Central Force

Central force is the force that is radially pointing and the magnitude is dependent on the distance from the source. The examples of central forces are gravitational force, electrostatic forces and spring force.

Following are the theorems that relate central force with angular momentum:

Theorem 1: For an object to have its angular momentum conserved, the object should be subjected only to the central force.

Theorem 2: For an object to have its motion on a plane, the object should be subjected only to the central force.

What is Central Force?

The central force in classical mechanics is defined as the force that is acting on an object which is directed along the line joining the object and the origin. The magnitude of the central force depends only on the distance of the object and the centre.

Central Force Equation

\(F=F(r)\hat{r}\)

Where,

  • F: conservative central force
  • r: vector magnitude \(\left | r \right |\) is the distance to the center of force
  • \(\hat{r}=\frac{r}{r}\)

Central force is a conservative force which is expressed as follows:

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

Where,

  • F(r) is the magnitude of a central force
  • U(r) is the time-independent potential energy

For a particle under central force to be in a uniform circular motion should have centripetal force as follows:

 \(\frac{mv^{2}}{r}=F(r)\)

Where,

  • r is the initial radius
  • v is the speed that satisfies the equation of centripetal force

Central Force Fields

Derivation of fields with the help of Lagrangian is as follows:

\(F=F(r)\hat{r}\) \(L=\frac{1}{2}m\dot{r}^{2}-V(r)\) (Lagrangian of an object with mass m) \(=\frac{1}{2}m(\dot{r}^{2}+r^{2}\Theta ^{2})-V(r)\) \(\frac{\partial L}{\partial \Theta }=0\) (Lagrangian has no Ө dependence) \(\frac{d}{dt}(\frac{\partial L}{\partial \dot{\Theta }})=0\) \(\dot{l}\equiv \dot{p}_{\Theta } =\frac{d}{dt}(mr^{2}\dot{\Theta })=0\) \(\frac{d}{dt}(\frac{\partial L}{\partial \dot{r}})-\frac{\partial L}{\partial r}=0\) \(\frac{d}{dt}(m\dot{r})-mr\dot{\Theta }^{2}+\frac{\partial V(r)}{\partial r}=0\) \(V_{eff}(r)=V(r)+\frac{1}{2}\frac{l^{2}}{mr^{2}}\)

Central Force Examples

Two familiar examples for central force are the gravitational force and Coulomb force with F(r) being proportional to 1/r2. For an object with such a force where F(r) is negative obeys Kepler’s laws of planetary motion. Using Bertrand’s theorem, \(F(r)=-\frac{k}{r^{2}}\) and \(F(r)=-kr\) are the possible central force fields where bounded orbits are stable closed orbits.