1 00:00:03,830 --> 00:00:05,990 Today we'd like to explore the idea 2 00:00:05,990 --> 00:00:08,360 of the center of mass, or the center of gravity, 3 00:00:08,360 --> 00:00:09,530 of a rigid object. 4 00:00:09,530 --> 00:00:11,330 For instance, take this rod. 5 00:00:11,330 --> 00:00:13,460 And if I try to balance it on my finger 6 00:00:13,460 --> 00:00:15,410 at a certain point in that rod, I'm 7 00:00:15,410 --> 00:00:18,620 balancing it via the gravitational force. 8 00:00:18,620 --> 00:00:20,690 And this point is often referred to as the center 9 00:00:20,690 --> 00:00:22,550 of gravity of the rod. 10 00:00:22,550 --> 00:00:26,270 Now, if we were in empty space with no gravitational field, 11 00:00:26,270 --> 00:00:29,300 then center of gravity doesn't make any sense. 12 00:00:29,300 --> 00:00:31,070 But this point still coincides with what 13 00:00:31,070 --> 00:00:34,020 we call the center of mass of the object. 14 00:00:34,020 --> 00:00:37,070 And now I'd like to define center of mass. 15 00:00:37,070 --> 00:00:40,100 So let's consider our rigid body. 16 00:00:40,100 --> 00:00:43,740 And we'll just describe it as some object-- 17 00:00:43,740 --> 00:00:47,510 we'll make it idealized-- and that there's 18 00:00:47,510 --> 00:00:51,470 going to be trying to find some point in this object. 19 00:00:51,470 --> 00:00:55,070 And we'll identify that point as the center of mass. 20 00:00:55,070 --> 00:00:58,040 Now, let's imagine that this rigid body is made up 21 00:00:58,040 --> 00:01:00,240 of a bunch of little pieces. 22 00:01:00,240 --> 00:01:02,060 So we have m, j. 23 00:01:02,060 --> 00:01:05,450 And this j piece is located from the center 24 00:01:05,450 --> 00:01:08,960 of mass, a vector rcm. 25 00:01:08,960 --> 00:01:13,730 Now how we want to define this particular point 26 00:01:13,730 --> 00:01:16,490 is that when we make the sum from j 27 00:01:16,490 --> 00:01:21,500 equals 1 to n over every single point in this body, 28 00:01:21,500 --> 00:01:25,250 then that will be 0. 29 00:01:25,250 --> 00:01:31,510 And this will be the definition of the center of mass, 30 00:01:31,510 --> 00:01:36,920 that when you add up the position vector with respect 31 00:01:36,920 --> 00:01:40,130 to this point weighted by the mass, 32 00:01:40,130 --> 00:01:43,460 and you add up all of those vectors, you'll get 0. 33 00:01:43,460 --> 00:01:47,210 Now, if you don't know where the center of mass is then 34 00:01:47,210 --> 00:01:49,039 this is difficult to calculate. 35 00:01:49,039 --> 00:01:52,140 So let's find a way where you choose an arbitrary point, 36 00:01:52,140 --> 00:01:56,890 and that's write that arbitrary point, say, over here. 37 00:01:56,890 --> 00:01:59,300 We'll write it like this-- s. 38 00:01:59,300 --> 00:02:01,340 And we'll treat that as our origin. 39 00:02:01,340 --> 00:02:02,990 And I'll draw a vector, rsj. 40 00:02:05,750 --> 00:02:08,090 And here, I'll draw the vector Rcm. 41 00:02:10,639 --> 00:02:15,140 And now what we have from our vector relationship 42 00:02:15,140 --> 00:02:20,390 is that the vector rsj equals the vector Rcm-- 43 00:02:20,390 --> 00:02:24,140 and that's what I want to find-- plus the vector rcmj. 44 00:02:27,890 --> 00:02:32,870 And now let's add up-- multiply each of these by the mass 45 00:02:32,870 --> 00:02:34,370 and make a sum. 46 00:02:34,370 --> 00:02:43,310 So we have mj rsj equals the sum of mj. 47 00:02:43,310 --> 00:02:46,790 Now, the vector Rcm, this vector, 48 00:02:46,790 --> 00:02:49,710 no matter where we picked a point in this object, 49 00:02:49,710 --> 00:02:51,620 the vector's always the same. 50 00:02:51,620 --> 00:02:53,480 And that's why I pulled it out of the sum. 51 00:02:53,480 --> 00:02:55,550 And over here, I have the sum from j 52 00:02:55,550 --> 00:03:00,284 goes from 1 to n, j 1 to n, of mj rcmj. 53 00:03:03,140 --> 00:03:05,780 Now, recall, this is precisely how 54 00:03:05,780 --> 00:03:10,240 we define the center of mass point, that this is 0. 55 00:03:10,240 --> 00:03:15,620 And so we can now conclude that the center of mass-- 56 00:03:15,620 --> 00:03:18,079 so you pick an arbitrary point, and if you want 57 00:03:18,079 --> 00:03:21,110 to find that vector to the center of mass, what you do 58 00:03:21,110 --> 00:03:27,579 is you make the sum from j goes from 1 to n of mj rsj, 59 00:03:27,579 --> 00:03:33,290 and you divide that by j goes from 1 to n, mj. 60 00:03:33,290 --> 00:03:37,070 And this is what we call the center of mass. 61 00:03:37,070 --> 00:03:39,680 So conceptually, the center of mass 62 00:03:39,680 --> 00:03:41,630 is the point in the object where, 63 00:03:41,630 --> 00:03:44,810 if you take a vector to any of the little mass elements, 64 00:03:44,810 --> 00:03:47,400 and you weight it by the mass element, and you add them up, 65 00:03:47,400 --> 00:03:48,890 you get 0. 66 00:03:48,890 --> 00:03:52,400 If you wanted to calculate the center of mass about any point, 67 00:03:52,400 --> 00:03:55,520 you choose a point, s, you draw the vector from s 68 00:03:55,520 --> 00:03:56,870 to the object. 69 00:03:56,870 --> 00:03:58,130 You sum up those vectors. 70 00:03:58,130 --> 00:04:01,010 We see by our vector triangle rule 71 00:04:01,010 --> 00:04:03,710 that we can now calculate that center of mass vector 72 00:04:03,710 --> 00:04:06,630 by this equation. 73 00:04:06,630 --> 00:04:13,100 And now let's just rewrite this because this is the total mass. 74 00:04:13,100 --> 00:04:20,640 And what we see is that the Rcm equals j goes from 1 to n, 75 00:04:20,640 --> 00:04:26,770 mj rsj divided by the total mass.