Suppose you have a cylinder on an ramp and you let it start rolling down. What will be its acceleration? Great question, right? I like this because it brings in many different concepts in introductory physics. Also, I'm not too fond of the way most textbooks solve this problem.

## Point Mass vs. Rigid Object

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:

Here both the momentum and the acceleration are for the center of mass of the object. Of course a point mass is ONLY a center - right? For the Work-Energy principle, a point mass can only have translational kinetic energy even though a system of the point mass and the Earth could also have gravitational potential energy. The point by itself only has translational kinetic energy.

What about a rigid object? A rigid is something that can clearly rotate. Suppose I have a meter stick. This stick can both rotate and have its center of mass move. That means two things. First, along with the momentum principle we also need the angular 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: