Re: Car acceleration

["John S. Denker" <jsd@MONMOUTH.COM>]

Bob Carlson wrote:

> Try looking at this document:
>
> http://www.mctc.mnscu.edu/~carlsoro/d2230.htm


Carl E. Mungan commented thereon:
> "The kinetic energy of the box is changing, so some force must be doing work."
> and later
> "But the vehicle has a linear acceleration, so there must be some
> force doing work."
>
> If you changed the last word in both cases to "pseudowork" these
> statements would be correct.

Actually, both statements (in their original context)
are correct without change.  This is a direct consequence
of the work-KE theorem.

In particular, we can approximate the truck as having
three components:
 -- drivetrain
 -- tires
 -- chassis
where the chassis is assumed to no moving parts
of any consequence.  The payload, if any, is assumed
rigidly attached to the chassis.

Then the work-KE theorem applies to the chassis
directly.  (Its applicability to the other components
is more complicated and less direct.)

We can even locate and identify the force in
question:  When the wheel first starts to rotate, it
rolls forward and its axle starts pushing against
its bearing.  This force is at the 3:00 position
on the bearing, if the truck is accelerating to
the observer's right.

> students wish it were so

Students ought to know it is so.  The work-KE
theorem is a theorem.  It is not very useful, and
it is probably applied wrongly more often than
rightly, but it is still a theorem.

But I may have taken Carl's student's statements out
of context.  I suspect that the intended meaning here
is that the students are wishing for an _external_ force
that supplies the kinetic energy, in analogy to the
mandatory external force that supplies the momentum.

Well, folks can wish for this all they want.
They're not going to get it.  Students sometimes
wish there wre 60 degrees in a radian.  Sometimes
they wish sqrt(a+b) would equal sqrt(a)+sqrt(b).
But it ain't gonna happen (for nonzero ab).

> (and this is why I like the intuitive
> concept of pseudowork)

And this is precisely why I don't like it.  Students
come in with only a vague notion of what "movingness"
is.  Their vague notion needs to be clarified in terms
of momentum and energy.  There are two concepts here,
momentum and energy.  I just can't imagine why anyone
would want to introduce a third concept (pseudowork)
which AFAICT just imposes extra brain-strain on the
students.

> Specifically, consider F dx.
> For ordinary "work", dx refers to the displacement of the point of
> application of the force. For our example of static friction acting
> on a car's tire, this is zero. What do we learn from this? That no
> energy is transferred from the road to the car.

Yes!!!

> Ordinary (ie.
> thermodynamic, first law) work tells us about energy transferred from
> one system to another. For "pseudowork", dx refers to the
> displacement of the center of mass of the car. This is nonzero
> indeed. What do we learn from this? That the kinetic energy of the
> car is changing. Pseudo (ie. mechanical, center of mass) work tells
> us about the change in kinetic energy of an object without
> distracting us with the thermodynamic question of the source of that
> energy.

But it would be so much easier to describe the
center-of-mass motion of the car in terms of momentum.
And there is no need to mention thermodynamics in
connection with this problem.

BTW, as I reported previously, I've asked a number
of real live physics researchers if they could
define "pseudowork" for me.  Precisely none of them
had ever heard the term (let alone used it).  They
assumed it was a trick question.  My buddy Paul did
have an answer, though:  "Pseudowork -- that's what
a department head does."

> "As the engine transmits power, forces are applied in a circular
> fashion to the axle as shown in the left part of the figure. If you
> add all these forces, you will get zero net force. Therefore, they
> cannot cause a linear acceleration. They do, however, provide a net
> torque as shown in the right part of the figure. This net torque
> causes an angular acceleration and does work as the wheel rotates
> through an angular displacement."

That statement (quoted from Bob's web page) is
misleadingly incomplete.  It identifies a bunch
of forces that do not contribute to the acceleration
of the chassis, but misleadingly overlooks the one
force that does contribute to the acceleration of
the chassis, namely the 3:00 force mentioned above.

> Caution! Friction also produces and contributes *nontrivially* to the
> net torque on the wheel. Everything you wrote in the above sentences
> would be equally true on ice, so it does not follow that the
> rotational work on the tires somehow leads to translational work on
> the car.

For that matter, no analysis of torques around the
axle will be able to describe the acceleration of the
chassis, since the chassis motion is non-rotational.
At some point you have to analyze the plain old forces.