+Benjamin Cordes Yep. GREAT lessons. Much thanks.

+Isaac Newton The integral sign is a part of the Leibnitz notation. Use somthing like a fluxion (o) or an accent to do justice to your profile name =P

Its common to keep M to see the forces

In general, you can define different centers of rotation and still get the same answer. With a different choice, the moment of inertia would be different, but the way you'd be measuring theta would also be different, and it would all work out the same way in the end. In this specific case, if you chose the point of contact as the center of rotation, your "rotation" would include movement of the center of mass of the disc, so you wouldn't get the simplifying benefit of separating the rotation and translation.

PhysicsHelps I have a quick question that been bugging me. Why isnt the potential energy: the potential energy of center of mass? In other words, if x=0 shouldnt PEcm =Mghcm= Mg(L+R)?? Why didnt we take h (from the bottom of the surface of the place) to the center of mass of the object? Thanks.

+Dillon Berger Great question. I don't actually think there are general rules. When I'm approaching a problem, I just try to find some set of variables that (1) can uniquely describe the state of the system and (2) is comprehensible to me. The variables I chose happen to be convenient because they separate the linear and rotational motion, but I probably only did this because I've used the technique before in earlier courses. There are other choices, and no choice is any more correct than another as long as you can uniquely describe the system. Also, this system really only needs one variable. It was just too difficult for my brain to directly write down the whole kinetic energy just in terms of the derivative of x, so I used theta and later eliminated it with the constraint. I could have started with three variables if it were easier, but I would have had two constraint equations relating the variables. Responding to your comment about this not seeming like an actual coordinate system: The variables x and theta may not together describe a spatial coordinate system, but they still make a coordinate system in a more abstract sense. You can think of a 2D coordinate plane with an x-axis and a theta-axis. This is similar to how you can have an x and time coordinate system. "Time" isn't something you can draw directly on the picture, but it's still usable as a coordinate/variable. Hopefully that helps, and thanks for watching and asking thoughtful questions.

+PhysicsHelps Regarding the speed X dot, if you choose the origin differently, says at the lower left corner you will get a different X and therefore different speed X dot. What should be adjusted in the equation then?

The term "coordinates" is used in a looser way here. There are no axes necessary. Sometimes they are called "general coordinates" to help make that clear, I think.

I teach people that the coordinates are simply enough pieces of required information to identify any possible points in the system.

so I think the potential term should have been m.g. [ l - x.sin(a) + R.cos(a) ] , please correct me if I'm wrong

Then again I guess it doesn't matter that much since it's just a constant

Exactly. But I should have been more explicit that I was using a simpler expression and explained why it was okay, so thank you for that feedback.

Iirc, it wouldn't rotate if there was no friction.

Good question. Since we're assuming the disk does not slip, we're dealing with static friction, and there is actually no energy converted to heat. The friction force does cause energy to change forms (to rotational kinetic energy), but this is still mechanical energy. The force of friction is conservative in this case.

In fact, you could substitute a smooth-surfaced toothed gear rolling on a smooth-surface toothed rail, and the equations would be the same, but there would be no friction involved at all.

@Ralph Dratman: Good point, but I think the real question is: what would the equations of motion look like for a square wheel rolling down a frictional hill made of inverted catenaries? What on Earth does a square wheel brachistochrone look like for that matter? What if the wheel is actually a fractal?

+Damian Anslik This is a method used for classical mechanics. While I imagine it could be extended to include relativistic effects, that would overly complicate the problem and unnecessarily confuse and already confusing problem.

+Liam Clink I dont think the speed of this disk would be high enough to take relativistic effects into account.......unless ofcourse the inclined plain was infinitely long :p

Yeah, but it should be possible to do the math regardless. It's just not really necessary.