Now that we've introduced
mechanical energy
in our potential
energy functions,
we're describing our
systems differently.
We talk about states.
We talk about the potential
energy of that state.
We talk about the mechanical
energy of that state.
Remember we're always
referring to a reference state
for a reference potential.
But in one dimension,
what we have
is that the
potential energy say,
in some final state, minus the
change of potential energies
from some initial state
was that integral x final
of the x component
of-- I'm going
to put c up there for
conservative force, dx.
And now, so the potential
energy difference
is the integral of the
force with the minus sign.
Now let's look at a
fundamental theorem
of calculus, which tells
us that any time you
take the difference of a
function between two end
points, then by
definition that's
the derivative integrated
with respect to dx.
So when we compare these two
pictures, this is a map here.
This is our physics,
how we define them.
That when we compare
these two pictures,
we see that we can recover
the conservative force
by taking the derivative
minus the derivative
of the potential function.
Here, force does not depend
on any reference point.
And when we differentiate
a constant, that's 0.
So this is independent
of the reference point.
And this enables us
to, when we think
about the potential function
and its first derivative,
then this tells us about forces.
Let's look at an example.
Suppose again we look
at our spring potential
where we're talking about
the potential energy
function of a spring
where at our zero
point where it was unstretched
was our reference point.
And if we plotted this
function-- so let's
plot that function.
So here is U of x versus x.
Now we can talk about
at any given point--
so suppose we're
at a point here.
Maybe our energy has
some fixed value.
Then the slope at this
point is equal to du dx,
and the force is
minus that slope.
So here you can see that
the slope is positive.
So the force is negative.
So I can write Fx like that.
So here Fx is negative,
so our actual force
is pointing inward.
When we're on this side
of the potential function,
my slope is negative.
So the x component of
the force is positive.
So my force is pointing
everywhere on this side
back to the unstretched length.
So knowledge of the
potential function,
also by knowledge of
its first derivative,
gives us information
about the force
at any point, any state
that the system is in.
So when we talk about
potential implicitly,
we also know what the force is.
And the potential function
is enough to tell us
what the force is at any point.
And let's just check
for this simple case.
This is minus du dx.
When you differentiate
that, you get minus kx,
and we know that's
the spring force.
So this is why we're
suddenly shifting our focus
to our state, our function u of
x, and it's first derivative.
