📚 Gravitation | 万有引力
Gravitation is the fundamental force of attraction between any two objects with mass. From the fall of an apple to the graceful orbits of planets around the Sun, this universal interaction shapes the cosmos. In IB Physics, the topic of gravitation bridges classical mechanics and field theory, providing essential tools to analyse planetary motion, satellite dynamics and the large-scale structure of the universe. Mastering gravitational principles deepens our understanding of both terrestrial and celestial phenomena.
引力是任意两个有质量物体之间的基本吸引力。从苹果落地到行星绕太阳的优美轨道，这种普遍的相互作用塑造了宇宙。在 IB 物理中，引力主题连接了经典力学和场论，为分析行星运动、卫星动力学以及宇宙的大尺度结构提供了重要工具。掌握引力原理能加深我们对地面与天文现象的理解。
1. The Concept of Gravitation | 引力的概念
Gravitation is a universal interaction that acts between all masses, however large or small. It is the weakest of the four fundamental forces, yet it dominates on astronomical scales because it is always attractive and has infinite range. In IB Physics, we explore how gravitational forces are described by Newton’s law and later deepened by the concept of fields.
引力是作用于一切质量之间的普遍相互作用，无论物体大小。它是四种基本力中最弱的，但在天文学尺度上占主导地位，因为它始终是吸引力且作用范围无限。在 IB 物理中，我们探讨如何用牛顿定律描述引力，并进一步通过场的概念加以深化。
Historically, the Aristotelian view held that objects fell towards the Earth's centre because it was their natural place. It was Isaac Newton who unified terrestrial and celestial mechanics by proposing that the same force pulling an apple downwards also keeps the Moon in orbit around the Earth. This synthesis marked a revolution in physics.
历史上，亚里士多德认为物体落向地心是因为那里是它们的天然位置。正是艾萨克·牛顿统一了地面力学与天体力学，他提出将苹果拉向地面的力同样使月球绕地球转动。这一综合标志着物理学的一次革命。
2. Newton's Law of Universal Gravitation | 牛顿万有引力定律
Newton's law states that every point mass attracts every other point mass with a force whose magnitude is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres:
牛顿定律指出，每一个质点都吸引其他质点，引力的大小与两质点质量的乘积成正比，与它们中心之间距离的平方成反比：
F = G m₁ m₂ / r²
avitational constant. The force acts along the line joining the two masses and is always attractive. The vector form includes a unit vector pointing from one mass to the other with a negative sign to indicate attraction.
其中 G = 6.67 × 10⁻¹¹ N m² kg⁻² 是引力常量。力沿两质点连线方向作用，总是吸引力。矢量形式包含从一个质量指向另一个质量的单位矢量，并带有负号表示吸引。
. Gravitational effects become significant only when at least one of the masses is astronomically large, like a planet or star.
由于 G 极小，日常物体间的引力可忽略。例如，两个相距 1 米的 100 kg 质量体之间的吸引力仅约 6.67 × 10⁻⁹ N。只有当至少有一个质量大到天文学尺度时，引力效应才会显著，如行星或恒星。
3. Gravitational Field Strength | 引力场强度
Gravitational field strength g at a point is defined as the gravitational force experienced per unit mass placed at that point: g = F/m. It is a vector quantity whose direction is the same as the direction of the force on a small test mass. Near a spherical mass, the field is radial and points towards the centre of the mass.
引力场强度 g 定义为放在该点处的单位质量所受的引力：g = F/m。它是一个矢量，其方向与作用在检验小质量上的力方向相同。在球对称质量附近，场呈辐射状并指向质量中心。
For a point mass M (or any spherically symmetric mass, from the outside), the magnitude of the field strength is:
对于点质量 M（或任何球对称质量，从外部看），场强大小为：
g = G M / r²
e distance reduces the field strength to one-quarter of its original value. Near the Earth's surface, g is approximately 9.81 N kg⁻¹, which can be derived directly from the Earth's mass (5.97 × 10²⁴ kg) and radius (6.37 × 10⁶ m).
其中 r 是到质量中心的距离。这种平方反比关系意味着距离加倍会使场强减为原来的四分之一。在地球表面附近，g 约为 9.81 N kg⁻¹，可直接由地球质量 (5.97 × 10²⁴ kg) 和半径 (6.37 × 10⁶ m) 推导得出。
4. Gravitational Potential Energy | 引力势能
The gravitational potential energy U of a system of two point masses m and M separated by a distance r is given by:
由两个点质量 m 和 M 组成的系统，其引力势能 U 表示为：
U = – G M m / r
The zero of potential energy is conventionally chosen at infinite separation, so U is always negative for bound systems. The negative sign indicates that work must be done against the gravitational force to pull the masses apart. As the separation decreases, the potential energy becomes more negative, meaning the system loses potential energy and speeds up as they fall together.
势能零点通常选在相距无限远处，因此对于束缚系统，U 始终为负。负号表示要克服引力将质量拉开需要做功。随着间距减小，势能变得更负，意味着系统失去势能，并在相互靠近时加速。
avitational potential energy when moving a mass from one position r₁ to another r₂ is ΔU = –GMm(1/r₂ – problems, such as calculating the speed of a meteorite approaching Earth from deep space.
将质量从位置 r₁ 移动到 r₂ 时，引力势能的变化为 ΔU = –GMm(1/r₂ – 1/r₁)。这一公式在能量守恒问题中至关重要，例如计算来自深空的陨石靠近地球时的速度。
5. Gravitational Potential | 引力势
ial energy per unit mass: V = U/m. For a point mass M (or outside a sphere), this becomes:
引力势 V 定义为每单位质量的引力势能：V = U/m。对于点质量 M（或球体外部），表达式为：
V = – G M / r
Potential is a scalar field, which simplifies calculations when multiple masses are present; the total potential at a point is the algebraic sum of the potentials due to each mass. Like potential energy, V is zero at infinity and negative at finite distances. The deeper the potential well, the more tightly an object is bound.
引力势是一个标量场，当存在多个质量时，这可以简化计算；某点的总势是各质量产生的势的代数和。与势能类似，V 在无穷远处为零，在有限距离处为负。势阱越深，物体被束缚得越紧密。
d strength and potential are related through the potential gradient. In one dimension, g = – he negative sign indicates that the field points in the direction of decreasing potential.
引力场强与势通过势梯度相关联。在一维情形下，g = –dV/dr。对于径向场，可导出 g = GM/r²，与之前场强的定义一致。负号表明场指向势降低的方向。
6. Equipotential Surfaces | 等势面
An equipotential surface is a surface on which the gravitational potential is constant. Because no change in potential occurs when moving along such a surface, no work is done by the gravitational field. This makes equipotentials powerful tools for visualising fields.
等势面是引力势恒定的曲面。由于沿这样的曲面移动时势不发生变化，引力场不做功。这使得等势面成为可视化场的强大工具。
Field lines are always perpendicular to equipotential surfaces. For a point mass, the equipotential surfaces are concentric spheres centred on the mass. Near the Earth's surface, where g is approximately uniform, the equipotentials are equally spaced horizontal planes. The spacing between equipotentials drawn for a fixed potential interval indicates the field strength: closer spacing means a stronger field.
场线始终与等势面垂直。对于点质量，等势面是以该质量为中心的同心球面。在地球表面附近，g 近似均匀，等势面为等间距的水平面。按固定势间隔绘出的等势面间距反映了场强：间距越密，场越强。
Contour lines on a map showing equal height above sea level are analogous to gravitational equipotentials; a steep slope corresponds to closely packed contours and a strong gravitational field.
地图上表示海拔高度相等的等高线类似于引力等势面；陡坡对应于密集的等高线和较强的引力场。
7. Orbital Motion and Kepler's Laws | 轨道运动与开普勒定律
Johannes Kepler formulated three empirical laws of planetary motion based on Tycho Brahe's observations. Newton later showed that these laws are direct consequences of his law of gravitation and his laws of motion.
约翰内斯·开普勒根据第谷·布拉赫的观测总结出行星运动的三条经验定律。牛顿后来证明这些定律是他的万有引力定律和运动定律的直接结果。
- First Law: Planets move in elliptical orbits with the Sun at one focus.
- 第一定律：行星沿椭圆轨道运行，太阳位于一个焦点上。
- Second Law: A line joining a planet to the Sun sweeps out equal areas in equal times.
- 第二定律：行星与太阳的连线在相等时间内扫过相等面积。
- Third Law: The square of the orbital period T is proportional to the cube of the semi-major axis r: T² ∝ r³.
- 第三定律：轨道周期 T 的平方与半长轴 r 的立方成正比：T² ∝ r³。
tripetal force required for circular motion: GMm/r² = mv²/r. This gives the orbital speed:
对于圆轨道这一特例，可令引力等于圆周运动所需的向心力：GMm/r² = mv²/r，从而得到轨道速率：
v = √G M / r
and the period:
以及周期：
T = 2π √r³ / G M
Notice that the orbital period depends only on the radius and the central mass M, not on the mass of the orbiting body. This reinforces the proportionality T² ∝ r³ with the constant 4π²/GM, allowing astronomers to determine the mass of the Sun or distant planets by measuring periods and radii of orbiting moons.
注意，轨道周期仅取决于半径和中心质量 M，与绕行天体本身的质量无关。这证实了 T² ∝ r³ 的比例关系，其比例常数为 4π²/GM，从而使天文学家能够通过测量绕行卫星的周期和半径来确定太阳或遥远行星的质量。
8. Satellite Orbits and Escape Velocity | 卫星轨道与逃逸速度
A satellite in a stable circular orbit has its weight providing exactly the centripetal force needed. The required orbital speed decreases with altitude: low-Earth satellites move at about 7.8 km s⁻¹, while geostationary satellites at an altitude of 35 800 km have a period of 24 hours, matching Earth's rotation. This special orbit lies in the equatorial plane and appears stationary from the ground, making it ideal for communications.
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