Let's consider an example of motion, in which we want
to use our energy concepts.
So suppose we have a ramp which is circular, radius r.
And we have an object here.
And now, here's a surface a certain distance,
d, that has friction, where the coefficients of friction
is non-uniform.
So we'll write it mu-0 And we'll write it as mu-1x.
We'll take x equals zero.
And then here, there's a spring and a wall.
And here there is no friction.
Now as we drop, assuming we have enough height here, h,
so that this block will slide across the friction,
make it all the way across and start compressing its spring.
What I'd like to find is how much
the spring has been compressed.
OK, now how do we analyze this?
Well the key is to use our energy principle,
where we have w-external equals the change
in mechanical energy.
And the key is to, what we're going
to do is the tool that we're going to use
is what we call energy diagrams for the initial.
So what does this mean?
We want to choose, we want to first identify
the initial and final states that we're referring to.
So, in our picture, here is our initial state.
And I'll draw the final state in when the spring has been
compressed a distance, x-final.
So I drew it on my diagram.
So we have initial and final states.
And now what we want to do is choose
reference points, zero-points, zero-point
for each potential function.
And show that on our diagram.
So for the potential energy of gravity, here,
if we chose this to be y, then this y equals 0.
And u at y equals 0 is 0.
And we'll call that the zero point for gravity.
And the zero point for the spring
is when this x, now I'm going to call here x equals 0.
So, let's call this a variable, I can call it anything I want.
I'll call it u-final.
And this is where u is zero.
So u-final just measures the stretch of the spring.
And so at y equals 0 is 0.
And u spring little u equal 0, 0.
So now I can identify my energies.
So let's talk about the initial energy
is all gravitational potential.
That's mgy.
It's starting at rest.
And e-final well, here, this is the distance
where it comes to rest, also.
So there's no final kinetic energy.
There's no gravitational potential energy,
because we're on the surface at y equals 0.
But our spring has been compressed
by 1/2 k little-u-final square.
Now, in terms of our external work
equals the change in mechanical energy,
we have now identified the right-hand side
using these tools of the energy diagrams.
And I can write my description u-final squared, minus mgy.
Now what I have to do is think about the friction
force, f kinetic friction, as the object moves.
This friction force is non-uniform.
If we were to draw n, and mg, then our friction force here
is equal to the integral minus the integral from x equals 0,
to x equals d.
Equals 0 to x equal d, of the friction force,
which is equal to the coefficient of friction, ukndx.
Now that 0 to d, notice that our coefficient of friction
is varying.
And I chose it intentionally to show you that friction
is really an integral.
So we have mu-0 plus mu-1 xmgdx.
Let's put our integration variable in there.
And these are just two separate integrals.
The first one is easy.
It's minus mu-0 mgd.
And the second one is -mu-1 mg.
And we're just integrating x-prime, dx prime
between 0 and d.
So that's simply d-squared over 2.
And that's equal to ku-final squared minus mgy.
Let's write that as y-initial.
And we're starting it at y-initial equals h.
And so, even though this is complicated,
I can now solve for how much this spring has been compressed
with a little bit of algebra.
And so I'm just going to bring a bunch of terms
over to the other side.
And take the square root of 2, divided by k of mgy-initial.
Minus mu-0 mgd.
Minus mu-1 mgd-squared over 2.
And that's how much the spring is compressed.
Notice what I did not do was divide this up
into a bunch of different motions.
I picked an initial state.
I picked a final state.
I drew my energy diagrams with my zero points.
I described the key parameters of initial and final states,
y-i and u-final.
I defined the initial mechanical energy,
the final mechanical energy.
And then applied the work energy,
work mechanical energy principle.
I had to integrate the friction force because it
was non-trivial and solve for how much the spring has
been compressed.