Central Force Orbit
المؤلف:
Sidney B. Cahn And Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 1 , p 10
25-7-2016
1800
Central Force Orbit
a) Find the central force which results in the following orbit for a particle:
b) A particle of mass is acted on by an attractive force whose potential is given by
Find the total cross section for capture of the particle coming from infinity with an initial velocity v∞.
SOLUTION
a) We see that the orbit describes a cardioid as shown in Figure 1.1. Invoking the orbit equation yet again, we may find the
Figure 1.1
force:
(1)
where u = 1/r Calculating the derivatives of u :
(2)
and substituting into (1), we obtain
b) The initial impulse to solve for the scattering angle as a function of the impact parameter leads one astray into the realm of elliptic integrals. Instead, realize that the operative word is “capture” and construct the effective potential of the particle, where
and A is a constant of proportionality, and l is the angular momentum. Those particles whose kinetic energy exceeds Ueff (r0) will be captured. At r0, dU(r0)/dr = 0, so we obtain
and
The condition for capture becomes
Where l = mv∞b, and is the impact parameter. Rearranging, we find that 
The cross section is given by πb2, so
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