The total mechanical energy, E sub mech
is defined as the sum of the kinetic energy
and the potential energy, so the sum
of K plus U, where U is the potential energy function
associated with the conservative force--
with an appropriate choice of zero point.
If there are multiple conservative forces acting
on the system, then U will be the sum
of individual potential energy functions
for each conservative force with an appropriate choice of zero
point for each individual function.
We've also seen that the change in the kinetic energy
plus the change in the potential energy
is equal to the change in the total mechanical energy.
And that's just basically the derivative of the equation
that I wrote above.
And this change in the total mechanical energy
will be 0 for conservative forces.
In addition, we've seen that the change in potential energy
is related to the conservative work done in the system
so that W sub C, the conservative work done,
is equal to the negative of the change in the potential energy.
But we also know that the total work done
includes both conservative and non-conservative forces.
So the total work done is equal to the sum of W sub C,
the conservative work done, plus the non-conservative work done,
and that that total work is equal to the change
in kinetic energy.
Now, I can rewrite that equation by rewriting
the conservative work in terms of the potential energy.
So I could write this, now, as the non-conservative work
minus the change in the potential energy
is equal to the change in the kinetic energy.
This is, again, just a restatement of the work
kinetic energy theorem.
Now, by rearranging this equation,
this means that the non-conservative work is equal
to delta K, the change in kinetic energy,
plus delta U, the change of potential energy.
But that's equal to the change in the total mechanical energy.
So the non-conservative work is equal to the change
in the total mechanical energy.
This is a sufficiently important result
that I'm going to write it by itself in a box.
So the non-conservative work is equal to the change
in the total mechanical energy.
So what we've shown here is that the result
of any non-conservative work on the system
is to change the total mechanical energy
of the system.
Now, if there is no non-conservative work,
if the forces acting on the system are all conservative,
then the total mechanical energy remains unchanged
now that Emech is 0.
And we say to the total mechanical energy is conserved.
Now if there is a non-conservative force,
it turns out that in most cases the work done
by non-conservative force is negative,
resulting in a negative change in
the total mechanical energy-- in other words,
in the removal of total mechanical energy
from the system.
It's lost from the system.
Now we've been talking here only about mechanical energy, which
we've defined as the sum of kinetic energy
and potential energy.
But one can also talk about non mechanical energy.
The most common example of which is loosely referred to as heat.
As an example, in the case of friction,
which is a non-conservative force,
we know that the action of friction generates heat.
What's happening is that the non-conservative work done
by friction is negative and, therefore, is reducing
the mechanical energy, the total mechanical energy,
according to this expression, but where does that energy go?
We said that it's removed from the system-- in terms
of mechanical energy, that's true.
But we can think of it as, at the same time,
causing an increase in the non mechanical energy, or the heat
energy, of the system.
So if one were to keep track of both mechanical and
non-mechanical energy, then it turns out
that the total energy, mechanical and non-mechanical,
the total energy of a system is conserved,
even if non-conservative forces are acting.
However, it's difficult to recover
the non-mechanical energy and change it back
into mechanical energy.
These non-conservative processes are not reversible.
This is a complicated topic, and it's
the province of more advanced courses in thermodynamics
or statistical mechanics, which you may encounter later.
In this course, we will confine our attention
to mechanical energy, the sum of kinetic energy
and potential energy, and will treat non-conservative force
as removing, or adding, mechanical energy
to the system.