9 Systems of Particles 1 2 3 4

9. Systems of Particles 1. 2. 3. 4. 5. 6. Center of Mass Momentum Kinetic Energy of a System Collisions Totally Inelastic Collisions Elastic Collisions 多質點系统 質心 動量 一個系统的動能 碰撞 完全非彈性碰撞

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�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。

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 3 rd 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。

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。

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 2 m. 兩模組的質量為 m，另一個為 2 m 。 Find the CM. 求質心。 Coord origin at m 2 = 2 m & y points downward. 座標原點在 m 2 = 2 m 且 y 朝下。 2: 2 m x 30 L CM 1: m 3: m y obtainable by symmetry 可以對稱性導得

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

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. Where’s its CM? 它的質量 m 分佈均勻。 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 y dx h w L By symmetry 由對稱性 , x

y b dy w/2 w x w/2 L

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。

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？

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 19 m 。 How far does the car move? 車廂移動多遠？ 1 t = 1 tonne 公噸 = 1000 kg Car not moving xcm 19+x. Ji Car moved 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 ) 。

9. 2. Momentum Total momentum: 總動量： M constant 不變 動量

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。

GOT IT 懂嗎! 9. 2. 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.

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 = 0. 開始時，總 p = 0. frictionless water p conserved 水無摩擦 p 守恆 After Nick catches it , total p = 0. 尼克接住它之後，總 p = 0. Kayak speed = 0 獨木舟的速率 = 0. Simple application of the conservation law. 祇是守恆定律的簡單應用。

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

Example 9. 5. Radioactive Decay 放射性衰變 A lithium-5 ( 5 Li ) nucleus is moving at 1. 6 Mm/s when it decays into 一個鋰-5 ( 5 Li ) 原子核以 1. 6 Mm/s 移動時衰變成 a proton ( 1 H, or p ) & an alpha particle ( 4 He, or ). [ Superscripts denote mass in AMU ] 一個質子( 1 H, 或 p ) 和一個阿爾發粒子( 4 He, 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 5 Li. 茲測得 以 1. 4 Mm/s 在與 5 Li 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

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

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： (a) momentum of individual skater. (b) momentum of basketball. 個別溜冰人的動量 籃球的動量 (c) 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 (d) 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

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… Thrust: 推力 …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

9. 3. Kinetic Energy of a System 一個系统的動能

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

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

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。

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？ elastic (a) a basketball rebounds off the backboard. 彈性 一個籃球從籃板反彈出來 elastic (b) two magnets approach, their north poles facing; they repel & reverse direction 彈性 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。 (c) 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。 inelastic 非彈性 (d) 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。 (e) 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。

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。

Example 9. 7. 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？

Example 9. 8. Fusion Consider a fusion reaction of 2 deuterium nuclei 2 H + 2 H 4 He 茲有二氘核子進行核融反應 2 H 核融 + 2 H 4 He. Initially, one of the 2 H is moving at 3. 5 Mm/s, the other at 1. 8 Mm/s at a 64 angle to the 1 st. 開始時，一個 2 H 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。

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

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 = 4 ( final state 終態 : v 1 fx , v 1 fy , v 2 fx , v 2 fy ) 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 : v 1 fx , v 1 fy , v 1 fz , v 2 fx , v 2 fy , v 2 fz ) number of equations 方程式個數 = 3 +1 = 4 2 more conditions needed. 欠二個條件方程。

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

Example 9. 10. Nuclear Engineering 核子 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 ( D 2 O ). 常用的減速子是重水( D 2 O ) 。 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 。

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？ 1 st one is lighter. 第一個較輕。

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。

Example 9. 11. 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? p cons 守恆: E cons 守恆: 另一球往那個方向走？

Center of Mass Frame 碰撞點 質心框