C2T (Mechanics)
Topic – Gravitation and Central Force Motion (Part – 2)
We have already discussed part 1 of this e-report.
Now let us continue part 2 of it.
Central Force Motion:
In this e-report we apply Newtonian Physics to the general problem of a central force motion. A central force between two bodies (or particles) is a radial force that depends only on the distance ( ) between the bodies and it is always directed along the line joining them. Mathematically this force can be expressed as
where is any function of the distance between the particles and is a unit vector along the line of centres. We shall start by looking at some of the general features of a system of two particles interacting with a central force of the form given above.
Reduction of Two-body Problem into One-body Problem. Let us consider an isolated system consisting of two particles interacting under a central force given by . The masses of the particles are and and their position vectors are and . According to Fig. 1, we have the separation vector as
and .
Therefore, the equations of motion of the two particles are
From our definition of , the force is attractive for and repulsive for . The equations of motion are coupled together by , as the behaviour of and depends on . Therefore, these equations cannot be solved directly, unless we decouple them. The decoupling is done by replacing the original coordinates and by the separation vector and the centre of mass vector defined as
and
.
If there is no external force, the equation of motion for is trivial as
.
This equation has a simple solution . The constant vectors and depend on the choice of coordinate system and the initial conditions. We therefore, find that the centre of mass under a central force field moves at a uniform velocity.
Now the equation for turns out to be like the equation of motion of a single particle and has a straightforward solution. By some elementary algebra, we find from the equations of motion
or
or Reduced Mass. Here we define
as the reduced mass of the two particles. Using this definition, we write
The above equation is identical to the equation of motion for a single particle of mass acted on by a force . So, no trace of the two-body problem remains. The two-body problem has been transformed or reduced to a one-body problem.
Consequences of the Central Force:
Solving the vector equation of motion
for depends on the particular form of , but some properties of central force motion hold true in general regardless of the form of . Constraints imposed by the conservation laws of energy and momentum provide a major step toward finding the complete solution. In this e-report we shall see how to use conservation laws to identify some consequences of the central force problems and the solutions.
(a) Conservation of Angular Momentum. We will show now that the angular momentum ( ) is constant for central force motion. for the motion is defined as
or
But is nothing but the force itself . So, we get , since and are directed along the same direction.
Since,
, therefore is independent of time and a constant of motion under central force field.
Fig. 2
(b) Confinement of Motion in a Plane. As a proof, , so it follows that is always perpendicular to by the properties of the cross product.
Because is a constant of motion, it is fixed in a particular direction. Therefore, the plane of the motion perpendicular to , is also fixed, and can only move in a plane perpendicular to (as shown in Fig. 2).
Introducing plane polar coordinates , in the plane of motion, the equation of motion
becomes
In particular, the 2nd equation a consequence of the fact that a central force has no tangential component, only radial component is present.
(c) Law of Equal Area. We need to prove that the rate at which area is swept out is constant, a result that leads directly to Kepler’s law of equal areas. In other way, it is said that the areal velocity is constant.
Fig. 3
The magnitude of the angular momentum is constant, and is given by
. We consider the position of the particle at and , when its polar coordinates are ( , ) and ( , ) respectively. The area swept out is shown in Fig. 3. For small values of , the area is approximately equal to the area of a triangle with base and altitude , as shown. So,
Therefore, the rate at which area is swept out or the areal velocity is
or
constant.
The areas swept out by are the same for equal time intervals. The law of equal areas holds for any central force and for both closed and open orbits. For the
would be like the orbit of a comet entering the solar system, sweeping around the sun, and heading back out to space, never to return.
Calculation of the Total Energy:
The kinetic energy ( ) of a particle or body under a central force is given by
where is the velocity of the body evaluated as . Therefore the velocity has both the radial component and the tangential component as shown in Fig. 4.
Fig. 4 So,
It can be shown that all central forces are conservative, so we can associate a potential energy with as
Without any loss of generality, we can assume the reference point to be at infinity, where the potential energy can be assumed to be zero. Therefore, we obtain .
From the work–energy theorem, we write
where , the total mechanical energy, is constant. We can eliminate from the previous equation by using the relation and we get
Effective Potential Energy. This equation looks like the energy equation for a particle moving in one dimension, as all reference to is gone. We can go further by introducing
such that . is called the effective potential energy (or effective potential). differs from the true potential energy by the term
, called the centrifugal potential energy. Introducing the effective potential is a convenient mathematical trick to make the previous equation look just like the energy equation for a particle in one dimension. However, the term
is not a true potential energy related to a force. This term is actually a kinetic energy, but grouping it with the true potential energy helps us write the formal solution of the energy equation more directly, and it will also help us use simple energy diagrams to describe central force motion qualitatively.
Energy Equations and Energy Diagrams:
We just found that the expression of energy , depends on the single coordinate . In fact, it is identical to the equation for the energy of a particle of mass constrained to move along a straight line with kinetic energy and potential energy . The coordinate is completely suppressed.
Now let us apply energy equations to the meatier problem of planetary motion to find the energy diagram. For the gravitational force between two masses and (which is always attractive), we write
So, . The effective potential energy is therefore
If , the repulsive centrifugal potential dominates at small values of , and the attractive gravitational potential dominates at large values of . Fig. 5 shows the energy diagrams with various values of the total energy .
Fig. 5
The repulsive centrifugal potential remains always positive and the attractive gravitational potential remains negative, but the sum becomes positive for low values of , but changes its sign to negative above a certain value of . As we will see later that the minimum possible value of will be equal to the
minimum allowed value of the total energy . The kinetic energy of radial motion is , and the motion is restricted to regions where . The nature of the motion is determined by the total energy. Here are the various possibilities, as shown in the Fig. 5.
1. : Here is unbounded for large values but cannot be less than a certain minimum if . So, the particles are kept apart by the centrifugal barrier.
2. : This is qualitatively similar to case 1 but on the boundary between unbounded and bounded motion.
3. : The motion is bounded for both large and small values of . Here the two particles form a bound system.
4. : Here is restricted to only one value. The particles stay a constant distance from one another.
All these cases also correspond to the shapes of various trajectories. We shall find that case 1 corresponds to motion in a hyperbola, case 2 corresponds to a parabola, case 3 to an ellipse and case 4 to a circle.
There is one last possibility for . In this case the particles accelerate towards each other along a straight line on a collision course, since when there is no centrifugal barrier to keep them apart.
This concludes part 2 of this e-report.
The discussion will be continuing in the part 3 of this e-report.
Reference(s):
An Introduction to Mechanics, Kleppner & Kolenkow, Cambridge University Press
A Treatise on General Properties of Matter, Chatterjee & Sengupta, New Central Book Agency
(All the figures have been collected from the above mentioned references)
