9. Systems of Particles 多質點系统

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1 9. Systems of Particles 多質點系统
Center of Mass 質心 Momentum 動量 Kinetic Energy of a System 一個系统的動能 Collisions 碰撞 Totally Inelastic Collisions 完全非彈性碰撞 Elastic Collisions 彈性碰撞

2 Ans. His center of mass (CM) 答： 他的質心
As the skier flies through the air, 當滑雪人在空中飛過時， most parts of his body follow complex trajectories. 他身體大部份的軌跡都很複雜。 But one special point follows a parabola. 可是一個特別的點走的是一條拋物線。 What’s that point, and why is it special? 那一點是甚麽，它為何特別? Ans. His center of mass (CM) 答： 他的質心 Rigid body: Relative particle positions fixed. 剛體： 質點間的距離固定。

3 9.1. Center of Mass 質心 N particles 質點 : 
= total mass 總質量 = Center of mass 質心 = mass-weighted average position 質量加權的平均位置 with 3rd law  第三定律 Cartesian coordinates: 卡氏座標 Extension: “particle” i may stand for an extended object with cm at ri . 延伸： “質點” i 可代表一個 CM 在 ri 的延綿物體。

4 Example 9.1. Weightlifting 舉重
Find the CM of the barbell consisting of 50-kg & 80-kg weights at opposite ends of a 1.5 m long bar of negligible weight. 一槓鈴的槓心長1.5 m ，重量可以忽略。在槓心兩端各掛 50-kg 和 80-kg 槓片。 求槓鈴的質心。 CM is closer to the heavier mass. 質心離重一點的質量較近。

5 Example 9.2. Space Station 太空站
A space station consists of 3 modules arranged in an equilateral triangle, connected by struts of length L & negligible mass. 一太空站由三個模組，以質量可忽略，長度為 L 的支柱，連成一等邊三角形。 2 modules have mass m, the other 2m. 兩模組的質量為 m，另一個為 2m 。 Find the CM. 求質心。 Coord origin at m2 = 2m & y points downward. 座標原點在 m2 = 2m 且 y 朝下。 2: 2m x 30 L CM obtainable by symmetry 可以對稱性導得 1: m 3:m y

6 Continuous Distributions of Matter 物質的連續分佈
Discrete collection 離散組合 : Continuous distribution 連續分佈: 原點隨便 Let  be the density of the matter. 設  為物質的密度

7 Example 9.3. Aircraft Wing 飛機翼
A supersonic aircraft wing is an isosceles triangle of length L, width w, and negligible thickness. 一架超音速飛機的翼是一個等腰三角形，長 L ，寬 w ，厚度則可忽略。 It has mass M, distributed uniformly. 它的質量 m 分佈均勻。 Where’s its CM? 它的質心在哪？ Density of wing 翼的密度 = . Coord origin at leftmost tip of wing. 座標原點在翼的最左端 By symmetry 由對稱性, y dx h w x L

8 y b dy w/2 w x w/2 L

9 CMfuselage 機身 CMplane 飛機 CMwing 機翼 A high jumper clears the bar, but his CM doesn’t. 一個跳高者躍過了橫桿，他的質心卻沒有。

10 Got it 懂嗎? 9.1. A thick wire is bent into a semicircle. 一條粗纜被扳成半圓形。
Which of the points is the CM? 那一點是質心？

11 Example 9.4. Circus Train 馬戲班火車
Jumbo, a 4.8-t elephant, is standing near one end of a 15-t railcar, 一頭 4.8-t 的大象，阿大，站在一部 15-t 的火車廂的一端。 which is at rest on a frictionless horizontal track. 火車廂停在一條無摩擦的水平鐵軌上 Jumbo walks 19 m toward the other end of the car. 阿大朝車廂的另一端走 19m 。 How far does the car move? 車廂移動多遠？ 1 t = 1 tonne 公噸 = 1000 kg Car not moving xcm  Car moved 19+xJi  Jumbo walks, but the center of mass doesn’t move (Fext = 0 ). 阿大有在走，可是質心不動 ( Fext = 0 ) 。

12 9.2. Momentum 動量 Total momentum: 總動量： M constant  不變

13 Conservation of Momentum 動量守恆
 Conservation of Momentum 動量守恆 : Total momentum of a system is a constant if there is no net external force. 如果沒有淨外力，一個系統的總動量是個常數。

14 GOT IT 懂嗎! 9.2. K.E. is not conserved. 動能沒有守恆。
A 500-g fireworks rocket is moving with velocity v = 60 j m/s at the instant it explodes. 一枚 500-g 的煙花火箭在爆炸當時以 v = 60 j m/s 的速度移動。 If you were to add the momentum vectors of all its fragments just after the explosion, 如果你把它所有碎片在爆炸後一剎那的動量向量都加起來， what would you get? 你會得甚麼？ K.E. is not conserved. 動能沒有守恆。 Emech = K.E. + P.E. grav is not conserved 沒有守恆. Etot = Emech + Uchem is conserved 守恆.

15 Conceptual Example 9.1. Kayaking 划獨木舟
Jess (mass 53 kg) & Nick (mass 72 kg) sit in a 26-kg kayak at rest on frictionless water. 潔西 (質量53 kg) 和尼克 (質量 72 kg) 坐在一條停在無摩擦的水上， 26-kg 的獨木舟上。 Jess toss a 17-kg pack, giving it a horizontal speed of 3.1 m/s relative to the water. 潔西拋出一個17-kg 的背包，給了它相對於水 3.1 m/s 的水平速率。 What’s the kayak’s speed after Nick catches it? 在尼克接住它之後，獨木舟的速率為何？ Why can you answer without doing any calculations ? 為甚麼你不用計算就可以回答 ? Initially, total p = 開始時，總 p = 0. frictionless water  p conserved 水無摩擦  p 守恆 After Nick catches it , total p = 尼克接住它之後，總 p = 0. Kayak speed = 獨木舟的速率 = 0. Simple application of the conservation law. 祇是守恆定律的簡單應用。

16 Making the Connection 連起來
Jess (mass 53 kg) & Nick (mass 72 kg) sit in a 26-kg kayak at rest on frictionless water. 潔西 (質量53 kg) 和尼克 (質量 72 kg) 坐在一條停在無摩擦的水上， 26-kg 的獨木舟上。 Jess toss a 17-kg pack, giving it a horizontal speed of 3.1 m/s relative to the water. 潔西拋出一個17-kg 的背包，給了它相對於水 3.1 m/s 的水平速率。 What’s the kayak’s speed while the pack is in the air & after Nick catches it? 當背包在空中，和在尼克接住它之後，獨木舟的速率為何？ Initially 開始時 While pack is in air 當背包在空中: Note: Emech not conserved 注： Emech 不守恆

17 Example 9.5. Radioactive Decay 放射性衰變
A lithium-5 ( 5Li ) nucleus is moving at 1.6 Mm/s when it decays into 一個鋰-5 ( 5Li ) 原子核以 1.6 Mm/s 移動時衰變成 a proton ( 1H, or p ) & an alpha particle ( 4He, or  ). [ Superscripts denote mass in AMU ] 一個質子( 1H, 或 p ) 和一個阿爾發粒子( 4He,或  ). [上標表示以 AMU為單位的質量]  is detected moving at 1.4 Mm/s at 33 to the original velocity of 5Li. 茲測得  以 1.4 Mm/s 在與 5Li 原來速度成 33 的方向移動。 What are the magnitude & direction of p’s velocity? p 的速度的大小和方向為何？ Before decay: 衰變之前 After decay: 衰變之後

18 Example 9.6. Fighting a Fire 救火
A firefighter directs a stream of water to break the window of a burning building. 一名消防員想用水柱把一幢着火房子的窗門打破。 The hose delivers water at a rate of 45 kg/s, hitting the window horizontally at 32 m/s. 水管的出水速率為 45 kg/s ，水則沿水平方向以32 m/s 打到窗門。 After hitting the window, the water drops vertically. 打到窗門後，水朝下畢直掉落。 What horizontal force does the water exert on the window? 水在水平方向施於窗門的力為何？ Momentum transfer to a plane  stream: 在  水流的平面上，動量轉移為： = Rate of momentum transfer to window 動量轉移至窗門的速率 = force exerted by water on window 水施於窗門的力

19 GOT IT 懂嗎 ? 9.3. Two skaters toss a basketball back & forth on frictionless ice. 兩名溜冰的人在無磨擦的冰上把一個籃球互相投擲。 Which of the following does not change: 下面那一項不會改變： momentum of individual skater. 個別溜冰人的動量 momentum of basketball. 籃球的動量 momentum of the system consisting of one skater & the basketball. 包含一個溜冰人和籃球的系统的動量 momentum of the system consisting of both skaters & the basketball. 包含兩個溜冰人和籃球的系统的動量

20 Application: Rockets 應用：火箭
High pressure gas pushes equally in opposite directions, & this “rocket” goes nowhere. 高壓氣體在相反方向的推力都相等，這“火箭”那裏都不能去。 Open an exhaust port …. 打開一個排氣孔… …and there’s now an unbalanced force on the front of the rocket. …火箭前端便有一個未被抵消掉的力 Thrust: 推力

21 9.3. Kinetic Energy of a System 一個系统的動能

22 9.4. Collisions 碰撞 Examples of collision 碰撞的例子:
Balls on pool table. 撞球桌上的撞球。 tennis rackets against balls. 網球拍碰到網球。 bat against baseball. 棒球棒碰到棒球。 asteroid against planet. 小行星撞上行星。 particles in accelerators. 加速器內的粒子。 galaxies 銀河群 spacecraft against planet 太空船遭遇行星 ( gravity slingshot 重力彈弓 ) Characteristics of collision 碰撞的特徵 : Duration: brief. 過程：短暫 Effect: intense 効果：激烈 (all other external forces negligible ) 其他的外力全都可以忽略

23 Momentum in Collisions 碰撞時的動量
External forces negligible  Total momentum conserved 外力都可以忽略  總動量守恆 For an individual particle 單一粒子 t = collision time 碰撞時間 impulse 衝力 More accurately,更精確的說法 Same size 同樣大小 Average 平均值 Crash test 撞車測試

24 Energy in Collisions 碰撞時的能量
Elastic collision: K conserved. 彈性碰撞： K 守恆 。 Inelastic collision: K not conserved. 非彈性碰撞： K 不守恆 。 Bouncing ball: inelastic collision between ball & ground. 彈跳中的球：球與地非彈性碰撞。

25 GOT IT 懂嗎? 9.4. Which of the following qualifies as a collision 以下何者可稱為碰撞？ Of the collisions, which are nearly elastic & which inelastic? 碰撞之中，那些差不多是彈性的，那些是非彈性的？ a basketball rebounds off the backboard. 一個籃球從籃板反彈出來 two magnets approach, their north poles facing; they repel & reverse direction without touching. 兩個磁鐵的北極互相靠近，它們互相排斥，沒有接觸就掉頭退開。 a basket ball flies through the air on a parabolic trajectory. 一個籃球在空中沿拋物線軌跡飛行。 a truck crushed a parked car & the two slide off together. 一輛貨車把一輛停好的汽車壓扁，然後兩者一齊滑走。 a snowball splats against a tree, leaving a lump of snow adhering to the bark. 一枚雪球碰上一棵樹，留下一團雪附在樹皮上。 elastic彈性 elastic 彈性 inelastic非彈性 inelastic非彈性

26 9.5. Totally Inelastic Collisions 完全非彈性碰撞
Totally inelastic collision: colliding objects stick together 完全非彈性碰撞：相撞的物體粘在一起。 maximum energy loss consistent with momentum conservation. 動量守恆所容許的最大能量消耗。

27 Example Hockey 冰上曲棍球 A Styrofoam chest at rest on frictionless ice is loaded with sand to give it a mass of 6.4 kg. 一個保麗龍盒子放在無摩擦的冰上，盒內的砂子使它的質量達到 6.4 kg。 A 160-g puck strikes & gets embedded in the chest, which moves off at 1.2 m/s. 一枚160-g 的球餅撞來，嵌在盒內，並使它以 1.2 m/s 走動。 What is the puck’s speed? 球餅的速率為何？

28 Example Fusion 核融 Consider a fusion reaction of 2 deuterium nuclei 2H + 2H  4He . 茲有二氘核子進行核融反應 2H + 2H  4He Initially, one of the 2H is moving at 3.5 Mm/s, the other at 1.8 Mm/s at a 64 angle to the 1st. 開始時，一個 2H 以 3.5 Mm/s 移動，另一個則以 1.8 Mm/s 與第一個成 64 移動。 Find the velocity of the Helium nucleus. 求氦核子的速度。

29 Example 9.9. Ballistic Pendulum 彈道單擺
The ballistic pendulum measures the speeds of fast-moving objects. 彈道單擺可用來測量快速物體的速率。 A bullet of mass m strikes a block of mass M and embeds itself in the latter. 一質量為 m 的子彈擊中並嵌入一質量為 M 的質塊。 The block swings upward to a vertical distance of h. 質塊上擺高度為 h。 Find the bullet’s speed. 求子彈的速率。  Caution小心: ( heat is generated when bullet strikes block ) 子彈撞擊質塊時產生熱量

30 9.6. Elastic Collisions 彈性碰撞
Momentum conservation: 動量守恆 Energy conservation: 能量守恆 Implicit assumption: particles have no interaction when they are in the initial or final states. ( Ei = Ki ) 不明文假定：在始和終態時粒子間無作用。 ( Ei = Ki ) 2-D case 二維系统 : number of unknowns 未知數個數 = 2  2 = ( final state 終態 : v1fx , v1fy , v2fx , v2fy ) number of equations 方程式個數 = 2 +1 = 3  1 more conditions needed. 欠一個條件方程。 3-D case 三維系统 : number of unknowns 未知數個數 = 3  2 = 6 ( final state 終態 : v1fx , v1fy , v1fz , v2fx , v2fy , v2fz ) number of equations 方程式個數 = 3 +1 = 4  2 more conditions needed. 欠二個條件方程。

31 Elastic Collisions in 1-D 一維彈性碰撞
1-D collision 一維碰撞 1-D case 一維系统: number of unknowns 未知數個數 = 1  2 = ( v1f , v2f ) number of equations 方程式個數 = 1 +1 = 2  unique solution. 解答獨一無二。 This is a 2-D collision 這是二維碰撞 

32   (a) m1 << m2 :  (b) m1 = m2 :  (c) m1 >> m2 : 

33 Example 9.10. Nuclear Engineering 核子工程
Moderator slows neutrons to induce fission. 減速子使中子慢下來以誘發核分裂。 A common moderator is heavy water ( D2O ). 常用的減速子是重水( D2O ) 。 Find the fraction of a neutron’s kinetic energy that’s transferred to an initially stationary D in a head-on elastic collision. 求在一迎頭碰撞中，中子的動能有幾分會轉移給一個本來是靜止的 D 。

34 GOT IT 懂嗎? 9.5. One ball is at rest on a level floor. 一個球停在一片水平地板上。
Another ball collides elastically with it & they move off in the same direction separately. 另一個球與它彈性碰撞，之後兩者分別朝同一方向走動。 What can you conclude about the masses of the balls? 你對這些球的質量有何結論？ 1st one is lighter. 第一個較輕。

35 Elastic Collision in 2-D 二維彈性碰撞
Impact parameter 撞擊參數 b : additional info necessary to fix the collision outcome. 要决定碰撞結果所需的額外資訊。

36 Example Croquet 槌球 A croquet ball strikes a stationary one of equal mass. 一個槌球撞上另一個和它質量相同的靜止槌球。 The collision is elastic & the incident ball goes off 30 to its original direction. 碰撞是彈性的，而且入射球後來的方向與原來的成 30 。 In what direction does the other ball move? 另一球往那個方向走？ p cons 守恆: E cons 守恆: 

37 Center of Mass Frame 質心框
碰撞點