Explore BrainMass

Explore BrainMass

    Rolling Cylinder

    Not what you're looking for? Search our solutions OR ask your own Custom question.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    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.

    © BrainMass Inc. brainmass.com March 4, 2021, 9:37 pm ad1c9bdddf


    Solution Preview

    See attached files.

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


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

    The potential energy is:


    Thus the general Lagrangian is:


    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:


    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.