Purchase Solution

# A bead of mass m slides without friction on a ring

Not what you're looking for?

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.

##### Solution Summary

This solution provides step by step calculations for a system where a bead of mass m slides without friction on a ring.

##### Solution Preview

See attached file for solution.

In spherical coordinates, the bead speed is:

And:

Thus, the energies of the system are:
(1.1)
(1.2)
And the Lagrangian is:

(1.3)

There is only one generalized coordinate (q), so we get only one equation of motion:

(1.4)
The Associated momentum is:
(1.5)
consequently,
(1.6)
The system's Hamiltonian is:

(1.7)

Recall the commuting relations between the Hamiltonian and system's parameter A:

Therefore:

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: ...

##### Introduction to Nanotechnology/Nanomaterials

This quiz is for any area of science. Test yourself to see what knowledge of nanotechnology you have. This content will also make you familiar with basic concepts of nanotechnology.

##### Variables in Science Experiments

How well do you understand variables? Test your knowledge of independent (manipulated), dependent (responding), and controlled variables with this 10 question quiz.

##### Classical Mechanics

This quiz is designed to test and improve your knowledge on Classical Mechanics.

##### The Moon

Test your knowledge of moon phases and movement.