Find the period of radial oscillation through effective potentials

In summary, the conversation discusses a circuit that is a circle and a central force. The effective potential is expressed as the sum of U(r) and L^2/2mr^2. The issue raised is that the angular momentum is a function of r, but in the solution, it is treated as a constant when differentiating the effective potential. The question is raised about the physical significance of treating angular momentum as a constant, as it is thought to depend on the position r.
  • #1
Richardbryant
24
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Homework Statement


Given circuit is a circle, force is a central force[/B]
Ueff(r)=U(r)+L^2/2mr^2

Homework Equations


the problem i find is, the angular momentum is a function of r
however, the solution when differentiate the effective potential, just treat angular momentum as a constant.
That's the point i am puzzle of, what is the physical sense of treating angular momentum as a constant? isn't it depends on the position r?

The Attempt at a Solution

 
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  • #2
Angular momentum is conserved if there are only radial forces - the particle can change its radius, that is no problem.
 

1. What is radial oscillation?

Radial oscillation refers to the back-and-forth motion of an object in a circular or spherical path, specifically in the radial direction.

2. What are effective potentials?

Effective potentials are mathematical models used to describe the motion of a particle in a specific system. They take into account both the kinetic and potential energies of the particle, allowing for a more accurate prediction of its behavior.

3. How is the period of radial oscillation calculated using effective potentials?

The period of radial oscillation can be calculated by finding the minimum of the effective potential function and using the equation T = 2π/ω, where T is the period and ω is the angular frequency.

4. What factors affect the period of radial oscillation in a system?

The period of radial oscillation can be affected by the mass of the particle, the strength of the potential, and the initial conditions of the system, such as the initial position and velocity of the particle.

5. How can the period of radial oscillation be used in scientific research?

The period of radial oscillation can be used to study the behavior of particles in various systems, such as planetary orbits, atomic structures, and molecular vibrations. It can also be used to make predictions and calculations in fields such as astrophysics, quantum mechanics, and chemistry.

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