HKDSE Physics – Gravitation

HKDSE Physics Core
Chapter 8: Gravitation | 萬有引力

Billy Sir’s Smart Notes: Master Newton’s Law of Gravitation, Orbital Motion, and Satellites.
由中大物理系碩士 Billy Sir 編寫，助你極速掌握萬有引力定律、軌道運動及人造衛星原理。

1. Newton’s Law of Gravitation | 牛頓萬有引力定律

Every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. 宇宙中任何兩個質點都會互相吸引，引力的大小與它們質量的乘積成正比，與它們中心距離的平方成反比。

🍎 Formula for Gravitational Force | 萬有引力公式

F = G \frac{m_1 m_2}{r^2}

\( F \): Gravitational force (N) 萬有引力
\( G \): Universal gravitational constant (\( 6.67 \times 10^{-11} \text{ N m}^2 \text{ kg}^{-2} \)) 萬有引力常數
\( m_1, m_2 \): Masses of the two objects (kg) 兩物體的質量
\( r \): Distance between their centers (m) 兩物體中心的距離

2. Gravitational Field Strength | 引力場強度

Gravitational field strength (\( g \)) at a point is defined as the gravitational force per unit mass acting on a small test mass placed at that point. 某點的引力場強度定義為放置在該點的單位質量所受的引力。

🌍 Formula for Field Strength | 引力場強度公式

g = \frac{F}{m} = G \frac{M}{r^2}

Note: On the surface of the Earth, \( r \) is the radius of the Earth, and \( g \approx 9.81 \text{ m s}^{-2} \) (or N kg\(^{-1}\)). 注意：在地球表面，\( r \) 為地球半徑，\( g \approx 9.81 \text{ m s}^{-2} \)。

3. Orbital Motion | 軌道運動

For a satellite or planet in circular orbit, the gravitational force provides the necessary centripetal force. 對於在圓形軌道上運行的衛星或行星，萬有引力提供了所需的向心力。

🚀 Derivation of Orbital Speed | 軌道速度推導

Equating Gravitational Force and Centripetal Force: (將萬有引力等同於向心力：)

G \frac{Mm}{r^2} = m \frac{v^2}{r}

Solving for orbital speed (\( v \)):

v = \sqrt{\frac{GM}{r}}

Important: The orbital speed depends only on the mass of the central body (\( M \)) and the orbital radius (\( r \)). It is independent of the mass of the satellite (\( m \)). 重要提示：軌道速度僅取決於中心天體的質量和軌道半徑，與衛星本身的質量無關。

4. Satellites | 人造衛星

Satellites are generally classified by their orbits. Two common types tested in HKDSE are Low Earth Orbit (LEO) satellites and Geostationary satellites. 衛星通常按軌道分類。DSE 常考的兩種是低地軌道衛星和地球同步衛星。

Feature (特徵)	Low Orbit Satellite (低地軌道衛星)	Geostationary Satellite (地球同步衛星)
Orbital Radius / Altitude 軌道半徑 / 高度	Small (Altitude ~ a few hundred km) 較小 (高度約數百公里)	Large (Altitude ~ 36,000 km) 較大 (高度約 36,000 公里)
Period (\( T \)) 週期	Short (~ 90 to 120 minutes) 短 (約 90 至 120 分鐘)	Exactly 24 hours (Same as Earth’s rotation) 精確為 24 小時 (與地球自轉相同)
Position relative to Earth 相對地球的位置	Moves across the sky quickly. 在天空中快速移動。	Appears stationary above the Equator. 在赤道上空看似靜止。
Common Applications 常見應用	Weather observation, spying, mapping. 氣象觀測、偵察、地圖繪製。	Telecommunications, GPS, broadcasting. 通訊、全球定位系統、廣播。

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