📚 A-Level CCEA Physics: Gravitation Key Concepts | A-Level CCEA 物理：万有引力考点精讲
Gravitation is one of the cornerstone topics in CCEA A‑Level Physics, linking celestial mechanics, satellite motion and the concept of fields. A clear understanding of Newton’s law, gravitational field strength, potential and orbital dynamics is essential for handling both quantitative problems and qualitative explanations in the exam.
万有引力是 CCEA A-Level 物理的基石课题之一，将天体力学、卫星运动和场的概念紧密地结合起来。透彻理解牛顿定律、引力场强、引力势以及轨道动力学，对于处理考试中的定量计算和定性解释都非常关键。
1. Newton’s Law of Universal Gravitation | 牛顿万有引力定律
Newton’s law of universal gravitation states that every point mass attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres.
牛顿万有引力定律指出：任何两个质点之间都存在相互吸引力，该力的大小与两个质量的乘积成正比，与它们质心之间距离的平方成反比。
F = Gm₁m₂ / r²
G is the universal gravitational constant, with a value of approximately 6.67 × 10⁻¹¹ N m² kg⁻². This law applies to point masses and spherically symmetric bodies, where r is the distance between their centres.
G 是万有引力常数，数值约为 6.67 × 10⁻¹¹ 牛·米²/千克²。该定律适用于质点和球对称物体，此处的 r 是两物体质心间的距离。
2. Gravitational Field Strength | 引力场强度
The 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.
引力场强度 g 定义为放置在一点的单位质量所受的引力。
g = F / m
For a spherical body of mass M, the field strength at a distance r from its centre is given by g = GM / r². Near the Earth’s surface, g ≈ 9.81 N kg⁻¹.
对于质量为 M 的球体，在距离其球心 r 处的场强为 g = GM / r²。在地球表面附近，g ≈ 9.81 牛/千克。
3. Variation of g with Altitude and Depth | 重力加速度随高度和深度的变化
Above the Earth’s surface, g decreases with the square of the distance from the centre: g = GM / (R + h)², where R is the Earth’s radius and h is the altitude.
在地表上方，g 随到地心距离的平方而减小：g = GM / (R + h)²，其中 R 是地球半径，h 为高度。
Below the surface (assuming uniform density), g decreases linearly and is proportional to the distance from the centre: g’ = g₀ (r / R), where r is the distance from the centre.
在地表以下（假设均匀密度），g 线性减小，并与到地心的距离成正比：g’ = g₀ (r / R)，r 为到地心的距离。
4. Gravitational Potential | 引力势
Gravitational potential V at a point is the work done per unit mass in bringing a test mass from infinity to that point. It is a scalar quantity and is always negative.
引力势 V 定义为单位质量从无穷远处移至该点所需做的功。它是标量，且恒为负值。
V = −GM / r
The zero of potential is taken at infinity. As r decreases, V becomes more negative, indicating that work is done by the field when a mass moves inwards.
引力势的零点取在无穷远处。随着 r 减小，V 变得更负，表明当质量向内移动时引力场做正功。
5. Gravitational Potential Energy | 引力势能
The gravitational potential energy U of a system of two point masses M and m separated by distance r is given by U = −GMm / r. This represents the work done to assemble the system from an infinite separation.
两个相距为 r 的质点 M 和 m 所构成的系统的引力势能为 U = −GMm / r。它代表从相距无穷远开始形成该系统所需做的功。
When dealing with a satellite of mass m orbiting a planet of mass M, the potential energy is negative and decreases as the orbital radius decreases.
对于绕质量为 M 的行星运行的卫星（质量为 m），势能为负，且随轨道半径的减小而变得更负。
6. Escape Velocity | 逃逸速度
Escape velocity is the minimum speed required for an object to leave a planet’s gravitational field without further propulsion. It is derived by equating kinetic energy and gravitational potential energy at the surface.
逃逸速度是物体无需后续推进即可脱离行星引力场所需的最小速率。可通过将表面处的动能与引力势能等效来推导。
vₑ = √(2GM / R) = √(2gR)
For Earth, vₑ ≈ 11.2 km s⁻¹. Note that escape velocity does not depend on the mass of the escaping object.
对于地球，vₑ ≈ 11.2 千米/秒。注意逃逸速度与逃逸物体的质量无关。
7. Orbital Motion and Kepler’s Third Law | 轨道运动与开普勒第三定律
For a satellite in a circular orbit, the centripetal force required is provided by the gravitational attraction. Equating these gives the orbital speed.
对于做圆周轨道运动的卫星，所需的向心力由万有引力提供。将此二力等同可求出轨道速率。
mv² / r = GMm / r² → v = √(GM / r)
The orbital period T can be found using v = 2πr / T, leading to Kepler’s third law: T² = (4π² / GM) r³. The square of the period is proportional to the cube of the orbital radius.
利用 v = 2πr / T 可求得轨道周期，从而得到开普勒第三定律：T² = (4π² / GM) r³。周期的平方与轨道半径的立方成正比。
8. Energy of Orbiting Satellites | 卫星轨道的能量
The total mechanical energy E of a satellite in a circular orbit is the sum of its kinetic and potential energies.
在圆轨道上运行的卫星的总机械能 E 是其动能与势能之和。
E = KE + PE = ½mv² − GMm / r
Substituting v² = GM / r yields E = −GMm / (2r). The total energy is negative, indicating a bound orbit. The kinetic energy is half the magnitude of the potential energy.
代入 v² = GM / r 可得 E = −GMm / (2r)。总能量为负值，表明这是一种束缚轨道。动能的大小等于势能绝对值的一半。
9. Geostationary Satellites | 地球同步卫星
A geostationary satellite orbits above the Earth’s equator with a period of 24 hours, appearing stationary relative to a point on the surface.
地球同步卫星在地球赤道上方运行，周期为 24 小时，相对于地面某点看起来静止不动。
Its orbital radius is approximately 42 300 km from the Earth’s centre (height ≈ 35 800 km). The satellite must orbit in the same direction as the Earth’s rotation and be in the equatorial plane.
其轨道半径距地心约 42 300 千米（高度约 35 800 千米）。卫星必须与地球自转同向，且轨道平面必须位于赤道平面内。
10. Weightlessness and Apparent Weight | 失重与表观重量
Astronauts in orbit experience weightlessness not because gravity is absent, but because both the astronaut and the spacecraft are in free fall towards the Earth with the same acceleration.
轨道中的宇航员体验到失重状态，并非因为引力消失，而是因为宇航员与航天器都以相同的加速度向地球自由下落。
Contact forces become zero, giving the sensation of weightlessness. Apparent weight is the normal reaction force; in free fall it is zero.
接触力变为零，从而产生了失重的感觉。表观重量即法向反作用力；在自由下落中为零。
11. Gravitational Field Lines and Equipotentials | 引力场线与等势面
Gravitational field lines indicate the direction of the field. For a spherical mass, they are directed radially inwards.
引力场线指示场的方向。对于球形体，场线沿径向指向质心。
Equipotential surfaces are surfaces of constant gravitational potential. For a point mass, they are concentric spheres. No work is done when moving a mass along an equipotential surface.
等势面是引力势保持恒定的曲面。对于点质量，等势面为同心球面。沿等势面移动质量时不做功。
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