PROFESSOR: What is a conservative force, and what
is a non-conservative force?
Let me start with an example of a conservative force:
gravity is a conservative force.
Let this be point A, and this is point B. We have a mass--
little m--
that's going to be moved from A to B. If the work done by
gravity is independent on routing that I choose from A
to B, which is the case, then the force is called a
conservative force, and that is the case.
If I go from A to B with that object little m, I go
vertically down-- gravity is in this direction--
and then I go horizontally, the work done by gravity will
be plus mgh if the vertical separation between A
and B equals h.
If I go this way, no work will be done by gravity.
If I move this object horizontally, and then I go
down, the work done by gravity will be mgh, but if I go this
way the work done by gravity will also be plus mgh.
If I go this way, and I finally arrive at B, the work
done by gravity will also be plus mgh.
Whenever the work done by a force in going from A to B is
independent of the [UNINTELLIGIBLE],
we call that force--
a conservative force, and in Newtonian mechanics, gravity
is a conservative force.
it
If the work done does depends on the routing, we call the
force non-conservative.
Friction, air drag--
which are resistive forces--
are non-conservative.
Let's look at this little bit closer.
I go from A to B. I have an object here--
I have my purse here, there it is.
There's friction, and I'm moving this along in a
horizontal direction, and I want to go from A to B.
I can go to in different way--
I can do this, and finally end up at point B. The work done
by friction now does depend on the path.
What is the work done by friction?
Let's first take some paths in this direction, and
we mark like so.
The frictional force always opposes the direction of the
velocity, so right here the frictional force would be in
this direction.
Right here, the frictional force would be in this
direction, and right here, the frictional force would be in
this direction.
The work done by the frictional force is the dot
product between the frictional force and dl is the
path that it takes.
There's a little element dl, and here, this is a little
element dl.
Work is a scalar.
I have the friction--
the force is constant in magnitude--
and it really obvious that the work done by the frictional
force depends very strongly on the path, whether I go this
way, or whether I would go this way.
In the case of my purse, the frictional force would be the
mass of my purse--
it's not too much, is it--
times g times the kinetic friction coefficients.
It's the normal force which the pad pushes up so that the
object stays in place.
A side view: this is my purse, there's a gravitational force
down, mg is normal [? force up ?]
these two are the same in magnitude.
The frictional force is normal force times the kinetic
friction coefficient, as long as this
object is moving around.
If the trajectory from A to B is a straight line, and if
this distance equals d, then the work done by the
frictional force would be minus f
frictional force times d.
It's negative because the l is in this direction, and the
force is in this direction--
friction always does negative work.
The work done by Walter Lewin--
if I push in a horizontal direction, I'm not pushing
down on it, I'm pushing in a horizontal direction.
That work done by me is plus frictional force times d.
If now I choose a different [UNINTELLIGIBLE]
so that the total length of this path equals 2d, then this
would equal minus 2 f friction times d.
The work done by Walter Lewin would have
been twice as much--
2 f friction times d.
The net work on this object going from A to B is 0--
there is no change in potential energy, and there's
no change in kinetic energy, because the net work done on
the object is 0.
I'm doing positive work--
what happens with that positive that I am doing?
That goes into heat--
the friction with the paper will produce heat, and maybe a
little bit of noise, but it'll be largely heat.
When we deal with air drag--
with cars and airplanes which move--
the engines do positive work.
The air drag, which is a resistive force, is doing
negative work.
If you drive your car from Boston to New York, and the
work done by friction is very, very different than if you go
from Boston to New York via Washington.
The engine has to do positive work, and it is immediately
obvious if you go from Boston to New York that the engine
does much less work than if you go from Boston to New York
via Washington.
I hope I have convinced you that frictional forces,
resistive forces in the form of air drag, are
non-conservative forces.
The work they do in moving an object from A to B
depends on the path.
If the work done going from A to B does not depend on the
path, or not depend on the routing, then that force is
called a conservative force.