Is this force a central field ?

In summary, the conversation discusses a particle moving in a spiral orbit and whether the force acting on it is a central field. It is determined that in order for a force to be considered central, it must only be a function of the scalar distance to the force center and its direction must be along the radius vector. The conversation also mentions that the angular momentum of a particle is constant under a central force. However, it is concluded that the given spiral orbit does not correspond to a central field due to the variable nature of the angular momentum. The conversation also suggests checking if the acceleration vector is radial to determine if a force is central.
  • #1
cj
85
0
Is this force a "central field" ?

A particle moves in a spiral orbit given by:

[tex]r=a\theta[/tex]

if [tex]\theta[/tex] increases linearly with time, is the force a central field? If not, how would [tex]\theta[/tex] have to vary with time for a central force?

I believe that a central force is a function only of the scalar distance, r, to the force center, and its direction is along the radius vector.

I also believe that the angular momentum of a particle is constant when it is moving under the action of a central force.

Even though I seem to remember the above, I'm at a loss see whether or not the above is a central force -- nor how to modify it to make it one??
 
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  • #2
What is the angular momentum, in terms of r and \theta ?
 
  • #3
Gokul43201 said:
What is the angular momentum, in terms of r and \theta ?

[tex]L = r \times mv = mr^2\dot{\theta}[/tex]

I think ...

So, this means

[tex]L = r \times mv = mr^2\dot{\theta} = ma^2\theta^2\dot{\theta}[/tex]

[tex]\dot{\theta}[/tex] is constant (since it varies linearly with t), but [tex]\theta^2[/tex] is not constant.

So -- this does not correspond to a central field? Or am I still missing something?
 
  • #4
From the trajectory, you can calculate the velocity vector and the acceleration vector. [Using Newton's Law, you can find the force vector.] You can express this as a vector field. Check if its curl is zero... (I believe this is necessary but not sufficient for a central force).

Of course, you can probably just check if the acceleration vector is radial.
 

1. What is a central field in physics?

A central field in physics is a type of force that acts on an object in a way that the direction of the force always points towards the center of the object's motion. This type of force is often found in systems with spherical or symmetrically shaped objects, such as planets orbiting around a star.

2. How is a central field different from other types of forces?

A central field differs from other types of forces in that its direction is always towards the center of the object's motion, rather than being influenced by the object's orientation or position. This type of force is also conservative, meaning that the work done by the force is independent of the path taken by the object.

3. What are some examples of central fields?

Some examples of central fields include the gravitational force between planets and stars, the electric force between charged particles, and the magnetic force between a moving charged particle and a magnetic field. These forces all act towards the center of the object's motion and are considered central fields.

4. How do you determine if a force is a central field?

To determine if a force is a central field, you can look at the direction of the force and see if it always points towards the center of the object's motion. You can also check if the work done by the force is independent of the path taken by the object. If both of these conditions are met, then the force can be considered a central field.

5. What are the applications of central fields in science?

Central fields have many applications in science, including the study of planetary motion, the behavior of charged particles in electric and magnetic fields, and the motion of objects in circular orbits. They are also used in engineering to design systems that require objects to move in a controlled and predictable manner.

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