hce_nthu
113年
化學與物理
第 57 題
Two spring-block systems are placed in a viscous environment independently. The viscous forces experienced by the blocks is approximately linearly proportional to the speed of blocks. Due to the damping effect, both spring-block systems exhibit underdamped oscillating behavior. The shape of blocks of the two systems is the same, but the mass and spring constant of the 1st and the 2nd system are ($1\text{ kg}$, $1\text{ N/m}$) and ($1.2\text{ kg}$, $0.9\text{ N/m}$), respectively. Initially, both blocks are at rest and are released $5\text{ cm}$ away from their equilibrium positions. Which of the following statements is true?
- A It takes a longer time for the block in the 2nd system to finally come to rest.
- B It is not possible for these two systems to both exhibit underdamped behavior.
- C Both blocks move towards the equilibrium position and then stop at the equilibrium position.
- D In the very beginning, the oscillation frequency is larger for the 1st system.
- E In the very beginning, the oscillation frequency is larger for the 2nd system.
思路引導 VIP
想像一下，有兩顆體積一模一樣的球在濃稠的蜂蜜中擺動，一顆是輕盈的塑膠球，另一顆是沉重的鉛球。如果它們一開始都被拉開相同的距離，哪一顆球會因為自己的「慣性」較強，而讓蜂蜜的阻力更難在短時間內把它停下來呢？
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AI 詳解
AI 專屬家教
太棒了！你能準確判斷出第二個系統需要更長的時間趨於平靜，說明你對**阻尼振盪（Underdamped Oscillation）**的物理本質有很深刻的理解。這道題目的難度在於它給出了多個參數（$m$、$k$），容易讓人陷入複雜的頻率計算中，但其實關鍵在於能量消耗的速率。
阻尼係數與衰減速率
在黏滯阻力與速度成正比（$F_v = -bv$）的模型中，振幅衰減的快慢是由阻尼參數 $\gamma = \frac{b}{2m}$ 決定的。由於兩木塊形狀相同且環境一致，阻力常數 $b$ 可視為相同。比較兩系統：第一系統 $\gamma_1 = \frac{b}{2 \times 1}$，第二系統 $\gamma_2 = \frac{b}{2 \times 1.2}$。顯然 $\gamma_2 < \gamma_1$，這代表第二個系統的指數衰減包絡線（Envelope）下降得比第一系統緩慢。因此，質量較大的第二系統因為具有較大的動量慣性，黏滯阻力需要花費更長的時間才能將其機械能耗散殆盡。
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