What's a conservative force?
Conservative forces are any force
wherein the work done by that force on an object
only depends on the initial and final positions of the object.
In other words, the work done by a conservative force on a mass
does not depend on the path taken by that mass.
If the work done by a force follows this rule,
then we call it a conservative force.
For instance, the gravitational force on a 5 kilogram mass
is 49 newtons.
If the mass moves downwards by an amount of 6 meters,
the work done by gravity is going to be 294 joules.
Now let's start over.
Say the mass again moves down 6 meters.
But then it moves up 6 meters, then down again 6 meters.
The work done by gravity for the first downwards trip
was 294 joules.
Then for the upwards trip, since the gravitational force
is pointing in the opposite direction of the motion
of the mass, the work done by gravity
is going to be negative 294 joules.
Then for the last trip downwards,
the work again is positive 294 joules.
That means that the total work done on the mass from gravity
is still 294 joules, just like it
was when the mass was lowered only once.
In other words, the work done by the gravitational force
doesn't depend on the specifics of the path taken by the mass.
The work done by gravity only depends
on the initial and final position of the mass.
In fact, you could allow the mass
to take any path from this initial point
to the final point, and the work done by gravity
is still just going to be 294 joules.
Because the work done by gravity doesn't
depend on the path taken, we call
gravity a conservative force.
The force exerted by a spring is another example
of a conservative force.
The total work done on a mass by a spring
does not depend on the path taken by the mass.
It only depends on the initial and final positions
of the mass.
The term conservative comes from the fact
that conservative forces conserve mechanical energy,
whereas non-conservative forces do not
conserve mechanical energy.
Mechanical energy is kinetic energy and potential energy.
An example of a non-conservative force is friction.
If I move a mass along a table from point A to point B,
friction does a certain amount of negative work
on the mass, which creates some thermal energy.
If instead of going straight from A to B,
I make the block go from A to B back to A over and over again,
the work done by friction will become larger and larger.
And it'll generate more and more thermal energy.
Because the work done by friction
depends on the path taken, friction
is not a conservative force.
Similarly, air resistance is not a conservative force
since the work done by air resistance
depends on the specifics of the path taken.
It's useful to note that if a force is conservative,
you could define a potential energy for that force.
That's why conservative forces like gravity and spring forces
have potential energies associated with them.
And non-conservative forces like friction
do not have potential energy associated with them.
This makes sense because if you do
work against the gravitational force
by lifting a mass in the air, you
can get that energy back out by letting the mass fall down,
turning potential energy into kinetic energy.
Similarly, if you do work against the spring force
by compressing a spring, you can get that energy back
out by letting the spring decompress, which
turns the stored potential energy into kinetic energy.
But if you do work against the force of friction,
you'll have a hard time trying to get that energy back out.
The energy's been dissipated into the form of thermal energy
and is now randomly distributed along the ground
and into the block.