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11. Rotational Vectors & Angular Momentum 轉動向量和角動量

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1 11. Rotational Vectors & Angular Momentum 轉動向量和角動量
Angular Velocity & Acceleration Vectors 角速度和加速度向量 Torque & the Vector Cross Product 力距和向量交叉乘積 Angular Momentum 角動量 Conservation of Angular Momentum 角動量的守恆 Gyroscopes & Precession 陀螺儀和進動

2 Earth isn’t quite round. 地球不是真的圓。
How does this affect its rotation axis, 這對它的轉軸有甚麼影响? and what’s this got to do with ice ages? 這又跟冰河期有甚麼關係? (The deviation from roundness is exaggerated.) (圖中的扁狀經過誇大) Axis precesses with period ~26,000 yr. 轉軸進動,週期約26,000年

3 Importance of rotation 轉動的重要:
Earth’s rotational axis  seasons. 地球的轉軸  四季 Angular momentum of protons in living tissues  MRI 活組織內質子的角動量  磁振造影 Rotating air  tornadoes. 轉動的空氣  龍捲風 Rotating wheel  stabilizes bicycle. 轉動的輪子  穩定腳車

4 11.1. Angular Velocity & Acceleration Vectors 角速度和加速度向量
Right-hand rule右手法則 Angular acceleration vector 角加速度向量: changes direction 改變方向  //     //   

5 11.2. Torque & the Vector Cross Product 力距和向量交叉乘積
手指捲過來 Right hand rule右手法則 拇指 cross product交叉乘積 從紙面出來

6 Cross Product 交叉乘積 Cross product C of vectors A & B: 向量 A & B 的叉積 C:
right-hand rule 右手法則 Dot product C of vectors A & B: 向量 A & B 的點積 C : Properties of cross product 叉積的性質 : Distributive 分配性 Anti-commutative 反交換性

7 GOT IT 懂嗎? Which numbered torque vector goes with each pair of force-radius vectors? 那一個有編號的力距向量可與那一對力-半徑向量配上? Neglect magnitudes. 不要管大小。 3 2 1 4 Note: Wolfson gave 6 as the answer to (b). 注:(b)題書上的答案是 6 。

8 11.3. Angular Momentum 角動量 Linear momentum: 線動量: Angular momentum:
角動量: particle 粒子 rigid body with axis of rotation along principal axis 剛體,主軸為轉軸 general case, I a tensor. 廣泛情況, I 為張量。  L &  can have different directions. L &  的方向可不同。

9 Example 11.1. Single Particle 單一粒子
A particle of mass m moves CCW at speed v around a circle of radius r in the x-y plane. 一質量為 m 的粒子以速率 v 繞着 x-y 面上一個半徑為 r 的圓圈依反時針方向走。 Find its angular momentum about the center of the circle, 求它對圓心的角動量, express the answer in terms of its angular velocity. 答案以其角速度表示。

10 Torque & Angular Momentum 力距和角動量
System of particles: 一組粒子: rotational analog of 2nd law. 第二定律的轉動比照。

11 11.4. Conservation of Angular Momentum 角動量的守恆
質量靠近軸: I 小, 大, L = I  一樣

12 Example Pulsars 脈衝星 A star rotates once every 45 days. 一星球每 45 天自轉一週。 It then undergoes supernova explosion, hurling most of its mass into space. 之後它發生超新星爆炸,把它大部份的質量拋到太空。 The inner core of the star, whose radius is initially 20 Mm, collapses into a neutron star only 6 km in radius. 星球本來半徑有 20Mm 的內核 ,崩潰成半徑祇有 6km 的中子星。 The rotating neutron star emits regular pulses of radio waves, making it a pulsar. 還在轉的中子星放出規律性的無線電波,使它成為一個脈衝星。 Calculate the pulse rate ( = rotation rate ). 求其脈衝速率 ( = 自轉速率 ) 。 Assume core to be a uniform sphere & no external torque. 可假設內核是個均勻的球,且無外加的力距。 Before collapse: 崩潰前: After collapse: 崩潰後:

13 Example 11.3. Playground 遊樂場
A merry-go-round of radius R = 1.3 m has rotational inertia I = 240 kg m2 & is rotating freely at 1 = 11 rpm. 一個旋轉台的半徑 R = 1.3 m ,轉動慣量 I = 240 kg m2 ,且正以 1 = 11 rpm 轉動。 A boy of mass mb = 28 kg runs straight toward the center at vb = 2.5 m/s & leaps on. 一質量 mb = 28 kg 的男孩畢直地朝中心以 vb = 2.5 m/s 的速度跳上。 At the same time, a girl of mass mg = 32 kg, running tangentially at speed vg = 3.7 m/s in the same direction as the merry-go-round also leaps on. 在同一時間,一質量 mg = 32 kg 的女孩順着轉台的轉向,以 vg = 3.7 m/s 朝切線方向跳上。 Find the new angular speed 2 once both children are seated on the rim. 求兩孩子在邊緣坐下來後,新的角速率 2 。 Before : After :

14 Demonstration of Conservation of Angular Momentum 角動量守恆示範
倒過來 靜止

15 GOT IT 懂嗎? If you step on a non-rotating table holding a non-rotating wheel. 如果你拿着一個不在轉的輪子踏上一個不在轉的桌子。 if you spin the wheel CCW as viewed from above, which way do you rotate? 如果你把輪子朝從上面看下來是反時針的方向轉,你往那裏轉? If you then turn the wheel upside down, will your rotation rate increase, decrease, or remain the same? 如果之後你把輪子上下顛倒,你的轉速會增加,減少,還是不變? What about your direction of rotation? 你旋轉的方向會怎樣? CW to keep L = 0. 順時以維持 L = 0 。 Same, CCW. 一樣,反時。

16 11.5. Gyroscopes & Precession 陀螺儀和進動
Gyroscope: spinning object whose rotational axis is fixed in space. 陀螺儀:自轉中的物體,其轉軸在空間內固定不變。 External torque required to change axis of rotation 改變轉軸需靠外加力距 Higher spin rate  larger L  harder to change orientation 自轉較快  L 較大  改變方向較難 Usage 用途: Navigation 航行。 Missile & submarine guidance. 飛彈和潛艇導航。 Cruise ships stabilization. 郵輪的平衡。 Space-based telescope like Hubble. 太空望遠鏡,如哈伯。

17 Precession 進動(旋進) Precession: Continuous change of direction of rotation axis, which traces out a circle. 進動:轉軸不停地改變方向,劃出一個圓來。 輸出軸 自轉軸 輸入軸

18 Earth’s Precession 地球的進動
13,000 年後 現時 地球 太陽 Earth’s precession (period ~ 26,000 y ) 地球的進動 ( 週期約26,000年 ) The equatorial bulge is highly exaggerated. 赤道處的鼓脹是故意誇大。

19 Perfect sphere 完美的球  = 0  = 0 Oblate spheroid 扁球體  < 
The equatorial bulge is highly exaggerated. 赤道處的鼓脹是故意誇大。

20 GOT IT 懂嗎? You push horizontally at right angles to the shaft of a spinning gyroscope. 一個陀螺正在自旋。你沿水平朝垂直於它轉軸的方向推。 Does the shaft move 轉軸會 upward, (a) 朝上, downward, (b) 朝下, in the direction you push, (c) 朝你推的方向, opposite the direction you push? (d) 朝你推的相反方向走?

21 Bicycling 騎腳踏車 Looking down at bike. 朝下看腳踏車 Direction of bike’s motion
腳踏車走的方向 L+  t wheel 輪子 Wheel turns 輪子轉向 L Einstein leans 愛因斯坦倚過去 points into paper 指入紙面 L //   wheel turns to Einstein’s left 輪子轉向愛因斯坦的左邊


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