Gravitation: Key Points for IGCSE WJEC Physics | 万有引力：IGCSE WJEC 物理核心考点

📚 Gravitation: Key Points for IGCSE WJEC Physics | 万有引力：IGCSE WJEC 物理核心考点

Gravitation is one of the fundamental forces that govern the motion of planets, moons, satellites and everyday objects on Earth. In the IGCSE WJEC Physics syllabus, understanding gravitation is essential for explaining weight, free fall, orbital motion and the structure of the Solar System. This article distils the core points you need to master, from Newton’s law of universal gravitation to the behaviour of satellites in orbit.

引力是支配行星、卫星以及地球上日常物体运动的基本力之一。在 IGCSE WJEC 物理大纲中，理解万有引力对于解释重量、自由落体、轨道运动和太阳系结构至关重要。本文提炼了你需要掌握的核心要点，从牛顿万有引力定律到卫星的轨道行为。

1. What is Gravitation? | 什么是万有引力？

Gravitation, or gravity, is a force of attraction that acts between any two masses. Unlike electric or magnetic forces, gravity is always attractive and never repulsive. It is the weakest of the four fundamental forces, yet it dominates on astronomical scales because it acts over infinite distances and all matter has mass.

万有引力，或称重力，是任何两个质量之间存在的吸引力。与电力或磁力不同，引力总是吸引，从不排斥。它是四种基本力中最弱的，但在天文尺度上占主导地位，因为它作用于无限远，且所有物质都有质量。

In IGCSE Physics, we distinguish between two related concepts: mass (the amount of matter in an object, measured in kg) and weight (the gravitational force on that object, measured in N). The weight of an object depends on the gravitational field strength at its location.

在 IGCSE 物理中，我们区分两个相关概念：质量（物体所含物质的量，单位 kg）和重量（作用在该物体上的引力，单位 N）。物体的重量取决于它所在位置的引力场强度。

2. Newton’s Law of Universal Gravitation | 牛顿万有引力定律

Isaac Newton proposed that every particle of matter in the universe attracts every other particle 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:

牛顿提出，宇宙中每一个物质粒子都会吸引其他每一个粒子，引力的大小与两个质量的乘积成正比，与它们中心之间距离的平方成反比：

F = G m₁ m₂ / r²

where F is the gravitational force (N), m₁ and m₂ are the two masses (kg), r is the distance between their centres (m), and G is the gravitational constant, approximately 6.67 × 10⁻¹¹ N m² kg⁻².

其中 F 是引力（N），m₁ 和 m₂ 是两个质量（kg），r 是它们中心之间的距离（m），G 是引力常数，大约为 6.67 × 10⁻¹¹ N m² kg⁻²。

The minuscule value of G explains why we do not notice gravitational attraction between everyday objects. However, when at least one mass is extremely large – like a planet – the force becomes significant.

G 的极小数值解释了为什么我们注意不到日常物体之间的引力。然而，当至少一个质量非常大——比如行星——时，这个力就变得显著。

This law is an inverse-square law: if the distance doubles, the force becomes one quarter; if the distance triples, the force becomes one ninth.

这是一个平方反比定律：如果距离加倍，力变为原来的四分之一；如果距离增加三倍，力变为九分之一。

3. Gravitational Field Strength (g) | 引力场强度 (g)

The gravitational field strength at a point is defined as the force per unit mass experienced by a small test mass placed at that point:

引力场强度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

The unit of g is N/kg. This is equivalent to m/s², so g is also the acceleration of free fall in that field. On the Earth’s surface, g ≈ 9.8 N/kg (often rounded to 10 N/kg for IGCSE calculations).

g 的单位是 N/kg。这等同于 m/s²，因此 g 也是在该场中自由落体的加速度。在地球表面，g ≈ 9.8 N/kg（IGCSE 计算中常取 10 N/kg）。

Using Newton’s law of gravitation, the field strength at the surface of a planet of mass M and radius R is given by:

利用牛顿引力定律，质量为 M、半径为 R 的行星表面的场强为：

g = G M / R²

This equation shows that g depends on the planet’s mass and radius, not on the mass of the object experiencing the force. That is why all objects in free fall (ignoring air resistance) accelerate at the same rate.

这个公式表明 g 取决于行星的质量和半径，而与受力物体的质量无关。这就是为什么在自由落体中（忽略空气阻力）所有物体的加速度相同。

4. Weight and Mass – Clearing the Confusion | 重量与质量——消除混淆

A common IGCSE pitfall is confusing mass and weight. Remember the clear distinction:

IGCSE 常见的陷阱是混淆质量和重量。记住明确的区别：

Property 属性	Mass 质量	Weight 重量
Definition	Quantity of matter in an object	Gravitational force acting on the object
Unit	kilogram (kg)	newton (N)
Varies with location?	No, constant throughout the universe	Yes, depends on g
Measured by	Beam balance or electronic balance	Spring balance or newton meter

For example, an astronaut of mass 80 kg has a weight of about 800 N on Earth (g = 10 N/kg) but only about 130 N on the Moon (g ≈ 1.6 N/kg). Her mass remains 80 kg.

例如，一名质量为 80 kg 的宇航员在地球上的重量约为 800 N（g = 10 N/kg），但在月球上仅约 130 N（g ≈ 1.6 N/kg）。她的质量始终是 80 kg。

IGCSE questions often ask you to calculate weight using W = m g, and to explain why mass is a more fundamental property than weight.

IGCSE 题目常常要求你使用 W = m g 计算重量，并解释为什么质量是比重量更基本的属性。

5. Variation of g on Earth and Other Planets | 地球及其他行星上 g 的变化

The value of g is not perfectly constant over the Earth’s surface. It varies slightly due to:

g 的值在地球表面并非完全恒定。它会因以下因素而略有变化：

Altitude: g decreases with height above the Earth’s surface because the distance from the centre increases. This follows from g = G M / (R + h)².
Latitude: The Earth is not a perfect sphere – it bulges at the equator. Points at the poles are closer to the centre, so g is slightly larger there. Additionally, the Earth’s rotation causes a small centrifugal effect that reduces apparent weight at the equator.
Local geology: Dense rock formations can cause tiny local increases in g.

海拔高度：g 随着离地球表面的高度增加而减小，因为到地心的距离增大。这由 g = G M / (R + h)² 得出。
纬度：地球并非完美球体——它在赤道处隆起。两极的点更靠近地心，因此那里的 g 略大。此外，地球自转产生微小的离心效应，减小了赤道处的表观重量。
当地地质：致密的岩石构造可能导致 g 的微小局部增加。

On other planets, the surface gravitational field strength can be very different. For example, Jupiter’s g ≈ 25 N/kg because of its huge mass, while Mars’ g ≈ 3.7 N/kg. You may be asked to compare weights of the same object on different planets using the g values provided.

在其他行星上，表面引力场强度可能差异很大。例如，木星的 g ≈ 25 N/kg，因为它质量巨大；而火星的 g ≈ 3.7 N/kg。你可能需要利用给出的 g 值，比较同一物体在不同行星上的重量。

6. Orbital Motion – The Role of Gravity | 轨道运动——引力的作用

Gravity provides the centripetal force that keeps planets in orbit around the Sun and satellites around the Earth. For a body moving in a circular orbit of radius r at speed v, the required centripetal force is:

引力提供了使行星绕太阳运行、卫星绕地球运行的向心力。对于在半径为 r 的圆形轨道上以速度 v 运动的物体，所需的向心力为：

F_c = m v² / r

If this force is supplied entirely by gravity, then:

如果这个力完全由引力提供，那么：

G M m / r² = m v² / r

Here M is the mass of the central body (e.g., the Sun), and m is the mass of the orbiting body (e.g., a planet). The m cancels out, showing that the orbital speed of a satellite does not depend on its own mass:

其中 M 是中心天体（如太阳）的质量，m 是轨道天体（如行星）的质量。m 被抵消掉，表明卫星的轨道速度与它自身的质量无关：

v² = G M / r so v = √(G M / r)

This means that for a given central mass, the orbital speed decreases with increasing orbital radius. Planets farther from the Sun move more slowly.

这意味着对于给定的中心质量，轨道速度随着轨道半径的增大而减小。离太阳越远的行星运动得越慢。

7. Kepler’s Laws of Planetary Motion | 开普勒行星运动定律

Johannes Kepler formulated three empirical laws that describe planetary motion, and Newton later showed they are a consequence of his law of gravitation. IGCSE WJEC often focuses on the third law, but an overview of all three is useful:

开普勒提出了三条描述行星运动的经验定律，后来牛顿证明它们是他的引力定律的推论。IGCSE WJEC 常常聚焦第三定律，但了解全部三条定律很有用：

First Law (Law of Ellipses): Planets move in elliptical orbits with the Sun at one focus. (For IGCSE, orbits are often approximated as circles.)
Second Law (Law of Equal Areas): A line joining a planet to the Sun sweeps out equal areas in equal intervals of time, meaning planets move faster when closer to the Sun.
Third Law (Law of Harmonies): The square of the orbital period T is proportional to the cube of the semi‑major axis r (average distance from the Sun).

第一定律（椭圆定律）：行星沿椭圆轨道运动，太阳位于一个焦点上。（在 IGCSE 中，轨道常被近似为圆形。）
第二定律（面积定律）：连接行星与太阳的线在相等时间内扫过相等面积，这意味着行星在靠近太阳时运动得更快。
第三定律（调和定律）：轨道周期 T 的平方与半长轴 r（到太阳的平均距离）的立方成正比。

T² ∝ r³ or T² / r³ = constant

By combining Kepler’s third law with Newton’s law of gravitation, the constant for solar‑system objects is found to be 4π²/(G M_Sun). This relationship allows us to calculate the mass of the central body if we know T and r.

将开普勒第三定律与牛顿引力定律结合，可得太阳系物体的常数为 4π²/(G M_太阳)。利用这一关系，如果我们知道 T 和 r，就可以计算中心天体的质量。

8. Types of Satellite Orbits | 卫星轨道的类型

Artificial satellites play a critical role in modern life, and their orbits are chosen according to their purpose. IGCSE WJEC expects you to be familiar with two main types:

人造卫星在现代生活中扮演着关键角色，它们的轨道根据用途选择。IGCSE WJEC 要求你熟悉两种主要类型：

Geostationary satellites: Orbit above the Earth’s equator at a height of about 36 000 km, with a period of exactly 24 hours. They appear fixed in the sky because their orbital period matches the Earth’s rotation. Used for telecommunications and weather monitoring.
Polar satellites: Orbit with a much lower altitude (typically 200–1000 km) and pass over the poles. Their orbital period is about 90–100 minutes, and as the Earth rotates beneath them, they scan the entire surface. Used for Earth observation, mapping, and spy satellites.

地球静止卫星：在地球赤道上方约 36 000 km 的高度运行，周期恰好为 24 小时。它们看似固定在天空中，因为轨道周期与地球自转一致。用于电信和天气监测。
极轨卫星：在低得多的高度（通常 200–1000 km）运行，并经过两极。轨道周期约为 90–100 分钟，随着地球在它们下方自转，它们可以扫描整个地表。用于地球观测、测绘和侦察卫星。

For a geostationary orbit, there is only one specific altitude that gives the required period of 24 h. This can be calculated by equating the centripetal force and gravity, then substituting v = 2π r / T.

对于地球静止轨道，只有一个特定的高度能产生所需的 24 小时周期。这可以通过令向心力等于引力，然后代入 v = 2π r / T 来计算。

9. Gravitational Potential Energy (Simple Treatment) | 引力势能（简单处理）

In IGCSE, gravitational potential energy (GPE) near the Earth’s surface is given by:

在 IGCSE 中，地球表面附近的引力势能（GPE）由下式给出：

ΔE_p = m g Δh

This formula is valid only when the change in height Δh is small enough that g can be considered constant. For large changes in altitude, g changes significantly, so the simple m g Δh does not apply, but this is beyond the IGCSE syllabus.

该公式仅在高度变化 Δh 足够小、g 可视为常数时有效。当高度变化很大时，g 变化显著，此时简单的 m g Δh 不适用，但超出了 IGCSE 大纲范围。

On a larger scale, the work done to move a mass against gravity changes the gravitational potential energy of the system. In an orbital context, a satellite in a higher orbit has more gravitational potential energy (less negative) than one in a lower orbit, but a lower kinetic energy because its speed is smaller. This energy trade‑off explains why satellites must fire rockets to move to a higher orbit.

在更大尺度上，克服引力移动质量所做的功会改变系统的引力势能。在轨道情境中，较高轨道上的卫星比较低轨道上的卫星具有更大的引力势能（负得更少），但由于速度较小，动能较低。这种能量权衡解释了为什么卫星必须点燃火箭才能移动到更高轨道。

10. Free Fall and Acceleration due to Gravity | 自由落体与重力加速度

An object is said to be in free fall when the only force acting on it is gravity. In the absence of air resistance, all objects near the Earth’s surface fall with the same acceleration, g ≈ 9.8 m/s². This is a direct consequence of the equivalence of gravitational and inertial mass, tested in IGCSE through the famous hammer‑and‑feather experiment on the Moon.

当作用在物体上的力只有重力时，称其处于自由落体状态。在没有空气阻力的情况下，地球表面附近的所有物体都以相同的加速度下落，g ≈ 9.8 m/s²。这是引力质量与惯性质量等效的直接结果，在 IGCSE 中通过著名的月球上锤子和羽毛实验加以验证。

Because weight is W = m g, applying Newton’s second law (F = m a) gives m g = m a, so a = g. The mass cancels, proving that the acceleration of free fall is independent of the object’s mass.

因为重量 W = m g，应用牛顿第二定律 (F = m a) 得 m g = m a，所以 a = g。质量被抵消，证明自由落体加速度与物体质量无关。

In real life, air resistance opposes the motion, causing objects with larger surface area relative to mass to fall more slowly. IGCSE exam questions often require you to explain terminal velocity: when air resistance equals weight, the net force is zero and the object stops accelerating, falling at constant speed.

在现实生活中，空气阻力阻碍运动，导致表面积相对于质量较大的物体下落得更慢。IGCSE 考试题目常要求你解释终端速度：当空气阻力等于重量时，合力为零，物体停止加速，以恒定速度下落。

11. Measuring g Experimentally | 实验测定 g

IGCSE WJEC practical assessments may include methods to determine g in the laboratory. One common method uses a simple pendulum. For small oscillations, the period T is given by:

IGCSE WJEC 实践考核可能包括在实验室测定 g 的方法。一种常见方法使用单摆。对于小幅度摆动，周期 T 由下式给出：

T = 2π √(L / g)

By measuring the period T for different lengths L, and plotting T² against L, a straight line through the origin is obtained. The gradient = 4π² / g, from which g can be calculated.

通过测量不同长度 L 对应的周期 T，并绘制 T² 对 L 的图，可以得到一条过原点的直线。斜率 = 4π² / g，由此可计算 g。

Another method is the free‑fall timing method: a ball bearing is dropped from a known height h and the time of fall t is measured using an electronic timer. Using s = ½ g t² (since u = 0), g can be found from g = 2h / t². A graph of h versus t² (or of 2h versus t²) gives a straight line whose gradient is g.

另一种方法是自由落体计时法：将钢球从已知高度 h 释放，用电子计时器测量下落时间 t。利用 s = ½ g t²（因 u = 0），可得 g = 2h / t²。绘制 h 对 t²（或 2h 对 t²）的图，得到一条直线，其斜率为 g。

Both methods require careful attention to uncertainties, such as reaction time when using a stopwatch, and ensuring small amplitudes for the pendulum.

两种方法都需要仔细关注不确定性，例如使用秒表时的反应时间，以及确保单摆摆幅较小。

12. Common Exam Pitfalls and Key Tips | 常见考试陷阱与关键提示

To score full marks on gravitation questions in IGCSE WJEC, keep these points in mind:

Always use the correct units: mass in kg, distance in m, force in N, g in N/kg (or m/s²).
When using F = G m₁ m₂ / r², remember that r is the distance between centres, not the altitude above the surface. If a satellite is at height h above Earth’s surface, r = R_Earth + h.
Do not confuse weight with mass. Weight is a force; it has both magnitude and direction (towards the centre of the planet).
In orbits, the gravitational force is the centripetal force; the orbiting object is continuously accelerating towards the centre, so its velocity changes direction, not speed (for circular orbits).
For Kepler’s third law, T must be in years (or seconds) and r in astronomical units (or metres) depending on the constant used. Be consistent with units.
In calculations of g on other planets, you may be given mass and radius in terms of Earth’s values; proportional reasoning often avoids tedious arithmetic.

要在 IGCSE WJEC 的引力问题中获得满分，请注意以下几点：

始终使用正确单位：质量用 kg，距离用 m，力用 N，g 用 N/kg（或 m/s²）。
使用 F = G m₁ m₂ / r² 时，记住 r 是中心之间的距离，而不是离地表的垂直高度。如果卫星在地球表面上方高度 h 处，r = R_地球 + h。
不要混淆重量与质量。重量是力；它有大小和方向（指向行星中心）。
在轨道中，引力就是向心力；轨道物体不断向中心加速，因此其速度方向改变，而大小不变（对于圆形轨道）。
对于开普勒第三定律，根据所用常数，T 必须以年（或秒）为单位，r 以天文单位（或米）为单位。单位要一致。
在其他行星上的 g 的计算中，可能会以地球的数值给出质量和半径；比例推理往往可以避免繁琐的算术。

Mastering gravitation not only secures marks in standalone questions but also supports topics like circular motion, energy, and astrophysics. Practise using the equations in varied contexts, and always draw a diagram showing forces and distances.

掌握引力不仅能在独立题目中拿分，还能支持圆周运动、能量和天体物理学等主题。在不同的情境中练习使用这些方程，并始终画出标有力和距离的示意图。

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Gravitation: Key Points for IGCSE WJEC Physics | 万有引力：IGCSE WJEC 物理核心考点