1 00:00:03,555 --> 00:00:05,680 I would now like to calculate the moment of inertia 2 00:00:05,680 --> 00:00:09,050 for a very thin disk. 3 00:00:09,050 --> 00:00:11,950 So we have a thin disk. 4 00:00:11,950 --> 00:00:14,920 And the radius of that disk is r. 5 00:00:14,920 --> 00:00:17,270 And it has a mass m. 6 00:00:17,270 --> 00:00:19,450 And I would like to calculate the moment of inertia 7 00:00:19,450 --> 00:00:20,830 for this disk. 8 00:00:20,830 --> 00:00:22,960 Now, let's just remind what point 9 00:00:22,960 --> 00:00:26,380 we're calculating it about, about the center of mass. 10 00:00:26,380 --> 00:00:28,720 So our definition of moment of inertia 11 00:00:28,720 --> 00:00:32,057 was take a small element, mass element to the disk. 12 00:00:32,057 --> 00:00:34,390 In fact, we're going to see it doesn't have to be small. 13 00:00:34,390 --> 00:00:37,390 Take a mass element to the disk that's useful, 14 00:00:37,390 --> 00:00:40,900 and multiply it by the perpendicular distance squared 15 00:00:40,900 --> 00:00:43,240 from the point we're calculating it. 16 00:00:43,240 --> 00:00:48,940 So the way I'll do it is I will choose a ring. 17 00:00:48,940 --> 00:00:53,860 I'm gonna choose a ring of radius r. 18 00:00:53,860 --> 00:00:58,180 And now I'll make the ring a certain thickness. 19 00:00:58,180 --> 00:01:02,830 And this thickness is dr. 20 00:01:02,830 --> 00:01:04,930 Now, in this calculation, we're going 21 00:01:04,930 --> 00:01:09,100 to take a limit as dr goes to zero. 22 00:01:09,100 --> 00:01:12,400 So even though the ring has some finite thickness, 23 00:01:12,400 --> 00:01:14,650 its radius-- we'll eventually treat 24 00:01:14,650 --> 00:01:16,560 treated as all of the mass element 25 00:01:16,560 --> 00:01:18,700 a distance r from the center. 26 00:01:18,700 --> 00:01:23,170 So r will be our integration variable. 27 00:01:23,170 --> 00:01:27,460 And that will be equal to rcm, what we're calling 28 00:01:27,460 --> 00:01:30,430 rcm in the abstract result. 29 00:01:30,430 --> 00:01:33,140 Now, the dm is the tricky part. 30 00:01:33,140 --> 00:01:40,720 So what is the mass that's contained in this area disk 31 00:01:40,720 --> 00:01:43,300 of radius r and thickness dr? 32 00:01:43,300 --> 00:01:47,080 Well, one way to think about that is it's-- here we 33 00:01:47,080 --> 00:01:53,380 didn't say this, but our disk is going to be uniform. 34 00:01:53,380 --> 00:01:56,860 And so we can describe the mass per unit area 35 00:01:56,860 --> 00:02:02,920 as the total mass divided by the area of the whole disk. 36 00:02:02,920 --> 00:02:07,000 And then we can say that the mass in that ring 37 00:02:07,000 --> 00:02:11,890 is equal to sigma mass per area times 38 00:02:11,890 --> 00:02:20,470 the area of the outer ring minus the area of the inner ring. 39 00:02:20,470 --> 00:02:26,980 Now, when we expand this out, dm, m over pi r squared, 40 00:02:26,980 --> 00:02:32,890 we get pi r squared plus 2rdr plus dr quantity 41 00:02:32,890 --> 00:02:36,280 squared minus pi r squared. 42 00:02:36,280 --> 00:02:39,440 And you can see those terms cancel. 43 00:02:39,440 --> 00:02:43,300 And so what I get is m times pi r squared. 44 00:02:43,300 --> 00:02:47,860 And in here I have 2 pi r dr. Now, 45 00:02:47,860 --> 00:02:51,760 this is only order dr, plus a second term 46 00:02:51,760 --> 00:02:55,300 that goes like pi dr squared. 47 00:02:55,300 --> 00:03:00,310 And so, when I take this limit as dr goes to 0, 48 00:03:00,310 --> 00:03:03,760 this term is much, much smaller than that term. 49 00:03:03,760 --> 00:03:08,770 And so I can say my mass element is m pi r squared times 50 00:03:08,770 --> 00:03:11,740 2 pi r dr. 51 00:03:11,740 --> 00:03:15,730 Now, let's think about this term, why it makes sense. 52 00:03:15,730 --> 00:03:22,329 When we're shrinking our ring, so taking a limit as dr 53 00:03:22,329 --> 00:03:27,100 goes to 0, and the ring just becomes an extremely thin ring 54 00:03:27,100 --> 00:03:33,610 at radius r, then this piece is a circumference, 55 00:03:33,610 --> 00:03:37,700 and this piece is just the width. 56 00:03:37,700 --> 00:03:42,040 And so it's no surprise that area is 2 pi r 57 00:03:42,040 --> 00:03:45,670 times d pi r dr in the limit. 58 00:03:45,670 --> 00:03:49,990 And now that enables us to write the moment of inertia 59 00:03:49,990 --> 00:03:53,950 about the center of mass, icm. 60 00:03:53,950 --> 00:03:58,301 Let's pull out these constants, m pi r squared. 61 00:03:58,301 --> 00:03:59,800 Now we're integrating over the body. 62 00:03:59,800 --> 00:04:01,960 Let's hold off on the limits for the moment, 63 00:04:01,960 --> 00:04:03,370 and put our values for dm. 64 00:04:03,370 --> 00:04:10,610 That's 2 pi r dr. And we have our distance squared, 65 00:04:10,610 --> 00:04:13,990 which was, again, the radius of r squared. 66 00:04:13,990 --> 00:04:16,160 And so the pis will cancel. 67 00:04:16,160 --> 00:04:22,840 I have 2m over r-squared times the integral of r cubed dr. 68 00:04:22,840 --> 00:04:25,420 Now, we're supposedly integrating over the body, 69 00:04:25,420 --> 00:04:29,030 but what does that body integral actually mean? 70 00:04:29,030 --> 00:04:33,820 Well, what we're doing is we're taking a series of rings 71 00:04:33,820 --> 00:04:38,890 and adding them up as we go from the origin 72 00:04:38,890 --> 00:04:41,620 out to the radius of the whole disk. 73 00:04:41,620 --> 00:04:44,380 So the limits of our body integral with respect 74 00:04:44,380 --> 00:04:47,860 to our integration variable, we start with rings 75 00:04:47,860 --> 00:04:51,230 that essentially have no width. 76 00:04:51,230 --> 00:04:53,500 And we're integrating these, we're adding up 77 00:04:53,500 --> 00:04:56,050 the contribution of every ring until we 78 00:04:56,050 --> 00:04:58,330 get to rings of radius r. 79 00:04:58,330 --> 00:05:02,080 And our integration variable, r cubed, dr. 80 00:05:02,080 --> 00:05:04,970 Now, this is an integral that's easy to do. 81 00:05:04,970 --> 00:05:10,930 That's r to the forth over 4 between 0 and r equals r. 82 00:05:10,930 --> 00:05:14,020 And when we put that in, the 2 cancels the 4. 83 00:05:14,020 --> 00:05:16,600 And oh, the pi we lost. 84 00:05:16,600 --> 00:05:19,900 So let's make sure this pi should be in m. 85 00:05:19,900 --> 00:05:26,410 So we have the 2 over the 4 is one half, and r squared. 86 00:05:26,410 --> 00:05:29,080 And that is the moment of inertia of it 87 00:05:29,080 --> 00:05:32,620 does about an axis passing through the center 88 00:05:32,620 --> 00:05:35,970 perpendicular to the plane of the disk.