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- - **Finding frequency of oscillation**
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Finding frequency of oscillationI am a little stuck and would love a hint or two if anyone has got any for me. There's a particle with mass m that is trapped in a potential. $$U(x) = \frac{-U_o}{(\frac{a}{a})^2 + 1}$$ where $$U_o > 1$$ and $$a>0$$. Assuming the amplitude is small, what would be the frequency of oscillation? My plan of attack was to use Taylor Expansion for $U(x)$ considering the particle remains close to equilibrium (x=a). $$U(x) = U(a) + U'(a)(x-a)+ \frac{1}{2}U''(a)(x-a)^2 + ... \approx U_o + \frac{1}{2}k(x-a)^2$$ My problem now is trying to figure out frequency from this information. Perhaps there is some relation with Simple Harmonic Oscillation and a potential function. Thanks for any tips! |

Looks like there is a typo. You've got a over a squared plus one in the denominator of the right-hand side. |

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Should be... $$U(x) = \frac{-U_o}{(\frac{x}{a})^2 + 1}$$ |

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$\dfrac{dU}{dx} \approx \dfrac{2 U_0 x}{a^2}$ and thus the restoring force is given by $\kappa = \dfrac{2U_0}{a^2}$ should be cookie cutter from here |

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Wouldn't the restoring force be equal to the derivative of the potential function? What happened to the x in the in $\kappa = \dfrac{2U_0}{a^2}$ ? compared to $\dfrac{dU}{dx} \approx \dfrac{2 U_0 x}{a^2}$ where there is the x. Do we remove the x since we are assuming the amplitude is small? |

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Yes, the force is the derivative of the potential. The "spring constant", $\kappa$ is what I gave you. With that constant you can determine the period of oscillation from the commonly known formula for simple harmonic oscillators with no damping. |

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$F = \frac{2U_o}{a^2} = -kx = ma$ Then considering... $\sqrt{\frac{k}{m}} = 2 \pi f$ $\Rightarrow \frac{k}{4\pi^2 f} = m$ I then multiply both sides by $a$ $\frac{ak}{4\pi^2 f} = ma = F$ Then plug back in and solve for f $ f = \sqrt{\frac{a^3 k}{8U_o \pi^2}}$ However, I have a constant k that is now in the formula that I did not have originally. |

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and $k = \dfrac{2U_o}{a^2}$ so $f = \sqrt{\dfrac{a}{4\pi^2m}}$ now you are back to your original constants.... (you dropped the $m$ in the original) |

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Crystal clear. Thank you for your help as always! |

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