# Central Force

Let us know about the Central Force. In classical mechanics , a centripetal force is a force on an object that is directed in the direction or away from a point called the center of force .

{\displaystyle {\vec {F}}=\mathbf {F} (\mathbf {r} )=\left\vert F(\mathbf {r} )\right\vert {\hat {\mathbf {r} } }}


where is power, f is a vector critical force function , f is a scalar critical force function, r is position vector , || R || is its length, and = r /|| R || The corresponding unit is the vector .

\vec{F} \hat{r}

Not all central force fields are conservatively or circularly symmetric . However, a centripetal force is conservative if and only if it is spherically symmetric or rotationally invariant.

## virtue

Centripetal forces which are conservative can always be expressed as the negative gradient of potential energy :-

\mathbf{F}(\mathbf{r}) = - \mathbf{\nabla} V(\mathbf{r})\text{, where }V(\mathbf{r}) = \int_{|\mathbf{ r}|}^{+~\infin} F(r)\,\mathrm{d}r

(The upper limit of integration is arbitrary, since the potential is defined up to an additive constant ).

In a conservative field, the total mechanical energy ( kinetic and potential) is conserved:

E = \frac{1}{2} m |\mathbf{\dot{r}}|^2 + V(\mathbf{r}) = \text{constant}


(where r means the derivative of r with respect to time, that is , velocity ), and is in a central force field, so is the angular momentum :

\mathbf{L} = \mathbf{r} \times m\mathbf{\dot{r}} = \text{constant}


Because the torque exerted by the force is zero. As a result, the body moves along a plane perpendicular to the angular momentum vector and contains the origin, and obeys Kepler’s second law . (If the angular momentum is zero, the body moves along the line joining the origin.)

It can also be shown that an object that moves under the influence of a centripetal force obeys Kepler’s second law. However, the first and third laws depend on the inverse-square nature of Newton’s law of universal gravitation and do not hold in general to other central forces.

As a consequence of being conservative, these specific central force fields are irrational, i.e. its curl is zero, except at the origin :

 \nabla\times\mathbf{F} (\mathbf{r}) = \mathbf{0}\text{.}

## Example

Gravitational force and Coulomb force are two familiar examples being only proportional to 1/ r2 . An object in such a force field is negative. With (corresponding to an attractive force) obeys Kepler’s laws of planetary motion .

{\displaystyle F(\mathbf {r} )}{\displaystyle F(\mathbf {r} )}


The force field of a spatial harmonic oscillator is central with which is only proportional and negative to r .

{\displaystyle F(\mathbf {r} )}


By Bertrand’s theorem , these two, and , are the only possible central force fields where all bounded orbitals are stationary closed orbitals. However, other force fields exist, some with closed orbits.

{\displaystyle F(\mathbf {r} )=-k/r^{2}}{\displaystyle F(\mathbf {r} )=-kr}


## Notes

This article uses the definition of centripetal force given in Taylor. Another general definition (Scienceworld) adds the constraint that the force is spherically symmetric, i.e.

 \vec{F} = \mathbf{F}(\mathbf{r}) = F( ||\mathbf{r}|| ) \hat{\mathbf{r}}