📚 Gravitation: Key Concepts for IB & AQA Physics | 万有引力：IB 与 AQA 物理考点精讲
Gravitation is a fundamental force of nature that governs the motion of celestial bodies and everyday objects on Earth. In both IB and AQA A-level Physics, mastering the law of universal gravitation, gravitational fields, potential energy, orbital mechanics, and Kepler’s laws is essential for exam success. This article distils the core concepts, formulas, and common pitfalls into a bilingual revision guide covering everything from Newton’s law to escape velocity.
万有引力是自然界的一种基本力，支配着天体运动以及地球上物体的日常行为。在 IB 和 AQA A-level 物理中，掌握万有引力定律、引力场、引力势能、轨道力学和开普勒定律是考试成功的关键。本文将核心概念、公式和常见易错点提炼为一篇双语复习指南，涵盖从牛顿定律到逃逸速度的所有内容。
1. Newton’s Law of Universal Gravitation | 牛顿万有引力定律
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. This is expressed by the equation:
每一个质点都会吸引其他每一个质点，引力的大小与两质量的乘积成正比，与它们中心之间距离的平方成反比。这一定律可以用以下公式表示：
F = Gm₁m₂ / r²
where F is the gravitational force between mass m₁ and m₂, r is the distance between their centres, and G is the universal gravitational constant, G ≈ 6.67 × 10⁻¹¹ N m² kg⁻². The force is attractive and acts along the line joining the centres of the masses.
其中 F 是质量 m₁ 和 m₂ 之间的引力，r 是两球心之间的距离，G 是万有引力常量，G ≈ 6.67 × 10⁻¹¹ N m² kg⁻²。该力是吸引力，方向沿着连接两质量中心的直线。
For extended spherical objects with uniform density, the law applies as if all their mass were concentrated at the centre. In IB and AQA exams, students are often asked to calculate the force between planets or between a planet and a satellite. Remember to use metres for distance and kilograms for mass.
对于均匀密度的球形大物体，这一定律同样适用，就好像所有质量都集中在球心。在 IB 和 AQA 考试中，常要求学生计算行星之间或行星与卫星之间的引力。务必使用米作为距离单位、千克作为质量单位。
2. Gravitational Field and Field Strength | 引力场与引力场强度
A gravitational field is a region of space in which a mass experiences a force. 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:
引力场是空间中的一个区域，置于其中的质量会受到力的作用。某点的引力场强度 g 定义为放置在该点的小检验质量所受到的每单位质量的力：
g = F / m
This vector quantity points towards the source of the field. For a point mass or a spherical body of mass M, the field strength at a distance r from the centre is:
这个矢量指向产生场的源。对于一个点质量或质量为 M 的球体，在距离中心 r 处的场强为：
g = GM / r²
Near the Earth’s surface, g is approximately 9.81 N kg⁻¹ (or m s⁻²). In IB and AQA contexts, it is crucial to distinguish between g as field strength and g as acceleration due to gravity, though they are numerically equivalent.
在地球表面附近，g 约等于 9.81 N kg⁻¹（或 m s⁻²）。在 IB 和 AQA 情景中，虽然数值相等，但区分作为场强的 g 和作为重力加速度的 g 很重要。
Field lines are used to represent gravitational fields: for a point mass they radiate inwards, and the spacing indicates field strength. Uniform fields are represented by parallel, equally spaced lines, as approximated over small regions near the Earth’s surface.
场线用来表示引力场：对于点质量，场线径向向内；场线的疏密表示场强的大小。匀强场由平行、等距的线表示，就像地球表面附近小范围内近似的那样。
3. Gravitational Acceleration and Variation with Height | 重力加速度及其随高度的变化
The acceleration of free fall g at a height h above the Earth’s surface is given by g = GM / (R + h)², where R is the Earth’s radius. As altitude increases, g decreases. This has important implications for satellite motion and weightlessness.
在地球表面上方高度 h 处的自由落体加速度 g 由 g = GM / (R + h)² 给出，其中 R 是地球半径。随着高度增加，g 减小。这对卫星运动和失重状态有重要影响。
In many exam problems, students need to calculate the value of g on another planet or compare gravitational accelerations using ratios. A common method is to write g ∝ M / R² and then form a ratio to avoid explicit use of G.
在许多考试题中，学生需要计算其他行星上的 g 值，或利用比值比较重力加速度。一种常见方法是写出 g ∝ M / R²，然后形成比例式，以避免直接使用 G。
The concept of ‘weightlessness’ in orbit arises not because g is zero, but because the spacecraft and everything inside it are in free fall together, experiencing the same centripetal acceleration. This is a key distinction in both IB and AQA syllabi.
轨道上的“失重”并不是因为 g 为零，而是因为航天器及其内部一切物体一起处于自由落体状态，具有相同的向心加速度。这是 IB 和 AQA 课程中的关键区别。
4. Gravitational Potential Energy | 引力势能
In a uniform field near the Earth’s surface, gravitational potential energy is given by Ep = mgh. However, for large distances where g varies, we must use the general expression for the potential energy of a system of two point masses:
在地表附近的均匀场中，引力势能用 Ep = mgh 表示。然而，在 g 变化的大尺度距离下，我们必须使用两个质点系统的势能通用表达式：
U = − GMm / r
This formula sets the zero of potential energy at infinity. The negative sign indicates that the force is attractive; energy is required to separate the masses to infinity. IB Physics and AQA both require using this formula for orbital calculations.
该公式将无穷远处的势能设为零。负号表明力是吸引力；需要能量才能将两个质量分离到无穷远。IB 物理和 AQA 都要求使用此公式进行轨道计算。
The change in potential energy when a satellite moves from one orbit to another is ΔU = −GMm (1/r₂ − 1/r₁). Be careful with the order of subtraction to avoid sign errors.
当卫星从一个轨道移动到另一个轨道时，势能的变化为 ΔU = −GMm (1/r₂ − 1/r₁)。请注意相减的顺序，以避免符号错误。
5. Gravitational Potential | 引力势
Gravitational potential V at a point is the gravitational potential energy per unit mass at that point. For a point mass M, it is given by:
某点的引力势 V 是该点每单位质量的引力势能。对于点质量 M，表示为：
V = − GM / r
Potential is a scalar quantity, which simplifies the calculation of fields from multiple sources: the total potential is the algebraic sum of individual potentials. The field strength g is related to the potential gradient: g = − dV/dr.
引力势是标量，这简化了多个源产生的场的计算：总势是各个势的代数和。场强 g 与势梯度相关：g = − dV/dr。
On a graph of V against r, the steepness of the curve gives the magnitude of g. Equipotential surfaces are spherical shells around a mass; field lines are perpendicular to these surfaces. These concepts feature in both AQA and IB multiple-choice and data-response questions.
在 V 对 r 的图上，曲线的陡峭程度给出了 g 的大小。等势面是围绕质量的球壳；场线垂直于这些表面。这些概念出现在 AQA 和 IB 的选择题和数据分析题中。
6. Escape Velocity | 逃逸速度
Escape velocity is the minimum speed an object must have at the surface of a planet to escape its gravitational field without further propulsion. By equating kinetic energy to the magnitude of gravitational potential energy, we derive:
逃逸速度是物体在行星表面必须具有的最小速度，使其无需进一步推进即可脱离行星的引力场。通过将动能等于引力势能的大小，我们推导出：
vesc = √(2GM / R)
where R is the radius of the planet. Note that escape velocity does not depend on the mass of the escaping object. For Earth, vesc ≈ 11.2 km s⁻¹. This derivation is frequently examined in both IB Paper 2 and AQA written papers.
其中 R 是行星的半径。注意逃逸速度与逃逸物体的质量无关。对于地球，vesc ≈ 11.2 km s⁻¹。这一推导过程在 IB Paper 2 和 AQA 笔试试卷中经常考查。
If a spacecraft’s speed exceeds the escape velocity, it will follow a hyperbolic trajectory and leave the planet permanently. Below escape velocity but above the circular orbital speed, the path is elliptical.
如果航天器的速度超过逃逸速度，它将沿双曲线轨道运行并永久离开行星。低于逃逸速度但高于圆形轨道速度时，路径为椭圆形。
7. Satellite Orbits and Centripetal Force | 卫星轨道与向心力
For a satellite in a circular orbit, the gravitational force provides the necessary centripetal force: GMm / r² = mv² / r. This leads to the orbital speed:
对于圆形轨道上的卫星，引力提供必要的向心力：GMm / r² = mv² / r。由此得出轨道速度：
v = √(GM / r)
The orbital period T is then found using v = 2πr / T, giving:
轨道周期 T 可由 v = 2πr / T 求得，得到：
T² = (4π² / GM) r³
This is the mathematical form of Kepler’s third law. It shows that T² is proportional to r³. Geostationary satellites, which have a period equal to Earth’s rotation period (24 hours), must orbit at a specific radius of about 42,300 km (from Earth’s centre) directly above the equator.
这就是开普勒第三定律的数学形式。它表明 T² 与 r³ 成正比。地球同步卫星的周期等于地球自转周期（24 小时），必须位于赤道正上方距地心约 42,300 km 的特定半径轨道上。
In problems, always set the centripetal force equal to the gravitational force. Do not confuse orbital radius r with altitude; r = R + h, where R is the planet’s radius.
解题时，始终令向心力等于万有引力。不要混淆轨道半径 r 与高度；r = R + h，其中 R 是行星半径。
8. Kepler’s Laws of Planetary Motion | 开普勒行星运动定律
First Law: Each planet moves in an elliptical orbit with the Sun at one focus. Second Law: A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. Third Law: The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.
第一定律：每颗行星沿椭圆形轨道运动，太阳位于一个焦点上。第二定律：行星与太阳的连线在相等时间内扫过相等的面积。第三定律：行星轨道周期的立方与其轨道半长轴的立方成正比。
While IB and AQA mainly focus on circular orbits (a special case of ellipses), the second law explains why planets move faster when closer to the Sun. The third law, T² ∝ r³, is used to compare periods of planets or to find the mass of a central body: M = (4π²r³) / (GT²).
尽管 IB 和 AQA 主要关注圆形轨道（椭圆的特殊情况），但第二定律解释了为什么行星在靠近太阳时运动得更快。第三定律 T² ∝ r³ 用于比较行星周期或求中心天体的质量：M = (4π²r³) / (GT²)。
Exam questions often provide data for two satellites or planets and ask for an unknown period or distance using a ratio form of Kepler’s third law: (T₁/T₂)² = (r₁/r₂)³.
考试题常提供两颗卫星或行星的数据，要求利用开普勒第三定律的比例形式 (T₁/T₂)² = (r₁/r₂)³ 来求未知的周期或距离。
9. Energy of an Orbiting Satellite | 轨道卫星的能量
The total mechanical energy E of a satellite in a circular orbit is the sum of its kinetic energy K and potential energy U. Using K = ½ mv² and U = − GMm/r, with v² = GM/r, we obtain:
圆形轨道上卫星的总机械能 E 是其动能 K 与势能 U 之和。利用 K = ½ mv² 和 U = − GMm/r，以及 v² = GM/r，我们得到：
E = − GMm / (2r)
The total energy is negative, indicating a bound system. The kinetic energy is half the magnitude of the potential energy. As the orbital radius increases, the total energy becomes less negative (increases), but more energy is needed to move a satellite to a higher orbit.
总能量为负，表明这是一个束缚系统。动能是势能大小的一半。随着轨道半径增加，总能量负得少一些（增加），但要把卫星移到更高的轨道需要输入更多能量。
This relationship is vital for understanding satellite transfers and fuel requirements. In AQA and IB exams, you may be asked to calculate the work done to change orbits.
这一关系对于理解卫星变轨和燃料需求至关重要。在 AQA 和 IB 考试中，可能要求计算变轨所需的功。
10. Gravitational Fields of Multiple Bodies and Neutral Points | 多天体的引力场与中性点
When two masses are present, there exists a point, usually on the line joining them, where the net gravitational field strength is zero. At this ‘neutral point’, the gravitational pulls from the two masses cancel: g₁ = g₂.
当存在两个质量时，通常在它们的连线上存在一个点，该点的合引力场强度为零。在这个“中性点”，两个质量的引力互相抵消：g₁ = g₂。
To find it, set GM₁ / x² = GM₂ / (d − x)² and solve for x, the distance from M₁. This concept is often tested in IB HL and AQA as an application of field addition. Remember that gravitational potential, being scalar, does not cancel in the same way; the potential at the neutral point is simply the algebraic sum.
要找到它，设 GM₁ / x² = GM₂ / (d − x)² 并解出 x，即距 M₁ 的距离。该概念常在 IB HL 和 AQA 中作为场叠加的应用进行考查。请记住，引力势是标量，不会以同样方式抵消；中性点的势只是代数和。
11. Common Misconceptions and Exam Tips | 常见误区与应试技巧
Weight vs mass: Weight is a force (mg) and varies with g; mass is invariant. In orbit, astronauts are weightless but not massless. Units of g: g can be expressed as N kg⁻¹ or m s⁻²; both are equivalent. Use N kg⁻¹ when speaking about field strength to avoid confusion.
重量与质量：重量是一种力（mg），随 g 变化；质量是不变的。在轨道上，宇航员重量为零，但质量不为零。g 的单位：g 可表示为 N kg⁻¹ 或 m s⁻²；两者等价。谈论场强时使用 N kg⁻¹ 可避免混淆。
Always draw a clear diagram showing forces and distances. When applying F = Gm₁m₂/r², ensure r is the distance between centres, not the surface separation. Practice interconverting between g, V, and U using calculus notation where required (especially for IB HL).
始终画出清晰的受力图和距离示意图。应用 F = Gm₁m₂/r² 时，确保 r 是中心间距，而不是表面间隔。练习根据需要使用微积分符号进行 g、V 和 U 之间的相互转换（尤其针对 IB HL）。
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