Environmental Engineering Reference
In-Depth Information
9.3 REDUCED MASS
The remainder of this chapter will focus upon solving problems involving the
motion of two bodies under their mutual gravitational interaction. There being two
bodies, we need two vectors (and hence six numbers) to specify their co-ordinates
at some instant in time and the motion of each particle is determined by solving
the corresponding equation of motion, i.e.
1
m
1
F
(
r
),
x
1
=−
¨
1
m
2
F
(
r
),
x
2
=+
¨
(9.29)
where
r
x
1
is the relative position vector and
m
1
and
m
2
are the masses of
the two bodies
1
. We assume that the system is isolated and that the force acting
upon the particles depends only upon their relative positions (this is of course true
for gravitational interactions). The general configuration is illustrated in Figure 9.6.
At first sight, these are two coupled second order differential equations (they are
coupled since the relative position depends upon
x
1
and
x
2
) and as such they
might require some effort to solve. However the situation can be simplified very
substantially once we appreciate that the centre of mass of the system moves with
a constant velocity since no external forces are acting. As a result, we can trade
off the six numbers which specify the co-ordinates of the two particles for three
numbers specifying the co-ordinates of the centre of mass,
R
, and three more
numbers specifying the relative positions of the particles,
r
. The motion of the
centre of mass is easy and all of the interesting dynamics resides in the behaviour
of the vector
r
. Let us put this intuition into mathematical language. What we
have described is a change of variables, i.e. we aim to recast Eq. (9.29) in terms
=
x
2
−
m
1
r
x
1
m
2
x
2
O
Figure 9.6
Two masses moving under the influence of their mutual gravitational interaction.
1
In this equation we have introduced a shorthand notation that is very common in dynamics. We
represent the derivatives with respect to time by placing 'dots' above the object being differentiated.
e.g.
d
2
x
d
t
2
.
d
x
d
t
x
≡
and
x
≡
