Central force - circular motion

[spacetime]

For what form of a central force is the motion of a body exactly circular?

 

[Kittel Knight]

Just keep
$$F = - M \cdot w^{2} \cdot r$$

 

[rayjohn01]

Good question. Spacetime -- I cannot speak for every form of force , but if you take gravity as an example it is not the force per se which gives you a circular orbit -- it is the initial velocity ( meaning speed and direction ) I do not think that there is a single planet in truly circular orbit.
If you write the equations balancing the force of gravity with the so called centrifugal force they can only balance exactly ( and hence cancel ) if the motion is tangential to gravity other wise you will be left with a component pointing somewhere else an an ellipse or some sort will occur.
I do not think the equation of gravity matters so long as it points outward from a point ( assuming you are outside the defind body ) ( i.e. for a gas you may not quite be totally inside ).
The previous answer is good but does not relate to type of force , in a more general sense if you had more than one source of gavity ( as an example two suns )
two protons etc etc . there maybe no solution at all -- even if they are close .
Example even if you can consider the Earth as a point source of force from a fair distance you could not ignore a mountain if you were in close Earth orbit , and probably there would be NO circular solution.
So circular implies highly regular force in the space , and a specific orientation to the start of motion.

Ray

 

[krab]

For what form of a central force is the motion of a body exactly circular?

Any form where the potential is a function only of r has a particular solution that is a circle.

So for example gravity has circular orbits: the general orbit is elliptical and circles are particular instances of ellipses. You get circles if the system is set up exactly right for it.

You can put a mass on a spring and swing it round in a plane (F is proportional to r-r_0). For every r, you can find a velocity where the centrifugal force is equal to F at some r. But deviate away from this r for the same v and you get a sine wave wrapped in a circle.

 

[Tide]

Kittel gave the correct answer since a central force was specified by Spacetime meaning that the force depends only on the separation and is directed from one object to the other.

 

[pervect]

For what form of a central force is the motion of a body exactly circular?

Any central force that's uniform with the angle (i.e. where the force is not a function of angle) that is attractive will allow a circular motion, but such an orbit may or may not be stable.

The stability of the force can be investiaged by looking at the plot of the "effective potential". It turns out you can reduce the two dimensional problem to a one dimensional problem for a uniform central force (by making use of the conservation of angular momentum). It turns out that the equations that govern the radial motion of a particle are exactly the same as a classical particle in a potential well. The stability analyis is then very simple - if the particle is "on a hill", it will slide off, and the circular orbit won't be stable. If the particle is "in a valley", the orbits will be stable. The marginal case is where the effective potential is flat - in this case, any small error will cause a circular orbit to drift away, but it won't be actively unstable as it would if the particle were on a hill.

An inverse cube law (f=-k/r^3) is an example of the marginally stable case where the "effective potential" is flat. Any power law higher than inverse cubic is actively unstable (i.e f = -k/r^4 , -k/r^5, etc are unstable).