Central Force

What is central force?

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 central force depends only on the distance of the object and the center.

Mathematical representation:

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

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.

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    Practise This Question

    If a light ray from air enters three different media in such a way that the refracted angles are 450, 300 and 600 respectively, Identify the right statement with respect to their density.


    In the figure, n1  represents refractive index of air.