Let's take an example that is not a conservative force
to help contrast this and bring out the differences.
Think about friction.
For nonconservative force, the work done by the force will be path dependent.
It will depend on the path that your object took.
So here what we have is an object, the box.
We have a green box and we have a red box,
both starting in the same spot which we've called Initial here.
They're both going to go to the final location,
as you can see here, but one of them is going to go up and then down
the slope and then go back to the final location
and the other one just goes straight across
and approaches the final location directly.
As you can imagine, the amount of work done by friction
as the objects slide across the path
is going to be more and more depending on how long the distance is,
because again, the work as the force times the distance
with the work and the force being in the same direction or in the same plane
if they're pointing on opposite directions as is the case with friction here.
So the red box which went up the path and then went down the path
and then back to our final location
took a much longer journey to get to the final location,
which means the work, which is the force of friction times the distance
is going to be of much greater quantity because the box took such a long path.
So the friction took a lot more energy,
a lot more of the energy that the box had, went into heat
if you took a longer path while friction
was opposing your motion the entire time.
We can actually calculate what the work done by friction
would be for any part of this path
just by remembering that the friction always opposes motion.
So wherever the box is moving,
whether it's up a slope or across a slope or down a slope,
the friction is always acting exactly opposite that direction of motion.
So the angle between friction and the displacement
is always going to be 180 degrees.
So the friction in this case will be always a contributing of a work
which is in the negative sense.
So it's always taking away any kinetic energy your object might have.
The distance as I said between path 1 and path 2
are very different in this situation,
meaning that the work done by gravity, sorry, the work in my friction
will be much greater for object 1 than it will be for object 2,
and this is what tells us that this is a nonconservative force,
that work done is very path dependent.
Just to compare and contrast, I always said is that for a conservative force,
the work done by, for example,
gravity only depends on the initial and final location of your object
regardless of the path such that the total energy is conserved,
whereas for a nonconservative force,
the work done by friction as an example
is certainly path dependent
because the longer a path you go on, the more work friction does.
The work that friction does is converting energy into heat.
So the energy of the universe is always still being conserved,
but in this case, some of the energy is leaving our object
and going to heat, unlike the case with a conservative force.