in this video we'll talk about the parallel axis theorem and we have chosen a system of a sphere this is a solid sphere and a rod the solid is feared and the rod are joined together but we have the different situations here in the first situation the entire system the solid sphere and the rod is rotating about this axis here this is the axis of rotation in the second case it is rotating about this axis of rotation and in the third case this is the axis of rotation so let me write down here this is the axis of rotation okay now we need to find out the moment of inertia in each cases the moment of inertia would be different because it depends the moment of inertia depends how the masses are distributed with respect to the axis of rotation and you can clearly see here clearly see in this particular case in this particular case from the this is the axis of rotation the the masses are distributed far from the axis of rotation so you can get sort of feeling that the moment of inertia would be greater in this case but in order to get a number the quantity the amount we have to do all the the mathematical process let's get into this one so we'll start with this case here the case one and again in order to solve this kind of combined moment of inertia of two systems we'll use the parallel axis theorem and i'll explain what the parallel axis theorem is the parallel axis theorem tells you so let's start with the sphere first so here i'm just case one and a sphere for the sphere and the axis of rotation is right here so what are the parallel parallel axis theorem tell you the parallel axis theorem tells you the moment of inertia is equal to the moment of inertia passing through the center of mass plus the mass times the distance square or in other words so here you have a sphere what is the center of mass of the sphere of course this passes through this line here so what is asking you to do first find out the moment of inertia of this sphere through the center of mass now plus then you have to add this term the mass of the sphere times this distance the distance between the center of mass and the axis of rotation that's what the parallel axis theorem tells you okay first you have to find out the moment of inertia through the center of mass plus the mass of the sphere times the distance between the axis of rotation and the center of mass so what is the distance the length of the rod the length of the rod is from this point to this point which is l and and this distance this is the radius so the total distance is l plus r so we have now calculated this this part here the moment of inertia the ms stands for the mass of this sphere that's what it is mass of the sphere and d is l plus r the moment of inertia of a sphere about the center of mass is 2 pi 2 over 5 ms r square so you have to keep this one in mind or you have to memorize this one okay or you can find this one in any book or you can look into my other videos to see how to get the moment of inertia of a sphere about the center of mass okay so this is the given and now we just calculated ms and d is the this distance l plus r so now life is a lot easier now two or five what is the mass of the sphere the mass of the sphere is given the mass of the sphere is 10 kilogram so 10 kilogram the radius of the sphere is 0.5 meter so just plugging all the values 0.5 meter again the mass of the sphere is 10 kilogram the l the length of the rod this is the length of the rod which is 2 meter and the radius is 0.5 so this is just all the numbers now so if if you do the math or use the calculator you'll get 63.5 so again remember this is just the moment of inertia of this sphere about this axis of rotation okay so it is rotating this way so you in half rotation the sphere will be here so that's the axis of rotation now let's again calculate the moment of inertia of the the rod now of this rod about this axis of rotation so this is the rod and this is the axis of rotation let me recall you again the parallel axis theorem what is the parallel axis theorem tell you first you need to find out the moment of inertia through the center of mass about the axis of rotation through the center of mass the cm stands for the center of mass second the mass of the rod times the d the distance between the axis of rotation and the center of mass the d is again the distance between center of mass and axis of rotation okay so in this case the center the the distance is l over 2 the total length is l and it's a homogeneous rod so the center of mass will lie exactly halfway so this distance will be l over 2 and now remember the moment of inertia of a rod about its center of mass is given by this formula so you have to memorize this one or you have to keep in mind you have to remember this one now if you solve these two expression what i get is the this form this expression here now this is the moment of inertia of the rod about this axis of rotation now just plug in all the values now one over three the mass of the rod is five kilogram the length is two meter so if you solve it you'll get 6.67 so the total moment of inertia of the of the sphere of this sphere and the rod together or the moment of inertia for this particular case will be now this number plus this number or the moment of inertia of the sphere plus moment of inertia of the rod and i got this number 70.2 kilogram meter square or square meter now let's calculate for the second case here in this case the axis of rotation is about here at the edge of the sphere so first we'll again calculate for this sphere again what you're going to do it's again not passing through the center of mass so remember to use the parallel axis theorem if it is not passing through the center of mass so the what is the parallel axis theorem tell you first you have to find the momentum inertia through the center of mass plus you have to add this term and this is the mass of the sphere and d is the the distance between axis of rotation and the center of mass which is r in this case d is exactly equal to r and what is the moment of inertia of a sphere through center of mass 2 5 over msr square so you have to again memorize these things which is exactly this thing here m s r square if you add them together you'll get seven over five ms r square plug in all the numbers the mass is ten the radius is point five if you solve it you'll get three point five compared to you see now the difference is here if the axis of rotation is at this point the moment of inertia is 63.5 and if the moment of inertia is here you'll get three and a half there's a whole difference now let's calculate the moment of inertia about this end so between this case in this case there's no difference at all so so here we have calculated about this end now here we're calculating about this end which is exactly the same so we'll again get the same value 6.67 it's exactly as case a so the now the total moment of inertia for this particular case where the axis of rotation is here what we're going to do now we're going to add these two numbers together three and a half and six point six seven and the total moment of inertia would be a sphere plus rod will be 10.17 which is about seven times smaller than than the previous number now we're going to calculate the moment of inertia passing through the center of mass of this sphere or about this axis let's do it that's the case third so in this case so so in this case as the moment the axis of rotation is passing through the center of mass so it will have only one term here because in this case d is zero even if you use the parallel axis theorem this axis theorem the d will be zero why because the distance between the center of mass and the axis of rotation is zero so we'll have only one term here two over five minus r squared and we already have calculated this term 2 over 5 m is r square this term gives you 1 so we'll have 1 and this makes sense because the moment of inertia in this case would be smaller than this case here in this case it was three and a half and we get one now let's calculate the moment of inertia for the rod about the center of mass through the sphere so in this case so we're going to use the parallel axis theorem which tells you the moment of inertia through the center of mass plus mr d square so the moment of inertia of the the rod through the center of mass is this icm and the d is the distance between the center of mass and about the axis of rotation and what is this length here this length is l over 2 because this total length is l from this end to this end is l and this is the center of mass and as the rod is homogeneous this length is l over 2 and this is the radius of the sphere so the total length or the distance will be l by 2 plus r so again the center of mass of homogeneous rod through its center of mass is 1 over 12 m m or l square so you have to keep this in mind you have to memorize this one and now for this part m r the d is l over 2 plus r square and plug in all the values 1 over 2 the mass of the rod is 5 is 2 mr is again 5 l is 2 over 2 plus r is 0.5 and if you do the math what you get is 12.92 and this is greater than this number which makes sense because in this case the axis of rotation was at one end but now in this case the axis of rotation is further so the moment of inertia has increased so the total moment of inertia in this case would be this number plus this number and together this is 13.92 what you can see here the moment of inertia is greatest in this case which makes sense because the intent the mass distribution more mass is distributed about this axis of rotation and the moment of inertia is the smallest for this case okay so this is how you calculate the moment of inertia of a combined system of two or more system using parallel axis theorem if you have any questions write down your questions in the comment section below or if you want me to make a video of a moment of inertia for finding out the moment of inertia of a different system write down in the questions in the comment section below and at the end do not forget to like share and subscribe the channel thank you