hce_nthu
114年
進階物理與線性代數
第 17 題
Two blocks (with the same mass $m$) connected by a spring of negligible mass are
at rest on a frictionless table, as shown in the upper part of figure. The separation
distance of the blocks when the spring is relaxed is $L$. During a time interval $\Delta t$,
a constant force of magnitude $F$ is applied horizontally to the left block, moving it
through a distance $x_1$ as shown in the lower part of figure. During this time interval,
the right block moves through a distance $x_2$. At the end of this time interval when
the force is removed, what is the maximum energy could be momentarily stored in
the spring?
at rest on a frictionless table, as shown in the upper part of figure. The separation
distance of the blocks when the spring is relaxed is $L$. During a time interval $\Delta t$,
a constant force of magnitude $F$ is applied horizontally to the left block, moving it
through a distance $x_1$ as shown in the lower part of figure. During this time interval,
the right block moves through a distance $x_2$. At the end of this time interval when
the force is removed, what is the maximum energy could be momentarily stored in
the spring?
- A $F(x_1 + x_2)$
- B $F(x_1 + x_2)/2$
- C $F(x_1 - x_2)/2$
- D $F(x_1 - x_2)$
- E It cannot be determined.
思路引導 VIP
我們知道外力 $F$ 對左側木塊做了 $F \cdot x_1$ 的功。想像一下，如果這是一個剛體（例如一根硬桿），這份功會全部變成質心的動能。但現在這是一個「可形變」的彈簧系統，除了質心會移動外，系統內部也會產生相對運動與形變。如果我們知道質心的總位移是兩木塊位移的平均值，那麼單純用來「推動整個系統質心前進」的那部分能量，應該如何計算？而剩下的能量又去哪裡了呢？
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AI 詳解
AI 專屬家教
太棒了！你能準確選出 (C) 說明你對質點系功能原理（Work-Energy Theorem for Systems）有著非常深刻的理解。這題在物理競賽或進階課程中極具鑑別度，因為學生很容易將外力做的總功直接誤認為是儲存的能量，卻忽略了系統整體動能的分配。
系統總能量與質心動能的分配
當恆力 $F$ 作用於左側質量為 $m$ 的木塊並移動 $x_1$ 時，外力對系統所做的總功為 $W = F \cdot x_1$。這份能量會轉換為系統的總動能（包含質心動能與相對動能）以及彈簧的位能。然而，根據質心運動定律，系統質心的位移為 $\Delta X_{cm} = \frac{m x_1 + m x_2}{2m} = \frac{x_1 + x_2}{2}$。外力作用在質心位移上所產生的能量，即為系統的質心動能：
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