# Question #8da2c

Nov 26, 2017

A Central Force is a special kind of force that originates from a single point in space. A particle under the influence of this force will have its force vector always directed towards that origin point and the magnitude of this force will only depend on the distance of the object from the origin.

It is convenient to work with polar coordinates with such forces, placing the source of the force at the coordinate origin. In such a coordinate system the central force has a magnitude that depends only on the radial coordinate $r$ and is always directed radially inward ($- \hat{r} \setminus \quad$; attractive) or radially outward ($+ \hat{r} \setminus \quad$; repulsive).

$\vec{F} \left(r\right) = \setminus \pm f \left(r\right) \hat{r}$ - positive sign for repulsion and negative sign for attraction.

Classic Examples of Central Force are:
[a] Gravitation: $\setminus q \quad {\vec{F}}_{g} \left(r\right) = - G \frac{M m}{r} ^ 2 \hat{r}$

[b] Electrostatic Force: $\setminus q \quad {\vec{F}}_{E} \left(r\right) = k \frac{{q}_{1} {q}_{2}}{r} ^ 2 \hat{r}$

Properties of Central Forces:
[1] Central Forces are always conservative forces and so the force vector can be written as the negative of the gradient of a scalar potential function.
$\vec{F} \left(r\right) = - \setminus \nabla U \left(r\right)$

[2] The motion of a particle under the influence of a central force is confined to a plane. This is a consequence of Angular Momentum Conservation.