📚 IB Physics: Newton’s Law of Universal Gravitation and Its Applications | IB物理：万有引力定律及其应用
Newton’s law of universal gravitation is a cornerstone of classical physics. It explains why an apple falls, why the Moon orbits the Earth, and why planets move around the Sun. In the IB Physics syllabus, this law forms the basis for understanding gravitational fields, orbital motion, and energy in space.
万有引力定律是经典物理学的基石。它解释了苹果为何下落、月球为何绕地球运动、行星为何绕太阳运行。在IB物理课程中，这一定律是理解引力场、轨道运动和空间能量的基础。
1. Statement of the Law | 定律的表述
Newton stated 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.
牛顿指出：宇宙中每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，与它们质心之间距离的平方成反比。
The mathematical form of the law is:
该定律的数学形式为：
F = G m₁m₂ / r²
Here, F is the gravitational force, G is the gravitational constant (6.674 × 10⁻¹¹ N m² kg⁻²), m₁ and m₂ are the two masses, and r is the distance between their centres.
其中，F是引力，G是万有引力常量（6.674 × 10⁻¹¹ N m² kg⁻²），m₁和m₂是两个物体的质量，r是它们质心间的距离。
The force is always attractive, acts along the line joining the two masses, and forms an action-reaction pair.
引力的方向总是相互吸引，沿两物体连线方向，并且构成一对作用力与反作用力。
2. Gravitational Field Strength | 引力场强度
A gravitational field is a region where a mass experiences a force. The gravitational field strength g at a point is defined as the gravitational force per unit mass placed at that point.
引力场是质量受到力的作用的区域。某点的引力场强度g定义为置于该点的单位质量所受的引力。
For a point mass M, the field strength at distance r is:
对于点质量M，在距离r处的场强为：
g = F / m = GM / r²
This equation shows that g is independent of the test mass m and decreases with the square of the distance. Near the Earth’s surface, g ≈ 9.81 N kg⁻¹, which is also called the gravitational acceleration.
该式表明g与试探质量m无关，并随距离的平方而减小。在地球表面附近，g ≈ 9.81 N kg⁻¹，也称重力加速度。
Gravitational field strength is a vector quantity. Its direction is always towards the mass creating the field.
引力场强度是矢量，方向始终指向产生该场的质量。
3. Gravitational Potential Energy | 引力势能
In a uniform gravitational field near the Earth’s surface, gravitational potential energy is approximated by Eₚ = mgh, where h is the height above a reference level.
在地球表面附近的均匀引力场中，引力势能近似为Eₚ = mgh，其中h是相对于参考平面的高度。
However, for radial fields far from the Earth, a more general definition is required. The gravitational potential energy of a mass m at distance r from mass M is:
然而，对于远离地球的径向引力场，需要更一般的定义。质量m在距质量M距离r处的引力势能为：
Eₚ = −GMm / r
The negative sign indicates that gravitational potential energy is zero at infinity and decreases (becomes more negative) as the masses move closer together.
负号表示引力势能在无穷远处为零，并且随着两物体靠近而减小（变得更负）。
This expression is essential for calculating the energy required to move satellites between orbits.
该表达式对于计算卫星在不同轨道之间移动所需的能量至关重要。
4. Gravitational Potential | 引力势
Gravitational potential V at a point in a field is the gravitational potential energy per unit mass:
引力场中某点的引力势V是单位质量的引力势能：
V = −GM / r
Its unit is J kg⁻¹. The potential is a scalar quantity, and it is negative because the field is attractive.
其单位是J kg⁻¹。引力势是标量，由于引力场是吸引性的，所以其值为负。
The relation between field strength and potential is: g = −ΔV/Δr, which in one dimension gives g = −dV/dr. A graph of V against r has a gradient equal to g (with a minus sign).
场强与势的关系为：g = −ΔV/Δr，在一维情形下即g = −dV/dr。V随r变化的图像斜率等于−g。
5. Kepler’s Laws of Planetary Motion | 开普勒行星运动定律
Kepler’s three laws describe the motion of planets around the Sun. They were derived empirically before Newton’s law and later explained by Newton’s gravitational theory.
开普勒三大定律描述了行星绕太阳的运动。这些定律在牛顿定律之前由经验总结而出，后来被牛顿引力理论所解释。
First law (ellipse law): every planet moves in an ellipse with the Sun at one focus.
第一定律（椭圆定律）：所有行星沿椭圆轨道运动，太阳位于椭圆的一个焦点上。
Second law (equal areas law): a line joining a planet and the Sun sweeps out equal areas in equal times, meaning the planet moves faster when closer to the Sun.
第二定律（面积定律）：行星与太阳的连线在相等时间内扫过相等的面积，即行星在靠近太阳时运动更快。
Third law (harmonic law): the square of the orbital period T is proportional to the cube of the semi-major axis a:
第三定律（周期定律）：轨道周期T的平方与半长轴a的立方成正比：
T² ∝ a³
For a circular orbit around a central mass M, the constant is 4π²/GM, so T² = (4π²/GM) a³.
对于绕中心质量M的圆轨道，比例常数为4π²/GM，因此T² = (4π²/GM) a³。
6. Satellite Orbits and Circular Motion | 卫星轨道与圆周运动
For a satellite moving in a circular orbit of radius r around a planet of mass M, the gravitational force provides the required centripetal force.
对于绕质量为M的行星沿半径为r的圆轨道运动的卫星，引力提供所需的向心力。
Setting GMm / r² = mv² / r gives the orbital speed:
令GMm / r² = mv² / r，即可得到轨道速度：
v = √(GM / r)
The orbital speed is independent of the satellite’s mass and decreases as r increases. This is why outer planets move more slowly than inner planets.
轨道速度与卫星质量无关，且随r增大而减小。这就是外行星比内行星运动更慢的原因。
The orbital period is derived from v = 2πr / T, giving:
由v = 2πr / T可得轨道周期：
T = 2π √(r³ / GM)
This confirms Kepler’s third law for circular orbits.
这证实了圆轨道情况下的开普勒第三定律。
7. Geostationary Satellites | 地球同步卫星
A geostationary satellite orbits the Earth directly above the equator, with the same angular speed as the Earth’s rotation. As a result, it appears stationary relative to an observer on the ground.
地球同步卫星在赤道正上方绕地球运动，角速度与地球自转相同。因此，相对于地面观察者，它看起来是静止的。
To achieve this, the satellite must satisfy:
要实现这一点，卫星必须满足：
GMm / r² = m (4π² / T²) r
With T = 24 hours (86,400 s), the radius r is about 42,200 km from the Earth’s centre, corresponding to an altitude of about 35,800 km above the surface.
当T = 24小时（86,400秒）时，轨道半径r约为4.22 × 10⁴ km（即距地心约42,200 km），对应地表上方约35,800 km的高度。
Geostationary satellites are used for telecommunications, weather monitoring, and broadcasting because they maintain a fixed position above one area.
地球同步卫星用于电信、气象监测和广播，因为它们保持在某一区域上方的固定位置。
8. Escape Velocity | 逃逸速度
Escape velocity is the minimum speed an object must have at a given distance from a planet to escape its gravitational field without further propulsion.
逃逸速度是指物体在离行星一定距离处，不需要进一步推进就能脱离该行星引力场所需的最小速度。
This is found by setting the total mechanical energy (kinetic plus gravitational potential) to zero:
这可以通过令总机械能（动能加引力势能）为零得到：
½mv² + (−GMm / r) = 0
Solving for v gives:
解出v得：
v_esc = √(2GM / r)
For Earth, r = 6.37 × 10⁶ m, so v_esc ≈ 11.2 km s⁻¹. Note that escape velocity does not depend on the mass of the escaping object.
对于地球，r = 6.37 × 10⁶ m，因此v_esc ≈ 11.2 km s⁻¹。注意逃逸速度与逃逸物体的质量无关。
9. Orbital Energy and Binding Energy | 轨道能量与结合能
A satellite in a circular orbit has both kinetic energy and gravitational potential energy. The kinetic energy is positive and equal to half the magnitude of the potential energy:
在圆轨道上的卫星同时具有动能和引力势能。动能为正值，且大小等于势能绝对值的一半：
Eₖ = GMm / (2r), Eₚ = −GMm / r
Therefore, the total mechanical energy is:
因此，总机械能为：
E = Eₖ + Eₚ = −GMm / (2r)
The negative total energy means the satellite is bound to the planet. To move to a higher orbit, energy must be added; to move to a lower orbit, energy is released.
负的总能量意味着卫星被行星束缚。要向更高的轨道移动，需要增加能量；向更低的轨道移动则会释放能量。
The binding energy is the energy needed to remove the satellite to infinity, which equals −E = GMm / (2r).
结合能是将卫星移至无穷远处所需的能量，等于−E = GMm / (2r)。
10. Weightless and Weight in Orbit | 轨道中的失重与重力
Astronauts in orbit appear weightless because they are in free fall. The only force acting on them is gravity, which provides the centripetal acceleration for their circular motion.
轨道中的宇航员看起来失重，是因为他们处于自由落体状态。作用于他们的唯一力是引力，该力提供圆周运动所需的向心加速度。
Although gravity is still significant at orbital altitudes (e.g., about 90% of Earth’s surface value at 300 km), the person and the spacecraft fall together, so no normal reaction force is felt.
尽管在轨道高度上引力仍然显著（例如300 km处约为地球表面的90%），但由于人与飞船一起下落，因此感受不到支持力。
Weightlessness is not the absence of gravity; it is the absence of a contact force supporting the body.
失重不是没有引力，而是没有支持人体的接触力。
11. Determining the Mass of Celestial Bodies | 测定天体质量
Newton’s law allows us to measure the mass of planets, stars, and the Sun using orbital data. If a body of mass m orbits a central mass M with period T and orbital radius r, then equating gravitational force to centripetal force gives:
万有引力定律使我们能够利用轨道数据测量行星、恒星和太阳的质量。如果质量为m的物体以周期T和轨道半径r绕中心质量M运动，则令引力等于向心力得：
M = 4π² r³ / (G T²)
For example, using the Earth’s orbital data (r = 1.496 × 10¹¹ m, T = 1 year = 3.156 × 10⁷ s), the Sun’s mass is calculated to be about 1.99 × 10³⁰ kg.
例如，利用地球的轨道数据（r = 1.496 × 10¹¹ m，T = 1年 = 3.156 × 10⁷ s），可算出太阳质量约为1.99 × 10³⁰ kg。
The same method is used to measure the mass of planets by observing the motion of their moons.
同样的方法也用于通过观测卫星运动来测量行星的质量。
12. Applications and Important Exam Tips | 应用与考试要点
Universal gravitation has many real-world applications, including satellite navigation, space exploration, and predicting planetary motion. In IB exams, common questions involve calculating orbital speed, period, gravitational field strength, and escape velocity.
万有引力有许多实际应用，包括卫星导航、太空探索和行星运动预测。在IB考试中，常见问题涉及计算轨道速度、周期、引力场强度和逃逸速度。
Key reminders:
关键提醒：
- Always use the distance from the centre of mass, not the altitude above the surface.
- 始终使用到质心的距离，而不是地表以上的高度。
- Distinguish between gravitational field strength g (N kg⁻¹) and gravitational acceleration (m s⁻²); numerically they are equal.
- 区分引力场强度g（N kg⁻¹）与重力加速度（m s⁻²）；它们的数值相等。
- Remember the sign conventions for gravitational potential energy and potential.
- 牢记引力势能与引力势的符号规定。
- For orbital problems, write down the equation \(GMm/r^2 = mv^2/r\) and solve systematically.
- 对于轨道问题，写出方程GMm/r² = mv²/r并系统求解。
Frequent exam mistakes include forgetting the minus sign in potential energy, using radius instead of altitude incorrectly, and mixing up the constants.
常见考试错误包括忘记势能的负号、错误地将高度当作半径，以及混淆常量。
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