In section 8.2, we showed that the motion of two bodies interacting only with each other by central forces could be reduced to an equivalent one-body problem. Show by explicit calculation that such a reduction is also possible for bodies moving in an external uniform gravitational field.
In section 8.2, we showed that the motion of two bodies interacting only with each other by central forces could be reduced to an equivalent one-body problem. Show by explicit calculation that such a reduction is also possible for bodies moving in an external uniform gravitational field.
Related questions
Question
Transcribed Image Text:8-1. In section 8.2, we showed that the motion of two bodies interacting only with each
other by central forces could be reduced to an equivalent one-body problem. Show
by explicit calculation that such a reduction is also possible for bodies moving in an
external uniform gravitational field.
Transcribed Image Text:8.2 Reduced Mass
Describing a system consisting of two particles requires the specification of six
quantities; for example, the three components of each of the two vectors r, and
ry for the particles. Alternatively, we may choose the three components of the
center-of-mass vector Rand the three components of r=r, - rg (see Figure 8-1la).
Here, we restrict our attention to systems without frictional losses and for which
"The orientation of the particles is assumed to be unimportant, that is, they are spherically symmet-
ric (or are point particles).
287
288
8/CENTRAL-FORCE MOTION
CM
CM
R0
(a)
(b)
FIGURE 8-1 Two methods to describe the position of two particles. (a) From an
arbitrary coordinate system origin, and (b) from the center of mas,
The position vectors are r and r the center-of-mass vector is R, and
the relative vector rr
the potential energy is a function only of r= r-rl. The Lagrangian for such
a system may be written as
L-
+t
(8.1)
Because translational motion of the system as a whole is uninteresting from
the standpoint of the particle orbits with respect to one another, we may choose
the origin for the coordinate system to be the particles' center of mass-that is,
R0 (see Figure 8-1b). Then (see Section 9.2)
(8.2)
This equation, combined with r -r, - r, yields
(8.3)
m, + m2
(8.3)
my + mg
Substituting Equation 8.3 into the expression for the Lagrangian gives
= alt - Ur)
(8.4)
where u is the reduced mass,
(8.5)
We have therefore formally reduced the problem of the motion of two bod-
ies to an equivalent one-body problem in which we must determine only the motion
of a "particle" of mass u in the central field described by the potential function
83 CONSERVATION THEOREMS-FIRST INTEGRALS OF THE MOTION
289
U(r). Once we obtain the solution for r(() by applying the Lagrange equations to
Equation 8,4, we can find the individual motions of the particles, r, () and r().
by using Equation 8.3. This latter step is not necessary if only the orbits relative
to one another are required.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 1 images
