so what exactly is inertia how would you describe it inertia is the tendency of an object to resist changes in its state of motion now if you recall newton's first law states that an object in motion will continue in motion and an object at rest will remain at rest unless acted on by net force so if you have an object it's going to remain at rest unless you try to push it and that's just the tendency of natural things things tend to remain in the state of motion that they're in so an object at rest wants to continue at rest and an object that's moving it wants to keep moving unless it's acted on by a net force to accelerate it or friction to slow it down so inertia is a property of an object that resists any changes to its state of motion so if an object is at rest inertia wants to keep it at rest now to illustrate the property of inertia we're going to use two objects the first object has a mass of 10 kilograms and the second object has a mass of 100 kilograms so in both cases we're going to apply a force of 50 newtons so which object do you think has more inertia the object with less mass or more mass intuitively you know it's the object with more mass it's easier to move a lighter object but it's more difficult to move a heavier object because the heavier object has more inertia so inertia is proportional to mass now based on newton's second law net force is the product of mass and acceleration so the acceleration is going to be the net force divided by the mass so for the first object it's a force of 50 divided by a mass of 10 which will yield an acceleration of 5 meters per second squared for the second mass it's 50 divided by 100 which will give us an acceleration of 0.5 meters per second squared so as you can see the small object has a very large acceleration the large object has a small acceleration so what this tells us is that the lighter object has less inertia because it was so easy to accelerate the object with a small amount of force the larger object has more inertia because even though we apply the same amount of force the acceleration of the object is a lot smaller so whenever you increase the mass of an object the inertia of that object will increase so inertia is proportional to mass it's harder to change the motion of a heavy object compared to a light object so an object with a lot of mass it's very difficult to get it to move so that's the basic concept between inertia as it relates to translational motion now what about rotational motion how does inertia play a role in that so let's compare two objects the first one is the thin hoop and the second one is a solid disc so for the thin hoop the mass is concentrated at the edge of the circle and for the solid disc the mass is distributed throughout the circle now let's say that the mass of the thin hoop is 10 kilograms and the mass of the solid disc is the same so they both have the same mass and let's say that the radius of both objects is the same we'll say the radius is 2 meters so which one has more inertia is it the thin hoop or is it the solid disk what would you say so if we took a string and attach it to each object and we apply the tension force that's the same let me draw this picture make sure that the angle is the same so let's say if we apply the same tension force of 100 newtons which one will be easier to rotate is it the thin hoop or the solid disc and keep in mind the radius is the same for both objects so which one has more inertia or is inertia the same now you might be thinking that the inertia is the same because it has the same mass and the radius is the same however the inertia is not the same because the distribution of mass is different it turns out that the thin hoop has more inertia than the disc with all else being equal the equation for the inertia of the thin hoop is mr squared it depends on the mass and also on the radius the inertia of the disc is the mass times r squared but multiplied by a half so this factor one half is based on the distribution of mass throughout the disc the fact that this is one is due to the fact that the mass is concentrated away from the center so whenever the mass is away from the central axis of rotation the inertia increase so if you can move the mass away from the axis of rotation you can increase the inertia of the thin hoop or whatever object you're dealing with if the mass is closer to the axis of rotation then the inertia will decrease so the inertia of an object that can rotate depends not only on the mass of the object but also on how that mass is distributed relative to the central axis of rotation so looking at the thin hoop all of the mass is concentrated away from the axis of rotation and that's why it has a higher inertia value with the disc some of it is at the edge some of it is in the middle some of it is close to the axis of rotation here you have no mass at the axis rotation but here you have some and so that's why the inertia of the disc is not zero it's not one but it's in between it's one half so if you can put all the mass away from the axis of rotation you can increase the inertia if you put it closer towards the axis of rotation then the inertia decreases so the thin hoop has more inertia which means it has more resistance to rotation so it's going to be harder to rotate the thin hoop the solid disc however because it has less inertia it has less resistance to rotation so it's easier to spin a solid disc but it's harder to spin the thin hoop so where does this equation come from mr squared where do we get that quantity so this is the inertia of a thin hoop and the inertia for a solid disc is one half m r squared and the inertia for a sphere is two fifths m r squared so we can see that this constant in front of the term mr squared has to do with the way the mass is distributed among the object and also the shape of the object as well so sometimes i tend to refer to that quantity as c c for a constant now let's talk about how we can derive m r squared for the inertia of an object so let's start with newton's second law f is equal to m a now let's multiply both sides by the radius r so on the left side f times r is equal to the torque the rotational equivalent of force just as force is a push or pull action that can cause an object to accelerate forward or decelerate torque can cause an object to rotate it can speed up while rotating or slowing down while rotating so let's think of torque as the rotational equivalent of a force now the acceleration is equal to the angular acceleration times the radius so let's replace a with alpha times r so the torque is equal to m r squared times alpha so therefore the sum of all the torques in the system is the sum of all of the mr square quantities times alpha and just as force is equal to mass times acceleration torque is equal to inertia times alpha so torque is the rotational equivalent of force alpha is the rotational equivalent of a a is the linear acceleration alpha is the angular acceleration now m the mass provides inertia to an object as you increase the mass the resistance of the object increases in the sense that it's harder to move a heavy object than a lighter object so this is the inertia of an object with reference to translational motion this quantity is rotational inertia is the resistance of an object it's the object's resistance to rotational motion so therefore we could say that inertia is the sum of mr squared and so that's how you can relate inertia to this equation using this process so this equation that we have here it's associated with newton's second law for rotational motion whereas this is newton's second law for translational motion or linear motion you