Central Force

Umiatin,M.Si

• The aim : to evaluate characteristic of motion under central force field

A. Introduction

• Central Force always directed along the line connecting the center of the two bodies

• Occurs in : motion of celestial bodies and nuclear interaction

Central Force Motion as One Body Problem

• Suppose isolated system consist two bodies and separated a distance r = |r| with interaction between them described by a central force F(r), we need six quantities used to describe motion of those particle :

Method 1 :

• To describe those, we need six quantities ( three component of r1 and three component of r2). The equation of motion of those particle are :

• If F(r) > 0: repusive , F(r) <0 : Attractive. Coupled by :

Method 2

• Describe a system using center mass (R) and relative position (r).

• R describes the motion of the center of mass and r describes the relative motion of one particle with respect to the other

No external forces are acting on the system, so the motion of the center of mass is uniform translational motion.

R** = 0.

• Where reduced mass define by :

• Two bodies problem has been simplified into one body problem.

• Solve the equation of motion :

• The center of mass moves with uniform velocity :

• By choosing the initial condition, vo, to, Ro = 0, the origin of coordinate coincides with center of mass R.

• So the position of m1 and m2 which measured from center of mass :

If m2 >> m1, then reduced mass:

The eq of motion :

Become :

• Hence the problem can be treated as a one body problem. Thus, whenever we use mass m instead of µ, we are indicating that the other mass is very large, whereas the use of µ indicates that either the two masses are comparable.

B. General properties of Central Force

1. Central Force is Confined to a Plane

If p is the linear momentum of a particle of mass µ, the torque τ about an axis passing through the center of force is :

• If the angular momentum L of mass µ is constant, its magnitude and direction are fixed in space. Hence, by definition of the cross product, if the direction of L is fixed in space, vectors r and p must lie in a plane perpendicular to L. That is, the motion of particle of mass µ is confined to a plane that is perpendicular to L.

• As we the force acting at body is central force, three dimensional problem can be reduced into two dimensional. Using polar coordinate system :

2. Angular Momentum and Energy are Constant

The angular momentum of a particle of mass µ at a distance r from the force center is :

• Since there are no dissipative systems and central forces are conservative, the total energy is constant :

3. Law of Equal Areas

Consider a mass µ at a distance r(θ) at time t from the force center O :

• Subtituting

C. Equation of Motion

From the previous description :

• If we know V(r), these equations can be solved for θ(t) and r(t). The set [θ(t), r(t)] describes the orbit of the particle.

• Solve the equation, we find :

• We will get t(r) then inverse r(t). But we are interested in the equation of the path in term r and θ

• We may write :

• And subtitute :

• Then :

Suppose the Force is F(r) = Krn

• K = constant

• If n = 1 the solution is motion of harmonic oscillator

• If n = -2 , eq : coulomb and gravitation force

Other method : Use Lagrangian

• Lagrangian of the system :

• We find

• To simplify, use other variable, for example : u in which = 1/r

• Next find

• Therefore :

• We can transform into :

Example

1. Find the force law for a central force field that allow a particle to move in logarithmic spiral orbit given by (k and α are constant) :

• Solution :

First determine :

• Now determine :

2. Find r(t) and θ(t) !

Solution :

3. What is the total energy ?

• Solution :

• We know that

D. Planetary Motion

The equation for the path of a particle moving under the influence of a central force whose magnitude is inversely proportional to the distance between the particle can be obtain from :

• If we define the origin of θ so that the minimum value of r occurs at θ = 0, so