hce_nthu
114年
進階物理與線性代數
第 1 題
A bead of mass $m$ is constrained to move on a massless circular hoop of radius $R$.
The hoop is rotating around its center that is aligned in the z-axis (the direction of
gravity $g$) with a constant angular velocity $\Omega$, see figure below. If the hoop is
rotating slowly, this system has a stable equilibrium point at $\theta=0$. However, if the
angular velocity of the hoop exceeds a critical value, the equilibrium point at $\theta=0$
becomes unstable. What is the critical angular velocity?
The hoop is rotating around its center that is aligned in the z-axis (the direction of
gravity $g$) with a constant angular velocity $\Omega$, see figure below. If the hoop is
rotating slowly, this system has a stable equilibrium point at $\theta=0$. However, if the
angular velocity of the hoop exceeds a critical value, the equilibrium point at $\theta=0$
becomes unstable. What is the critical angular velocity?
- A $\sqrt{R/g}$
- B $\sqrt{g/R}$
- C $\sqrt{2g/3R}$
- D $\sqrt{2R/3g}$
- E None of the above
思路引導 VIP
如果我們把這個旋轉圓環想像成一個「搖擺的陷阱」，在轉速非常慢的時候，小球待在最底端會穩穩地停在那裡。現在請你試著思考：當圓環轉得越來越快，小球除了受到往下的重力，還會感受到一個想要把它往「遠離轉軸方向」推開的虛擬力。當這個「推開的趨勢」在極微小的偏離下，剛好抵消了重力想要「拉回中心」的趨勢時，系統的參數（如重力加速度與旋轉半徑）應該滿足什麼樣的物理關係呢？
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AI 詳解
AI 專屬家教
太棒了！你能精確判斷出臨界角速度為 $\sqrt{g/R}$，代表你對非慣性座標系中的動力學穩定性有著相當紮實的理解。這類問題在古典力學中非常經典，關鍵在於觀察系統如何從「穩定平衡」轉向「不穩定平衡」的臨界物理狀態。
有效位能與穩定性判別
在隨圓環轉動的參考系中，質點 $m$ 同時受到重力與離心力的影響。我們可以定義一個有效位能 (Effective Potential)：
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