in this video we're going to derive the formula for the inertia of a hollow cylinder so let's begin with a picture so let's say this is our cylinder and let's say this is the inside part of that cylinder and then this is going to be the axis of rotation R1 is going to be the radius of the inside part of the hollow cylinder R2 is the outer radius so we have the inner radius the outer radius and let's say this is some generic point the distance between the axis of rotation and that generic Point that's going to be lowercase R not sure if you can see that but I'll try to put it here now the distance between the inner radius and that same point that's going to be Dr and the length of the cylinder is going to be L now for a generic cylinder we know the volume of a cylinder the volume of the cylinder is pi r 2 * height where this is r and this is H but for this example H is the equivalent of L so the volume of a cylinder is going to be Pi r^ 2 * L now density is mass over volume if you multiply both sides by V you get that mass is density time volume if we take the differential of both sides we get that DM is p DV now what I want to do is I want to get DV from this equation in order to do that I need to differentiate both sides on the left we'll have just DV on the right L is a constant so we're not going to differentiate that R is what changes in this equation the derivative of R 2 is going to be 2 r d r so we get that DV is 2i r l Dr R so I'm going to replace DV with this expression here so we get that DM is p * 2 pi RL Dr now in order to get the inertia we can use this this formula the inertia is equal to the integral of R 2 DM so what we can do at this point is replace dm with this expression so we get that the inertia is going to be the integral we're going to integrate it since the mass is in this region it's between R1 and R2 we are going to integrate it from R1 to R2 from the inner radius to the outer radius so we're going to integrate it from R1 R2 and then we have R 2 and then DM is p * 2i RL Dr R so I'm going to move all the constants to the front so 2 pi is a constant lowercase p or technically row that's the density so we're assuming we have a uniform density throughout the cylinder so the density is going to be constant the length of the cylinder is constant as well and then we have the integral from R1 to R2 we have r^ 2 * R which is R the 3 power and then Dr so now at this point we could find the integral of R the 3 so that's going to be R 4th over 4 integrated from R1 to R2 so I'm going to move the four to the front so we have 2 pi row * L over 4 and then R to 4th evaluated from R1 to R2 2 over 4 is 12 so we have Pi row * L over two now plugging in the top one into R to 4th we get R2 to the 4th power and then minus plug in the lower limit of integration R1 into R4 we get R1 to the 4th power now we could use the difference of perfect squares formula to factor R 2 to the 4th power minus I mean R2 to 4th power minus R1 4th power so using the difference of squares formula we have Pi row * L over two and then it's going to be to go from a squ to a we need to take the square root so the square root of R2 to the 4th power is going to be R2 2 to go from B squ to B we got to take the square root the square root of R1 to the 4th power is just r1^ 2 one of them is going to be plus the other is going to be minus so this is what we have at this point now let's talk about the volume of a cylinder we know for a regular cylinder with the radius R height H the volume is pi r 2 * H but what is the volume of a hollow cylinder let's break up the cylinder into two parts so we have the large cylinder which the radius of the large cylinder we Define it to be R2 and it has the length L and then we have the small cylinder inside the large cylinder with radius R1 and length l so V2 let's call that the volume of the large cylinder that's pi R2 2 and H is L now the volume of the small cylinder V1 is pi r1^ 2 L now in order to calculate the volume of the hollow cylinder so just the parts in between the inner and the outer cylinder to get the volume of the hollow cylinder we need to subtract these two and we'll get what's in in the middle so the volume of the hollow cylinder is going to be V2 minus V1 so V2 is pi R2 ^ 2 * L and V1 is pi r1^ 2 * l so what I'm going to do is factor out the constants pi and we're left with r 2 R2 2us r1^ 2 now we mentioned that density is mass over volume multiplying both sides by V we know that mass is density times volume we could use lowercase M or capital M so capital M the mass is density time volume and we have the volume of the hollow cylinder it's that formula so I'm going to replace V with p l * R2 2 - r1^ 2 now let's focus our attention back on this equation I'm going to rewrite this equation such that we can insert M into the equation so here we have Pi * P * l or P Pi l so as you can see this part is the same as that part and then we have this part which I'll write it here R2 2us r1^ 2 so right now I wrote p i mean row Pi L and R2 2us R1 squ so now I'm going to write what's left over so this is times we have the two in the bottom so that's going to be2 and then what we have left over here that's R2 2 plus R1 2 now this part as we could see it's equal to M so we could replace it with M so we have the inertia is M * 12 R2 2 + r1^ 2 now let's make it look better so I'm just going to switch M in2 and then we'll have our formula so the inertia for a hollow disc I mean a hollow cylinder is 12 M * R2 2+ r1^ 2 so that's how you can derive the formula for the inertia of a hollow cylinder now once you have this formula you can easily calculate or determine the formula for the inertia of a solid cylinder so for the solid cylinder there's no inner radius we only have the outer radius for a solid cylinder which we'll Define it as R2 L is still the same so for the solid cylinder because there's no inner radius that means that R1 is zero because it's non-existent if R1 is zero this disappears and so the inertia of a solid cylinder becomes 12 M R2 2 now there's really no point in using R2 if there's only one radius so we can drop the subscript 2 and thus we get this formula the inertia for solid disc is 12 Mr 2 so that's how you can get the formula for the inertia of a solid cylinder and a hollow cylinder