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.


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{.}


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}


This article uses the definition of centripetal force given in Taylor. [1]Another general definition (Scienceworld[3]) adds the constraint that the force is spherically symmetric, i.e.

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