WEBVTT 00:00:03.555 --> 00:00:05.680 I would now like to calculate the moment of inertia 00:00:05.680 --> 00:00:09.050 for a very thin disk. 00:00:09.050 --> 00:00:11.950 So we have a thin disk. 00:00:11.950 --> 00:00:14.920 And the radius of that disk is r. 00:00:14.920 --> 00:00:17.270 And it has a mass m. 00:00:17.270 --> 00:00:19.450 And I would like to calculate the moment of inertia 00:00:19.450 --> 00:00:20.830 for this disk. 00:00:20.830 --> 00:00:22.960 Now, let's just remind what point 00:00:22.960 --> 00:00:26.380 we're calculating it about, about the center of mass. 00:00:26.380 --> 00:00:28.720 So our definition of moment of inertia 00:00:28.720 --> 00:00:32.057 was take a small element, mass element to the disk. 00:00:32.057 --> 00:00:34.390 In fact, we're going to see it doesn't have to be small. 00:00:34.390 --> 00:00:37.390 Take a mass element to the disk that's useful, 00:00:37.390 --> 00:00:40.900 and multiply it by the perpendicular distance squared 00:00:40.900 --> 00:00:43.240 from the point we're calculating it. 00:00:43.240 --> 00:00:48.940 So the way I'll do it is I will choose a ring. 00:00:48.940 --> 00:00:53.860 I'm gonna choose a ring of radius r. 00:00:53.860 --> 00:00:58.180 And now I'll make the ring a certain thickness. 00:00:58.180 --> 00:01:02.830 And this thickness is dr. 00:01:02.830 --> 00:01:04.930 Now, in this calculation, we're going 00:01:04.930 --> 00:01:09.100 to take a limit as dr goes to zero. 00:01:09.100 --> 00:01:12.400 So even though the ring has some finite thickness, 00:01:12.400 --> 00:01:14.650 its radius-- we'll eventually treat 00:01:14.650 --> 00:01:16.560 treated as all of the mass element 00:01:16.560 --> 00:01:18.700 a distance r from the center. 00:01:18.700 --> 00:01:23.170 So r will be our integration variable. 00:01:23.170 --> 00:01:27.460 And that will be equal to rcm, what we're calling 00:01:27.460 --> 00:01:30.430 rcm in the abstract result. 00:01:30.430 --> 00:01:33.140 Now, the dm is the tricky part. 00:01:33.140 --> 00:01:40.720 So what is the mass that's contained in this area disk 00:01:40.720 --> 00:01:43.300 of radius r and thickness dr? 00:01:43.300 --> 00:01:47.080 Well, one way to think about that is it's-- here we 00:01:47.080 --> 00:01:53.380 didn't say this, but our disk is going to be uniform. 00:01:53.380 --> 00:01:56.860 And so we can describe the mass per unit area 00:01:56.860 --> 00:02:02.920 as the total mass divided by the area of the whole disk. 00:02:02.920 --> 00:02:07.000 And then we can say that the mass in that ring 00:02:07.000 --> 00:02:11.890 is equal to sigma mass per area times 00:02:11.890 --> 00:02:20.470 the area of the outer ring minus the area of the inner ring. 00:02:20.470 --> 00:02:26.980 Now, when we expand this out, dm, m over pi r squared, 00:02:26.980 --> 00:02:32.890 we get pi r squared plus 2rdr plus dr quantity 00:02:32.890 --> 00:02:36.280 squared minus pi r squared. 00:02:36.280 --> 00:02:39.440 And you can see those terms cancel. 00:02:39.440 --> 00:02:43.300 And so what I get is m times pi r squared. 00:02:43.300 --> 00:02:47.860 And in here I have 2 pi r dr. Now, 00:02:47.860 --> 00:02:51.760 this is only order dr, plus a second term 00:02:51.760 --> 00:02:55.300 that goes like pi dr squared. 00:02:55.300 --> 00:03:00.310 And so, when I take this limit as dr goes to 0, 00:03:00.310 --> 00:03:03.760 this term is much, much smaller than that term. 00:03:03.760 --> 00:03:08.770 And so I can say my mass element is m pi r squared times 00:03:08.770 --> 00:03:11.740 2 pi r dr. 00:03:11.740 --> 00:03:15.730 Now, let's think about this term, why it makes sense. 00:03:15.730 --> 00:03:22.329 When we're shrinking our ring, so taking a limit as dr 00:03:22.329 --> 00:03:27.100 goes to 0, and the ring just becomes an extremely thin ring 00:03:27.100 --> 00:03:33.610 at radius r, then this piece is a circumference, 00:03:33.610 --> 00:03:37.700 and this piece is just the width. 00:03:37.700 --> 00:03:42.040 And so it's no surprise that area is 2 pi r 00:03:42.040 --> 00:03:45.670 times d pi r dr in the limit. 00:03:45.670 --> 00:03:49.990 And now that enables us to write the moment of inertia 00:03:49.990 --> 00:03:53.950 about the center of mass, icm. 00:03:53.950 --> 00:03:58.301 Let's pull out these constants, m pi r squared. 00:03:58.301 --> 00:03:59.800 Now we're integrating over the body. 00:03:59.800 --> 00:04:01.960 Let's hold off on the limits for the moment, 00:04:01.960 --> 00:04:03.370 and put our values for dm. 00:04:03.370 --> 00:04:10.610 That's 2 pi r dr. And we have our distance squared, 00:04:10.610 --> 00:04:13.990 which was, again, the radius of r squared. 00:04:13.990 --> 00:04:16.160 And so the pis will cancel. 00:04:16.160 --> 00:04:22.840 I have 2m over r-squared times the integral of r cubed dr. 00:04:22.840 --> 00:04:25.420 Now, we're supposedly integrating over the body, 00:04:25.420 --> 00:04:29.030 but what does that body integral actually mean? 00:04:29.030 --> 00:04:33.820 Well, what we're doing is we're taking a series of rings 00:04:33.820 --> 00:04:38.890 and adding them up as we go from the origin 00:04:38.890 --> 00:04:41.620 out to the radius of the whole disk. 00:04:41.620 --> 00:04:44.380 So the limits of our body integral with respect 00:04:44.380 --> 00:04:47.860 to our integration variable, we start with rings 00:04:47.860 --> 00:04:51.230 that essentially have no width. 00:04:51.230 --> 00:04:53.500 And we're integrating these, we're adding up 00:04:53.500 --> 00:04:56.050 the contribution of every ring until we 00:04:56.050 --> 00:04:58.330 get to rings of radius r. 00:04:58.330 --> 00:05:02.080 And our integration variable, r cubed, dr. 00:05:02.080 --> 00:05:04.970 Now, this is an integral that's easy to do. 00:05:04.970 --> 00:05:10.930 That's r to the forth over 4 between 0 and r equals r. 00:05:10.930 --> 00:05:14.020 And when we put that in, the 2 cancels the 4. 00:05:14.020 --> 00:05:16.600 And oh, the pi we lost. 00:05:16.600 --> 00:05:19.900 So let's make sure this pi should be in m. 00:05:19.900 --> 00:05:26.410 So we have the 2 over the 4 is one half, and r squared. 00:05:26.410 --> 00:05:29.080 And that is the moment of inertia of it 00:05:29.080 --> 00:05:32.620 does about an axis passing through the center 00:05:32.620 --> 00:05:35.970 perpendicular to the plane of the disk.