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Situation without a varying mass system:particles. Therefore,
Consider our system of interest to be the 6 moving
the sum of external forces = 0 = (total mass)*(accelerationof center of
mass) and we conclude v_cm is constant.particles, but at time
Situation with a varying mass system:
Again the system of interest will be the 6 moving
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 systemexample),
we see that around t=t_o the velocity of the cm of oursystem of interest
changed and we must conclude that the acceleration of thecm is non-zero.
interval
If we try to write an equation of motion, valid, say, for a small
of time around t=t_o, we might try to get an equation form:If I understand you correctly the a_cm has been caused simply
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
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?
<snip>
In the rocket in deep space propulsion problem, you dealwith a somewhat
similar situation. Your system of interest is losing massrather than
gaining mass. We typically utilize conservation ofmomentum 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.
or maybe "the
In this situation we don't hestitate to call the added terms
{-U_exhaust* |dm/dt|, in the rocket example} "the thrust "
force of the thrust".interaction
We even might be able to identify them has a physical force of
between parts of molecules of the unburnt fuel acting on each otherduring
the burning process.
It seems to me that this is exactly the cause of the force.
the system of
In both cases, we arbitrarily change what we view as being
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 naturalmore and more
division, but we never-the-less are arbitrarily excluding
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.