(^72) Systems of Point Masses
Tj — rfe, we therefore find
^f = tJ x Ff> = £T&, = T&, (2.2.18)
where TCM, is the external torque of Fj ' about the center of mass and
TQM ^S tne total torque by the external forces. Using (2.2.16) above, we
find
^ = ±mjvj,ij=TcSM. (2.2.19)
Equations (2.2.5) and (2.2.19) provide a complete description of the motion
of the system of point masses in an inertial frame.
As an EXAMPLE illustrating (2.2.5) and (2.2.19), let us consider BI-
NARY STARS. A binary, or double, star is a system of two relatively close
stars bound to each other by mutual gravitational attraction. Mathemati-
cally, a binary star is a system of n = 2 masses mi and mi that are close
enough for the mutual gravitational attraction to be much stronger than
the gravitational attraction from the other stars. In other words, we have
F$ = -F^ and F[B) = F( 2 E) = 0. Using the equation for the linear
momentum (2.2.5), page 68, we conclude that rcM = 0 and TCM is con-
stant relative to every inertial frame O. We can therefore choose an inertial
frame with origin at the center of mass of the two stars. Applying (2.2.19)
in this frame, we find dLcAi/dt — 0, and by (2.2.16),
mi ti x ti + m.212 x x-2 = 0. (2.2.20)
EXERCISE 2.2.4.B Assume that mi = m^. Show that the two stars move
in a circular orbit around their center of mass. Hint: you can complete the
following argument. By (2.2.1), rcM = (1/2)(TI --T2). SO CM is the midpoint
between m\ and m.2. Hence, ti = —12, ii = —12, ti = —12- By (2.2.20) above,
2ri x Vi =0. This implies that ti and vi are parallel (assuming ti ^= 0). Since
dLcM/dt = 0, LCM is constant. By (2.2.14) on page 70, 2m\X\ x ti is constant
as well. Together with X\ x ti = 0, this is consistent with equations (1.3.27),
(1.3.28), and (1.3.29), page 36, for uniform circular motion.
Binary stars provide one of the primary settings in which astronomers
can directly measure the mass. It is estimated that about half of the fifty
stars nearest to the Sun are actually binary stars. The term "binary star"
was suggested in 1802 by the British astronomer Sir WILLIAM HERSCHEL
coco
(coco)
#1