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varying mass systems: rockets included (medium length)



While I'm sure the weighty subject thread will continue, its time to start a
new topic. It'll give our multi-tasking capabilities some excercise. I'm
rethinking my presentation of rocket motion in my introductory class. The
rethinking process in (you name the topic); always reveals shortcomings in
my personal understanding. Hence, my bringing the topic up for discussion.

Consider the following situation (We will analyze the situation in an
inertial frame of reference using a Newtonian paradigm): We go down to the
local gedanken physics supply house and obtain 8 non-interacting isolated
particles. We contrive to have two of them at rest at two arbitrary
locations and the other six distributed arbitrarily, but all having
identical velocity vectors. This means, for any individual particle, its
acceleration is zero and the particles continue in their state of uniform
motion.

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

We need to include some terms to take into account the changing mass of the
system. The question is what to call those terms.

They're not forces, they're not psuedo forces (inertial forces) or are they?

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|

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.

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


Joel Rauber
Joel_Rauber@sdstate.edu