A compound disk of outside diameter 140.0 cm is made up of a uniform solid disk of radius 50.0 cm and area density 3.00 g/cm^2 surrounded by a concentric ring of inner radius 50.0 cm, outer radius 70.0 cm, and area density 2.00 g^cm^2. Find the moment of inertia of this object about an axis perpendicular to the plane of the object and passing through its center.

Video transcript

Hello everyone. So in this problem we have coins which can be treated as disks. We have a dummy coin with an inner uniform solid disc with a diameter of 120 centimeters in an area density of four g per centimeter squared which is encircled by a concentric disk or ring rather with inner and outer diameters of 1 20 centimeters and 1 60 centimeters in an area density of 2.5 g per centimeter squared. And we want to find the coins moment of inertia around an axis that passes through its center and is perfectly suited to play. So we can start by drawing a schematic Where the discs radius is just half the diameter 60 cm surrounded by concentric ring. The inner diameter is the 60 cm but the or the inner radius rather is the 60cm While the outer radius is equal to the outer diameter divided by two. So 160 divided by two is 80. Sad Makers. You know with the area density of this disc road D. Which we do not is equal to four g per centimeter square. The role of the ring Is equal to 2.5 g per centimeter. And so to find the total moment of inertia of this entire system where the axis of rotation is perpendicular to the center of mass of the system have to add up the moment of inertia of the this plus the moment of nurture of the ring. All the moment version of the disk through its center it's just one half M o. M o. Of the disk times the radius of the disk squared. And we have to add the moment on our show the ring, which is one to have the mass of the ring times radius, the inner radius squared plus the outer radius squared. Damn inner radius as we can see 60 centimeters which we can divide by centimeters to get this value in terms of meters 10000.6 m we can do the same for the outer radius where 80 centimeters Almost . m. We also know that the radius of the disk is equal to the inner radius which is .06 m. So in this total moment of inertia operation we have the radius of the disk, the inner radius and the outer radius. We need to find the mass of the disk and the mass of the ring. We could use this area density which were given you know that the area density is able to the mass divided by the area. So you can directly solve with a mass of the disk multiplying this area density times the area. We can do the same for the ring as well. The mass of the ring is equal to of the ring times the area of the ring. So now we can make this substitution Massive. The disk is equal to roll over the disk which is four g over cm squared. Now the area of the disk is simply pi times the radius of the disk which is .06 m squared. We get that the massive of this is simply equal to 21, Which we can convert into kg by dividing 1000 g. We get that massive disk is equal to 21.991 kg. We could do the same for the ring Where the row of the ring is equal to 2.50 g per centimeter squ. And the area of the ring is just pi times the inner radius squared plus the outer radius squared, Which is .06 m squared Plus . m Squared. All times is five times 2.5. And we get the mass of the ring. Or rather this value is the mass of the ring. This calculation actually gives us g which gives us a Mass of 45.239 kg. And this mass of the ring is actually the 21,991 g. We do the same conversion, we divide by 1000 g. The ring is 21.991 kg. So now we have all of the steps for all of the values we need to solve for this total moment of nursing. So I total is equal Q1 half times the mass of this which is 45.239 kg Times The radius of this, which is .06 m swell. And now we can calculate the moment of the show the ring, which is one half times the mass of the ring, which is 21.991 kg Times the inner radius squared, which is .06 m Squared, plus the battery squared, which is .08 m squared. Doing this entire calculation, we get that the total moment of inertia is 19.1 kg times meters square gives us an answer choice of c. All the stops were great.