Calculating angular velocity of rolling object with just gravity?
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In most of the introductory physics course, students deal with point masses. Oh, sure - they aren't really point masses. A baseball isn't a point mass and neither is a car. But if you are only looking at the motion of the center of mass, then it is essentially a point mass. For a point mass, we have the momentum principle:


Torque and angular momentum are actually pretty complicated. Maybe this look at the weight of Darth Vader will at least help with the idea of torque. For the other parts, let's focus on two things: the moment of inertia (I) and the angular acceleration (α). The angular acceleration tells you how the angular velocity changes with time. It's just like plain acceleration is to plain velocity. I like to call the moment of inertia the "rotational mass". 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). The moment of inertia plays the same role as mass in the momentum principle. For now, I will just say that the moment of inertia depends on the shape, mass, and size of the object.
Second, rigid objects need a change in the work-energy principle. A point mass can't rotate. Well, maybe it can. However, if it is really just a point, how would you know it's rotating? A rigid object can clearly rotate. There is a difference between a stick moving in a translational motion and a rotating stick. This means that we need another type of kinetic energy, rotational kinetic energy.

Ok, now we can get to work.
Block Sliding Down Plane ————————
Before looking at rolling objects, let's look at a non-rolling object. Suppose that I have some frictionless block on an inclined plane.

The block can only accelerate in the direction along the plane. This means that if I put the x-axis in this direction, the net forces in the x-direction will be mass*acceleration and the net forces in the y-direction will be zero. The only force acting in the x-direction is a component of the gravitational force. This means that the forces in the x-direction will be:

I skipped some steps, but that problem isn't too complicated.
Rolling Disk Using Work-Energy ——————————
Now we replace the frictionless block with a disk (actually frictionless disks are hard to come by and thus in a large demand). Suppose the disk has a mass M and a radius R. Without deriving it, I will just say that the moment of inertia for this disk would then be:

In order to use the work-energy principle, I need two things. First I need to declare the system that I will be looking at. 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). Second, I need to pick two points over which to look at the change in energy. Let me just pick one at the top of the incline and the other point at the bottom of the incline.

In order to use the work-energy principle, I need to first consider any forces that do work on the system. There are three forces on the disk. There is the gravitational force, but it doesn't do any work. Why? Because it's actually the gravitational force between the disk and the Earth. Since it's part of the system, it doesn't do any work (and we have the gravitational potential energy instead). Next, there is the normal force. This normal force pushes up on the disk perpendicular to the incline. This force also doesn't do any work because the angle between the force and the displacement is 90°. Remember, the definition of work by a force is:

The cosine of 90° is zero. Finally, there is a frictional force that is parallel to the incline. Since we are dealing with a rigid object, this force actually doesn't have any displacement (I know that sounds crazy). But just look at a rolling wheel, the frictional force is at the point of contact, but this force doesn't move. Instead the wheel turns and there is a new contact point. In short, you can either have a rigid object OR work done by friction, but not both.
This leaves us with the following work-energy equation. Remember that the work is zero and the disk starts at position 1 from rest and not rotating.

Now I can add to this two ideas. First, I know the expression for the moment of inertia of disk. Second, the disk is rolling and not sliding. Since the disk is rolling, the speed of the center of mass of the disk is equal to the angular speed times the radius of the disk. Putting this all together, I can solve for the velocity at the bottom.

Solving for v2, I get:

.


Torque and angular momentum are actually pretty complicated. Maybe this look at the weight of Darth Vader will at least help with the idea of torque. For the other parts, let's focus on two things: the moment of inertia (I) and the angular acceleration (α). The angular acceleration tells you how the angular velocity changes with time. It's just like plain acceleration is to plain velocity. I like to call the moment of inertia the "rotational mass". 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). The moment of inertia plays the same role as mass in the momentum principle. For now, I will just say that the moment of inertia depends on the shape, mass, and size of the object.
Second, rigid objects need a change in the work-energy principle. A point mass can't rotate. Well, maybe it can. However, if it is really just a point, how would you know it's rotating? A rigid object can clearly rotate. There is a difference between a stick moving in a translational motion and a rotating stick. This means that we need another type of kinetic energy, rotational kinetic energy.

Ok, now we can get to work.
Block Sliding Down Plane ————————
Before looking at rolling objects, let's look at a non-rolling object. Suppose that I have some frictionless block on an inclined plane.

The block can only accelerate in the direction along the plane. This means that if I put the x-axis in this direction, the net forces in the x-direction will be mass*acceleration and the net forces in the y-direction will be zero. The only force acting in the x-direction is a component of the gravitational force. This means that the forces in the x-direction will be:

I skipped some steps, but that problem isn't too complicated.
Rolling Disk Using Work-Energy ——————————
Now we replace the frictionless block with a disk (actually frictionless disks are hard to come by and thus in a large demand). Suppose the disk has a mass M and a radius R. Without deriving it, I will just say that the moment of inertia for this disk would then be:

In order to use the work-energy principle, I need two things. First I need to declare the system that I will be looking at. 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). Second, I need to pick two points over which to look at the change in energy. Let me just pick one at the top of the incline and the other point at the bottom of the incline.

In order to use the work-energy principle, I need to first consider any forces that do work on the system. There are three forces on the disk. There is the gravitational force, but it doesn't do any work. Why? Because it's actually the gravitational force between the disk and the Earth. Since it's part of the system, it doesn't do any work (and we have the gravitational potential energy instead). Next, there is the normal force. This normal force pushes up on the disk perpendicular to the incline. This force also doesn't do any work because the angle between the force and the displacement is 90°. Remember, the definition of work by a force is:

The cosine of 90° is zero. Finally, there is a frictional force that is parallel to the incline. Since we are dealing with a rigid object, this force actually doesn't have any displacement (I know that sounds crazy). But just look at a rolling wheel, the frictional force is at the point of contact, but this force doesn't move. Instead the wheel turns and there is a new contact point. In short, you can either have a rigid object OR work done by friction, but not both.
This leaves us with the following work-energy equation. Remember that the work is zero and the disk starts at position 1 from rest and not rotating.

Now I can add to this two ideas. First, I know the expression for the moment of inertia of disk. Second, the disk is rolling and not sliding. Since the disk is rolling, the speed of the center of mass of the disk is equal to the angular speed times the radius of the disk. Putting this all together, I can solve for the velocity at the bottom.

Solving for v2, I get:

.
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