,280 to all right so I'll take M 1 which you 32 00:01:11,280 --> 00:01:13,200 could take either one as M 1 but I 33 00:01:13,200 --> 00:01:14,760 already colored this one red so we'll 34 00:01:14,760 --> 00:01:17,580 just say the 2 kilogram mass is M 1 and 35 00:01:17,580 --> 00:01:20,400 we're gonna have to multiply by X 1 the 36 00:01:20,400 --> 00:01:23,009 position of mass 1 and at this point you 37 00:01:23,009 --> 00:01:24,960 might be confused you might be like the 38 00:01:24,960 --> 00:01:27,119 position I don't know what the position 39 00:01:27,119 --> 00:01:29,159 is there's no coordinate system up here 40 00:01:29,159 --> 00:01:31,740 well you get to pick so you get to 41 00:01:31,740 --> 00:01:33,450 decide where you're measuring these 42 00:01:33,450 --> 00:01:35,820 positions from and wherever you decide 43 00:01:35,820 --> 00:01:37,950 to measure them from will also be the 44 00:01:37,950 --> 00:01:39,689 point where the center of mass is 45 00:01:39,689 --> 00:01:41,939 measured from in other words you get to 46 00:01:41,939 --> 00:01:44,670 choose where x equals 0 let's just say 47 00:01:44,670 --> 00:01:46,380 for the sake of argument the left-hand 48 00:01:46,380 --> 00:01:48,930 side over here is x equals 0 let's say 49 00:01:48,930 --> 00:01:51,990 right here is x equals 0 on our number 50 00:01:51,990 --> 00:01:53,490 line and then it goes this way it's 51 00:01:53,490 --> 00:01:55,439 positive this way so if this is x equals 52 00:01:55,439 --> 00:01:58,770 0 half way would be x equals 5 and then 53 00:01:58,770 --> 00:02:01,079 over here would be x equals 10 we're 54 00:02:01,079 --> 00:02:03,210 free to choose that in fact it's kind of 55 00:02:03,210 --> 00:02:05,670 cool because if this is x equals 0 the 56 00:02:05,670 --> 00:02:08,940 position of mass 1 is 0 meters so it's 57 00:02:08,940 --> 00:02:10,679 going to be this term is just going to 58 00:02:10,679 --> 00:02:12,900 go away which is okay we're going to add 59 00:02:12,900 --> 00:02:13,830 to that 60 00:02:13,830 --> 00:02:16,710 - which is six kilograms times the 61 00:02:16,710 --> 00:02:18,900 position of M - again we can choose 62 00:02:18,900 --> 00:02:20,520 whatever point we want but we have to be 63 00:02:20,520 --> 00:02:22,920 consistent we already chose this is x 64 00:02:22,920 --> 00:02:25,320 equals zero for mass one so that still 65 00:02:25,320 --> 00:02:27,630 has to be x equals zero for mass two 66 00:02:27,630 --> 00:02:29,940 that means this has to be 10 centimeters 67 00:02:29,940 --> 00:02:31,590 now and then those are our only two 68 00:02:31,590 --> 00:02:33,150 masses so we stop there and we just 69 00:02:33,150 --> 00:02:35,400 divide by all the masses added together 70 00:02:35,400 --> 00:02:37,860 which is going to be two kilograms for 71 00:02:37,860 --> 00:02:42,000 m1 plus six kilograms for m2 and what we 72 00:02:42,000 --> 00:02:44,550 get out of this is two times zero zero 73 00:02:44,550 --> 00:02:48,120 plus 6 times 10 is 60 kilograms 74 00:02:48,120 --> 00:02:51,420 centimeters divided by 2 plus 6 is going 75 00:02:51,420 --> 00:02:54,810 to be 8 kilograms which gives us seven 76 00:02:54,810 --> 00:02:57,480 point five centimeters so it's going to 77 00:02:57,480 --> 00:02:59,910 be seven point five centimeters from the 78 00:02:59,910 --> 00:03:02,460 point we called x equals zero which is 79 00:03:02,460 --> 00:03:05,040 right here that's the location of the 80 00:03:05,040 --> 00:03:06,960 center of mass so in other words if you 81 00:03:06,960 --> 00:03:09,540 connect these two spheres by a rod a 82 00:03:09,540 --> 00:03:12,210 light rod and you put a pivot right here 83 00:03:12,210 --> 00:03:14,340 they would balance at that point right 84 00:03:14,340 --> 00:03:16,200 there and just to show you you might be 85 00:03:16,200 --> 00:03:18,360 like wait we can choose any point as x 86 00:03:18,360 --> 00:03:20,250 equals zero won't we get a different 87 00:03:20,250 --> 00:03:22,530 number you will so let's say you did 88 00:03:22,530 --> 00:03:24,239 this so instead of instead of picking 89 00:03:24,239 --> 00:03:26,489 that as x equals zero let's say we pick 90 00:03:26,489 --> 00:03:28,890 this side as x equals zero let's say we 91 00:03:28,890 --> 00:03:32,310 say x equals zero is this six kilogram 92 00:03:32,310 --> 00:03:34,410 masses position what are we going to get 93 00:03:34,410 --> 00:03:36,570 then we'll get that the location of the 94 00:03:36,570 --> 00:03:38,730 center of mass for this calculation is 95 00:03:38,730 --> 00:03:40,110 going to be well we'll have two 96 00:03:40,110 --> 00:03:42,270 kilograms but now the location of the 97 00:03:42,270 --> 00:03:44,940 two kilogram mass is not zero it's going 98 00:03:44,940 --> 00:03:46,530 to be if this is zero and we're 99 00:03:46,530 --> 00:03:48,450 considering this way is positive it's 100 00:03:48,450 --> 00:03:50,400 going to be negative 10 centimeters 101 00:03:50,400 --> 00:03:51,959 because it's 10 centimeters to the left 102 00:03:51,959 --> 00:03:53,580 so there's going to be negative 10 103 00:03:53,580 --> 00:03:57,480 centimeters plus 6 kilograms times now 104 00:03:57,480 --> 00:03:59,550 the location of the six kilogram mass is 105 00:03:59,550 --> 00:04:02,010 zero using this convention and we divide 106 00:04:02,010 --> 00:04:03,900 by both of the masses added up so that's 107 00:04:03,900 --> 00:04:06,540 still two kilograms plus six kilograms 108 00:04:06,540 --> 00:04:08,370 and what are we going to get we're going 109 00:04:08,370 --> 00:04:10,890 to get two times negative 10 plus 6 110 00:04:10,890 --> 00:04:12,720 times zero well that's just zero so 111 00:04:12,720 --> 00:04:14,970 there's going to be negative 20 kilogram 112 00:04:14,970 --> 00:04:17,540 centimeters divided by eight kilograms 113 00:04:17,540 --> 00:04:20,640 gives us negative two point five 114 00:04:20,640 --> 00:04:22,470 centimeters so you might be worried you 115 00:04:22,470 --> 00:04:23,820 might be like what we got a different 116 00:04:23,820 --> 00:04:26,510 answer the location can't change Bay 117 00:04:26,510 --> 00:04:28,610 dawn where we're measuring from and it 118 00:04:28,610 --> 00:04:30,290 didn't change it's still in the exact 119 00:04:30,290 --> 00:04:32,660 same position because now this negative 120 00:04:32,660 --> 00:04:35,600 2.5 centimeters is measured relative to 121 00:04:35,600 --> 00:04:38,720 this x equals 0 so what's negative 2.5 122 00:04:38,720 --> 00:04:40,610 centimeters from here it's 2.5 123 00:04:40,610 --> 00:04:43,730 centimeters to the left which lo and 124 00:04:43,730 --> 00:04:46,070 behold is exactly at the same point 125 00:04:46,070 --> 00:04:48,500 since this was seven point five and this 126 00:04:48,500 --> 00:04:50,690 is negative two point five and the whole 127 00:04:50,690 --> 00:04:52,220 thing is ten centimeters it gives you 128 00:04:52,220 --> 00:04:53,930 the exact same location for the center 129 00:04:53,930 --> 00:04:57,230 of mass it has to it can't change based 130 00:04:57,230 --> 00:04:59,030 on whether you're calling this point 131 00:04:59,030 --> 00:05:01,460 zero or this point zero but you have to 132 00:05:01,460 --> 00:05:03,260 be careful and consistent with your 133 00:05:03,260 --> 00:05:05,510 choice any choice will work but you have 134 00:05:05,510 --> 00:05:07,400 to be consistent with it and you have to 135 00:05:07,400 --> 00:05:09,230 know at the end where is this answer 136 00:05:09,230 --> 00:05:11,480 measured from otherwise you won't be 137 00:05:11,480 --> 00:05:13,370 able to interpret what this number means 138 00:05:13,370 --> 00:05:15,620 at the end so recapping you can use the 139 00:05:15,620 --> 00:05:17,900 center of mass formula to find the exact 140 00:05:17,900 --> 00:05:19,880 location of the center of mass between a 141 00:05:19,880 --> 00:05:22,400 system of objects you add all the masses 142 00:05:22,400 --> 00:05:24,740 times their positions and divide by the 143 00:05:24,740 --> 00:05:27,290 total mass the position can be measured 144 00:05:27,290 --> 00:05:29,630 relative to any point you call x equals 145 00:05:29,630 --> 00:05:31,670 zero and the number you get out of that 146 00:05:31,670 --> 00:05:34,160 calculation will be the distance from x 147 00:05:34,160 --> 00:05:37,070 equals zero to the center of mass of 148 00:05:37,070 --> 00:00:00,000 that system