Re: positive and negative work

[John Mallinckrodt <ajmallinckro@CSUPOMONA.EDU>]

On Sun, 11 Nov 2001, Brian Whatcott wrote:

> At 09:11 AM 11/11/01 -0800, you wrote:
>
> > So the car exerts a force on itself?
>
> ... Yes, the car engine exerts a force which can resist this
> tendency to accelerate down hill.

I disagree.  It is the road that exerts that force.  Take away the
ability of the engine to do its thing (whatever that is) and I can
still call up the necessary force by, for instance, applying the
brakes.  Take away the ability of the road to do its thing (by,
for instance, coating it with glare ice) and you are in decidedly
more serious trouble.

> I identified the site of operation as the torque converter, in a case of
> interest.
>
> There does seem to be a real interest in how the work done by the drive shaft
> on the torque converter is reflected by the tire road interaction.
> I'm not sure I heard an explicit answer yet   :-)

I guess I don't see it as being all that mysterious.  Let's
confine our attention for now to the case of a car moving *down* a
slope at a slowly *decreasing* speed *without* any tire slippage.

 From a simple force standpoint here is what is going on:

Some internal thing--either a complicated thing that makes the
fundamental mechanism a bit hard to understand like an
engine/drive shaft/torque converter combination or a very simple
thing like a brake mechanism--acts to restrict the freedom of the
tires to roll.  As a result they exert a down-the-slope force on
the road and, by Newton's third law, the road exerts an
up-the-slope force on the tires.  The speed of the car is
decreasing *because* the up-the-slope force of the road is
greater than the down-the-slope component of the gravitational
force.

We can also look at the problem from *many* different work-energy
perspectives: (See the excerpts referred to earlier for more
info.)

1. From the frame of the road, both the road and gravity
contribute to (what I like to call) the "pseudowork" done on the
car. The contribution from the road is negative and the
contribution from gravity is positive, but the road's contribution
is larger in magnitude so the pseudowork is negative.
Accordingly, the bulk translational kinetic energy of the car
decreases.

2. From the frame of the road, only gravity does (what I like to
call) "external frame-specific work."  That work is positive and
can be shown to equal the change in the total energy of the car.
(Note that I do not consider gravitational potential energy to
"belong to" the car and that I *do* consider gravity simply to be
another external force on the car.) The car's total energy is the
sum of its bulk translational kinetic energy and its internal
energy.  In this case, the bulk translational kinetic energy
decreases and the internal energy increases, but the increase in
the internal energy is greater than the decrease in the bulk
translational kinetic energy.

3. Within the system of the car itself only the road does (what I
like to call) "external system-specific work."  That work is
positive and is equal to the increase in the internal energy of
the car.

The final two cases are intended to illustrate the frame-
dependence of the forms of work discussed in cases 1 and 2.  Both
adopt the reference frame of a truck that is moving down the road
at a constant velocity equal to the *initial* velocity of the car.

4. From the frame of the truck, both the road and gravity
contribute to the "pseudowork" done on the car. The contribution
from the road is positive (in contrast to case 1) and the
contribution from gravity is negative (also in contrast to case
1), but the road's contribution is larger in magnitude (as it was
in case 1) so the pseudowork is positive (in contrast to case 1).
Accordingly, the bulk translational kinetic energy of the car
increases (in constrast to case 1).

5. From the frame of the truck, both the road and gravity
contribute to the "external frame-specific work."  The road does
positive work and gravity does negative work, but the magnitude of
the work done by the road is now *much* greater than that done by
gravity (because the road force is larger AND it acts through a
larger distance.)  Again, this work is equal to the change in the
total energy of the car, but in this case that energy increases by
more than it did in case 2 because *both* the bulk translational
kinetic energy and the internal energy increase.

John Mallinckrodt      mailto:ajm@csupomona.edu
Cal Poly Pomona          http://www.csupomona.edu/~ajm