# Rolling Cylinder

A cylinder of mass m and radius a is rolling on a stationary larger cylinder of radius R.

At what angle will the smaller cylinder will leave the surface and what is the friction force between the cylinders (assume coefficient mu).

Please see the attached file for full problem description.

Â© BrainMass Inc. brainmass.com March 4, 2021, 9:37 pm ad1c9bdddfhttps://brainmass.com/physics/torques/cylinder-mass-radius-rolling-stationary-larger-259421

#### Solution Preview

See attached files.

The energy of the motion of the cylinder's center of mass is:

(1.1)

The energy of the cylinder's rotation about its own center of mass is:

(1.2)

The potential energy is:

(1.3)

Thus the general Lagrangian is:

(1.4)

Now we get to the constraints that are imposed on the coordinates.

The first constraint is the fact that the cylinder rolls on the surface of the lower cylinder:

(1.5)

Or this can be written as a homonymic constraint

The generalized force in the direction of the r coordinate is (see appendix):

This is a force that is applied on the small cylinder by the large cylinder in the radial direction. In ...

#### Solution Summary

The solution uses Lagrangian mechanics to solve all the equations of motion and show how to use constraints in conjunction with the standard Lagrangian.

The solution is 6 pages long with full derivations and appendix.