Analytical mechanics- working with Lagrangian and holonomic constraints

[ronniegertman]

Homework Statement

on top of a static cylinder with a radius of R there is a cylinder with a radius of r, it is free to move, and conducts a rolling without slipping motion. When will the small cylinder detach from the larger cylinder?(R>r)

Relevant Equations

I want to solve this problem using effective potential energy. I think that since there is conservation of angular momentum, J always equals 0.

The tutor solved the problem using kinetic spinning energy though I find it very difficult and confusing to do so, therefore, I would like to know if there is a way to solve the problem using effective potential energy,
V_eff = J²/(2mr²

below is a sketch of the problem

 

[Orodruin]

If I understand your problem description correctly, there is only one configuration parameter. Effective potentials appear when you use conserved quantities to rewrite a higher dimensional problem as a lower dimensional one. It is not really applicable to a problem that already has only one parameter.

It is also unclear to me why you think angular momentum would be conserved. The angular momentum of what and why would it be conserved?

 

[ronniegertman]

I am still new to this kind of material,

If I understand your problem description correctly, there is only one configuration parameter. Effective potentials appear when you use conserved quantities to rewrite a higher dimensional problem as a lower dimensional one. It is not really applicable to a problem that already has only one parameter.

It is also unclear to me why you think angular momentum would be conserved. The angular momentum of what and why would it be conserved?

I’m still quite new to such material, but I believe that the angular momentum of the top cylinder is conserved (in the center of mass point). Moreover, could you please explain what do you mean by “higher dimensional problem” and why my problem is one dimensional?

 

[kuruman]

I’m still quite new to such material, but I believe that the angular momentum of the top cylinder is conserved (in the center of mass point).

As I understand this problem, the bottom cylinder is "static" which means that it is not allowed to move in any way. The only object that moves is the top cylinder. An external torque, about the axis of contact between cylinders, is generated by gravity and acts on the top cylinder. The external torque results in angular acceleration which means that angular momentum is not conserved.

 

[Orodruin]

I am still new to this kind of material,

I’m still quite new to such material, but I believe that the angular momentum of the top cylinder is conserved (in the center of mass point). Moreover, could you please explain what do you mean by “higher dimensional problem” and why my problem is one dimensional?

Your problem is one-dimensional because the configuration of the system you describe can be determined with a single parameter, eg, where along the big cylinder is the small one.