Center
of Mass
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Consider
an arbitrary system of particles that interact under forces that obey the weak
form of Newton’s Third Law, i.e. the weak law of action and reaction,
i.e.
The
total force, Fi,
on a particle is related to the particle’s momentum, pi,
as
It
is assumed in Eq. (2) the mass of the particle is constant. Let F(e)i
be the external force acting on the ith particle. This is the force that
is caused by an influence that is external to the system, i.e. it is that part
of the total force acting on the particle that is not a result of the force
impressed upon it by any particle that is considered part of the system. Let Fji
be the force on the ith particle due to the jth particle. Then the
total force on the ith particle is the sum of these two forces, i.e.
Sum all
the forces to get
The term
on the left side of Eq. (4) can be expressed as
The
first term on the right side of Eq. (4) is simply the total external force, F(e)i,
acting in the system, i.e.
The second term on the right side
of Eq. (4) vanishes due to the weak law of action and reaction. The center of
mass, R, is defined as
where
is the total
mass of the system and is constant since the mass of each particle is
constant by assumption. Eq.(5) then reduces to
Thus the
center of mass moves as if the total external force was acting on a particle of
mass M and located at the position of the center of mass. If F(e)
= 0 then
the center of mass moves with uniform velocity. Thus is the center of mass was
initially at rest then it will remain at rest, i.e. R = constant. This is
known as the center-of-mass theorem or the conservation of the center
of mass law.
Note:
The center of mass, a vector quantity, should not be confused with the zero
momentum frame which, as the name indicates, is a frame of reference. The
later is sometimes called the center of mass frame and as such can be the source
of some confusion.