📚 Gravitation: Key Points for IB & CIE Physics | IB CIE 物理：万有引力考点精讲
Gravitation is a central topic in both IB and CIE Physics, bridging mechanics, energy and astronomical motion. Understanding Newton’s law of universal gravitation, gravitational fields, potential energy and orbital dynamics is essential for tackling both structured problems and extended analytical questions. This article systematically covers the key concepts, equations and exam techniques you need to master gravitation, with direct comparisons between the two syllabuses where relevant.
万有引力是 IB 与 CIE 物理共同的核心主题，它将力学、能量与天体运动紧密联结。深入理解牛顿万有引力定律、引力场、势能以及轨道动力学，不仅有助于解答结构化试题，更是应对综合分析题的基石。本文系统梳理关键概念、核心方程与应试技巧，并适时对比两大课程体系的侧重，助你彻底掌握万有引力考点。
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. The vector equation is F = G·m₁·m₂ / r², where G = 6.67 × 10⁻¹¹ N m² kg⁻² is the universal gravitational constant. The force acts along the line joining the two masses and is always attractive.
牛顿万有引力定律指出，任意两个质点间的引力大小与两质点质量的乘积成正比，与它们之间距离的平方成反比。矢量表达式为 F = G·m₁·m₂ / r²，其中引力常量 G = 6.67 × 10⁻¹¹ N·m²·kg⁻²。力的方向沿两质点连线，且始终为吸引力。
In problems involving extended bodies, a spherically symmetric mass can be treated as if all its mass were concentrated at the centre, as long as you are outside the body. This simplification is crucial for calculating the gravitational force on satellites or planets.
对于具有球对称分布的延展体，若考察点位于物体外部，则可将该物体的全部质量视为集中于球心进行计算。这一简化是分析卫星或行星受力的重要前提。
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: g = F / m. It is a vector quantity, measured in N kg⁻¹, which is equivalent to m s⁻². For a point mass M, the field strength at a distance r from its centre is g = G·M / r².
引力场强度 g 定义为放入场中某点的单位质量试探物体所受的引力，即 g = F / m。它是矢量，单位为 N·kg⁻¹，与加速度单位 m·s⁻² 等效。对于点质量 M，在其外部距离 r 处的场强大小为 g = G·M / r²。
Near the Earth’s surface, g is approximately 9.81 N kg⁻¹ and can be assumed constant for small altitude changes. The variation of g with height and depth is frequently examined. Inside a uniform solid sphere, g decreases linearly with distance from the centre, becoming zero at the centre.
在地球表面附近，引力场强度 g 约为 9.81 N·kg⁻¹，并可在高度变化不大时视为恒量。g 随高度和深度的变化是常见考点。在一均匀实心球体内部，g 随距球心的距离线性减小，至球心处减为零。
3. Gravitational Potential Energy | 引力势能
The gravitational potential energy U of a two-body system is defined with zero potential at infinite separation. For two point masses M and m separated by a distance r, U = – G·M·m / r. The negative sign indicates that work must be done against the gravitational field to separate the masses to infinity.
双质点系统的引力势能 U 以无穷远为零势点进行定义。对于相距为 r 的两个点质量 M 和 m，有 U = – G·M·m / r。负号表示要将两质点分离至无穷远，必须克服引力做功。
When a satellite moves in an elliptical or circular orbit, its total mechanical energy is negative, confirming it is bound to the central body. In IB and CIE exam questions, you often need to apply the relation between kinetic energy, potential energy and total energy for orbiting objects.
当卫星沿椭圆或圆轨道运动时，其总机械能为负值，表明卫星与中心天体构成束缚系统。在 IB 与 CIE 考试中，经常需要运用轨道物体动能、势能与总能量间的关系进行求解。
4. Gravitational Potential | 引力势
Gravitational potential V at a point is the gravitational potential energy per unit mass experienced by a test mass at that point: V = U / m. For a point mass M, V = – G·M / r. Potential is a scalar, making it easier to sum contributions from multiple masses.
引力势 V 定义为单位质量试探物体在场中某点具有的引力势能，即 V = U / m。对于点质量 M，有 V = – G·M / r。引力势是标量，因此方便叠加多个质量的贡献。
The field strength is related to the potential gradient: g = – dV / dr. CIE candidates may encounter numerical potential gradients, while IB students are expected to understand the meaning of equipotential surfaces and the direction of the field relative to them.
引力场强度与势的梯度相关：g = – dV / dr。CIE 考纲可能涉及数值型的势梯度计算，而 IB 课程则要求学生理解等势面的意义及场方向与等势面的垂直关系。
5. Kepler’s Laws of Planetary Motion | 开普勒行星运动定律
Kepler’s three laws describe planetary motion: (1) Each planet moves in an ellipse with the Sun at one focus. (2) A line segment joining a planet and the Sun sweeps out equal areas in equal time intervals. (3) The square of the orbital period is proportional to the cube of the semi‑major axis: T² ∝ a³.
开普勒三大定律描述了行星运动规律：（1）所有行星沿椭圆轨道运行，太阳位于椭圆的一个焦点上；（2）行星与太阳的连线在相等时间内扫过相等的面积；（3）公转周期的平方正比于轨道半长轴的立方：T² ∝ a³。
For circular orbits, Kepler’s third law can be derived from Newton’s law: T² = (4π² / G·M) · r³, where r is the orbital radius. This equation is fundamental in calculating the mass of a central body from orbital data.
对于圆轨道，开普勒第三定律可由牛顿定律导出：T² = (4π² / G·M) · r³，其中 r 为轨道半径。该方程是根据轨道数据推算中心天体质量的核心依据。
6. Circular Orbits and Satellite Motion | 圆轨道与卫星运动
For a satellite in a stable circular orbit, the required centripetal force is provided entirely by gravity: G·M·m / r² = m·v² / r. Solving for orbital speed gives v = √(G·M / r). The orbital period is then T = 2π·r / v = 2π · √(r³ / G·M).
对于稳定圆轨道上的卫星，所需向心力完全由万有引力充当：G·M·m / r² = m·v² / r。由此解得轨道速率 v = √(G·M / r)，轨道周期为 T = 2π·r / v = 2π · √(r³ / G·M)。
These relationships show that orbital speed decreases with increasing radius, while the period increases. Both IB and CIE exams often ask you to compare the velocities and periods of satellites at different altitudes.
上述关系表明，轨道速率随轨道半径增大而减小，而周期则随之增大。IB 与 CIE 考试经常要求比较不同高度卫星的速度与周期。
7. Energy of Orbiting Satellites | 轨道卫星的能量
For a satellite of mass m in a circular orbit of radius r around a mass M, the kinetic energy is K = ½·m·v² = G·M·m / (2r). The potential energy is U = – G·M·m / r. Therefore the total mechanical energy is E = K + U = – G·M·m / (2r).
对于在半径 r 的圆轨道上绕质量 M 运行的卫星，其动能 K = ½·m·v² = G·M·m / (2r)，势能 U = – G·M·m / r，因此总机械能为 E = K + U = – G·M·m / (2r)。
This result shows that the total energy equals half the potential energy, and that magnitude of total energy equals the kinetic energy. To move to a higher orbit, the satellite’s total energy must increase (become less negative); work must be done on the system.
这个结果说明总能量等于势能的一半，且总能量的绝对值等于动能。若要转移到更高轨道，卫星的总能量必须增加（负得较少），这意味着系统需克服引力做功。
8. Escape Velocity | 逃逸速度
Escape velocity is the minimum speed a body must have at a given distance from a mass M in order to escape its gravitational field and reach infinity with zero kinetic energy. Setting total energy to zero gives ½·m·vₑₛ² – G·M·m / R = 0, so vₑₛ = √(2·G·M / R).
逃逸速度是指物体在距离质量 M 某处必须具有的最小速率，以使其恰好能挣脱引力场并到达无穷远处且动能为零。令总能量为零可得 ½·m·vₑₛ² – G·M·m / R = 0，因此 vₑₛ = √(2·G·M / R)。
It is often useful to compare escape velocity with circular orbital velocity: vₑₛ = √2 · v_orbital. The escape velocity depends only on the mass and radius of the central body, not on the projectile’s mass.
常常需要将逃逸速度与圆轨道速率进行对比：vₑₛ = √2 · v_orbital。逃逸速度仅取决于中心天体的质量与半径，与抛射体的质量无关。
| Body 天体 | Escape velocity (km/s) 逃逸速度 |
|---|---|
| Earth 地球 | 11.2 |
| Moon 月球 | 2.4 |
| Jupiter 木星 | 59.5 |
9. Geostationary Satellites | 地球同步卫星
A geostationary satellite orbits Earth in the equatorial plane with a period exactly equal to one sidereal day (23 h 56 min 4 s), so it appears stationary relative to the Earth’s surface. Its orbital radius is fixed: using Kepler’s third law, r ≈ 4.22 × 10⁷ m, giving an altitude of about 3.58 × 10⁷ m above the surface.
地球同步卫星在赤道平面内运行，其公转周期恰好等于一个恒星日（23时56分4秒），因此在地面上看来似乎静止不动。其轨道半径是固定的：由开普勒第三定律得 r ≈ 4.22 × 10⁷ m，距地表高度约 3.58 × 10⁷ m。
For both IB and CIE, you must be able to justify that such satellites must be in equatorial orbits and at a specific height. They are vital for communications and weather monitoring.
无论 IB 还是 CIE 均要求能论证同步卫星必须位于赤道上空且具有特定高度。这类卫星对通信和气象监测至关重要。
10. Weightlessness and Apparent Weight | 失重与视重
Weightlessness does not mean the absence of gravity; rather, it occurs when there is no normal contact force opposing gravity. Astronauts in an orbiting spacecraft experience apparent weightlessness because both they and the spacecraft are in free fall towards Earth, with the same acceleration g.
失重并非意味引力消失，而是指不存在与引力抗衡的法向接触力。在轨道航天器中的宇航员会感受到视重为零，因为他们和航天器一同相对地球做自由落体运动，均具有相同的加速度 g。
IB exam questions often ask students to interpret scale readings in accelerating lifts, while CIE may extend this to rotating space stations and artificial gravity. In all cases, apparent weight = m(g – a) locally, where a is the acceleration of the reference frame.
IB 考题常要求学生分析加速电梯中秤的读数，CIE 则可能将其延伸至旋转空间站与人造重力。所有情况均可通过视重公式 m(g – a) 来理解，其中 a 为参考系加速度。
11. Exam Focus: Common Pitfalls and Data Analysis | 考试聚焦：常见误区与数据分析
Many students confuse gravitational field strength g with universal constant G or forget that potential energy is negative. When dealing with multiple bodies, always remember that forces are vectors but potential is scalar. In IB data‑based questions, you may be required to linearise relationships, e.g., plotting T² against r³ to find the central mass.
许多学生混淆引力场强度 g 与万有引力常量 G，或忘记引力势能为负值。处理多体问题时，务必牢记力是矢量而势是标量。在 IB 数据分析题中，可能需要将关系线性化，例如绘制 T²–r³ 图线以求解中心天体质量。
CIE structured questions frequently embed the gravitational constant G in numerical calculations, so be careful with unit conversions. Always check whether a scenario involves a uniform field (g constant) or a radial field (g ∝ 1/r²).
CIE 结构化试题常将引力常量 G 嵌入数值计算，务必留意单位转换。判断问题情景究竟是匀强场（g 恒定）还是辐射场（g ∝ 1/r²）至关重要。
- For radial fields: use F = G M m / r², g = G M / r², V = – G M / r
- 对于辐射场：使用 F = G M m / r²，g = G M / r²，V = – G M / r
- For uniform fields (near Earth’s surface): F = m g, ΔU = m g Δh
- 对于匀强场（近地表）：使用 F = m g，ΔU = m g Δh
- In energy problems, set zero of potential at infinity for radial fields, but at a chosen reference level for uniform fields.
- 在处理能量问题时，辐射场以无穷远为零势点，匀强场可选取参考平面为零势面。
12. Summary of Key Equations | 核心公式速查
| Quantity 物理量 | Equation 方程 |
|---|---|
| Gravitational force 万有引力 | F = G·m₁·m₂ / r² |
| Field strength (radial) 辐射场强 | g = G·M / r² |
| Potential energy 引力势能 | U = – G·M·m / r |
| Gravitational potential 引力势 | V = – G·M / r |
| Orbital speed 轨道速率 | v = √(G·M / r) |
| Kepler III (circular) 开普勒第三定律 | T² = (4π² / G·M) · r³ |
| Total energy (circular) 总能量 | E = – G·M·m / (2r) |
| Escape velocity 逃逸速度 | vₑₛ = √(2·G·M / R) |
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