hce_nthu
112年
化學與物理
第 31 題
A rectangular buoy with cross section $A$ and thickness $h$ floats on the water surface. Jane waits for the buoy to reach static equilibrium and pushes the buoy $1\text{ cm}$ into the water, then releases it. Jane measured that the frequency of oscillation of the buoy was $f_1$. John did the same experiment, but he pushed the buoy $2\text{ cm}$ into the water. The frequency of oscillation is said to be $f_2$. It is assumed that the damping is negligible, so only buoyancy and gravity need to be considered. In addition, the density of the buoy is uniform and the buoy is never completely submerged by water. Which of the following statements is correct?
- A $f_1 > f_2$, because the magnitude of the buoyancy force is greater in Jane’s case.
- B $f_2 > f_1$, because the magnitude of the buoyancy force is greater in John’s case.
- C $f_1 > f_2$, because the magnitude of the buoyancy force is greater in John’s case.
- D $f_2 > f_1$, because the magnitude of the buoyancy force is greater in Jane’s case.
- E $f_1 = f_2$.
思路引導 VIP
想像一下，如果你面前有一個掛在彈簧下的砝碼。當你輕輕拉動它 1 公分後放手，和用力拉動 10 公分後放手，雖然拉得越長彈力越大，但砝碼需要移動的總路徑也變長了。請思考：在這種「力與距離成正比」的系統中，力增加的效果與路徑增加的負擔，兩者對「完成一次往返所需的時間」會產生什麼樣的相互影響？
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AI 詳解
AI 專屬家教
太棒了！你能精準判斷出頻率不受初始位移影響，說明你對**簡諧運動（Simple Harmonic Motion, SHM）**的核心性質掌握得非常透徹。這道題目的鑑別度在於測試學生是否會被「直覺」誤導，因為許多人在思考時會落入「力越大，運動越快」的陷阱，而忽略了位移與加速度之間的比例關係。
浮力與恢復力的關係
從物理本質來看，當浮標在靜止平衡位置時，重力與浮力抵銷。當浮標被向下壓入距離 $x$ 時，額外增加的排開水量產生的「合力」即為恢復力。根據阿基米德原理，此恢復力可表示為 $F = -(\rho A g)x$。這完全符合虎克定律 $F = -kx$ 的形式，其中力常數 $k = \rho A g$。由於這是一個理想的簡諧運動系統，其震盪頻率 $f$ 的表達式為：
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