Energy

CONSERVATIVE AND NON-CONSERVATIVE FORCES

a)Conservative force :
*If work done by the force around a closed path is zero and it is independent of the path, such a force is called conservative force.
*Workdone by the conservative force is stored in the form of potential energy.
Example: Gravitational force, electrostatic force, and spring force
For a conservative force change in potential energy is equal to the negative of the workdone by the force dU = -\( \overrightarrow F .\overrightarrow {dr} \)
                      \( \int\limits_{U_i }^{U_f } {dU} = - \int\limits_{r_i }^{r_f } {\left( {\overrightarrow F .\overrightarrow {dr} } \right)} \)

Note : 1) By definition the negative of the work done by gravitational force is defined as the change in gravitational potential energy of the body.
        dU = -\( \overrightarrow F .\overrightarrow {dr} \)
          \( \therefore \int\limits_{U_i }^{U_f } {dU} = - \int\limits_0^h {\left( {\overrightarrow F .\overrightarrow {dr} } \right)} = mg\int\limits_0^h {dr} \)
              \( U_f - U_i = mg(h - 0) = mgh \)
2) Workdone, potential energy and kinetic energy depend on frame of reference selected to observe the motion of the body.

Example-1: 
In the absence of air resistance, a body is projected vertically up, then workdone by the gravitational force in moving the body through a height 'h' is W mgh and in return journery the work done by the gravitational force is  W_g=+mgh. On reaching the ground the net work done by the gravitational force in a round trip is zero.

Energy

b) Non conservative force :
* If work done by the force around a closed path is not equal to zero and it depend on the path, such a force is called non-conservative force Ex: Frictional force, viscous force
*Work done by the non conservative force will not be stored in the form of potential energy. 

Example-1:
In the presence of air resistance, when a body is projected up then it reaches a maximum height 'h'. Work done by the air resistance in upward journey is -fh and in return journey is also, it is -fh. On reaching the ground, net work done by the air friction is negative. so work done by the air resistance in a round trip is non-zero.

Example-2: 
A block of mass 'm' is dragged on a rough horizontal surface through distance 's' from the point 'p' to point 'q' and then back to the point 'p'. Work done by the frictional force from p to q is negative and from q to p is negative. So the work done by the frictional force around a closed path is negative not equal to zero.
                            \( \begin{gathered} W_1 \ne W_2 \ne W_3 \hfill \\ \end{gathered} \)  

Example-3:
In the above case instead of moving the block from p to q along a straight line, if it follow different paths, the work done by the friction is still negative but its magnitude is different in different paths. So work done by the frictional force is dependent on the path.