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Consider a particle of reduced mass orbiting in a central
Chapter 8, Problem 8.14(choose chapter or problem)
Consider a particle of reduced massµ orbiting in a central force with \(U=k r^{n}\) where kn > 0.
(a) Explain what the condition kn > 0 tells us about the force. Sketch the effective potential energy \(U_{\text {eff }}\) for the cases that \(n=2,-1\), and \(-3\).
(b) Find the radius at which the particle (with given angular momentum \(\ell\)) can orbit at a fixed radius. For what values of n is this circular orbit stable? Do your sketches confirm this conclusion?
(c) For the stable case, show that the period of small oscillations about the circular orbit is \(\tau_{\mathrm{osc}}=\tau_{\mathrm{orb}} / \sqrt{n+2}\). Argue that if ,\(\sqrt{n+2}\) is a rational number, these orbits are closed. Sketch them for the cases that \(n=2,-1\), and 7.
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QUESTION:
Consider a particle of reduced massµ orbiting in a central force with \(U=k r^{n}\) where kn > 0.
(a) Explain what the condition kn > 0 tells us about the force. Sketch the effective potential energy \(U_{\text {eff }}\) for the cases that \(n=2,-1\), and \(-3\).
(b) Find the radius at which the particle (with given angular momentum \(\ell\)) can orbit at a fixed radius. For what values of n is this circular orbit stable? Do your sketches confirm this conclusion?
(c) For the stable case, show that the period of small oscillations about the circular orbit is \(\tau_{\mathrm{osc}}=\tau_{\mathrm{orb}} / \sqrt{n+2}\). Argue that if ,\(\sqrt{n+2}\) is a rational number, these orbits are closed. Sketch them for the cases that \(n=2,-1\), and 7.
ANSWER:Step 1 of 8
(a) The central force is given as,
\(F=-\frac{\partial U}{\partial r}\)
Here, \(r\) is the position.
The potential energy of the particle in the central force field is given as,
\(U=k r^{n}\)
Here, \(k\) is the constant and \(k n>0\).
Substitute \(k r^{n}\) for \(U\) in equation (1.1)
\(\begin{aligned} F & =-\frac{\partial\left(k r^{n}\right)}{\partial r} \\ & =-n k r^{n-1} \end{aligned}\)
The force in the vector form,
\(\mathbf{F}=-n k r^{n-1} \mathbf{r}\)
If \(n k>0\), then the force vector \(F\) is opposite to \(r\). This means that the force is a restoring force.
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