Is this force a central field ?

[cj]

Is this force a "central field" ?

A particle moves in a spiral orbit given by:

$$r=a\theta$$

if $$\theta$$ increases linearly with time, is the force a central field? If not, how would $$\theta$$ 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??

 

[Gokul43201]

What is the angular momentum, in terms of r and \theta ?

 

[cj]

What is the angular momentum, in terms of r and \theta ?

$$L = r \times mv = mr^2\dot{\theta}$$

I think ...

So, this means

$$L = r \times mv = mr^2\dot{\theta} = ma^2\theta^2\dot{\theta}$$

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

So -- this does not correspond to a central field? Or am I still missing something?

 

[robphy]

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.