Part Two: Oscillations, Waves, & Fluids 振盪，波，和流體 Examples of oscillations & waves: 振盪和波的範例： Earthquake – Tsunami 地震 – 海嘯 Electric guitar – Sound wave 電結他 – 聲波 Watch – quartz crystal 手錶 – 石英晶體 Radar speed-trap 雷達超速監視區 Radio telescope 無線電望遠鏡 Examples of fluid mechanics: 流體力學的範例： Flow speed vs river width 流速對河寬 Plane flight 飛機飛行 High-speed photo : spreading circular waves on water. 高速相片：在水面散開的圓形波。

13. Oscillatory Motion 振盪性運動 (振動) Describing Oscillatory Motion 描述振盪性運動 Simple Harmonic Motion 簡單和諧運動 Applications of Simple Harmonic Motion 簡諧運動的應用 Circular & Harmonic Motion 圓周與諧和運動 Energy in Simple Harmonic Motion 簡諧運動的能量 Damped Harmonic Motion 阻尼諧動 Driven Oscillations & Resonance 受驅振盪和共振

帶個圈計劃的舞者在垂直的面上表演，執行幽雅的慢動作跳躍。 Wilberforce Pendulum Dancers from the Bandaloop Project perform on vertical surfaces, executing graceful slow-motion jumps. 帶個圈計劃的舞者在垂直的面上表演，執行幽雅的慢動作跳躍。 What determines the duration of these jumps? 甚麼東西决定這些跳躍的持續時間？ pendulum motion: rope length & g 單擺運動：繩長和 g 。

Disturbing a system from equilibrium results in oscillatory motion. 在平衡狀態中的系统受到干擾後就會進行振盪運動。 穩定平衡 振盪 Absent friction, oscillation continues forever. 若無摩擦，會一直振盪下去。 Examples of oscillatory motion 振盪運動的範例 : Microwave oven : Heats food by oscillating H2O molecules in it. 微波爐： 以振盪食物中的水分子來把它加熱。 CO2 molecules in atmosphere absorb heat by vibrating  global warming. 大氣中的二氧化碳分子以振動來吸熱  地球暖化。 Watch keeps time thru oscillation ( pendulum, spring-wheel, quartz crystal, …) 手錶以振動來定時間 ( 單擺，彈簧轉輪，石英晶體，… ) Earth quake induces vibrations  collapse of buildings & bridges . 地震引起振動  建物和橋樑倒塌。 Oscillation

13.1. Describing Oscillatory Motion 描述振盪性運動 Characteristics of oscillatory motion 振盪性運動的特徵： Amplitude A = max displacement from equilibrium. 振幅 A = 離平衡點的最大位移。 Period T = time for the motion to repeat itself. 週期 T = 運動重複自已所需時間。 Frequency f = # of oscillations per unit time. 頻率 f = 每單位時間的振盪次數 same period T 同樣的週期 T same amplitude A 同樣的振幅 A [ f ] = hertz (Hz) = 1 cycle / s 赫茲 (赫) 周 / 秒 A, T, f do not specify an oscillation completely. 不能完全指定一個振盪。 Oscillation

Example 13.1. Oscillating Ruler 振盪中的尺 An oscillating ruler completes 28 cycles in 10 s & moves a total distance of 8.0 cm. 一把振盪的尺 10 s 振 28 次，每次移動總距離 8.0 cm。 What are the amplitude, period, & frequency of this oscillatory motion? 這振盪運動的振幅，週期，和頻率為何？ Amplitude = 8.0 cm / 2 = 4.0 cm.

13.2. Simple Harmonic Motion 簡單和諧運動 Simple Harmonic Motion (SHM) : 簡單和諧運動 (簡諧運動) ： Ansatz: 擬設： angular frequency 角頻  

A, B determined by initial conditions 由初始條件確定   ( t )  2 x  2A

Amplitude & Phase 振幅和相位 C = amplitude 振幅  = phase 相位  Note:  is independent of amplitude only for SHM. 注：簡諧運動的  與振幅無關 Curve moves to the right for  < 0.  < 0 時，曲線往右移。 Oscillation

Velocity & Acceleration in SHM 簡諧運動的速度和加速度 位移 |x| = max at v = 0 速度 |v| = max at a = 0 加速度

GOT IT 懂嗎? 13.1. Two identical mass-springs are displaced different amounts from equilibrium & then released at different times. 兩個一樣的彈簧經過不同的壓伸後，在不同的時間放開。 Of the amplitudes, frequencies, periods, & phases of the subsequent motions, which are the same for both systems & which are different? 在後繼運動的振幅，頻率，週期，和相位中，那些在兩個系统都一樣，那些不一樣？ Same: frequencies, periods 一樣： 頻率，週期 Different 不一樣 : amplitudes ( different displacement ) 振幅 (壓伸不同) phases ( different release time ) 相位 (放開時間不同 )

Application: Swaying skyscraper 搖擺的摩天樓 Tuned mass damper 調諧質塊阻尼器 : Damper highly damped , 阻尼器的阻尼值大 Overall oscillation overdamped. 整體擺動為過阻尼型 Taipei 101 TMD : 台北 101 的調諧質塊阻尼器： 41 steel plates, 41 塊鋼板， 660 tonne, d = 550 cm, 660 公噸， d = 550 cm ， 87th-92nd floor. 在 87-92 層樓間 Movie Tuned Mass Damper Also used in: 也用於： Tall smokestacks 高的煙囪 Airport control towers. 機場控制塔 Power-plant cooling towers. 發電廠冷卻塔 Bridges. 橋樑 Ski lifts. 滑雪上山吊椅

Example 13.2. Tuned Mass Damper 調諧質塊阻尼器 The tuned mass damper in NY’s Citicorp Tower consists of a 373-Mg (vs 101’s 3500 Mg) concrete block that completes one cycle of oscillation in 6.80 s. 紐約花旗銀行大樓的調諧質塊阻尼器是一塊 373-Mg (101 的是 3500 Mg) 的水泥塊，它振盪一週要 6.80 s 。 The oscillation amplitude in a high wind is 110 cm. 在一次強風中它的振幅是110 cm。 Determine the spring constant & the maximum speed & acceleration of the block. 找出彈簧系數，與水泥塊的最高速率和加速度。 

13.3. Applications of Simple Harmonic Motion 簡諧運動的應用 The Vertical Mass-Spring System 垂直式質塊彈簧系统 The Torsional Oscillator 扭力振盪器 The Pendulum 單擺 The Physical Pendulum 物理擺 (複擺)

The Vertical Mass-Spring System 垂直式質塊彈簧系统 Spring stretched by x1 when loaded. 負載後彈簧伸長 x1。 mass m oscillates about the new equil. pos. 質塊在新平衡點處振盪 with freq 其頻率為

The Torsional Oscillator 扭力振盪器 = torsional constant 扭力常數  Used in timepieces 用於鐘錶

The Pendulum 單擺 Small angles oscillation: 小角度振盪： 支點 Small angles oscillation: 小角度振盪： Simple pendulum (point mass m): 單擺 (質點 m) :

Example 13.3. Rescuing Tarzan 拯救泰山 Tarzan stands on a branch as a leopard threatens. 泰山被斑豹趕到一條樹枝上。 Jane is on a nearby branch of the same height, holding a 25-m-long vine attached to a point midway between her & Tarzan. 亞珍在附近另一根同樣高的樹枝上，拿着一條縛在她和泰山的中點處，長 25-m 的籐條。 She grasps the vine & steps off with negligible velocity. 她抓緊籐條便以可忽略的速度盪開。 How soon can she reach Tarzan? 要多久她才能到泰山那裡？ Time needed 所需時間 :

GOT IT 懂嗎? 13.2. What happens to the period of a pendulum if 如果單擺 no change 不變 What happens to the period of a pendulum if 如果單擺 its mass is doubled, (a) 質量加倍 it’s moved to a planet whose g is ¼ that of Earth, (b) 被移到 g 祇有地球1/4的行星上 its length is quadrupled? (c) 長度拉長 4 倍 它的週期會怎樣？ doubles 加倍 doubles 加倍

Conceptual Example 13.1. Nonlinear Pendulum 非線性鐘擺 A pendulum becomes nonlinear if its amplitude becomes too large. 鐘擺在振幅太大時會變成非線性。 As the amplitude increases, how will its period changes? 振幅增加時，它的週期有何變化? If you start the pendulum by striking it when it’s hanging vertically, 如果鐘擺的起動是當它在正下方時打它， will it undergo oscillatory motion no matter how hard it’s hit? 是否不管打多重它都會在擺動? If it’s hit hard enough, motion becomes rotational. 打得夠重，運動會變成轉動。 (a) sin increases slower than  sin 增加比  慢  smaller      較小  longer period  週期較長

The Physical Pendulum 物理擺 (複擺) Physical Pendulum = any object that’s free to swing 物理擺 = 任何能自由擺動的物體 支點 Small angular displacement  SHM 小角度位移  簡諧運動 重心

Example 13.4. Walking 行走 When walking, the leg not in contact of the ground swings forward, 行走時，不與地面接觸的腿往前擺， acting like a physical pendulum. 就像個物理單擺。 Approximating the leg as a uniform rod, find the period for a leg 90 cm long. 把腿當成一根均勻桿子，求腿長 90 cm 的週期。 Table 表 10.2 Forward stride 往前跨 = T/2 = 0.8 s

13.4. Circular & Harmonic Motion 圓周與諧和運動 Circular motion 圓周運動 : 2  SHO with same A &  but  = 90 兩互相垂直的簡諧振盪器， A &  相同但  = 90 x =  R x = R x = 0 Lissajous Curves

GOT IT 懂嗎? 13.3. The figure shows paths traced out by two pendulums swinging with different frequencies in the x- & y- directions. 圖示兩個 x- 和 y- 頻率不相等的單擺在擺動時劃出的軌跡。 What are the ratios x : y ? x : y 的比值是甚麽？ Lissajous Curves 1 : 2 3: 2

13.5. Energy in Simple Harmonic Motion 簡諧運動的能量 能量 時間 位置 SHM: 簡諧運動 = constant Energy in SHM 平衡點

Potential Energy Curves & SHM 位能曲線和簡諧運動 “最恰”拋物線 Linear force: 線性力： parabolic potential energy: 拋物線位能 位能 位移 Taylor expansion near local minimum : 在局部低點附近的泰勒展式： Small disturbances near equilibrium points  SHM 平衡點附近的小干擾  簡諧運動

GOT IT 懂嗎? 13.4. Two different mass-springs oscillate with the same amplitude & frequency. 兩個不同的質塊彈簧以同樣的振幅和頻率振動。 If one has twice as much energy as the other, how do 如果其中一個的能量是另一個的兩倍，它們的 their masses & (b) their spring constants compare? (a) 質量 和 (b) 彈簧係數 相比如何？ (c) What about their maximum speeds? 它們的最大速率又如何？ The more energetic oscillator has 能量較多的振盪器有 twice the mass (a) 兩倍的質量 twice the spring constant (b) 兩倍的彈簧係數 (c) Their maximum speeds are equal. (c) 它們的最大速率相等

13.6. Damped Harmonic Motion 阻尼諧動 sinusoidal oscillation 正弦振盪 Damping (frictional) force: 阻(摩擦)力： Damped mass-spring: 阻尼質塊彈簧： Amplitude exponential decay 振幅指數式遞減 Ansatz 擬設 :  

At t = 2m / b, amplitude drops to 1/e of max value. t = 2m / b 時，振幅掉到最大值的 1/e。 is real, motion is oscillatory ( underdamped ) 是實數，運動為振盪式(欠阻尼) (a) For (c) For is imaginary, motion is exponential ( overdamped ) 是虛數，運動呈指數式衰減(過阻尼) (b) For = 0, motion is exponential ( critically damped ) = 0，運動呈指數式衰減(臨介阻尼) Damped & Driven Harmonic Motion

Example 13.6. Bad Shocks 爛避震器 A car’s suspension has m = 1200 kg & k = 58 kN / m. 一部車的懸掛系统的 m = 1200 kg 且 k = 58 kN / m。 Its worn-out shock absorbers provide a damping constant b = 230 kg / s. 它已磨損的避震器提供一個阻尼系數 b = 230 kg / s。 After the car hit a pothole, how many oscillations will it make before the amplitude drops to half its initial value? 當車子碰上坑洞時，要振盪幾次振幅才能掉到開始值的一半？ Time  required is 所需時間  是  # of oscillations: 振盪次數： bad shock ! 好爛的避震器！

13.7. Driven Oscillations & Resonance 受驅振盪和共振 External force  Driven oscillator 外力  受驅振盪器 Let d = driving frequency 驅動頻率 ( long time ) ( 長期 ) Prob 75: 習題 振幅 = natural frequency 自然頻率 Resonance: 共振： Damped & Driven Harmonic Motion 驅動頻率

Buildings, bridges, etc have natural freq. 建物，橋樑，等都有自然頻率。 If Earth quake, wind, etc sets up resonance, disasters result. 如果地震，風，等形成共振，結果就是災難。 Collapse of Tacoma bridge is due to self-excitation described by the van der Pol equation. 塔科馬橋的倒塌源於自我激發，可以范德蒲方程描述。 Tacoma Bridge Resonance in microscopic system 微系统的共振 : electrons in magnetron  microwave oven 磁控管內電子  微波爐 Tokamak (toroidal magnetic field)  fusion 托卡馬克 ( 環形磁場 )  核融 CO2 vibration: resonance at IR freq  Green house effect 二氧化碳振盪：共振於紅外線頻率  温室効應 Nuclear magnetic resonance (NMR)  NMI for medical use. 核子磁性共振 (核磁共振)  醫療用核磁造像