📚 IB WJEC Physics: Gravitation Key Points | IB WJEC 物理：万有引力 考点精讲
Gravitation plays a central role in both IB and WJEC physics syllabuses, linking celestial mechanics to everyday phenomena. This article unpacks the essential concepts, formulas, and exam skills you need, with matched explanations in English and Chinese to reinforce understanding. We cover Newton’s law, field strength, potential, Kepler’s laws, satellite motion, escape velocity, and more, all tuned to the requirements of IB and WJEC assessments.
万有引力是 IB 和 WJEC 物理大纲的共同核心，它将天体力学与日常现象联系在一起。本文拆解必备概念、公式和考试技巧，采用中英对照讲解以加深理解。我们涵盖牛顿引力定律、场强、引力势、开普勒定律、卫星运动、逃逸速度等内容，完全匹配 IB 与 WJEC 的考查要求。
1. Newton’s Law of Universal Gravitation | 牛顿万有引力定律
Any two point masses attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of their separation. This is expressed as F = Gm₁m₂/r², where G = 6.67 × 10⁻¹¹ N m² kg⁻².
任何两个质点都会相互吸引，引力大小与它们的质量乘积成正比，与它们之间距离的平方成反比。公式为 F = Gm₁m₂/r²，其中 G = 6.67 × 10⁻¹¹ N m² kg⁻²。
The law applies strictly to point masses, but can also be used for spherical bodies where the distance r is measured between their centres. In IB exams, you may need to explain why a uniform sphere acts as if all its mass is concentrated at the centre.
这一定律严格适用于质点，但也可用于球体，此时距离 r 为两球心之间的距离。在 IB 考试中，你可能需要解释为什么均匀球体的引力效果相当于所有质量集中在球心。
WJEC questions often require direct calculation of force between two objects, such as the Earth and the Moon, or between two small laboratory masses. Remember to convert all units to SI and use standard form for G.
WJEC 试题常要求直接计算两个物体之间的引力，比如地球和月球，或实验室中的两个小质量物体。记住所有单位都要转换为国际单位，引力常数 G 用科学记数法表示。
F = G m₁ m₂ / r²
2. Gravitational Field Strength | 引力场强度
The gravitational field strength g at a point is defined as the force per unit mass experienced by a small test mass placed at that point: g = F/m. For a spherical mass M, the field strength at distance r from its centre is g = GM/r².
引力场强度 g 定义为单位质量在该点受到的引力：g = F/m。对于一个质量为 M 的球体，距离其球心 r 处的场强为 g = GM/r²。
Near the Earth’s surface, g is approximately 9.81 N kg⁻¹, but this value decreases with altitude. IB HL candidates must be able to derive g = GM/r² from Newton’s law and discuss how g varies inside and outside a planet.
地表面附近 g 约为 9.81 N kg⁻¹，但随海拔升高而减小。IB 高阶考生必须能直接从牛顿定律推导出 g = GM/r²，并能讨论 g 在地球内部和外部如何变化。
WJEC papers frequently test the relationship between field strength and distance, often in the context of binary stars or comparing the gravitational field at different heights above a planet.
WJEC 试卷频繁考查场强与距离的关系，常见于双星问题或比较行星表面不同高度的引力场。
g = GM / r²
3. Gravitational Potential | 引力势
Gravitational potential V at a point is the work done per unit mass in bringing a small test mass from infinity to that point. It is given by V = –GM/r. The negative sign indicates that gravity is attractive; potential increases with distance, reaching zero at infinity.
引力势 V 是单位质量从无穷远移至该点外力所做的功。表达式为 V = –GM/r。负号表明引力是吸引力；势随距离增加而增大，在无穷远处为零。
IB students must understand the significance of the zero reference at infinity and why potentials are always negative near massive bodies. Gravitational potential energy of two masses is U = –GMm/r.
IB 学生必须理解无穷远处势为零的设定，以及为什么大质量天体附近的势总是负值。两质点间的引力势能为 U = –GMm/r。
In WJEC, potential is less frequently examined but may appear in energy conservation problems involving escape or orbital mechanics. Be prepared to sketch V against r and explain the gradient gives field strength.
在 WJEC 中，引力势考查较少，但可能出现在涉及逃逸或轨道力学的能量守恒问题中。准备好画出 V–r 图并解释斜率即场强。
V = –GM / r
4. Kepler’s Laws of Planetary Motion | 开普勒行星运动定律
Kepler’s three laws describe planetary motion: (1) Planets move in elliptical orbits with the Sun at one focus. (2) A line segment joining a planet and the Sun sweeps out equal areas in equal times. (3) The square of the orbital period T is proportional to the cube of the semi‑major axis r: T² ∝ r³.
开普勒三定律描述行星运动：（1）行星沿椭圆轨道运行，太阳位于一个焦点上。（2）行星与太阳的连线在相等时间内扫过相等的面积。（3）轨道周期 T 的平方与半长轴 r 的立方成正比：T² ∝ r³。
For circular orbits, the third law can be derived from equating gravitational force to centripetal force: T² = (4π²/GM)r³. Both IB and WJEC syllabi require you to perform this derivation.
对于圆轨道，第三定律可通过万有引力等于向心力推导得出：T² = (4π²/GM)r³。IB 和 WJEC 大纲都要求掌握该推导。
Exam questions often involve comparing periods and radii of different planets or moons. Always state the constant of proportionality depends on the central mass M.
考试题经常要求比较不同行星或卫星的周期与半径。务必指出比例常数取决于中心天体质量 M。
T² = (4π² / GM) r³
5. Orbital Motion and Satellite Dynamics | 轨道运动与卫星动力学
A satellite in circular orbit has its centripetal force provided by gravity: GMm/r² = mv²/r. This gives orbital speed v = √(GM/r) and period T = 2πr/v = 2π√(r³/GM). Notice that orbital speed decreases with increasing radius.
圆轨道上的卫星靠引力提供向心力：GMm/r² = mv²/r。由此可得轨道速度 v = √(GM/r)，周期 T = 2πr/v = 2π√(r³/GM)。注意轨道速度随半径增大而减小。
For geostationary satellites, the orbital period matches the Earth’s rotation (24 hours) and the orbit lies in the equatorial plane. Both IB and WJEC assess understanding of this specific orbit, including its altitude of about 3.6 × 10⁷ m.
对于地球同步卫星，轨道周期等于地球自转周期（24 小时），轨道位于赤道平面上。IB 和 WJEC 都考查对这一特定轨道的理解，包括其高度约为 3.6 × 10⁷ m。
When tackling energy questions, total mechanical energy E = –GMm/(2r) for circular orbits; this is half the potential energy. You may be asked to calculate energy changes when a satellite moves between orbits.
处理能量问题时，圆轨道的总机械能 E = –GMm/(2r)，是势能的一半。你可能会被要求计算卫星在不同轨道之间转移时的能量变化。
v = √(GM / r) E = –GMm / (2r)
6. Escape Velocity | 逃逸速度
Escape velocity is the minimum speed required for an object to escape a planet’s gravitational field without further propulsion. By energy conservation, ½mv² – GMm/r = 0, giving vₑ = √(2GM/r). At Earth’s surface, this is about 11.2 km s⁻¹.
逃逸速度是物体无需后续推力就能脱离行星引力场的最小速度。由能量守恒，½mv² – GMm/r = 0，可得 vₑ = √(2GM/r)。在地球表面，这约为 11.2 km s⁻¹。
Note that escape velocity does not depend on the mass of the escaping object. It is a scalar, independent of direction (though air resistance would in practice require vertical launch). IB often links this to dark matter or black holes conceptually.
注意逃逸速度与逃脱物体的质量无关。它是一个标量，与方向无关（不过实际中空气阻力要求垂直发射）。IB 常将其与暗物质或黑洞的概念联系起来。
WJEC may feature calculations comparing escape velocities on different planets or asking why the Moon lacks an atmosphere (its lower escape velocity allows gas molecules to escape).
WJEC 可能会出现比较不同行星逃逸速度的计算题，或问及月球为何没有大气层（其较低的逃逸速度使气体分子逃逸）。
vₑ = √(2GM / r)
7. Gravitational Potential Energy | 引力势能
The gravitational potential energy of a system of two point masses is U = –GMm/r. This equation represents the work done to assemble the system from infinite separation. It is negative, meaning work must be done against gravity to separate the masses.
双质点系统的引力势能为 U = –GMm/r。该方程表示从相距无穷远处组装系统所做的功。它为负值，意味分离两质点需要克服引力做功。
In an IB context, you should be able to derive this from W = ∫ F dr, integrating from infinity to r. HL students are expected to handle non‑uniform fields using calculus; SL may be given the formula directly.
在 IB 中，你应该能够从 W = ∫ F dr 推导该公式，积分从无穷远积至 r。高阶学生需用微积分处理非匀强场；标准级别通常会直接给出公式。
WJEC specifications often combine potential energy with kinetic energy in conservation problems, especially for comets moving in highly elliptical orbits where total energy remains constant.
WJEC 考纲常将势能与动能结合在能量守恒问题中，尤其对于高度椭圆轨道上的彗星，其总能量保持不变。
U = –GMm / r
8. Comparison Between Gravitational and Electric Fields | 引力场与电场的比较
Gravitational and electric fields share many mathematical similarities: both obey inverse square laws and have potentials that vary as 1/r. However, gravitational forces are always attractive, while electric forces can be attractive or repulsive.
引力场和电场有许多数学上的相似之处：都遵循平方反比律，势能均随 1/r 变化。但引力永远表现为吸引力，而电力可以是吸引力或排斥力。
IB students are often asked to compare the two fields, noting that gravitational field strength g is analogous to electric field strength E, and that the gravitational constant G parallels 1/(4πε₀) in Coulomb’s law.
IB 学生常被要求比较两种场，指出引力场强 g 相当于电场强度 E，引力常数 G 与库仑定律中的 1/(4πε₀) 作用类似。
WJEC may also test this comparison, especially when linking planetary models to atomic models historically. Remember that the gravitational force between protons is vastly weaker than the electric force, which is why gravity is negligible at small scales.
WJEC 也可能考查这种比较，特别是在将行星模型与历史上的原子模型联系时。记住质子间的引力远弱于电力，这就是为什么在微观尺度引力可忽略。
9. Geostationary Satellites and Applications | 地球同步卫星及其应用
A geostationary satellite orbits at a fixed position relative to the Earth’s surface, which requires an equatorial orbit at a radius of about 4.2 × 10⁷ m from Earth’s centre. Its period is exactly one sidereal day (23 h 56 min).
地球同步卫星相对于地球表面固定在一个位置上空，这要求轨道在赤道平面内，距地心约 4.2 × 10⁷ m。其周期正好是一个恒星日（23 时 56 分）。
Both IB and WJEC examine the conditions: orbit must be equatorial, circular, and have a period equal to Earth’s rotation. Common uses include communications, weather monitoring, and GPS (though GPS satellites are not geostationary).
IB 和 WJEC 都考查条件：轨道必须位于赤道面、圆形，且周期等于地球自转周期。常见应用包括通信、气象监测和 GPS（尽管 GPS 卫星并非同步）。
Calculations frequently involve r = (GMT²/(4π²))^(1/3). Be able to substitute T = 86400 s (or the more accurate 86164 s) and show the radius is about 6.6 Earth radii from the centre.
计算中常用 r = (GMT²/(4π²))^(1/3)。要能代入 T = 86400 s（或更准确的 86164 s）并证明轨道半径约为 6.6 个地球半径（距地心）。
r = ³√(GMT² / 4π²)
10. Common Pitfalls and Exam Tips | 常见易错点与考试技巧
Many candidates confuse gravitational field strength g (N kg⁻¹) with acceleration due to gravity (m s⁻²) – numerically identical near a planet’s surface but distinct concepts. Always use m for mass of the orbiter and M for central mass.
许多考生混淆引力场强度 g（N kg⁻¹）与重力加速度（m s⁻²）——数值上相同但在行星表面附近概念不同。始终用 m 表示绕行体质量，M 表示中心天体质量。
When applying Kepler’s third law, ensure that the semi‑major axis is measured in metres and the period in seconds. For elliptical orbits, a is half the longest diameter, but for circular orbits it is simply the radius.
应用开普勒第三定律时，确保半长轴以米为单位，周期以秒为单位。对于椭圆轨道，a 为最长直径的一半，但在圆轨道中就是半径。
Sign errors in potential energy are common: U = –GMm/r, and total energy in a circular orbit is negative, indicating a bound system. In escape velocity problems, set total energy equal to zero at infinity.
势能符号错误很常见：U = –GMm/r，圆轨道总能量为负，表明系统是束缚态。在逃逸速度问题中，设定无穷远处总能量为零。
Finally, show all steps in derivations: equate centripetal and gravitational force, cancel m, and rearrange. IB examiners award marks for clear reasoning; WJEC mark schemes similarly value method even if the final answer has a numerical slip.
最后，推导过程中要展示所有步骤：令向心力等于万有引力，约去 m，然后整理。IB 考官会给清晰的推理过程打分；WJEC 评分方案同样重视方法，即使最终答案有计算错误。
Check units, signs, and whether the question asks for force, field, potential or energy.
检查单位、正负号，并确认题目要求的是力、场、势还是能量。
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