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AbigailM
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Just preparing for a physics prelim and working through previous exam questions.
A mass m moving in a circular orbit about the origin is attracted by a three dimensional harmonic potential,
[itex]U(r)=\frac{1}{2}kr^{2}[/itex]
What is the frequency of the orbit? If a small kick is supplied in the radial direction, what will be the frequency of the ensuing small oscillations in r?
[itex]k(r-r_{0})=m\omega^{2}_{0}r[/itex]
[itex]\frac{k}{m}\frac{(r-r_{0})}{r}=\omega^{2}_{0}[/itex]
[itex]\mathbf{Frequency\hspace{1 mm} of\hspace{1 mm} orbit\hspace{1 mm}}\omega_{0}=\sqrt{\frac{k}{m}\frac{(r-r_{0})}{r}}[/itex]
[itex]\ddot{r}=-\omega^{2}r \hspace{5 mm} \omega^{2}=\frac{k}{m}[/itex]
[itex]\omega^{2}=\frac{k}{m}=\frac{r\omega^{2}_{0}}{r-r_0}[/itex]
[itex]\mathbf{Frequency\hspace{1 mm} of\hspace{1 mm} oscillation\hspace{1 mm}}\omega=\omega_0\sqrt{\frac{r}{r-r_{0}}}[/itex]
Just wondering if my solution looks correct. Thanks for the help.
Homework Statement
A mass m moving in a circular orbit about the origin is attracted by a three dimensional harmonic potential,
[itex]U(r)=\frac{1}{2}kr^{2}[/itex]
What is the frequency of the orbit? If a small kick is supplied in the radial direction, what will be the frequency of the ensuing small oscillations in r?
The Attempt at a Solution
[itex]k(r-r_{0})=m\omega^{2}_{0}r[/itex]
[itex]\frac{k}{m}\frac{(r-r_{0})}{r}=\omega^{2}_{0}[/itex]
[itex]\mathbf{Frequency\hspace{1 mm} of\hspace{1 mm} orbit\hspace{1 mm}}\omega_{0}=\sqrt{\frac{k}{m}\frac{(r-r_{0})}{r}}[/itex]
[itex]\ddot{r}=-\omega^{2}r \hspace{5 mm} \omega^{2}=\frac{k}{m}[/itex]
[itex]\omega^{2}=\frac{k}{m}=\frac{r\omega^{2}_{0}}{r-r_0}[/itex]
[itex]\mathbf{Frequency\hspace{1 mm} of\hspace{1 mm} oscillation\hspace{1 mm}}\omega=\omega_0\sqrt{\frac{r}{r-r_{0}}}[/itex]
Just wondering if my solution looks correct. Thanks for the help.