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A particle of mass m moves with angular momentum in the
Chapter 8, Problem 8.23(choose chapter or problem)
A particle of mass m moves with angular momentum \(\ell\) in the field of a fixed force center with
\(F(r)=-\frac{k}{r^{2}}+\frac{\lambda}{r^{3}}\)
where \(k\) and \(\lambda\) are positive.
(a) Write down the transformed radial equation (8.41) and prove that the orbit has the form
\(r(\phi)=\frac{c}{1+\epsilon \cos (\beta \phi)}\)
where c, \(\beta\), and \(\epsilon\) are positive constants.
(b) Find c and \(\beta\) in terms of the given parameters, and describe the orbit for the case that \(0<\epsilon<1\).
(c) For what values of \(\beta\) is the orbit closed? What happens to your results as \(\lambda \rightarrow 0\)?
Questions & Answers
(1 Reviews)
QUESTION:
A particle of mass m moves with angular momentum \(\ell\) in the field of a fixed force center with
\(F(r)=-\frac{k}{r^{2}}+\frac{\lambda}{r^{3}}\)
where \(k\) and \(\lambda\) are positive.
(a) Write down the transformed radial equation (8.41) and prove that the orbit has the form
\(r(\phi)=\frac{c}{1+\epsilon \cos (\beta \phi)}\)
where c, \(\beta\), and \(\epsilon\) are positive constants.
(b) Find c and \(\beta\) in terms of the given parameters, and describe the orbit for the case that \(0<\epsilon<1\).
(c) For what values of \(\beta\) is the orbit closed? What happens to your results as \(\lambda \rightarrow 0\)?
ANSWER:Step 1 of 5
(a)
The force on the particle is given as,
\(F(r)=-\frac{k}{r^{2}}+\frac{\lambda}{r^{3}}\)
Here, k and \(\lambda\) are positive constants.
Let, \(u=\frac{1}{r}\). Therefore,
\(F=-k u^{2}+\lambda u^{3}\)
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