
185Motion Under a Central Force
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or because f(r) = −μg(r)
3
0
ρ
ff() ()+
′
< (6.67)
which is identical to Equation 6.62. In particular, if the force law is a power law
f(r) = −cr
n
where c is a positive constant, then the condition for stability reads
ρ
n−1
(3 + n) > 0. (6.68)
In order for the condition 6.68 to hold, n is restricted to values greater than −3. Hence, we con-
clude that, of the central forces of the form −cr
n
, only those for which n > −3 will provide stable
circular orbits. We notice that the inverse square law (n = −2) gives stable circular orbits as does the
law of direct distance (n = 1). The lat