![]() There is only one force that produces a torque about the center of mass of the disk - that's the frictional force. The angular momentum principle says that the net torque (about the center) is equal to the moment of inertia times the angular acceleration of the disk (about the center). Let's look at the other place this frictional force matters - in the torque. If I could only find this frictional force, I would have an answer for the acceleration. Here is the equation for the net forces in the x-direction (I am calling down the incline as the positive direction): It exerts whatever force it needs such that the disk rolls instead of slides - up to some maximum value. The static frictional force is called a constraint force. ![]() What if the frictional force was larger than the component of the gravitational force in the direction of the plane? This would make the disk accelerate UP the plane. For a case like this, it's possible the frictional force is quite large. I know that seems crazy, but imagine a super-rough surface for the disk and plane. If I know the normal force, then I can calculate the MAXIMUM frictional force, but not the exact frictional force. It depends on the two types of materials interacting. Here μ s is the coefficient of static friction. We can model the magnitude of this force with the following equation. Since the disk rolls without slipping, the frictional force will be a static friction force. This frictional force is what prevents the disk from slipping. Three forces, this should be simple - right? The disk only accelerates along the x-direction (along the plane) so this should be a simple problem. Remember, the definition of work by a force is: This force also doesn't do any work because the angle between the force and the displacement is 90°. This normal force pushes up on the disk perpendicular to the incline. Since it's part of the system, it doesn't do any work (and we have the gravitational potential energy instead). Why? Because it's actually the gravitational force between the disk and the Earth. There is the gravitational force, but it doesn't do any work. In order to use the work-energy principle, I need to first consider any forces that do work on the system. Let me just pick one at the top of the incline and the other point at the bottom of the incline. Second, I need to pick two points over which to look at the change in energy. For this case, I will choose the system to consist of the disk along with the Earth (that way I can have gravitational potential energy). First I need to declare the system that I will be looking at. In order to use the work-energy principle, I need two things. Without deriving it, I will just say that the moment of inertia for this disk would then be: Suppose the disk has a mass M and a radius R. Now we replace the frictionless block with a disk (actually frictionless disks are hard to come by and thus in a large demand). I skipped some steps, but that problem isn't too complicated. Suppose that I have some frictionless block on an inclined plane. Block Sliding Down Planeīefore looking at rolling objects, let's look at a non-rolling object. This means that we need another type of kinetic energy, rotational kinetic energy. There is a difference between a stick moving in a translational motion and a rotating stick. ![]() However, if it is really just a point, how would you know it's rotating? A rigid object can clearly rotate. Second, rigid objects need a change in the work-energy principle. For now, I will just say that the moment of inertia depends on the shape, mass, and size of the object. The moment of inertia plays the same role as mass in the momentum principle. This is a property of a rigid object (with respect to some rotational axis) such that the greater the moment of inertia, the lower the angular acceleration (for a constant torque). I like to call the moment of inertia the "rotational mass". It's just like plain acceleration is to plain velocity. The angular acceleration tells you how the angular velocity changes with time. For the other parts, let's focus on two things: the moment of inertia ( I) and the angular acceleration (α). Maybe this look at the weight of Darth Vader will at least help with the idea of torque. Torque and angular momentum are actually pretty complicated. ![]()
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