Central Force

Any force is called a central force when it is always directed towards a fixed point, and its magnitude is dependent on the distance r of the particle from a fixed point (O).

Mathematically, this force can be described as: 

\[F=f(r)\hat{r}\]

Here, 

F = Central conservative force,

\[\hat{r}=\frac{\vec{r}}{\mid \vec{r}\mid }\] can be described as a unit vector in the direction of r. 

F represents conservative central force,

f (r) represents vector magnitude | r |

As well as the conservative force can be represented further as

\[\vec{\triangledown }\times \vec{F}(r)=0\]

When force is positive, it can be considered an attractive force directed towards O. However, when fris negative, the force is deemed to be negative. The same can be illustrated in graphical form below: 

(Image will be uploaded soon)

Where,

\[F(r)=-\frac{dU}{dr}\]

Here, 

F(r) is the magnitude of a central force, and

U(r) is the potential energy (that is time-independent)

Now, for a particle under central force making a uniform circular motion, it should have centripetal force in the following manner:   

 \[\frac{mv^2}{r}=F(r)\]…..(a)

Here, 

r = the initial radius

v = the speed that satisfies the equation (a) of centripetal force

Relevance of the Central Force Motion

The relevance of the central force motion of a particle:

Many natural phenomenons define the central force in Physics. Some of them are observed in:

Planets that work around the sun

Natural satellites that move around the earth

Two charged particles in terms of the movement against each other

Different Motions in a Central Force

There are broadly two different motion types observed in a central force field. They are:

Bounded Motion: In this case, the distance between two bodies or objects keeps a similar value, and never gets more than the set figures. Examples of such motion are the motion of the planets around the sun.

Unbounded Motion: In this case, the distance between the two bodies or objects is infinite in its initial and final stages. An example of such a movement is the scattering of alpha particles in the Rutherford experiment.

Properties of Central Force

There are several distinct properties that occur in a central force; some of them are:

The overall motion of the particle can take place in a plane curve, and that can be understood from the equation:  

The overall motion of the particle can take place in a plane curve, and that can be understood from the equation:  

F = F (r) = ma, where m refers to mass and a refers to the acceleration

Also, r × a = 0,  

\[r\ast \frac{dv}{dt}=0\]

\[\frac{d}{dt}(r\ast v)=0\]

r × v = h = constant vector; 

Where r and v lie in the same plane and h stays perpendicular to the same plane for every value of t. Therefore, the path stays in a plane. 

Taking from the above equation,  the angular momentum of the conserved particle is as follows;

m (r x v) = mh

L = mh is a constant, and the angular momentum is constant. 

The position vector r with respect to the central force covers equal areas in equal time periods; thus, having a constant areal velocity. In the following graph, in a small time interval dt, the radius r covers an area of dA. Therefore, the covered area can be likened to that of half the area of a parallelogram with the sides r and dr. 

\[dA=\frac{1}{2}\mid r\times dr\mid \]

\[dA=\frac{1}{2}\mid r\times vdt\mid \]

\[\frac{dA}{dt}=\frac{1}{2}\mid r\times v\mid \]

\[\frac{dA}{dt}=\frac{h}{2}\]

(Image will be uploaded soon)

Motion in a Central Force Field - Equations

One of the most widely used coordinate methods to represent particle motion under a central force is through the application of a polar coordinate system. If the central force is represented in the r direction, the equation can be defined as: 

\[F=F(r)\]

\[L=\frac{1}{2}mr^2-V(r)\]

\[F=F(r)=f(r)\hat{r}\] and \[F=-\triangledown V(r)\] where V(r) is the central potential.

We know that, the equation of motion for objects moving a central field using “lagrangian formalism”. The lagrangian of system is basically the difference in the kinetic and potential energy of the system, it is given by:

\[L=K.E-P.E\]

\[L=\frac{1}{2}mr^2-V(r)\]

Potential Energy of a Central Force

For a particle moving from points to P1 to P2, the central force can act on its origin, and the path of the particle is taken as a combination of radial and curved path segments. Therefore, the primary force can act on the direction of the radial segments and lie in perpendicular to the curved parts' displacement. Thus, the total work done via the central force across the curved section can be considered zero, and the whole work done can be defined as:  

(Image will be uploaded soon)

If the central force is a conservative force, then the magnitude F(r) of a central force can always be expressed as the derivative of a time-independent potential energy function U(r) i.e., \[F(r)=-\frac{dU}{dr}\]. To show this, it suffices that the work W done by the force depends only on initial and final positions, not on the path taken between them.

\[W=\int_{r_1}^{r_2}F.dr=\int_{r_1}^{r_2}f(r)\hat{r}dr=\int_{r_1}^{r_2}Fdr=U(r_2)-U(r_1)\]

Total Energy of the Particle in Motion

As F is a conservative force, the total energy can be described as: 

\[E=\frac{1}{2}mv^2+U(r)\]

\[v^2r^2+r^2\Theta ^2\]

Where \[v^2\] can be defined as \[[(\dot{r})^2+r^2(\dot{\Theta })^2]\]

\[E=\frac{1}{2}m((\dot{r})^2+r^2(\dot{\Theta })^2)+U(r)\]

\[E=\frac{1}{2}m(h^2(\frac{du}{dt})^2\frac{1}{u^2}(hu^2)^2)+U(r)\]

This equation gives the expression for total Energy of the Particle in Motion under the effect of central force.