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Re: varying mass systems: rockets included (medium length)

Cliff,
thanks for taking interest in this, I'll try to respond in the middle of the
text below

Situation without a varying mass system:

Consider our system of interest to be the 6 moving
particles. Therefore,
the sum of external forces = 0 = (total mass)*(acceleration
of center of
mass) and we conclude v_cm is constant.

Situation with a varying mass system:

Again the system of interest will be the 6 moving
particles, but at time
t_o
we will call the system of interest all 8 particles.
Hence, our system
of
interest now is a system whose mass varies with time.
Excusing the use
of a
step function change in mass, (we could make it a continous system
example),
we see that around t=t_o the velocity of the cm of our
system of interest
changed and we must conclude that the acceleration of the
cm is non-zero.

If we try to write an equation of motion, valid, say, for a small
interval
of time around t=t_o, we might try to get an equation form:

a_cm = stuff

or

(total mass) * a_cm = stuff

(this is what one does for the acceleration of a rocket in a rocket
problem.)

This equation clearly can't be:

(total mass)*a_cm = (sum of external forces)=0
If I understand you correctly the a_cm has been caused simply
by redefining
the
system of interest not because some "real" particles have
accelerated. You
are
describing a "make believe" acceleration and "make believe"
accelerations
require no real force. Is that correct?

Yes, you understand me correctly here.

<snip>

In the rocket in deep space propulsion problem, you deal
with a somewhat
similar situation. Your system of interest is losing mass
rather than
gaining mass. We typically utilize conservation of
momentum to obtain an
equation for the motion of rocket that typically appears as follows:

(total mass) a_cm = -U_exhaust* |dm/dt|

I think I am beginning to understand your equation now. Are
you once again
saying that we may consider a_cm to be caused by changing the
system of
interest
from the Rocket, fuel & exhaust to just the exhaust? But
this is not true.
The
center of mass of the system does not accelerate as rockets are fired.

The center-of-mass of the "rocket", (rocket + unburnt carried fuel) is
accelerating. As time progresses, the system of interest has less and less
mass, i.e. the circle we draw around stuff to define the system of interest
is arbitrarily including fewer of the particles; the reverse of the first
example where includes more particles.

What I
will call a "make believe" force has been created out of this "make
believe"
acceleration similar to what was described previously (I
think this "make
believe" force is what you are driving at). Consider the following
equation
which I believe is "really" true.

(mass of rocket and fuel) a_rocket & fuel = -V_exhaust* |dm/dt|


I believe the above as well, it is what I called the equation of motion for
the rocket, rocket equation for short.

My (mass of rocket and fuel) does have a "real" acceleration.
Your (total
mass)
has zero acceleration.

Just to clarify again, what I'm calling total mass is the same thing you are
calling "mass of rocket and fuel".

Your (total mass) and my (mass of
rocket and fuel)
may
be considered the same thing for very small changes in time.
So while the
acceleration of the center of mass remains zero the
acceleration of the
rocket
is not zero.


In this situation we don't hestitate to call the added terms
{-U_exhaust* |dm/dt|, in the rocket example} "the thrust "
or maybe "the
force of the thrust".

We even might be able to identify them has a physical force of
interaction
between parts of molecules of the unburnt fuel acting on each other
during
the burning process.

It seems to me that this is exactly the cause of the force.

I'd agree!




In both cases, we arbitrarily change what we view as being
the system of
interest at varying times

We may only do so when changes in time are very small.

I'm not sure why you said this, I think I agree; and I think both my
examples are viewing things this way.


(in the rocket case, it seems to be a more natural
division, but we never-the-less are arbitrarily excluding
more and more
burnt fuel from being a part of the system of interest.).
This is why
both
are examples of varying mass situations. What similarity,
if any, is
there
between the extra terms necessary for computing the motion
(acceleration)
of
the system of interest, and what are the nature of those terms.

Interesting thoughts. Let me know if any of my ramblings are in the
ballpark of
what you are thinking.

We're in the same ball park, I'm a little troubled that in one case, this
arbitrary shifting of what is the "system of interest", leads to an
acceleration that we seem to be able to identify with physical causes, but
in the first case not; maybe the answer is as simple as the fact that they
are two different examples. Some of my ramblings here is brought on by how
my graduate text in mechanics is treating varying mass problems, more
generally than just looking at rockets, (which has some special features to
it, which simplify matters); they look at the dangling string through a hole
in a table, as the string falls through the whole more and more mass as a
net force on it and you can look at the problem as a varying mass problem,
the system of interest being that part of the string which is dangling.

Joel