Central Force Planar Motion

A central force is, by definition, produced by a center of force and acts on a particle along the radial direction from the center:

F = F_r e_r

where, F_r in general can be a function of all the three co-ordinates, (r, θ, Ø).

To begin with, we assume the center of force to be fixed in space and set up a co-ordinate system with origin at the above center. In practice, such an assumption is justified if we are considering the motion of a plane/artificial satellite in the gravitational force of Sun/earth; the Sun or earth can be taken as fixed centers of force for the study of motion of relatively small objects like a planet or an artificial satellite, respectively. Similarly, a relatively heavy nucleus forms a fixed center of Coulomb force to study the motion of an electron in an atom.

Let us consider a particle P with initial position and velocity vectors r₀ and v₀ in a central force (with a fixed center at O). Note that the initial vectors r₀ and v₀ defines a plane which passes through the origin of the co-ordinate system fixed at center O. Now, since the force F acts on the particle along r (or r₀ initially), the acceleration (and hence d v) of the particle remains confined to this plane, and consequently particle always moves in this plane.

Thus, we get an important result: the motion of a part particle in a central force (produced by a fixed center) is a motion in a plane given by initial position and velocity vectors of the particle. In particular, if v₀ = 0, then the motion is a rectilinear motion along the line defined by vector r₀.

It is therefore natural that we use plane polar co-ordinates (r, θ) to describe the motion of a particle in a central force produced by a fixed center. The position, velocity and acceleration of the particle in polar co-ordinates are given by,

R = r e_r



For a central force, F (or a) has no component along eθ. Hence, we have

This implies that, r² d θ/dt = constant of motion for a particle moving in a central force. The above expression is the statement of conservation of orbital angular momentum of the particle in a central force. We shall now prove this in a more formal way.

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