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A particle of mass m moves with angular momentum in the

Chapter 8, Problem 8.23

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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 \(\lambda \rightarrow 0\)?

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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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