1 00:00:03,777 --> 00:00:05,860 We would now like to compare the moment of inertia 2 00:00:05,860 --> 00:00:06,820 for a rigid body. 3 00:00:06,820 --> 00:00:11,950 Let's take an arbitrary rigid body about the center of mass. 4 00:00:11,950 --> 00:00:16,770 So let's say the rigid body is rotating about this axis. 5 00:00:16,770 --> 00:00:18,940 And what we'd like to compare that is 6 00:00:18,940 --> 00:00:22,990 to the moment of inertia, say, about a parallel axis that's 7 00:00:22,990 --> 00:00:25,270 also going through the rigid body. 8 00:00:25,270 --> 00:00:28,720 Now, let's recall how we define moment of inertia. 9 00:00:28,720 --> 00:00:31,750 We first choose a mass element dm. 10 00:00:31,750 --> 00:00:33,670 What I'd like to show is if this object 11 00:00:33,670 --> 00:00:37,990 is rotating about this axis, then what is that object doing. 12 00:00:37,990 --> 00:00:40,150 Well, that object is undergoing a little bit 13 00:00:40,150 --> 00:00:41,650 of circular motion. 14 00:00:41,650 --> 00:00:45,940 And this distance here is what we call the perpendicular 15 00:00:45,940 --> 00:00:49,250 distance about that axis. 16 00:00:49,250 --> 00:00:53,590 And let's indicate this is for our element dm. 17 00:00:53,590 --> 00:00:56,260 So this perpendicular distance is 18 00:00:56,260 --> 00:01:00,580 what shows up in our definition for the center of mass moment 19 00:01:00,580 --> 00:01:05,980 of inertia about that axis-- it's the interval of dm r cm 20 00:01:05,980 --> 00:01:07,030 perp. 21 00:01:07,030 --> 00:01:09,594 Now again, quantity squared. 22 00:01:09,594 --> 00:01:10,510 What is this distance? 23 00:01:10,510 --> 00:01:13,570 This is the perpendicular distance from our dm 24 00:01:13,570 --> 00:01:15,850 and to the axis of rotation. 25 00:01:15,850 --> 00:01:18,160 Imagine it's doing a circle and that's 26 00:01:18,160 --> 00:01:20,480 the radius of that circle. 27 00:01:20,480 --> 00:01:23,230 So if we were to calculate the moment of inertia 28 00:01:23,230 --> 00:01:26,860 about another axis, then about this 29 00:01:26,860 --> 00:01:31,430 axis the perpendicular distance here 30 00:01:31,430 --> 00:01:35,332 that I'll write as rs perp. 31 00:01:35,332 --> 00:01:37,539 And you can see these perpendicular distances are not 32 00:01:37,539 --> 00:01:38,660 the same. 33 00:01:38,660 --> 00:01:40,390 And the moment of inertia about that 34 00:01:40,390 --> 00:01:48,190 other axis is equal to the integral of dm rs perp quantity 35 00:01:48,190 --> 00:01:49,700 squared. 36 00:01:49,700 --> 00:01:54,759 Now, how do we relate those perpendicular distances? 37 00:01:54,759 --> 00:01:56,830 Well, there's a couple of ways to do it. 38 00:01:56,830 --> 00:02:01,420 And notice that the distance between these axes 39 00:02:01,420 --> 00:02:03,460 is given by d. 40 00:02:03,460 --> 00:02:08,560 And I'm going to call this the distance r cm. 41 00:02:08,560 --> 00:02:15,780 Now, let's just call this-- the x direction-- I'll call that x. 42 00:02:15,780 --> 00:02:19,980 So how do I relate these distances? 43 00:02:19,980 --> 00:02:25,960 Well, d is a fixed distance. 44 00:02:25,960 --> 00:02:29,990 And you can see from my diagram that rs perp 45 00:02:29,990 --> 00:02:35,280 is equal to d plus r cm x. 46 00:02:35,280 --> 00:02:44,310 And if I square this, I get d-squared plus 2d rcm x 47 00:02:44,310 --> 00:02:48,390 plus rcm x-squared. 48 00:02:48,390 --> 00:02:51,990 And that r cm x-squared is precisely what we're 49 00:02:51,990 --> 00:02:56,800 calling perpendicular distance. 50 00:02:56,800 --> 00:03:01,290 So when I put those into my moment of inertia Is, 51 00:03:01,290 --> 00:03:16,079 I get dm times d-squared plus 2d times rcm x plus, parentheses, 52 00:03:16,079 --> 00:03:20,340 rcm perp squared. 53 00:03:20,340 --> 00:03:23,170 Now, I'll separate this into three terms. 54 00:03:23,170 --> 00:03:27,120 The first term is dm times d-squared. 55 00:03:27,120 --> 00:03:29,950 This is an integral over the body. 56 00:03:29,950 --> 00:03:34,260 The second term is 2d-- and I'm going 57 00:03:34,260 --> 00:03:36,240 to hold off on the interval, because the 2d is 58 00:03:36,240 --> 00:03:41,490 the same for every piece-- dm rcm x. 59 00:03:41,490 --> 00:03:49,876 And the third piece is integral over the body of dm 60 00:03:49,876 --> 00:03:54,880 r-- since rcm x is the r perp, I'll 61 00:03:54,880 --> 00:03:57,600 write it as r perp squared. 62 00:03:57,600 --> 00:04:00,690 And you can see that this term is precisely 63 00:04:00,690 --> 00:04:03,930 the moment of inertia about the center of mass. 64 00:04:03,930 --> 00:04:06,000 Now, what I'd like to focus on is 65 00:04:06,000 --> 00:04:11,280 this terribly, in particular, dm rcm 66 00:04:11,280 --> 00:04:14,280 x that appears in our integral expression. 67 00:04:14,280 --> 00:04:16,740 Recall, that we define center of mass. 68 00:04:16,740 --> 00:04:24,330 We had the condition that the sum of mj rcmj was 0. 69 00:04:24,330 --> 00:04:27,120 Now for an integral relationship, 70 00:04:27,120 --> 00:04:34,170 this is dm rcmj cm equal to 0. 71 00:04:34,170 --> 00:04:37,409 So when you sum up the position of every object with respect 72 00:04:37,409 --> 00:04:41,590 to the vector from the center of mass to your dm element, 0. 73 00:04:41,590 --> 00:04:44,220 What does this say in terms of components? 74 00:04:44,220 --> 00:04:47,040 In terms of components, each component 75 00:04:47,040 --> 00:04:51,780 separately vanishes so we have the condition that cm x is 0. 76 00:04:51,780 --> 00:04:57,530 So that term is 0, which is precisely this term-- that's 0. 77 00:04:57,530 --> 00:05:01,710 And so we can conclude that Is-- now 78 00:05:01,710 --> 00:05:06,460 in this term, where d is the same piece for every object-- 79 00:05:06,460 --> 00:05:09,130 so we're just pulling out the total mass. 80 00:05:09,130 --> 00:05:12,250 So it's m total d-squared. 81 00:05:12,250 --> 00:05:14,880 And let's remind ourselves that d 82 00:05:14,880 --> 00:05:17,760 is the distance between the two parallel axes 83 00:05:17,760 --> 00:05:22,710 plus the moment of inertia about the center of mass. 84 00:05:22,710 --> 00:05:29,660 And this result is called the parallel axis theorem.