Earth's Gravity and Gravitational Effects | 地球引力与重力作用

📚 Earth’s Gravity and Gravitational Effects | 地球引力与重力作用

The concept of gravity is fundamental to our understanding of the physical universe. From the falling of an apple to the orbital motion of planets, gravity governs the large-scale structure and behaviour of everything around us. In this article, we will explore the principles of Earth’s gravity, the gravitational field, and the various effects that arise from this universal force, with a particular focus on the CIE A-Level Physics syllabus.

引力概念是我们理解物理宇宙的基础。从苹果落地到行星的轨道运动，引力支配着我们周围一切事物的宏观结构和行为。在本文中，我们将探讨地球引力的原理、引力场以及这一普遍力所产生的各种效应，并特别聚焦于CIE A-Level物理考纲。

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 in the universe 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 gravitational force between two masses m₁ and m₂ separated by distance r is given by:

牛顿万有引力定律指出，宇宙中每个质点都以与两质量乘积成正比、与两质量中心间距离的平方成反比的力吸引其他每个质点。两质量分别为 m₁ 和 m₂、相距 r 的物体之间的引力为：

F = G·m₁·m₂ / r²

where G is the universal gravitational constant, equal to 6.67 × 10⁻¹¹ N·m²·kg⁻². The force is always attractive, acting along the line joining the two masses. It is important to note that this law applies to point masses; for extended objects, we consider the mass to be concentrated at the centre of mass.

其中 G 是万有引力常量，等于 6.67 × 10⁻¹¹ N·m²·kg⁻²。引力始终是吸引力，沿两质量连线方向作用。需注意该定律适用于质点；对于扩展物体，我们将其质量视为集中在质心处。

2. Gravitational Field Strength | 引力场强度

A gravitational field is a region of space in which a mass experiences a gravitational force. 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:

引力场是空间中质量受到引力作用的区域。某点的引力场强度 g 定义为放置在該点的单位质量所受到的引力：

g = F / m

The unit of gravitational field strength is N·kg⁻¹, which is equivalent to m·s⁻². For a point mass M, the gravitational field strength at distance r from the mass is:

引力场强度的单位是 N·kg⁻¹，等同于 m·s⁻²。对于质量 M 的质点，距离 r 处的引力场强度为：

g = G·M / r²

This expression applies to any spherically symmetric mass distribution when r is measured from the centre, which is why we can treat the Earth as a point mass when considering objects outside its surface.

该表达式适用于任何球对称质量分布，此时 r 从球心测量，这就是为什么当我们考虑地球表面之外的物体时，可以将地球视为质点。

3. Acceleration Due to Gravity at the Earth’s Surface | 地球表面的重力加速度

At the Earth’s surface, the gravitational field strength is approximately 9.81 N·kg⁻¹, which corresponds to the acceleration due to gravity g = 9.81 m·s⁻². This value can be obtained from the formula:

在地球表面，引力场强度约为 9.81 N·kg⁻¹，对应重力加速度 g = 9.81 m·s⁻²。该值可由以下公式得出：

g = G·Mₑ / Rₑ²

where Mₑ = 6.0 × 10²⁴ kg is the mass of the Earth and Rₑ = 6.4 × 10⁶ m is its mean radius. The value of g varies slightly across the Earth’s surface due to factors such as altitude, latitude (because of the Earth’s rotation and its equatorial bulge), and local variations in the density of the Earth’s crust.

其中 Mₑ = 6.0 × 10²⁴ kg 是地球质量，Rₑ = 6.4 × 10⁶ m 是地球平均半径。由于海拔、纬度（缘于地球自转及其赤道隆起）以及地壳密度局部变化等因素，g 值在地球表面各处略有差异。

4. Variation of Gravity with Altitude | 重力随高度的变化

As we move away from the Earth’s surface, the gravitational field strength decreases. At a height h above the Earth’s surface, the gravitational field strength is given by:

当我们离开地球表面时，引力场强度减小。在地面上方高度 h 处，引力场强度为：

g(h) = G·Mₑ / (Rₑ + h)²

This inverse-square relationship means that at one Earth radius above the surface (h = Rₑ, i.e. an altitude of approximately 6400 km), the value of g falls to one-quarter of its surface value, or about 2.45 m·s⁻². For small heights, we can approximate the change using a binomial expansion, giving g(h) ≈ g₀(1 − 2h/Rₑ), which is valid when h ≪ Rₑ.

这种平方反比关系意味着在一个地球半径的高度处（h = Rₑ，即约6400公里的高海拔），g 值降至地面值的四分之一，约为 2.45 m·s⁻²。对于较小的高度，我们可以用二项式展开近似，得到 g(h) ≈ g₀(1 − 2h/Rₑ)，该式在 h ≪ Rₑ 时成立。

5. Gravity Inside the Earth | 地球内部的重力

Inside the Earth, the situation is more complex. For a point at depth d below the Earth’s surface, only the mass within the sphere of radius (Rₑ − d) contributes to the net gravitational force; the outer shell exerts no net gravitational field at that point. If we assume the Earth has uniform density, the gravitational field strength at depth d is:

地球内部的情况更为复杂。对于地面以下深度 d 处的一点，只有半径 (Rₑ − d) 球体内包含的质量对该点产生净引力；外层球壳在该点不产生净引力场。如果我们假设地球密度均匀，则深度 d 处的引力场强度为：

g(d) = g₀(1 − d/Rₑ) = g₀ · (distance from centre) / Rₑ

Thus, the gravitational field strength decreases linearly from its maximum at the surface to zero at the Earth’s centre. At the centre, the gravitational field strength is zero because the gravitational forces from all directions cancel out, although the pressure there is immense.

因此，引力场强度从表面最大值线性减小到地球中心的零值。在地心处，引力场强度为零，因为来自四面八方各方向的引力相互抵消，尽管那里的压力极其巨大。

6. Gravitational Potential Energy | 重力势能

The gravitational potential energy of a system of two masses m₁ and m₂ is the work done in bringing them from infinity to their current separation. For a point mass m at distance r from the centre of a spherically symmetric mass M, the gravitational potential energy is:

两质量系统 m₁ 和 m₂ 的重力势能是将它们从无穷远移动到当前距离所需做的功。对于距球对称质量 M 中心距离 r 处的一个质点 m，其重力势能为：

U = −G·M·m / r

The negative sign indicates that the potential energy is zero at infinity and becomes increasingly negative as the masses approach each other, meaning that work must be done against the gravitational force to separate them. The gravitational potential at a point is defined as the potential energy per unit mass, V = −G·M/r, measured in J·kg⁻¹.

负号表示无穷远处势能为零，随着质量靠近而变得越来越负，这意味着要克服引力做功才能将它们分离。某点的引力势定义为每单位质量的势能，V = −G·M/r，单位为 J·kg⁻¹。

7. Escape Velocity | 逃逸速度

The escape velocity is the minimum speed that an object must have in order to escape from a gravitational field completely. To escape from the surface of the Earth, an object must have sufficient kinetic energy to overcome the gravitational potential energy. Setting the total energy to zero (the condition for just escaping to infinity with zero speed):

逃逸速度是物体完全脱离引力场所需的最小速度。要从地球表面逃逸，物体必须具有足够的动能来克服重力势能。令总能量为零（恰好以零速度逃逸到无穷远的条件）：

½·m·vₑ² = G·Mₑ·m / Rₑ

Solving for vₑ gives:

解得 vₑ：

vₑ = √(2·G·Mₑ / Rₑ) = √(2·g·Rₑ)

Substituting the Earth’s values, we obtain an escape velocity of approximately 11.2 km·s⁻¹. This is independent of the mass of the escaping object — any object, from a molecule to a spacecraft, requires the same speed to escape from the same altitude.

代入地球数值，我们得到逃逸速度约为 11.2 km·s⁻¹。逃逸速度与逃逸物体的质量无关——从分子到航天器，任何物体从同一高度逃逸所需的速度都相同。

8. Satellite Motion and Orbits | 卫星运动与轨道

A satellite in orbit around the Earth experiences a centripetal force provided by the Earth’s gravitational attraction. For a satellite of mass m orbiting at radius r with speed v, equating the gravitational force to the centripetal force gives:

环绕地球运行的卫星所受的向心力由地球引力提供。对于质量为 m、轨道半径为 r、速度为 v 的卫星，将引力与向心力相等可得：

G·Mₑ·m / r² = m·v² / r

Therefore, the orbital speed is:

因此轨道速度为：

v = √(G·Mₑ / r)

Notice that the orbital speed is independent of the satellite’s mass and decreases with increasing orbital radius. Conversely, satellites closer to the Earth must travel faster. A geostationary satellite, which always remains above the same point on the Earth’s equator, must have an orbital period of 24 hours and orbit in the equatorial plane at a height of approximately 35,800 km above the surface.

注意轨道速度与卫星质量无关，并且随轨道半径增大而减小。相反，更靠近地球的卫星必须以更快的速度运行。地球同步卫星始终位于赤道同一位置上方，必须具有24小时的轨道周期，并在赤道平面上距地面约35,800公里的高度运行。

9. Kepler’s Laws and Orbital Period | 开普勒定律与轨道周期

Kepler’s third law, which can be derived from Newton’s law of gravitation and circular motion, states that the square of the orbital period T is directly proportional to the cube of the mean orbital radius r. Combining v = 2πr/T with v² = G·Mₑ/r, we obtain:

开普勒第三定律可由牛顿万有引力定律和圆周运动推导得出，它指出轨道周期 T 的平方与平均轨道半径 r 的立方成正比。将 v = 2πr/T 与 v² = G·Mₑ/r 结合，我们得到：

T² = (4π²/G·Mₑ) · r³

This relationship is of great practical importance. By measuring the orbital period and radius of a satellite or a moon, we can determine the mass of the central body. This is how astronomers determine the masses of planets and stars.

这一关系具有重要的实际意义。通过测量卫星或月球绕转的轨道周期和半径，我们可以确定中心天体的质量。天文学家正是以此方法来确定行星和恒星的质量。

10. Tidal Effects of Gravity | 引力的潮汐效应

The Moon’s gravitational pull on the Earth is not uniform; it is stronger on the side of the Earth facing the Moon and weaker on the opposite side. This differential gravitational pull creates tidal bulges in the Earth’s oceans, giving rise to high and low tides approximately every 12 hours. The Sun also contributes to tidal effects, and when the Sun, Earth and Moon are aligned (at new moon and full moon), the tidal range is greatest — these are called spring tides. When the Sun and Moon are at right angles to each other, the tidal range is smallest — these are neap tides.

月球对地球的引力并不均匀；面对月球一侧的引力更强，背对月球一侧的引力更弱。这种差异引力在地球海洋中产生潮汐隆起，形成大约每12小时一次的高低潮。太阳也对潮汐有贡献，当太阳、地球和月球排列成一线时（新月和满月时），潮差最大——这称为大潮。当太阳和月球彼此成直角时，潮差最小——这称为小潮。

11. Gravitational Field Lines and Equipotential Surfaces | 引力场线与等势面

Gravitational fields can be represented graphically by field lines (also called lines of force). In a radial field, such as that surrounding the Earth, the field lines are straight lines pointing radially inward toward the centre. In a uniform field, such as near the Earth’s surface over a small region, the field lines are parallel and evenly spaced. Equipotential surfaces are surfaces of constant gravitational potential. No work is done in moving a mass along an equipotential surface, and these surfaces are always perpendicular to the field lines.

引力场可以用场线（也称为力线）图形化表示。在径向场中，如地球周围的场，场线是沿径向指向球心的直线。在均匀场中，如地球表面附近的小区域内，场线是平行且等间距的。等势面是引力势恒定的面。沿等势面移动质量不做功，且等势面始终垂直于场线。

12. Exam-Focused Summary | 考试要点总结

For CIE A-Level Physics, the key points to remember about Earth’s gravity are:

针对CIE A-Level物理考试，关于地球引力的关键要点如下：

Newton’s law: F = Gm₁m₂/r², with G = 6.67 × 10⁻¹¹ N·m²·kg⁻². Always attractive, acting along the line joining centres.
牛顿定律：F = Gm₁m₂/r²，其中 G = 6.67 × 10⁻¹¹ N·m²·kg⁻²。始终为吸引力，沿两中心连线方向作用。

Gravitational field strength: g = F/m = GM/r². Outside a spherically symmetric mass, the mass behaves as a point mass at the centre.
引力场强度：g = F/m = GM/r²。在球对称质量外部，质量等效于集中于球心的质点。

Inside the Earth: g increases linearly from zero at the centre to its surface value; outside, it decreases according to the inverse-square law.
地球内部：g 从地心处的零值线性增加到地面值；在地球外部，g 按平方反比定律递减。

Gravitational potential: V = −GM/r, a scalar quantity; gravitational potential energy: U = −GMm/r. Define the zero of potential at infinity.
引力势：V = −GM/r，是标量；重力势能：U = −GMm/r。定义无穷远处势为零。

Escape velocity: vₑ = √(2GM/R) = 11.2 km·s⁻¹ for Earth at ground level.
逃逸速度：vₑ = √(2GM/R)，在地面处地球的逃逸速度为 11.2 km·s⁻¹。

Satellite orbits: v = √(GM/r); T² = (4π²/GM)·r³; geostationary satellites orbit at 35,800 km altitude in the equatorial plane with period 24 hours.
卫星轨道：v = √(GM/r)；T² = (4π²/GM)·r³；地球同步卫星在赤道平面上、高度35,800公里处轨道运行，周期24小时。

Mastering these principles will enable you to solve problems involving gravitational fields, satellite motion, and energy considerations with confidence. Remember to always check whether you are asked for gravitational field strength or gravitational potential — the former is a vector quantity while the latter is a scalar.

掌握这些原理将使您能够自信地解决涉及引力场、卫星运动和能量相关的问题。请务必注意题目要求的是引力场强度还是引力势——前者是矢量，后者是标量。

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Earth’s Gravity and Gravitational Effects | 地球引力与重力作用