Conservation Laws for Particles in Circular Cylinder

In summary, the conversation discusses the motion of a particle in a conservative field produced by a mass distribution. It is noted that the force generated by a volume element is derived from a potential that is proportional to the mass and distance from the element. The conversation then explores the conserved quantities in the motion of the particle, specifically in the case of a uniformly distributed circular cylinder of finite and infinite length, as well as a uniform wire in the geometry of an infinite helical solenoid. It is concluded that in the infinite length cases, the linear momentum along the z axis is conserved due to the uniformity of the space and rotation symmetry.
  • #1
Magister
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0

Homework Statement



A particle moves in a conservative field of force produced by a mass distribution. In each instance the force generated by a volume element of the distribution is derived from a potential that is proportional to the mass of the volume element and is a function only of the scalar distance from the volume element.
Supose that the mass is uniformly distributed in a circular cylinder of finite length, with axis along the z axis. What are the conserved quantities in the motion of the particle?

Homework Equations



The fact that if the space is uniform then the linear momentum is conserved and that if the space is invariant to rotations then the angular momentum is conserved.


The Attempt at a Solution



The linear momentum is not conserved in any direction and the angular momentum is only conserved along the z axis. Am I wrong?

But let's now supose that the cylinder has infinite length. In this case the space is uniform along the z axis and hence the linear momentum along z is also conserved. Am I wrong?

Thanks for your time.
 
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  • #2
I think your reasoning is correct.
 
  • #3
What about if the mass is in the form of a uniform wire wound in the geometry of an infinite helical solenoid, with axis along the z axis?
The only symmetry in the system is a translation together with a rotation along the z axis. What does that means in terms of conserved quantities?
 

1. What are conservation laws for particles in a circular cylinder?

The conservation laws for particles in a circular cylinder refer to the principles that govern the behavior of particles within a closed cylinder. These laws include the conservation of energy, momentum, and angular momentum.

2. How do these laws apply to particles in a circular cylinder?

The conservation laws for particles in a circular cylinder apply by ensuring that the total energy, momentum, and angular momentum of the particles remain constant over time, regardless of any internal interactions or external forces acting on the particles.

3. What is the significance of conservation laws for particles in a circular cylinder?

The conservation laws for particles in a circular cylinder are significant because they provide a fundamental understanding of the behavior and interactions of particles within a closed system. This knowledge is essential for various fields of study, including physics, engineering, and materials science.

4. How are these laws derived?

The conservation laws for particles in a circular cylinder are derived from fundamental physical principles, such as Newton's laws of motion and the law of conservation of energy. They can also be derived mathematically using the Lagrangian formalism, which describes the dynamics of a system based on its energy and forces.

5. Can these laws be violated?

No, the conservation laws for particles in a circular cylinder are fundamental physical principles that have been extensively tested and proven to hold true in all known cases. Violations of these laws would contradict our understanding of the laws of physics and require a significant reevaluation of our current theories.

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