We're now going to introduce the concept of potential energy.
Let's begin by considering a system where
a conservative force is acting.
So I'll consider a conservative force, which I'll call F sub c.
So force set to force, the work integral is path independent.
So the work integral for this force is the integral of F sub
c .ds, which is the path going from point A to point B.
And for a conservative force, this integral does not depend
upon the path from A to B. It's independent of A and B.
So it depends only upon the endpoints.
So this is a path independent integral.
And since it depends only upon the endpoints,
I can write it, since it's going to be an integral from point A
to point B-- this integral must be
equal to some function of the final point.
So some function of r sub B minus some function
of the initial point r sub A. And to just get this integral
from A to B, in our usual way of evaluating a definite integral,
it's going to be equal to some function of r
B minus some function of r A, since the integral depends only
upon the endpoints.
Now, let's call this function-- I'm
going to make a sort of funny choice here-- so let's
call this function minus U as a function of position factor r.
And we'll see the reason for this funny choice of minus sign
in just a moment.
So now with this definition, my work integral, which again,
is the integral of F sub c .ds from point A to point B is now
minus U of r B minus minus U of r A. So in other words,
that's minus U of r B-- so minus minus gives me a plus U of r
sub A.
For shorthand, I can write that as minus U sub B plus U sub A.
And since we start out at point A and go to point B,
notice that I can also write this
as the negative of the change in U. Since the final value of U
is r U at B and the initial value is U at A,
so this minus U B plus U A is equal to minus delta
U, the change in U as we go from the initial
to the final position.
And note that in addition to that, given that this
is the work integral, I can summarize that
by writing that-- so I'll say, note
that delta U is equal to the negative of the work done
going from point A to point B. Now,
let's write the work kinetic energy theorem using this newly
introduced U function.
So the work kinetic energy theorem,
which tells us that the work done, which we've seen
is minus U sub B plus U sub A is equal to the change
in kinetic energy delta k, which I can write as K sub B minus K
sub A. Or I could also write that as 1/2 M V B squared
minus 1/2 M V A squared.
So this is just me stating that the work done on the system
is equal to the change in kinetic energy.
And I can write the work in terms of my function U
that I've introduced here to minus U sub B plus U sub A.
So I'm going to rearrange this equation now--
basically the one involving U's and the one involving
kinetic energies-- so that I have all the terms involving
point A on one side and all the terms involving
point B on the other side.
So rearranging, I get that at point A 1/2 M V
A squared plus U sub A is equal to at point B 1/2 M V B squared
plus U sub B. Now, notice however,
that there is nothing special about how
I chose the points A and B. They're completely arbitrary.
So that means that this equation must
be true for any points A and B.
And what that means is that each side
must be equal to the same constant
for any point in the system.
So in fact, we can write that K plus U for any point
must be able to some constant, which
I'm going to call E sub mech.
So K here is the kinetic energy.
U is my function that I introduced,
and we're going to call it the potential energy.
And E sub mech-- and remember, this E sub mech
here is a constant.
E sub mech is something that we call
the total mechanical energy.
Now, what we've done here is that we've
shown that the total mechanical energy, which
is the sum of the kinetic energy and the potential energy,
is a constant under the action of a conservative force.
In other words, if we look at this equation
and look at how it changes with time,
the change in the kinetic energy, plus the change
in the potential energy is equal to the change
in the total mechanical energy.
And this is 0 for our conservative force.
So in other words, the change in kinetic energy
is balanced by the change in potential energy, such
that the sum is 0 when the force acting is conservative.
Now, we've now introduced the very important concept
of the potential energy that is associated
with the conservative force.
And we see that the change in the potential energy,
the way we defined it, the change in potential energy
is equal to the negative of the work integral
for our conservative force going from point A
to point B. Now in fact, it's actually
only the change in the potential energy
that has physical significance.
We'll be concerned with potential energy differences
or changes.
The actual value of the potential energy
itself doesn't matter.
We're free to choose any convenient reference
point, or 0 point, for measuring the potential energy.
It's equivalent to choosing a coordinate origin when
we're talking about positions.
Now, the potential energy change is
related to the work done by conservative forces.
But we know that in general, work
can also be done by non-conservative forces.
Although, that work by non-conservative forces
will depend upon the path taken from point A to point B.
So in general, the total work is given
by the sum of the conservative work--
the work done by conservative forces, which
we can relate to a potential energy change,
and the non-conservative work done.
And it's this total work that tells us
what the change in the kinetic energy is.
Now we'll soon see that in the presence
of non-conservative forces, the total mechanical energy, which
is K plus U, is not a constant.