Center of Mass
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Consider
a system of particles, all of which
move at constant speed in the inertial frame of reference S.
Let mk represent the mass of the kth particle whose
proper mass is m0k and whose velocity is vk.
If m0k ¹0
then it can be shown that
If
m0k = 0 then it can be shown that
where
E is the inertial energy of the particle. The total momentum, P,
of the system is defined as
rk is the position vector of the kth particle. The vector defined as
is known as the center
of mass [2]. Since mass is proportional to energy this vector is sometimes
referred to as the center of energy vector. The quantity
is
the total mass of the system. With these definitions the total momentum
can be expressed as
The vector
is the velocity of the
center of mass. Thus
Thus
if one is given a system of particles for which the velocity and mass of each
particle is known then the total momentum of the system can be found. If the
magnitude of the velocity of the center of mass is less than c then one can
transform to the frame S’ moving with velocity Vcm
with respect to S. In this frame the new center of mass vector will be constant
and therefore the new Vcm
in that system will be zero. The total momentum in S’ will be zero. This new
system is called the zero-momentum (ZM) frame. The ZM frame is sometimes
called the center of mass (C-O-M) frame since the center of mass is at
rest in this frame.
Note:
It’s possible that the system of particles could be such that the
magnitude of the center of mass frame is the speed of light. In such cases the
center of mass, as defined above, would still be definable. However, according
to the special theory of relativity, no observer can be at rest in that frame.
Therefore no ZM frame can exist in such a cases. For this, although the center
of mass is defined, the center of mass frame will be undefined while the center
of mass vector will be well defined.
For a system having a finite and continuous mass distribution the center of mass is defined using the stress-energy momentum tensor Tab. The energy density is defined as
A
mass element is then given by dm =
r dV.
Therefore the center of mass becomes
where
Notice
this definition allows for a finite about of radiation to have a finite and
well-defined center of mass. A similar method can be used for a system
composed of both discrete and continuous matter distributions [3].
For a detailed derivation
of the center of mass vector as applied to an EM field and its relation to
conservation of momentum see Momentum
Conservation.
[1] The
quote a the top of this page Light carries mass with it was
stated in a letter written by Einstein in 1905 to his friend Conrad Habicht (The
latter was written from Bern, Switzerland somewhere between June to Sept 1905).
The entire letter states
The
value of my time does not weigh heavily these days; there aren't always subjects
that are ripe for rumination... A consequence of the study on electrodynamics
did cross my mind. Namely, the relativity principle requires that the mass be a
direct measure of the energy contained in a body. Light carries mass with it.
[2] Relativity:
Special, General and Cosmological, Rindler, Oxford Univ., Press,
(2001), page 126
[3] The Principle of Conservation of Motion of the Center of Gravity and the Inertia of Energy, Albert Einstein, Annalen der Physik 20 (1906) - It was in this paper that Einstein first postulated the concept that radiation has mass and defined what is essentially a center of mass relation which included electromagnetic energy. I.e. Einstein wrote ...we assign to the electromagnetic field too a mass density (re)..