Newton’s Law of Universal Gravitation and Its Applications | 牛顿万有引力定律及其应用

📚 Newton’s Law of Universal Gravitation and Its Applications | 牛顿万有引力定律及其应用

Newton’s law of universal gravitation is a cornerstone of classical physics. It explains the motion of planets, moons, satellites, and even falling apples. This article reviews the key concepts, formulas, and applications that appear in A-level physics exams.

牛顿万有引力定律是经典物理学的基石。它解释了行星、月球、卫星乃至落下的苹果的运动。本文系统复习A-level物理中考到的核心概念、公式与应用。

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1. Statement of Newton’s Law of Universal Gravitation | 牛顿万有引力定律的表述

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 centers.

牛顿定律指出：任何两个质点之间都存在相互吸引力，引力大小与它们质量的乘积成正比，与它们质心之间距离的平方成反比。

The magnitude of this gravitational force is given by:

该引力的大小由下式给出：

F = G m₁m₂ / r²

• F is the gravitational force (N).
• G is the gravitational constant (6.674 × 10⁻¹¹ N m² kg⁻²).
• m₁, m₂ are the masses of the two objects (kg).
• r is the distance between their centers (m).

• F 为引力（N）。
• G 为万有引力常量（6.674 × 10⁻¹¹ N m² kg⁻²）。
• m₁、m₂ 为两物体质量（kg）。
• r 为两物体质心间距离（m）。

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2. The Gravitational Constant G | 万有引力常量 G

The constant G was first measured by Henry Cavendish in 1798 using a torsion balance. Its value is extremely small, which means gravitational forces are only noticeable when at least one of the masses is large.

常量 G 由卡文迪许在1798年用扭秤首先测量。它的数值极小，这意味着只有在至少一个物体质量很大时，引力才会明显。

Quantity	Value	Unit
G	6.674 × 10⁻¹¹	N m² kg⁻²

In calculations, G is often provided in the exam, but you should know its approximate value and the fact that it is a universal constant – the same everywhere in the universe.

在计算中，G 通常会在考试中给出，但应知道其近似值，并且它是普适常量——在宇宙任何地方都一样。

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3. Gravitational Field Strength | 引力场强度

A gravitational field is a region of space where a mass experiences a gravitational force. The gravitational field strength g is defined as the force per unit mass:

引力场是空间中对质量产生引力的区域。引力场强度 g 定义为单位质量所受的引力：

g = F / m

For a point mass M, the field strength at a distance r from its center is:

对于质点 M，在距离其中心 r 处的场强为：

g = G M / r²

This shows that gravitational field strength decreases with the square of the distance. On Earth’s surface, g ≈ 9.81 N kg⁻¹.

这说明引力场强度随距离的平方减小。在地球表面，g ≈ 9.81 N kg⁻¹。

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4. Acceleration due to Gravity and Surface Gravity | 重力加速度与地表重力

Near the surface of a planet, the gravitational force on an object of mass m is F = mg. Equating this with Newton’s law gives:

在行星表面附近，质量为 m 的物体所受引力为 F = mg。与牛顿定律联立可得：

g = G M / R²

where M is the planet’s mass and R is its radius.

其中 M 为行星质量，R 为行星半径。

• Earth: M = 5.97 × 10²⁴ kg, R = 6.37 × 10⁶ m, g = 9.81 m s⁻².
• Moon: M = 7.35 × 10²² kg, R = 1.74 × 10⁶ m, g ≈ 1.62 m s⁻².

• 地球：M = 5.97 × 10²⁴ kg，R = 6.37 × 10⁶ m，g = 9.81 m s⁻²。
• 月球：M = 7.35 × 10²² kg，R = 1.74 × 10⁶ m，g ≈ 1.62 m s⁻²。

Note that g varies with altitude and latitude. At height h above the surface, g(h) = G M / (R + h)².

注意 g 随海拔和纬度变化。在地表上方高度 h 处，g(h) = G M / (R + h)²。

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5. Planetary Motion and Kepler’s Laws | 行星运动与开普勒定律

Newton’s universal gravitation explains Kepler’s laws of planetary motion. Kepler’s third law can be derived for circular orbits by equating gravitational force to centripetal force.

牛顿万有引力定律解释了开普勒行星运动定律。对于圆轨道，将引力与向心力相等即可推导出开普勒第三定律。

For a planet of mass m orbiting a star of mass M in a circular orbit of radius r:

对于质量为 m 的行星绕质量为 M 的恒星在半径为 r 的圆轨道上运动：

G M m / r² = m v² / r

This leads to T² = (4π² / G M) r³, which is Kepler’s third law.

这导致 T² = (4π² / G M) r³，即开普勒第三定律。

• Kepler’s first law: planets move in elliptical orbits with the Sun at one focus.
• Kepler’s second law: a line joining a planet and the Sun sweeps equal areas in equal times.
• Kepler’s third law: T² ∝ r³ for orbits around the same central body.

• 开普勒第一定律：行星沿椭圆轨道运动，太阳位于一个焦点上。
• 开普勒第二定律：行星与太阳的连线在相等时间内扫过相等的面积。
• 开普勒第三定律：绕同一中心天体运动的 T² ∝ r³。

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6. Circular Motion of Satellites | 卫星的圆周运动

A satellite orbits the Earth because gravity provides the required centripetal force. For a satellite of mass m at a distance r from Earth’s center, moving with speed v:

卫星绕地球运动是因为引力提供了所需的向心力。对于质量为 m、距地心 r、速度为 v 的卫星：

G M m / r² = m v² / r

Therefore the orbital speed is:

因此轨道速度为：

v = √(G M / r)

The orbital period is:

轨道周期为：

T = 2π √(r³ / G M)

Notice that orbital speed is independent of satellite mass. A lower orbit means higher speed and shorter period.

注意轨道速度与卫星质量无关。轨道越低，速度越快，周期越短。

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7. Geostationary Satellites | 同步卫星

A geostationary satellite appears stationary over a fixed point on Earth’s equator. It has the same angular velocity as Earth’s rotation.

同步卫星在地球赤道某一点上方看起来是静止的。它的角速度与地球自转角度相同。

Conditions for a geostationary orbit:

同步轨道的条件：

• Orbit lies in the plane of the equator.
• Direction of revolution is the same as Earth’s rotation.
• Orbital period equals Earth’s sidereal day: T = 24 h = 86 400 s.
• Orbital radius is approximately 42,200 km from Earth’s center (about 35,800 km above the surface).

• 轨道位于赤道平面内。
• 公转方向与地球自转方向相同。
• 轨道周期等于地球恒星日：T = 24 h = 86 400 s。
• 轨道半径约为42,200 km（距地心），即地表上方约35,800 km。

Geostationary satellites are used for communication, weather monitoring, and broadcasting.

同步卫星用于通信、天气预报和广播。

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8. Weightlessness and Apparent Weight | 失重与视重

In a freely falling satellite or spacecraft, the only force acting is gravity. Objects inside appear weightless because the normal reaction force is zero.

在自由下落的卫星或航天器中，唯一作用的力是引力。由于支持力为零，内部物体看起来失重。

True weight is the gravitational force mg. Apparent weight is the normal force. In orbit, both satellite and astronaut accelerate toward Earth at the same rate, so the astronaut feels weightless.

真实重量是引力 mg，视重是支持力。在轨道上，卫星和宇航员以相同加速度朝向地球，因此宇航员感觉失重。

N = m(g – a)

When a = g, N = 0, hence weightlessness.

当 a = g 时，N = 0，即失重。

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9. Gravitational Potential Energy | 引力势能

For a point mass m at a distance r from a mass M, the gravitational potential energy is:

对于距离质量 M 为 r 处的质点 m，引力势能为：

Eₚ = – G M m / r

The negative sign means the potential energy is zero at infinity and decreases (becomes more negative) as r decreases. This is because gravity is attractive.

负号表示在无穷远处势能为零，随着 r 减小，势能减小（更负）。这是因为引力是吸引力。

Gravitational potential V is the potential energy per unit mass:

引力势 V 是单位质量的势能：

V = – G M / r

At Earth’s surface, V ≈ -6.25 × 10⁷ J kg⁻¹.

在地球表面，V ≈ -6.25 × 10⁷ J kg⁻¹。

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10. Escape Velocity | 逃逸速度

The escape velocity is the minimum speed an object must have at a given point to escape a planet’s gravitational field without further propulsion. It is found by equating kinetic energy to the gravitational potential energy:

逃逸速度是物体在某一位置为逃出行星引力场而不再推进所需的最小速度。令动能等于引力势能可得：

½ m v² = G M m / R

Thus:

因此：

v = √(2 G M / R) = √(2 g R)

For Earth, v ≈ 11.2 km s⁻¹. Note that escape velocity does not depend on the mass or direction of the object (as long as it path avoids the planet).

对地球，v ≈ 11.2 km s⁻¹。注意逃逸速度与物体质量或方向无关（只要路径不撞上行星）。

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11. Applications of Universal Gravitation | 万有引力的实际应用

Universal gravitation is essential for many real-world technologies and discoveries:

万有引力在许多现实技术和发现中至关重要：

• Global Positioning System (GPS): Relies on precise orbital mechanics of satellites; gravitational time dilation corrections also involve gravity.
• Spacecraft trajectories: Slingshot maneuvers use the gravity of planets to change a probe’s velocity.
• Tides: The Moon’s and Sun’s gravitational pulls cause ocean tides.
• Mass determination: The mass of celestial bodies can be found using orbital periods and radii.

• 全球定位系统（GPS）：依赖卫星的精确轨道力学；引力时间膨胀修正也涉及引力。
• 航天器轨道：引力弹弓利用行星引力改变探测器速度。
• 潮汐：月球和太阳的引力引起海洋潮汐。
• 质量测定：通过轨道周期和半径可以求出天体质量。

Example: If a moon orbits a planet with period T and radius r, the planet’s mass is M = 4π²r³ / (G T²).

示例：若卫星以周期 T 和半径 r 绕行星运动，则行星质量 M = 4π²r³ / (G T²)。

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12. Common Exam Points and Pitfalls | 常考考点与易错点

When solving gravitational problems, watch out for the following:

在解决引力问题时，注意以下几点：

Key Idea	Common Mistake
Use r as the distance between centers, not surface-to-surface.	Using height above surface instead of distance from center.
g is a vector pointing toward the center of mass.	Treating g as scalar in vector problems.
Gravitational potential energy is negative.	Forgetting the negative sign.
Orbital speed is slower for higher orbits.	Thinking speed increases with distance.

核心要点	常见错误
r 是质心间距，不是表面距离。	误将离地高度当作地心距离。
g 是指向质心的矢量。	在矢量题中将 g 当标量。
引力势能是负值。	遗漏负号。
轨道越高，速度越慢。	误以为距离越远速度越快。

Always write down the formula before substituting numbers, and check units carefully. For multi-step problems, clearly identify which law or principle applies.

代入数值前先写出公式，并仔细检查单位。对于多步问题，明确用哪条定律或原理。

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Published by TutorHao | Physics Revision Series | aleveler.com

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