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we should talk some more about the moment of inertia because this is something that people get confused about a lot so remember first of all this moment of inertia is really just the rotational inertia in other words how much something's going to resist being angular ly accelerated so being sped up in its rotation or slowed down so if it has a if this system has a large moment of inertia it's going to be very difficult to try to get this thing accelerating but if the moment of inertia is small it should be very easy relatively easy to get this thing angular ly accelerating so that's what this number is good for the reason why you want to know the moment of inertia is because it'll let you determine how difficult it will be to angular ly accelerate something and remember it shows up in the angular version of Newton's second law that says that the angular acceleration is going to be equal to the net torque divided by the moment of inertia or the rotational inertia since they're the same thing so that should make sense we're dividing by the moment of inertia we're dividing by the rotational inertia because that means if this rotational inertia is big look at this is in the denominator you've got a big denominator you're going to have a small value that means this alpha is going to be small it's gonna be a small angular acceleration but if this moment of inertia were small then it's going to be easier to rotate and you'll get a relatively larger angular acceleration because you're now dividing by a smaller number so it does serve the same role that mass did it serves as this inertia term for angular acceleration and we figured out how to determine the moment of inertia for a point mass and you'll hear people say this a lot point mass I'm going to say it a lot by point mass I just mean a mass you could treat as if all the mass were rotating at the same distance from the axis and that's what's happening here if you've got a heavy ball connected to a string a very light string that has very little mass you can neglect the mass here if all the mass is rotating at the same radius like this is we determined the last time that the moment of inertia of a point mass going in a circle is just the mass times how far that mass is from the axis squared this is the term for a point mass going in a circle for what the moment of inertia is how difficult it's going to be too angular Lee accelerate this is the rotational inertia mr squared but you get more complicated problems too so you could be like alright what happens if we don't have a single point mass we've got the three well we did this last time as well if you have multiple point masses all you need to do is say that alright for multiple point masses just add up all the contributions from each individual point mass so if we're careful here mathematically we should put a I subscript but don't let that freak you out this just really means add them all up so this would be M one times R one squared so you take the mass one times its distance from the axis squared plus M 2 times R two squared you take mass two times its distance from the axis squared and then you do the same for M 3 and if you had more masses you would just keep adding them up if you have a whole bunch of point masses that you can treat as if all the mass were rotating at the same distance from the axis and you might object you might say wait wait different masses here rotating at different distances from the axis but all of that particular mass all of M one is rotating at the same radius from the axis so we can use this formula for point masses we can add them up the total amount is going to be the total rotational inertia so in other words for this case here if we really wanted to do it we would say that the moment of inertia for these objects and this system in total would be our I chose to take them on order M 1 is going to contribute M 1 times its distance from the x squared would be a so we do a squared and let's say B is just the length of this string so B just represents that length and similarly C represents that length we're going to assume the radii of these masses are small I had to draw big so we can see them but it's easiest if you consider them to be small because then we don't have to take into account they are actual radius so we'd add to this that's this m1 a squared is just the contribution to the moment of inertia that's being contributed by just m1 so we have to figure out the contributions from each of these other masses so we'll have M two times its distance from the axis isn't going to be B it's going to be all the way so it's going to be a plus B squared and if you wanted to find the contribution from m3 so that you'd get the total you'd have em three times well it'd be a plus B plus C squared this would be the total moment of inertia for the entire system which says it's going to be more difficult right the more mass you add into the system the more sluggish it is to acceleration the more difficult it is to rotate so how can we make this three mass system easier to rotate let's say you're tired of requiring so much torque to move this thing you want to make it easier to rotate one thing you can always do is just take your masses and move them toward the axis ie just move these toward the center if you do that notice all of these RS are going to get smaller if you reduce the R you're going to get less moment of inertia and that object is going to be easier to rotate easier to angular Li accelerate you can whip this thing around easier if the mass is more toward the axis so this makes sense think about a baseball bat if you had a baseball bat so if you've got this baseball bat this is not the best drawing of a baseball bat but you've got a baseball bat if you swing it from this end where this is the axis it's hard to rotate right you've got all this heavy mass over here at the end but if you swing it instead where this is the axis if you just turn it around and swing it from this end where this is the axis now you've made it so most of the mass is near the axis and if you do that the radius of that mass is going to be smaller and if the radius is smaller it's going to contribute less to the moment of inertia less to the rotational inertia it's going to be easier to swing so you can swing a baseball bat really easy if you hold it by the fat end compared to the actual end you're supposed to hold you can swing this faster it's probably not a good idea your PI not going to hit the ball very very far but you'll be able to swing it much faster because that moment of inertia is going to be smaller and then the other thing we could do we could always just reduce the masses you can make the mass less you reduce the moment of inertia and if you if you can move those masses toward the axis you reduce the are you reduce the moment of inertia or the rotational inertia but what if you don't have point masses at all I mean we don't always have situations where the thing that's rotating are a bunch of point masses what if you had something more like this where was like a rod that had its mass evenly distributed throughout the entire rod and it rotated in a circle I mean we couldn't use this formula now because this assumes that all the mass is rotating at some radius R but for this rod only the mass at the end of the rod is rotating at the full length of the rod the mass that's closer to the axis it's going to have a smaller radius it will only be rotating it part of the length this would only have a radius of L over 2 and this part right here would only have a radius of maybe L over 8 so how do we figure this out we can't just say the total mass of this rod if this rod has a total mass m and a total length L we cannot say that the moment of inertia of this rod about its end is going to be M L squared that's just a lie this total mass is not rotating all that a radius of length L only the little piece at the end is rotating with a radius of length L the rest of this mass is having its contribution to the rotational inertia diminished by the fact that these masses are getting closer and closer to the axis so what do we do well we can't we can't use this let's get rid of this that's that's not possible the truth is you have to use calculus to derive the formula for these continuous objects and it's fun you can do integrals and you can solve for these moments of inertia that's one of my favorite calculations to do it's kind of like a puzzle you can solve for the moments of inertia but if you don't know calculus that would just look like witchcraft to you so I suggest you learn calculus and try it because it's really fun but I'm just going to give you the result it turns out the moment of inertia for this rod is going to be and without knowing the exact answer we should be able to say is it going to be bigger than less than or equal to ml squared we should be able to say it's got to be less than ml squared it's not going to be ml squared it's going to be less than this because ml squared would be if all of the mass were at the full length of the rod for their radius then you would put ml squared if if you could melt this rod down into just a ball and put that ball at the very far end you'd be maximizing its rotational inertia because you put all of the mass with the same largest radius R but some of this mass is in here some of this mass is only at L over to where at L over for L over 8 so those little pieces of mass or having their rotational inertia contribution diminished so we're going to have less than ml squared how much less turns out for a rod about its end it's one-third ml squared and if you do the integral that's where this 1/3 comes from so this is for a rod with the axis at the end of the rod so that's the moment of inertia for a rod rotating about an axis that's at one of the ends of the rod but what if we move this axis to the center what if we move the axis here so that this whole rod rotates around a point in its center do you think the moment of inertia of this rod that's the same mass and length that it was we're just rotating about the center do you think this moment of inertia is going to be bigger than smaller than or equal to what the moment of inertia was for a rod rotated about the end the way I would think about it I just asked myself this question is more of the mass farther away now or closer to the axis because we know if we can decrease these R's we decrease the moment of inertia and in this case we did decrease the RS think about it the farthest some piece of mass will be from the axis now as L over 2 it's L over 2 this way an L over 2 that way whereas before where the axis was at one end some of the mass was at L away so I'd be l squared but now you're only going to have L over 2 squared for the farthest some piece of this mass is going to be and that's going to decrease the moment of inertia even more because more of this mass is closer to the axis when you move it to the center so it's going to be less than one-third ml squared turns out if you do the integral you get 1/12 ml squared so this is for a rod with the axis at its center so what's what's another common geometry well if we get rid of that another case that comes up a lot is the cylinder or sometimes it's called the disk so let's say you have a cylinder a solid cylinder of mass m and it has a radius R what would this moment of inertia be we can probably tell by now all right so it's not going to be the total M R squared and it's not going to be the total M R squared because all of the mass is not rotating at the full radius of the cylinder right so it's going to be less than this how much less if you do that integral it turns out that you get one-half M R squared so it turns out the fact that some of these masses are closer to the axis then the full radius of the cylinder makes it so that the total moment of inertia is one-half the total mass of the cylinder times the total radius of the cylinder squared that this is for a cylinder with the axis through the center so the center's rotating around a point right here so it's rotating like this around this point here and that's important to note it's not enough to just say hey I gave you a rod what's the moment of inertia because you've got to know where's the axis if someone just hands you something and says what's the moment of inertia of this you can't give them an answer until they've specified where they want you to rotate the object around if you rotate the rod about its end it's one-third ml squared for the moment of inertia if you rotate the rod about the center its 1/12 and again the reason for that is because by rotating it around different axes you've made it so some of the masses at different ARS from other axes that you could choose so this was for a cylinder also called a disk sometimes a sphere comes up so this is another common example say you had a sphere also rotating around an axis like the earth rotating on its axis and let's say it also has a mass m and a radius R again because some of this mass is closer to the axis look at this mass right here is only rotating in a circle like that as opposed to at the full radius of the sphere it's going to have less than M R squared how much less will for a sphere rotating about an axis that goes through its center you get that the moment of inertia is two-fifths M R squared so that was for a sphere rotating about an axis that goes through its center and at this point you might object you might say wait a minute we had spheres when we had spheres before we did M R squared but that was for spheres that were rotating where all of their mass was rotating at the same radius so if you have a sphere in other words if you have a sphere you're going to rotate this whole sphere around in a circle like this if that's the case you're talking about then yeah that total mass is all rotating at the same radius but here that's not the case this is a sphere rotating around its center so if you just have a sphere that spins in place that's not the same case as this mass that's being whirled around around some common axis all at the same radius it's the difference between this is like the moon rotating around the earth if you want to talk about the moment of inertia of the moon rotating about the earth you could treat the moon as a point mass and you'd use M R squared but if you're talking about the earth rotating on its axis right not the earth going around the Sun but the earth rotating on its axis then you'd have to say that the moment of inertia for that amount of rotation is 2/5 M R squared because it's a sphere rotating through an axis that goes through its center all right so recapping the moment of inertia or the rotational inertia gives you a number that tells you how difficult it will be to angular ly accelerate an object if you've just got a point mass where all the mass rotates at the same radius you could use M R squared if you've got a collection of point masses you can just add up all the M R Squared's if you've got a rod rotating about its end you could use one-third ml squared a rod rotating about its center is 1/12 ml squared a cylinder rotating about its center is one-half M R squared and a sphere rotating with an axis through its center is 2/5 M R squared the reason why all these shapes that have mass distributed through them have factors that make their moment of inertia less than M R squared or ml squared is because some of that mass for a distributed object has mass closer to the axis than a case where all the mass is at the end so the fact that you've got some of these masses that are closer to the axis for a uniform object reduces the total moment of inertia since it reduces the R and if you ever forget any of these formulas there's often a chart in your textbook or look up the chart online they're all over the place lists of all the moments of inertia of commonly shaped objects and the axis got to check that it's the axis that you're concerned with as well