Conservative force and non conservative force

Conservative force: There are the following points that describe the conservative force-

1. The conservative force depends on the position of the particle and does not depend on the path of the particle.
2. In conservative force, The kinetic energy of the particle does not change between the positions.
   
   Let us consider a particle is moving from position $A$ to position $B$ under the conservative force. If the kinetic energy of the particle at position $A$ and $B$ is $K_{i}$ and $K_{f}$ then for conservative force-
   
   

   $K_{i}=K_{f}$



3. In conservative force, The work done by the force in completing one round between any two positions is zero.
   
   
   According to the work energy theorem-
   
   
   $W=\Delta{K}$
   
   
   $W=K_{f}-K_{i}.......(1)$
   
   
   For conservative force, the kinetic energy of the particle does not change between the positions of the particle. i.e.$K_{f}=K_{i}$. So from equation $(1)$
   
   

   $W=0$

   
   
   Alternative Method:
   
   
   The above statement can also be proven by the following method-
   
   
   The work is done by the force to move a particle from position $A$ to position $B$ = $W_{AB}$.
   
   
   The work done by the force to move a particle from position $B$ to position $A$ = $W_{BA}$
   
   
   We know that the kinetic of the particle does not change between the positions so work done by force between the positions will be equal and opposite. i.e
   
   
   $W_{AB}=-W_{BA}$
   
   

   $W_{AB}+W_{BA}=0$

   
   
   So from above, we can conclude that the net work done by force between the two positions in completing the one round is zero.
4. The central force is also known as the conservative force.
5. The conservative force is always equal to the negative gradient of potential energy.
   
   

   $\overrightarrow{F}=-\overrightarrow{\nabla}U$

   
   

   $\overrightarrow{F}= -\frac{dU}{dr}$

Non-conservative force:

1. The non-conservative force depends on the path of the particle between the two positions and does not depend on the positions of the particle.
2. The kinetic energy of the particle does not the same at different positions. i.e. the kinetic energy is different at different positions. The change in kinetic energy is happen due to friction.
3. For non-conservative force, the work done by the force in completing one round between two positions is not zero.
4. This is not a central force.
5. The non-conservative force does not equal the negative gradient of potential energy.