Short Version : 8. Gravity 短版: 8. 重力 mp4

Retrograde Motion 逆行 Retrograde motion of Mars. 火星的逆行。 As explained by Ptolemy. 托勒密的解析。

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

Cassini Apparent 凱西尼表觀 Sun 太陽 Venus 金星

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

Cavendish Experiment: Weighing the Earth 卡文迪什實驗 : 秤地球 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。

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 8.3. 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

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

Energy in Circular Orbits 圓形運動的能量 圓形軌道  r > r K K  E K E U K U Higher K or v  Lower E & orbit (r) . To catch the satellite, the shuttle needs to lose energy. 太空梭若要追上衛星，必需減少能量。 It does so by turning to fire its engine opposite its direction of motion. 要這樣的話，它掉頭朝運動的反方向啓動引擎就可以。 It drops lower, 它往下掉， turns again , 再次掉頭， and fires its engine to achieve a circular orbit, now faster and lower than before. 然後啓動引擎以達成一較低，也較快的圓形軌道， 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

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：不用管場如何產生 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