Central Force

Central Force Definition

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)\[r_{1}\] = f(r)\[\widehat{r}\]

Here, \[\widehat{r}\] = \[\frac{r}{r}\] that can be described as a unit vector in the direction of r. 

F represents conservative central force,

frrepresents vector magnitude | r |

As well as the conservative force can be represented further as


\[\overrightarrow{\bigtriangledown}\]  × \[\overrightarrow{F}\](r) = 0


When fris 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: 


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


The relevance of the central force motion of a particle:


Many natural phenomenons define 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 particle 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:  

F = f(r)\[r_{1}\] = ma, where m refers to mass and a refers to acceleration


r × a = 0,  


r × \[\frac{dv}{dt}\] = 0


\[\frac{d}{dt}\]  (r × 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 = \[\frac{1}{2}\]  | r × dr |

dA = \[\frac{1}{2}\]  | r × vdt |

\[\frac{dA}{dt}\] = \[\frac{1}{2}\] | r × v |

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


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Motions 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(r) \[\widehat{r}\] L = \[\frac{1}{2}\] m \[r^{2}\] - V(r) (Here V (r) refers to the Langrangian Constant)


= \[\frac{1}{2}\] m ( \[r^{.2}\] - \[\theta^{.}\] \[r^{2}\]) - V (r)  \[\frac{\Delta l}{\Delta \theta}\]


= \[\frac{d}{dt}\]  \[\left ( \frac{\Delta L}{\Delta \theta } \right )\] = 0\[\dot{1}\]


= \[\frac{d}{dt}\]  (\[mr^{2}\] \[\theta^{.}\]) = 0 


Therefore, 


\[\frac{d}{dt}\] (m \[r^{.}\]) - mr \[\Theta^{.2}\] + \[\frac{\Delta V(r)}{\Delta r}\] 


\[V_{eff}\] (r)  = V (r)  + \[\frac{1}{2}\]   \[\frac{1^{2}}{mr^{2}}\]

Potential Energy of a Central Force

For a particle moving from points \[P_{1}\] to \[P_{2}\], 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:  


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The entire work done in the movement of the particle from \[P_{1}\] to \[P_{2}\] is:


W = - \[\Delta\] U


W = \[\int_{P_{1}}^{P_{2}}\] F. dr = \[\int_{r_{i}}^{r_{f}}\] f(r) \[r_{1}\]. dr = \[\int_{r_{i}}^{r_{f}}\] f(r)\[\frac{r}{r}\] . dr

 

\[\Delta\] U = \[U_{f}\] - \[U_{i}\] = - \[\int_{r_{i}}^{r_{f}}\] f(r) dr

Total Energy of the Particle in Motion

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


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


Where \[v^{2}\] can be defined as:


\[r^{.2}\] + \[r^{2}\] \[\theta^{.2}\] 


Therefore, E = \[\frac{1}{2}\] m ( \[r^{2}\] + \[r^{2}\] \[\theta^{.2}\] ) + U ®   

E = \[\frac{1}{2}\] m  \[\left ( h^{2} \left ( \frac{du}{d\theta} \right )^{2} + \frac{1}{u^{2}} (hu^{2})^{2}  \right )\] + U


FAQ (Frequently Asked Questions)

1. What are some central force examples?

Amongst the several popular examples of a central force, the most widely known are the gravitational force and Coulomb force, respectively, where the force F(r) is directly proportional to 1/r2 . For any object that represents such a force F(r) in negative values abides by Kepler's Law of Planetary motion. Therefore, according to Bertrand’s theorem, 


F (r) = - k/r2 and -kr respectively; for all bounded and closed orbits.

2. What is the Central Force? Are all Conservative forces central?

In mechanics, the central force is often defined as a force (mostly negative) that acts on a particle focussed towards a fixed point or center. This force is only proportional to the distance of the object from the center. No, not all central force fields are conservative or have spherical symmetry. But, a central force is only observed to be conservative when it is spherically symmetric.