A bead of mass m slides without friction on a ring. The ring rotates with constant angular velocity w about a rotational axis that is aligned with a ring diameter, as shown in Figure 1. Find the Lagrange equations of motion, and the Hamiltonian for the bead. Is the Hamiltonian a constant of motion? Does it coincide with the energy of the system? Interpret the Hamiltonian as the sum of the kinetic energy of a bead rotating on a fixed ring and an effective potential resulting from the gravitational potential and a centrifugal potential. Plot the effective potential as a function of the angle theta for different values of w and discuss the motion of the bead for all cases. Under which conditions does the bead reach an equilibrium condition where theta remains constant?
See attached file for diagrams.© BrainMass Inc. brainmass.com December 15, 2022, 7:17 pm ad1c9bdddf
See attached file for solution.
In spherical coordinates, the bead speed is:
Thus, the energies of the system are:
And the Lagrangian is:
There is only one generalized coordinate (q), so we get only one equation of motion:
The Associated momentum is:
The system's Hamiltonian is:
Recall the commuting relations between the Hamiltonian and system's parameter A:
But our Hamiltonian is not explicitly dependent on time, so we get
Which means that the Hamiltonian is a constant of motion and it represents the system energy.
It has two components:
• A kinetic energy of a bead rotating on a fixed ring: ...
This solution provides step by step calculations for a system where a bead of mass m slides without friction on a ring.