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[Computer] Consider a particle with mass m and angular
Chapter 8, Problem 8.25(choose chapter or problem)
[Computer] Consider a particle with mass m and angular momentum \(\ell\) in the field of a central force \(F=-k / r^{5 / 2}\). To simplify your equations, choose units for which \(m=\ell=k=1\). (a) Find the value \(r_{0}\) of \(r\) at which \(U_{\mathrm{eff}}\) is minimum and make a plot of \(U_{\text {eff }}(r)\) for \(0<r \leq 5 r_{0}\). (Choose your scale so that your plot shows the interesting part of the curve.) (b) Assuming now that the particle has energy \(E=-0.1\), find an accurate value of \(r_{\text {min }}\), the particle's distance of closest approach to the force center. (This will require the use of a computer program to solve the relevant equation numerically.) (c) Assuming that the particle is at \(r=r_{\min }\), when \(\phi=0\), use a computer program (such as "NDSolve" in Mathematica) to solve the transformed radial equation (8.41) and find the orbit in the form \(r=r(\phi)\) for \(0 \leq \phi \leq 7 \pi\). Plot the orbit. Does it appear to be closed?
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QUESTION:
[Computer] Consider a particle with mass m and angular momentum \(\ell\) in the field of a central force \(F=-k / r^{5 / 2}\). To simplify your equations, choose units for which \(m=\ell=k=1\). (a) Find the value \(r_{0}\) of \(r\) at which \(U_{\mathrm{eff}}\) is minimum and make a plot of \(U_{\text {eff }}(r)\) for \(0<r \leq 5 r_{0}\). (Choose your scale so that your plot shows the interesting part of the curve.) (b) Assuming now that the particle has energy \(E=-0.1\), find an accurate value of \(r_{\text {min }}\), the particle's distance of closest approach to the force center. (This will require the use of a computer program to solve the relevant equation numerically.) (c) Assuming that the particle is at \(r=r_{\min }\), when \(\phi=0\), use a computer program (such as "NDSolve" in Mathematica) to solve the transformed radial equation (8.41) and find the orbit in the form \(r=r(\phi)\) for \(0 \leq \phi \leq 7 \pi\). Plot the orbit. Does it appear to be closed?
ANSWER:Step 1 of 4
(a) The effective potential energy \(U_{eff}(r)\) is
\(U_{\mathrm{eff}}=U_{\mathrm{ef}}+U=\frac{\ell^{2}}{2 m r^{2}}-\frac{2 k}{3 r^{3 / 2}}=\frac{1}{2 r^{2}}-\frac{2}{3 r^{3 / 2}}\)
where the last expression results from the choice of units with \(m=\ell=k=1\)
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