8. Gravity 重力 Toward a Law of Gravity 邁向一個重力定律 Universal Gravitation 萬有引力 Orbital Motion 軌道運行 Gravitational Energy 重力能 The Gravitational Field 重力場

This TV dish points at a satellite in a fixed position in the sky. 這碟型電視天線指向在天空中固定一點的人造衛星。 How does the satellite manage to stay at that position? 衛星怎麼能停在那位置？ period 週期 = 24 h

Ptolemaic (Geo-Centric) System 托勒密(地心)系统 Epicycle 本輪 Equant 等點  Deferent 均輪  swf

8.1. Toward a Law of Gravity 邁向一個重力定律 1543: Copernicus – Helio-centric theory. 哥白尼 – 日心理論 1593: Tycho Brahe – Planetary obs. 泰戈‧布拉赫 – 行星觀察 1592-1610: Galileo – Jupiter’s moons, sunspots, phases of Venus. 伽利略 –木星的月亮，日斑，金星的相。 1609-19: Kepler’s Laws 克卜勒定律 1687: Newton – Universal gravitation. 牛頓 – 萬有引力。 Phases of Venus: Size would be constant in a geocentric system. 金星的相：在一個以地球為中心的系统裏，它的大小應該不變

Kepler’s Laws 克卜勒定律 Explains retrograde motion 解釋行星逆行 First law: The orbit is elliptical, with the Sun at one focus … 第一定律：軌道為橢圓，太陽在其中一焦點上… Explains retrograde motion 解釋行星逆行 Third law: The square of the orbital period is proportional to the cube of the semimajor axis. 第三定律：軌道週期的平方與半長軸的立方成正比。 Second law: If the shaded areas are equal, so is the time to go from A to B and from C to D 第二定律：如果有色的面積相等，則從A到B的時間與從C到D的相等。 Mathematica

8.2. Universal Gravitation 萬有引力 Newton’s law of universal gravitation 牛頓的萬有引力定律: m1 & m2 are 2 point masses 質點 . r12 = position vector from 1 to 2. 從 1 到 2 的位置向量 F12 = force of 1 on 2. 1 加於 2 的力 G = Constant of universal gravitation 重力常數 = 6.67  1011 N m2 / kg2 . F12 m2 r12 m1 Law also applies to spherical masses. 定律亦可用於球形質量。

Example 8.1. Acceleration of Gravity 重力加速度 Use the law of gravitation to find the acceleration of gravity 用重力定律來找下列的重力加速度 at Earth’s surface 在地球表面。 at the 380-km altitude of the International Space Station 在高 380-km 的國際太空站中。 on the surface of Mars. 在火星表面。  (a) (b) (c) see App.E

TACTICS 8.1. Understanding “Inverse Square” 策略 8.1.了解 ”反平方” Given Moon’s orbital period T & distance R from Earth, 知道月球的週期 T 和它離地球的距離 R ， Newton calculated its orbital speed v and hence acceleration a = v2 / R. 牛頓便能算出它的運行速率 v 和加速度 a = v2 / R 。 He found 他得到 a ~ g / 3600. Moon-Earth distance is about 60 times Earth’s radius. 月-地球的距離約為地球半徑的 60 倍。  Inverse square law 反平方定律

Cavendish Experiment: Weighing the Earth 卡文迪什實驗 :稱地球 小球因重力而被吸向大球 ME can be calculated if g, G, & RE are known. 知道 g, G, & RE 就能算 ME 。 Cavendish 卡文迪什 : G determined using two 5 cm & two 30 cm diameter lead spheres. G 用兩個 5 cm 和兩個 30 cm直徑的鉛球量得。 Gravity is weakest & long ranged 重力最弱且長程 always attractive 總是吸引的  dominates at large range. 在遠距離佔優勢 EM is strong & long ranged, 電磁力強且長程 can be attractive & repulsive 能吸引也能排斥  cancelled out in neutral objects. 中性物體內會抵消掉 Weak & strong forces: very short-range. 弱和強作用：非常短程

8.3. Orbital Motion 軌道運行 g = 0 projectiles orbit 拋物體 軌道 Orbital motion: Motion of object due to gravity from another larger body. 軌道運行 ：物體因另一較大物體的重力而做成的運動。 E.g. Sun orbits the center of our galaxy with a period of ~200 million yrs. 如：太陽繞銀河中心而轉，週期 ~兩億年。 Newton’s “thought experiment” 牛頓的 “假想實驗” g = 0 Condition for circular orbit 圓形軌道的條件 Speed for circular orbit 圓形軌道的速率 projectiles 拋物體 orbit 軌道 Orbital period 軌道週期 Kepler’s 3rd law 克卜勒定律第三定律

Example 8.2. The Space Station 國際太空站 The ISS is in a circular orbit at an altitude of 380 km. 國際太空站的軌道是圓的，高度為 380 km 。 What are its orbital speed & period? 它軌道的速率和週期為何？ Orbital speed: 軌道速率 Orbital period: 軌道週期 Near-Earth orbit 近地軌道 T ~ 90 min. Moon orbit 月球軌道 T ~ 27 d. Geosynchronous orbit 地球同步軌道 T = 24 h.

Example 8.3. Geosynchronous Orbit 地球同步軌道 What altitude is required for geosynchronous orbits? 地球同步軌道的高度為何？ Altitude 高度 = r  RE Earth circumference 地球的圓週長 Earth not perfect sphere  orbital correction required every few weeks. 地球不是完美的球  軌道每隔幾個星期就要調整。

Elliptical Orbits 橢圓形軌道 Projectile trajectory is parabolic only if curvature of Earth is neglected. 拋體的軌跡祇有在地球的曲度可以忽略時才是一條拋物線。 這一段差不多是條拋物線 ellipse 橢圓 Orbits of most known comets, are highly elliptical. 已知的彗星軌道，大部份都是非常橢圓的。 Perihelion: closest point to sun. 近日點：靠太陽最近那點。 Aphelion: furthest point from sun. 遠日點：離太陽最遠那點。 焦點是地球的中心

Open Orbits 開放型軌道 Open 開放型 (hyperbola 雙曲線) Closed 密閉型 (circle 圓) Borderline 兩者之間 (parabola 拋物線) Closed 密閉型 (ellipse 橢圓) Mathematica

8.4. Gravitational Energy 重力能 How much energy is required to boost a satellite to geosynchronous orbit? 把衛星送上地球的同步軌道要多少能量？ on this path U = 0 在此路徑上 U = 0 U12 depends only on radial positions. U12 祇與徑向位置有關。 … so U12 is the same as if we start here. …所以 U12 跟在這裡開始算一樣

Example 8.4. Steps to the Moon 到月球的步驟 Materials to construct an 11,000-kg lunar observatory are boosted from Earth to geosyn orbit. 用來建做一個 11,000-kg 月球天文臺的材料已從地球送上同步軌道。 There they are assembled & launched to the Moon, 385,000 km from Earth. 材料先在那裡裝組，再送到離地球 385,000 km 的月球。 Compare the work done against Earth’s gravity on the 2 legs of the trip. 比較這兩段路程上，對地球重力所作的功。 1st leg: 第一段 2nd leg: 第二段

Zero of Potential Energy 位能的零(原)點 Gravitational potential energy 重力位能  E > 0, open orbit Open 開放的 Closed 密閉的 E < 0, closed orbit Bounded motion 有限運動 Turning point 轉折點

Example 8.5. Blast Off ! 發射 A rocket launched vertically at 3.1 km/s. How high does it go? 它能達多高？ Initial state : 初態： Final state : 終態： Energy conservation : 能量守恆： Altitude = r  RE 高度

Escape Velocity 逸速 Open Closed  Body with E  0 can escape to  Moon trips have v < vesc . 到月球的 v < vesc .

Energy in Circular Orbits 圓形運動的能量 圓形軌道 :  K  E K U Higher K or v  Lower E & orbit (r) . 較大 較低 軌道

Conceptual Example 8.1. Space Maneuvers 太空操作 Astronauts heading for the International Space Station find themselves in the right circular orbit, but well behind the station. 去國際太空站的太空人發現他們在正確的軌道上，卻遠遠落在站的後面。 How should they maneuver to catch up? 他們應該怎樣操作來追上？ Fire rocket backward to decrease energy & drop to lower, & faster orbit. 往後發動火箭以減低能量，並降到低和快一點的軌道。 Fire to circularize orbit. 發動火箭以達成圓周軌道。 After catching up with the station, fire to boost to up to its level. 追上太空站後，發動火箭以升至它的高度。 Mathematica

energy Altitude  K K > K E = K+U = U / 2 h h < h E = K+U = U / 2 E = K+U < E ( K < K ) U U < U UG

GOT IT 懂嗎 ? 8.3. Spacecrafts A & B are in circular orbits about Earth, with B at higher altitude. 太空船 A & B 都在繞地球的圓形軌道上，但 B 的比較高。 Which of the statements are true? 下面的表述中，那些是對的？ B has greater energy. B 的能量較大。 B is moving faster. B 跑的比較快。 B takes longer to complete an orbit. B 要較長的時間來繞一圈。 B has greater potential energy. B 的位能較大。 a larger proportion of B’s energy is potential energy. B 的能量裏位能占的成份較高     

8.5. The Gravitational Field 重力場 Two descriptions of gravity: 重力的兩種看法 body attracts another body (action-at-a-distance) 物體吸引另一物體 ( 遠距作用 ) Body creates gravitational field. 物體產生重力場 Field acts on another body. 場對另一物體作用 Near Earth: 地球附近 near earth地球附近 Large scale: 大尺寸 Action-at-a-distance  instantaneous messages 遠距作用  即時訊息 Field theory  finite propagation of information 場論 資訊以有限速度傳播 Only field theory agrees with relativity. 祇有場論與相對論吻合 Great advantage of the field approach: No need to know how the field is produced. 場論一大好處： 用管場如何產生 in space 太空中

Application: Tide 應用：潮汐 Moon’s tidal (differential) force field at Earth’s surface 在地球表面的月潮汐 (差異) 場 Moon’s tidal (differential) force field near Earth 在地球附近的月潮汐 (差異) 場 Two tidal bulges 兩個漲潮 Mathematica Sun + Moon  tides with varying strength. 日+月  潮汐的強度有變化 Tidal forces cause internal heating of Jupiter’s moons. 潮汐力使得木星的衛星的內部熱起來 They also contribute to formation of planetary rings. 它也幫忙形成行星的環