Conservative Forces

A conservative force is one that does work independent of the path of the object. The work done by a conservative force thus only depends on the final and initial points of the object. Examples of conservative forces are gravity, elastic force, and electric force.

In contrast, a non-conservative force is one whose work done is dependent on the path taken by the object. An example of non-conservative forces is friction.

Let's look at the smiley face on the images above. If the force doing work in both of these images to erase the smiley face were conservative, then there is equal work done in both images, because the final and initial points that the force acts on the smiley face are the same. Yes, the path is different but conservative forces act independent of path.

Now, if the force is non-conservative(which it is, because you need friction to erase something with an eraser), then the path matters. The path taken by the eraser(and thus the length at which friction acts) is longer in the right image. This means more energy(as thermal energy) is dissipated as the eraser erases. This makes the friction non-conservative.

Let's look at the smiley face on the images above. If the force doing work in both of these images to erase the smiley face were conservative, then there is equal work done in both images, because the final and initial points that the force acts on the smiley face are the same. Yes, the path is different but conservative forces act independent of path.

Now, if the force is non-conservative(which it is, because you need friction to erase something with an eraser), then the path matters. The path taken by the eraser(and thus the length at which friction acts) is longer in the right image. This means more energy(as thermal energy) is dissipated as the eraser erases. This makes the frictional force non-conservative.

To add friction or any other non-conservative forces to the conservation of mechanical energy equation, we do this:

The third term on the left side of the equation refers to the work done by a non-conservative force like friction.

The effects of non-conservative forces can be very noticeable in some scenarios. The system on the left of the image above has only conservative forces(the elastic force and gravity) acting on the rock. The rock can, in theory, just keep falling and going back up.

However, on the system on the right, there is no spring to keep the object back up in the air. When the rock hits the ground, it will be subject to non-conservative forces and lose all of its mechanical energy, which would be in the form of kinetic energy. All of this energy is dissipated as sound, heat, and in the deformation of the ground below the rock.