hce_nthu
114年
進階物理與線性代數
第 9 題
A ball of mass $m$ is connected by a string of length $L$ to a pivot fixed in place. A.
The ball is originally at rest. A force $F$ applied horizontally to the right on the ball.
The force barely balances the gravity and the string tension so that the ball has
negligible kinetic energy when it slowly swings up. The string finally makes an
angle $\theta$ with the vertical line What is the work done by the force $F$ during this
process?
The ball is originally at rest. A force $F$ applied horizontally to the right on the ball.
The force barely balances the gravity and the string tension so that the ball has
negligible kinetic energy when it slowly swings up. The string finally makes an
angle $\theta$ with the vertical line What is the work done by the force $F$ during this
process?
- A $mgL(\sin\theta)$
- B $mgL(1-\cos\theta)$
- C $mgL(\cos\theta-1)$
- D $mgL(\cos\theta)$
- E $0$
思路引導 VIP
請試著思考：如果一個物體在移動過程中，它的速度始終保持微小到可以忽略，那麼根據功能定理，作用在它身上的「合力功」應該是多少？另外，除了外力 $F$ 之外，還有哪一個力在物體上升時會對它作功？這個功與物體上升的高度有什麼幾何關係呢？
🤖
AI 詳解
AI 專屬家教
恭喜你準確鎖定了正確選項！你能敏銳捕捉到題目中「動能幾乎為零」這個關鍵提示並選出 (B)，顯示你對功能轉換的觀念掌握得非常扎實。
功能原理與幾何關係
這題的核心在於應用功與能量定理（Work-Energy Theorem）。由於球在過程中是以極慢的速度移動，代表動能變化量 $\Delta K$ 趨近於零，這意味著作用在球體上所有力所做的總功為零。在移動過程中，繩子張力恆與運動位移垂直而不作功，因此外力 $F$ 所做的功 $W_F$ 必須完全用來補償重力所做的負功。從幾何圖形觀察，球從最低點上升到角度 $\theta$ 時，其垂直高度增加了 $h = L - L\cos\theta = L(1-\cos\theta)$。依據能量守恆，外力所作的功正好等於重力位能的增加量，即 $W_F = \Delta U_g = mgL(1-\cos\theta)$。
▼ 還有更多解析內容