AS Physics: Gravitation | AS 物理：万有引力考点精讲

📚 AS Physics: Gravitation | AS 物理：万有引力考点精讲

Gravitation is one of the most fundamental topics in AS Physics. It describes the universal attractive force between masses, explains the motion of planets and satellites, and builds the foundation for understanding gravitational fields. In this revision guide, we will go through all the essential concepts, formulas, and problem-solving techniques you need to master for your exam.

万有引力是 AS 物理中最基础的主题之一。它描述了质量之间的普适吸引力，解释了行星和卫星的运动，并为理解重力场奠定了基础。在这份考点精讲中，我们将梳理所有重要的概念、公式和解题技巧，帮助你为考试做好准备。

1. 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 that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres. The formula is F = G M m / r², where G is the gravitational constant, 6.67 × 10⁻¹¹ N m² kg⁻².

牛顿定律指出，每个质点都吸引其他质点，引力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。公式为 F = G M m / r²，其中 G 是引力常量，6.67 × 10⁻¹¹ N m² kg⁻²。

The force acts along the line joining the centres of the two masses. It is always attractive, never repulsive. This is a universal law applicable to all objects with mass, from apples to galaxies.

这个力沿着两质量中心的连线作用，且总是吸引力，从不排斥。这是一条普适定律，适用于从苹果到星系的所有有质量物体。

Remember that r is measured from centre to centre. For a uniform sphere, the entire mass can be considered to be concentrated at its centre, which is a crucial simplification when calculating the force on a particle outside the sphere.

请记住，r 是从中心到中心测量的。对于一个均匀球体，它的全部质量可以视作集中在球心，这是计算球外质点受力时一个关键的简化。

2. Gravitational Field and Field Strength | 重力场与场强

A gravitational field is a region of space where a mass experiences a force. 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, with units N kg⁻¹.

重力场是空间中使质量受到引力的区域。某点的重力场强度 g 定义为放置在该点的小检验质量所受的引力与质量的比值：g = F / m，单位是 N kg⁻¹。

For a point mass M or a spherical body, the magnitude of g at a distance r from the centre (outside the body) is given by: g = G M / r². This shows that g decreases with the square of the distance.

对于点质量 M 或球形天体，在距离其中心 r 处（球外）的 g 的大小为：g = G M / r²。这表明 g 随距离的平方减小。

Near the Earth’s surface, g is approximately 9.81 N kg⁻¹. Field strength is a vector quantity, always pointing towards the centre of the mass creating the field.

在地球表面附近，g 约为 9.81 N kg⁻¹。场强是矢量，其方向总是指向产生场的质量中心。

Note that N kg⁻¹ is dimensionally equivalent to m s⁻², so g is also the acceleration due to gravity in free fall.

注意，N kg⁻¹ 在量纲上等同于 m s⁻²，因此 g 也是自由落体中的重力加速度。

3. Representing Gravitational Fields: Field Lines | 重力场的表示：场线

Gravitational fields are visualised using field lines (lines of force). The direction of the field line at any point shows the direction of the gravitational force on a small mass, i.e., radially inward towards the centre of the mass causing the field.

重力场用场线（力线）来表示。场线在任何点的切线方向表示放在该点的小质量所受引力的方向，即径向指向产生场的质量中心。

For an isolated point mass or a uniform sphere, the field lines are radial and inward. The spacing of the lines indicates the strength of the field: the closer together the lines, the stronger the field.

对于一个孤立的点质量或均匀球体，场线呈径向并向内。场线的间距表示场的强弱：线越密集，场越强。

A uniform gravitational field is represented by parallel and equally spaced field lines, all pointing in the same direction. This is a good approximation for the field close to the Earth’s surface over a small region.

匀强重力场用平行且等距的场线表示，它们都指向同一方向。这是地球表面附近小区域内场的良好近似。

Make sure you can sketch field line patterns for radial and uniform fields, and understand that field lines never cross.

确保你能画出径向场和匀强场的场线图，并理解场线永不相交。

4. Gravitational Potential | 引力势

Gravitational potential V at a point is defined as the work done per unit mass in bringing a small test mass from infinity to that point. It is a scalar quantity and is always negative on the surface and outside a mass, as work is done by the field in attracting the mass.

某点的引力势 V 定义为将单位质量从无穷远移动到该点所做的功。它是一个标量，并且在质量表面及外部总是负值，因为场吸引质量时对外做功。

The formula for the potential due to a point mass M (or a spherical body) at a distance r from its centre is: V = – G M / r. The zero of potential is taken at infinity (r→∞, V→0).

由点质量 M（或球体）在距离其中心 r 处产生的引力势公式为：V = – G M / r。势的零点取在无穷远（r→∞，V→0）。

Potential becomes more negative as you get closer to the mass. The potential gradient ΔV/Δr is closely related to field strength: g = – ΔV / Δr (in one dimension), pointing in the direction of decreasing potential.

当你靠近质量时，势变得更负。势梯度 ΔV/Δr 与场强密切相关：g = – ΔV / Δr（一维情况），指向势减小的方向。

In AS exams, you may be asked to explain why potential is negative, or to calculate the change in gravitational potential energy as a satellite moves in its orbit.

在 AS 考试中，你可能需要解释为什么势是负值，或者计算卫星在轨道上移动时引力势能的变化。

5. Gravitational Potential Energy | 引力势能

The gravitational potential energy (U) of a two-mass system is the work done to assemble the system from an infinite separation. For two point masses M and m separated by a distance r, U = – G M m / r.

两个质量系统的引力势能 U 是将它们从无穷远移至相距 r 所做的功。对于相距 r 的两个点质量 M 和 m，U = – G M m / r。

This energy is negative, indicating a bound system. To completely separate the masses to infinity, you must provide energy equal to |U|. This concept is vital for understanding escape velocity.

这个能量是负值，表明系统是束缚的。要将质量完全分离到无穷远，你必须提供等于 |U| 的能量。这一概念对于理解逃逸速度至关重要。

In a uniform field near Earth’s surface, we use the simpler approximation ΔU = m g Δh, where Δh is the vertical displacement. But remember this only applies when g is constant over the distance.

在地表附近的匀强场中，我们使用更简单的近似 ΔU = m g Δh，其中 Δh 是竖直位移。但请记住，这仅在 g 在该距离上恒定时适用。

6. Satellite Orbits: Circular Motion | 卫星轨道：圆周运动

For a satellite in a stable circular orbit around a planet, the gravitational force provides the necessary centripetal force: G M m / r² = m v² / r = m ω² r. Here M is the mass of the central body, m the satellite mass, and r the orbital radius.

对于绕行星做稳定圆周运动的卫星，引力提供所需的向心力：G M m / r² = m v² / r = m ω² r。这里 M 是中心天体的质量，m 是卫星质量，r 是轨道半径。

This relationship allows you to derive many useful expressions for orbital speed, period, and radius. The orbital speed v = √(G M / r). Notice that the satellite’s mass cancels, so orbital speed depends only on the mass of the central body and the orbital radius.

由这一关系可以推导出许多关于轨道速度、周期和半径的有用表达式。轨道速度 v = √(G M / r)。注意，卫星的质量被消掉了，因此轨道速度只取决于中心天体的质量和轨道半径。

The orbital period T can be found using T = 2π r / v. Substituting v yields Kepler’s third law: T² = (4π² / G M) r³, or T² ∝ r³. This law shows that planets farther from the Sun have longer orbital periods.

轨道周期 T 可通过 T = 2π r / v 求得。代入 v 可得到开普勒第三定律：T² = (4π² / G M) r³，即 T² ∝ r³。这一定律表明，离太阳越远的行星公转周期越长。

Geostationary satellites have a period of exactly 24 hours and are positioned above the equator. Using T = 24 h, you can calculate their orbital radius, which is about 42 300 km from the Earth’s centre.

地球同步卫星的周期恰好为 24 小时，并位于赤道上空。利用 T = 24 h，可以计算出它们的轨道半径，约为距地心 42 300 km。

7. Kepler’s Laws | 开普勒定律

Johannes Kepler formulated three laws of planetary motion based on observations. The first law states that planets move in elliptical orbits with the Sun at one focus. In AS, we often approximate orbits as circular for simplicity.

开普勒根据观测总结出行星运动三定律。第一定律指出，行星沿椭圆轨道运动，太阳位于其中一个焦点上。在 AS 中，为简化处理，我们常将轨道近似为圆形。

Kepler’s second law (law of equal areas) tells us that a line joining a planet to the Sun sweeps out equal areas in equal intervals of time. This means planets move faster when closer to the Sun and slower when farther away.

开普勒第二定律（面积定律）表明，行星与太阳的连线在相等时间内扫过相等的面积。这意味着行星在靠近太阳时运动较快，远离时较慢。

The third law, as derived above, is T² ∝ r³ for circular orbits. This relationship is extremely useful for comparing the orbits of different planets or satellites around the same central body.

如上所述，第三定律对于圆轨道为 T² ∝ r³。这一关系在比较绕同一中心天体的不同行星或卫星的轨道时非常有用。

When solving problems, always ensure you use consistent units: T in seconds, r in metres, and M in kg when using the full formula.

解题时，务必保证单位一致：使用完整公式时，T 用秒、r 用米、M 用千克。

8. Energy in Orbits | 轨道能量

The total mechanical energy of an orbiting satellite is the sum of its kinetic energy and gravitational potential energy. For a circular orbit, Ek = ½ m v² = G M m / (2r), and U = – G M m / r. Thus the total energy E_total = Ek + U = – G M m / (2r).

沿轨道运行的卫星的总机械能是其动能和引力势能之和。对于圆轨道，Ek = ½ m v² = G M m / (2r)，而 U = – G M m / r。因此总能量 E_total = Ek + U = – G M m / (2r)。

The total energy is negative, indicating a bound orbit. The magnitude of the total energy is half the magnitude of the potential energy, which is a key result for circular orbits.

总能量为负值，表明这是一个束缚轨道。总能量的量值是势能量值的一半，这是圆轨道的一个关键结论。

If a satellite loses energy (e.g., due to atmospheric drag), it will move to a lower orbit and actually move faster, contrary to everyday intuition. Its total energy becomes more negative, but kinetic energy increases.

如果卫星因大气阻力等原因损失能量，它将移至较低轨道，并且实际上移动得更快，这与日常直觉相反。它的总能量变得更负，但动能增加了。

9. Escape Velocity | 逃逸速度

Escape velocity is the minimum speed an object must have at the surface (or from a point in space) to completely escape the gravitational field of a planet or body, without further propulsion. At escape velocity, the total mechanical energy is zero: Ek + U = 0.

逃逸速度是物体在行星或天体表面（或空间某点）为完全脱离其引力场且无需进一步推进所需的最小速度。在逃逸速度下，总机械能为零：Ek + U = 0。

Setting ½ m v_esc² – G M m / R = 0 gives v_esc = √(2 G M / R), where R is the radius of the planet. Notice escape velocity is independent of the mass of the escaping object.

令 ½ m v_esc² – G M m / R = 0，可得 v_esc = √(2 G M / R)，其中 R 是行星半径。注意逃逸速度与逃离物体的质量无关。

For Earth, escape velocity is about 11.2 km s⁻¹. If a body’s speed is greater than escape velocity, it will follow a hyperbolic trajectory and leave the planet forever.

对于地球，逃逸速度约为 11.2 km s⁻¹。如果物体的速度大于逃逸速度，它将沿双曲线轨迹运动并永远离开行星。

This concept is also related to black holes: the escape velocity exceeds the speed of light, hence nothing can escape.

这一概念也与黑洞相关：逃逸速度超过光速，因此没有任何东西能逃脱。

10. Variations in g: Factors Affecting Surface Gravity | g 的变化：影响表面重力的因素

The value of g on the surface of a planet depends on its mass and radius: g_surface = G M / R². Different planets have different surface gravities due to their varying density and size.

行星表面的 g 值取决于其质量和半径：g_surface = G M / R²。由于密度和大小不同，不同行星的表面重力也不同。

On Earth, g is not exactly constant. Factors causing slight variations include: altitude (g decreases with height above the surface), latitude (Earth is not a perfect sphere; it bulges at the equator, so g is slightly less at the equator than at the poles), and local geological features.

在地球上，g 并不完全恒定。引起微小变化的因素包括：海拔（g 随高度增加而减小）、纬度（地球并非完美球体；它在赤道处隆起，因此赤道的 g 略小于两极），以及局部地质特征。

Free-air correction and Bouguer correction are used in geophysics to account for these variations during surveys, but for AS level, a simple understanding of the trends is sufficient.

地球物理勘探中使用自由空气校正和布格校正来解释这些变化，但在 AS 级别，简单了解这些趋势就足够了。

11. Apparent Weight and Weightlessness | 视重与失重

Weight is the gravitational force on an object, but apparent weight is the force exerted by the object on its support. In many situations, such as in an accelerating lift or an orbiting spacecraft, apparent weight differs from true weight.

重量是作用在物体上的引力，而视重是物体施加在其支撑物上的力。在许多情况下，如加速的升降机或沿轨道运行的航天器中，视重与真实重量不同。

Astronauts in the International Space Station feel weightless not because there is no gravity, but because they are in continuous free fall towards Earth, along with their spacecraft. Gravity still acts on them (about 90% of surface g at ISS altitude), but they do not experience a supporting force.

国际空间站中的宇航员感到失重，并非因为没有引力，而是因为他们和航天器一起持续向地球自由下落。引力仍然作用在他们身上（在国际空间站高度约为地表 g 的 90%），但他们没有受到支撑力。

This state is analogous to the brief weightlessness experienced in a freely falling elevator. Understanding the difference between true weight and apparent weight is important for exam questions.

这种状态类似于在自由下落的电梯中短暂体验到的失重。理解真实重量与视重之间的区别对考试题目很重要。

12. Problem-Solving Techniques and Common Pitfalls | 解题技巧与常见误区

When tackling gravitation problems, first identify the central mass (usually the one providing the field) and the orbiting or test mass. Write down the fundamental equation: F = G M m / r² = m v²/r if the motion is circular.

处理万有引力问题时，首先要确定中心质量（通常是提供场的那个）以及轨道或检验质量。写出基本方程：如果运动是圆周的，F = G M m / r² = m v²/r。

Always check your units. G is 6.67 × 10⁻¹¹ N m² kg⁻², masses in kg, distances in m. Convert periods to seconds and speeds to m s⁻¹ before substituting.

务必检查单位。G = 6.67 × 10⁻¹¹ N m² kg⁻²，质量用 kg，距离用 m。代入之前，将周期转换为秒，速度转换为 m s⁻¹。

Be careful with r: it is the distance from the centre of the central mass, not the altitude above the surface unless specified. If the problem gives altitude h, then r = R + h, where R is the radius of the planet.

注意 r：它是距中心质量中心的距离，除非特别说明，不是距表面的高度。如果题目给出高度 h，那么 r = R + h，其中 R 是行星半径。

A common mistake is to forget that gravitational potential and potential energy are negative. This sign matters when calculating total energy or escape speed.

一个常见错误是忘记引力势和势能为负值。这个符号在计算总能量或逃逸速度时很重要。

Finally, when comparing two satellites or planets, use ratios derived from Kepler’s third law or other proportionalities to avoid unnecessary calculations.

最后，当比较两颗卫星或行星时，使用从开普勒第三定律或其他比例关系中推导出的比值，以避免不必要的计算。

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AS Physics: Gravitation | AS 物理：万有引力 考点精讲