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

[Cliff Parker <cparker@EMPOWERING.COM>]

Here is my crack at it.

Joel Rauber wrote:

> 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?

> 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|

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.  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|

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

> 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.

> (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.

Cliff Parker