# Planarity of Central Force Motion

## Main Question or Discussion Point

Hi There!!,
I'm here just to know an answer to a question that is bothering me for a while now. We know that motion of a body under the influence of central force with given center [say A] is planar . I was thinking whether this is possible even when body is allowed to move under the influence of two centers corresponding to the central forces.
Also, if it's not , does that mean that motion of planets is not a planar one as these are also moving under the influence of multiple centers .

Any help is appreciated .

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jbriggs444
Homework Helper
If [all of] the gravitating centers lie in the same plane, if this plane contains the objects's initial position and if the object's initial velocity also lies within the plane then all displacement, velocity, force and acceleration vectors will stay within the plane.

The solar system is only approximately planar.

• Chandra Prayaga
The statement that the motion under a central force is planar is a consequence of the fact that the central force conserves angular momentum. Angular momentum is conserved if the net force on the body is central, giving zero net torque. Once you have two distinct centers of force (such as a planet moving in the gravitational field of two stars), The net force is no longer central and does not conserve angular momentum, and the motion is not planar.

jbriggs444
Homework Helper
The statement that the motion under a central force is planar is a consequence of the fact that the central force conserves angular momentum. Angular momentum is conserved if the net force on the body is central, giving zero net torque. Once you have two distinct centers of force (such as a planet moving in the gravitational field of two stars), The net force is no longer central and does not conserve angular momentum, and the motion is not planar.
Conservation of angular momentum and planar motion are separate concepts. Neither one implies the other.

For instance, angular momentum is strictly conserved for a system of objects interacting under internal force pairs. Their motion is nonetheless not confined to a plane.

For instance, a puck on an air hockey table undergoes planar motion. Despite this, angular momentum is not conserved as such a puck is batted around under the influence of external forces.

Chandra Prayaga
I would appreciate being corrected here. My thinking runs like this.
If a particle is moving under a central external force, then its angular momentum is certainly conserved, both in magnitude and direction. So the position vector and linear momentum vector remain confined to a plane perpendicular to the angular momentum. That looks like planar motion.
Your example of a puck on a table certainly shows that planar motion in general does not imply conservation of angular momentum. The forces on the puck in this example are not central.
In the case of a system of objects, the net force on any one object is also not central, and the angular momentum of any one object is not conserved.

jbriggs444
Homework Helper
In the case of a system of objects, the net force on any one object is also not central, and the angular momentum of any one object is not conserved.
Yes, if you focus your attention on a single object that lies in a plane containing the point about which angular momentum is being calculated and if that object's initial angular momentum is [non-zero and] perpendicular to that plane then conservation of angular momentum means that its velocity vector will be at all times confined to the plane.

[There is a loophole if the angular momentum vector is zero. The object could follow a path straight to the center and then up or down on a perpendicular while retaining zero angular momentum at all times]

However, the fact that conservation of angular momentum is sufficient to ensure planar motion does not mean that it is necessary.

Chandra Prayaga
Yes, if you focus your attention on a single object that lies in a plane containing the point about which angular momentum is being calculated and if that object's initial angular momentum is [non-zero and] perpendicular to that plane then conservation of angular momentum means that its velocity vector will be at all times confined to the plane.

[There is a loophole if the angular momentum vector is zero. The object could follow a path straight to the center and then up or down on a perpendicular while retaining zero angular momentum at all times]

However, the fact that conservation of angular momentum is sufficient to ensure planar motion does not mean that it is necessary.
Agreed.

But still my question is unanswered....
(Or I think so )
Does the superposition of two central forces results into another central force
With some other center beside the given ones????

If [all of] the gravitating centers lie in the same plane, if this plane contains the objects's initial position and if the object's initial velocity also lies within the plane then all displacement, velocity, force and acceleration vectors will stay within the plane.

The solar system is only approximately planar.
Because if superposition of two centers generate another one (I hope u understand what I mean) then it would imply that if a particle is free to move under multiple centers no matter whether they lie in same plane or not and whichever direction does the velocity of free particle points .It's motion must be confined to the plane.

jbriggs444
Homework Helper
Because if superposition of two centers generate another one (I hope u understand what I mean) then it would imply that if a particle is free to move under multiple centers no matter whether they lie in same plane or not and whichever direction does the velocity of free particle points .It's motion must be confined to the plane.
The superposition of two centers does not generate another one. For instance, the gravitational field from two objects cannot be accurately simulated by a single gravitating object.

Any proof??

Chandra Prayaga
The proof is actually quite simple. First draw the plane containing one star, the planet, and the velocity or momentum vector of the planet relative to the star. With that single star attracting the planet, the force on the planet is central, the torque of the force about the star taken as origin is zero, and the angular momentum of the planet is constant. Now place a second star which is not in that same plane. The net force on the planet is no longer pointing toward the first star, so it is not central. The torque of this new force about the first star is not zero.

" Now place a second star which is not in that same plane. The net force on the planet is no longer pointing toward the first star, so it is not central."
How can we say that resulting force will not point in the direction of an hypothetical center ie. 'two centers will not superpose to generate another one .That other one is not there really but direction of resultant force conveys the knowledge of its existence'

DrGreg
Gold Member
The superposition of two centers does not generate another one. For instance, the gravitational field from two objects cannot be accurately simulated by a single gravitating object.
Any proof??
If you are very close to the first gravitating object, the net force will point almost directly towards the first object.

If you are very close to the second gravitating object, the net force will point almost directly towards the second object.

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