Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: POE summary--long! (but *last* from me)



On Fri, 11 Apr 1997, Mark Shapiro wrote:

I'd like to get back to the original question for a moment. As I
understand it you have an observer on a moving train who is looking at
tree someplace on the ground. The observer on the train calculates the
K.E. of the tree. The train is then decelerates to a stop, and the
observer calculates the K.E. of the tree again and finds that it is zero.
Someone then wanted to know where the lost K.E. went.

My naively classical mind said that it didn't go anywhere - that
the observer on the train decelerated so the change in KE was not measured
in an "inertial" frame so it was a physically meaningless number.

I don't see how this changes if you view the problem from a GR
point of view. Give the guy on the train an accelerometer and give a guy
handcuffed to the tree an identical accelerometer. Let the accelerometers
be designed in such a way that the measure the "local" acceleration in
conformity with GR.

O.K. I am, therefore, justified in understanding that the accelerometer
of the person handcuffed to the tree correctly indicates (as it *must*) an
acceleration of 9.8 m/s^2 upward.

The observer handcuffed to the tree will say that the
reading on his accelerometer never changed (and neither did his KE).

Well yes and no. Of course, the accelerometer indicates a *time-
independent* acceleration--again, of 9.8 m/s^2 upward. If he takes that
reading literally he must conclude that the speed of the tree--and,
therefore, its kinetic energy--are changing. But the more general
question is, "How *does* the handcuffed observer make sense of his
accelerometer reading?" Here are three possibilities:

1) He can take the accelerometer's reading at face value. That is,
notwithstanding the abundant psycho-visual indications that he is *not*
accelerating, he accepts that he and the tree are, indeed, accelerating
upward at 9.8 m/s^2. After awhile, this makes sense to him because when
he adds up all of the *measurable* forces acting on him (of which the
contact force with the ground is by far the largest), the net force turns
out to be precisely the amount needed to produce the indicated
acceleration.

He is happy; physics makes sense (in spite of the misleading visible
cues.)

2) He can deny the direct indication of the accelerometer, stubbornly
insist that he is *not* accelerating, and *interpret* the accelerometer's
reading as an error produced by the existence of a mysterious force that
he chooses to call a "gravitational force." This force *cannot* be
directly detected, but it can be easily inferred from the requirement
that Newton's Second law be satisfied. Doing so he finds that he needs a
"gravitational force" of 9.8 N of "downward" on every kilogram of mass in
order to bring himself and the tree into static equilibrium.

He is happy; physics makes sense (once it is patched up with the
introduction of this unmeasurable force.)

3) He can choose a superposition of options 1 and 2 in which a *part* of
the indicated acceleration is real and the *other* part is an error
produced by a gravitational force. There are an infinite number of such
arrangements and with each one... You guessed it.

He is happy; physics makes sense (once he superposes correctly.)

I believe that option 2 is your preferred viewpoint. Fine; Newton agrees
with you. All three options work, but understand that option 1 is the
twentieth century viewpoint. (BTW, doesn't that option 3 just sound
*ridiculous*?)

The observer on the train will say that the reading on his accelerometer did
change at least momentarily (and thus his calculation of the change in KE
for the tree needs to be interpreted vary carefully.)

Don't get hung up, as others have, on the red herring of time-dependence.
We can always easily integrate over time if we need to.

He thinks for a minute and realizes that he did not measure a
change in KE for the tree, but rather his own change in KE as he was
decelerated.

That is one possibility. But hold on.

First, note that the accelerometer now indicates an acceleration which is
directed somewhere between "upward" and "backward." For the sake of
illustration, I'll assume an instantaneous acceleration of the train wrt
the ground of 5.0 m/s^2 (Wow!). Then the accelerometer will show 11.0
m/s^2 at an angle of 27 degrees "behind straight up."

Now let's put the three possible interpretations into parallel form with
those delineated for the observer handcuffed to the tree.

1') He can take the accelerometer's reading at face value. That is,
notwithstanding the abundant psycho-visual indications that he is only
accelerating horizontally toward the rear of the train, he accepts that he
and the train are, indeed, accelerating at 11.0 m/s^2 at an angle of 27
degrees "behind straight up." After awhile, this makes sense to him
because when he adds up all of the *measurable* forces acting on him (of
which the upward contact force and backward frictional force from his seat
are by far the largest), the net force turns out to be precisely the
amount needed to produce the indicated acceleration.

In the case of the tree, he adds its *apparent* acceleration with respect
to him (5.0 m/s^2 "forward") to his acceleration as indicated by the
accelerometer to find that its acceleration is 9.8 m/s^2 "upward." This
also makes sense since the *measurable* net force on the tree is
precisely the amount needed to produce this acceleration.

He is happy; physics makes sense (in spite of the misleading visible
cues.)

2') He can deny the direct indication of the accelerometer, stubbornly
insist that he is *not* accelerating, and *interpret* the accelerometer's
reading as an error produced by the existence of a mysterious force that
he chooses to call a "gravitational force." This force *cannot* be
directly detected, but it can be easily inferred from the requirement
that Newton's Second law be satisfied. Doing so he finds that he needs a
"gravitational force" of 11 N directed 27 degrees "forward of straight
down" on every kilogram of mass in order to bring himself into
instantaneous static equilibrium.

In the case of the tree, he adds his mysterious and unmeasurable
"gravitational force" to the measurable forces and finds a net force of
5.0 N directly "forward" for every kilogram of the tree's mass. This net
force precisely explains the observed acceleration of 5.0 m/s^2 forward
(not to mention the loss of KE by the tree.)

He is happy; physics makes sense (once it is patched up with the
introduction of this unmeasurable force.)

3') He can choose any one of an infinite number of superpositions of
options 1' and 2'. For instance, he might consider the "backward"
*component* of the indicated acceleration real and the "upward" component
an error produced by a downward "gravitational force." When he adds the
required downward "gravitational force" on him to the upward and backward
force from the seat, he finds a net force of 5.0 N "backward" acting on
every kg of his mass. This precisely explains the horizontal component
of his accelerometer reading, the part that he is willing to accept as
real.

In the case of the tree, when he adds his "gravitational force" to the
measurable forces, he finds a net force of zero. This makes sense since
the *apparent* "forward" acceleration of the tree is precisely accounted
for by his understanding that it is, in fact, *he* who is accelerating
"backward."

He is happy; physics makes sense (once he superposes correctly.)

Now I believe that option 3' is your preferred viewpoint. (Wait a
minute, didn't you agree with me before that option 3 was just
*ridiculous*?) That's O.K.; they all work. You liked the "total denial"
option before; now you like the "partial denial" option. No problem;
you've changed your mind; and Newton *still* agrees with you. But you
*must* acknowledge the inconsistency of your (and of Newton's) two
interpretations of the accelerometer readings. And again, understand
that option 1 is the (consistent) twentieth century viewpoint.

And now, at long last, I am very tired of this struggle. If you don't
agree with me, fine, maybe I'm wrong.

John
----------------------------------------------------------------
A. John Mallinckrodt http://www.intranet.csupomona.edu/~ajm
Professor of Physics mailto:mallinckrodt@csupomona.edu
Physics Department voice: 909-869-4054
Cal Poly Pomona fax: 909-869-5090
Pomona, CA 91768 office: Building 8, Room 223