📚 Gravitational Fields and Satellites | 引力场与卫星
Gravitational fields are one of the fundamental concepts in A Level Physics. They govern the motion of planets, moons, and artificial satellites, and provide a rich source of exam questions involving both calculation and conceptual understanding. This topic test guide covers everything you need for the OxfordAQA International A Level Physics specification on gravitational fields and satellites.
引力场是A Level物理中的核心概念之一。它支配着行星、月球和人造卫星的运动，也是考试中计算题与概念题的重要来源。本专题测试指南涵盖了牛津AQA国际A Level物理大纲中关于引力场与卫星的所有必备知识点。
1. What Is a Gravitational Field? | 什么是引力场？
A gravitational field is a region of space in which a mass experiences a gravitational force. Any object with mass creates a gravitational field around itself, and any other mass placed within that field experiences a force of attraction.
引力场是空间中一个质量体会受到引力作用的区域。任何具有质量的物体都会在自身周围产生引力场，而置于该场中的其他质量体则会受到吸引力。
A gravitational field is a vector field — it has both magnitude and direction at every point.
The field lines (or lines of force) point towards the mass that creates the field, showing the direction of the force on a small test mass.
For a point mass or a spherical object, the field is radial: field lines spread out uniformly in all directions.
引力场是矢量场——它在每一点上同时具有大小和方向。
场线（或力线）指向产生场的质量体，表示作用在测试质量上的力的方向。
对于质点或球形物体，场是径向的：场线均匀地向所有方向发散。
For a uniform field, such as near the surface of the Earth over a small area, the field lines are parallel and equally spaced. This approximation is commonly used in projectile motion problems.
对于匀强场，例如地球表面小范围内的引力场，场线是平行且等间距的。这种近似常用于抛体运动问题中。
2. Newton's Law of Gravitation | 牛顿万有引力定律
Newton's law of gravitation states that the gravitational force between two point masses is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres.
牛顿万有引力定律指出，两个质点之间的引力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。
F = Gm₁m₂ / r²
d m₂ are the two masses in kilograms, and r is the separation between their centres in metres.
其中F是引力，单位为牛顿（N）；G是万有引力常量（6.67 × 10⁻¹¹ N m² kg⁻²）；m₁和m₂是两个质量体的质量，单位为千克；r是它们质心之间的距离，单位为米。
The force is always attractive — gravity never repels.
The law applies to point masses, but can be extended to uniform spheres by treating all mass as concentrated at the centre.
G is a universal constant — it has the same value everywhere in the universe.
引力总是吸引力——引力永远不会排斥。
该定律适用于质点，但可以推广到均匀球体，将全部质量视为集中在球心。
G是普适常量——它在宇宙任何地方都具有相同的值。
G = 6.67 × 10⁻¹¹ N m² kg⁻²
Exam tip: Always check units carefully. The gravitational constant has units that combine N, m², and kg⁻², which can be rewritten as m³ kg⁻¹ s⁻².
考试提示：务必仔细检查单位。万有引力常量的单位结合了N、m²和kg⁻²，也可以改写成m³ kg⁻¹ s⁻²。
3. Gravitational Field Strength | 引力场强度
Gravitational field strength g at a point in a gravitational field is defined as the gravitational force per unit mass acting on a small test mass placed at that point.
引力场强度g定义为放置在引力场中某点的单位质量所受到的引力。
g = F/m
The units of g are N kg⁻¹, which is equivalent to m s⁻². This is why gravitational field strength and acceleration due to gravity are numerically equal — both are approximately 9.81 on Earth's surface.
g的单位是N kg⁻¹，与m s⁻²等价。这就是为什么引力场强度与重力加速度在数值上相等——两者在地球表面都约为9.81。
gravitational field strength at distance r is found by combining F = GmM/r² with g = F/m:
对于质点M，在距离r处的引力场强度可以通过F = GmM/r²和g = F/m联立得到：
g = GM / r²
g is a vector — it always points towards the centre of the mass creating the field.
g decreases with the square of the distance from the centre of mass.
On Earth's surface (r = RE ≈ 6.37 × 10⁶ m), g ≈ 9.81 N kg⁻¹.
g是矢量——它始终指向产生场的质量体的中心。
g随距质心距离的平方而减小。
在地球表面（r = RE ≈ 6.37 × 10⁶ m），g ≈ 9.81 N kg⁻¹。
This inverse-square relationship means that doubling the distance from the centre of the Earth reduces g to one quarter of its original value. Tripling the distance reduces it to one ninth.
这种平方反比关系意味着，距离地心加倍时，g减小为原来的四分之一；距离增至三倍时，g减小为原来的九分之一。
4. Gravitational Potential and Energy | 引力势与引力势能
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 measured in J kg⁻¹.
引力势V定义为将单位质量的测试质量从无穷远处移动到该点所做的功。它是一个标量，单位为J kg⁻¹。
V = −GM / r
The negative sign indicates that gravitational potential is zero at infinity and becomes increasingly negative as we approach the mass. This reflects the fact that gravity does positive work when a mass moves towards the attracting body.
负号表示引力势在无穷远处为零，并随着靠近质量体而变得越来越负。这反映了当质量体向吸引体移动时，引力做正功的事实。
Gravitational potential energy U of a mass m at a point where the potential is V is given by:
质量为m的物体在引力势为V的点处的引力势能U为：
U = mV = −GMm / r
Gravitational potential is scalar — it has no direction, only a value at each point.
Equipotential surfaces are surfaces of constant V; no work is done moving along an equipotential surface.
ifference between two points determines the work done moving a mass between them: W = mΔV.
引力势是标量——它没有方向，每一点只有一个数值。
等势面是具有恒定V值的面；沿等势面移动不做功。
两点间的势差决定了在两点间移动质量的做功量：W = mΔV。
Note that for small height changes near the Earth' h is valid. This is because the gravitational field is approximately uniform over small distances.
注意，在地球表面附近的小高度变化中，近似ΔU = mgΔh是有效的。这是因为在小距离范围内引力场近似均匀。
5. Orbital Motion in Gravitational Fields | 引力场中的轨道运动
When a satellite orbits a planet, the gravitational force provides the centripetal force required for circular motion. This is the key equation that links orbital motion to gravitational fields.
当卫星绕行星运行时，引力提供了圆周运动所需的向心力。这是将轨道运动与引力场联系起来的关键方程。
GMm / r² = mv² / r
Cancelling m from both sides and rearranging gives the orbital speed of a satellite at radius r:
两边消去m并整理，得到卫星在半径r处的轨道速度：
v = √GM / r
The orbital speed is independent of the mass of the satellite — only the central mass M and orbital radius r matter.
Orbital speed decreases with increasing orbital radius: satellites further out travel more slowly.
The direction of the gravitational force is always perpendicular to the satellite's velocity, so gravity does no work on the satellite and its kinetic energy remains constant.
轨道速度与卫星本身的质量无关——只取决于中心质量M和轨道半径r。
轨道速度随轨道半径增大而减小：越远的卫星运行越慢。
引力的方向始终垂直于卫星的速度，所以引力对卫星不做功，其动能保持不变。
For a satellite at height h above the Earth's surface, the orbital radius is r = RE + h, where RE is the radius of the Earth.
对于距地球表面高度为h的卫星，轨道半径为r = RE + h，其中RE是地球半径。
6. Kepler's Laws of Planetary Motion | 开普勒行星运动定律
Johannes Kepler formulated three laws describing planetary motion, all of which can be derived from Newton's laws of gravitation. For the A Level syllabus, the most important is the third law.
开普勒提出了描述行星运动的三条定律，这些定律都可以从牛顿万有引力定律推导出来。在A Level大纲中，最重要的是第三定律。
The first law states that planets move in elliptical orbits with the Sun at one focus. In most A Level problems, orbits are treated as circular for simplicity.
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。在大多数A Level问题中，为简化计算，轨道被视为圆形。
The third law states that the square of the orbital period T is proportional to the cube of the semi-major axis (or radius r for circular orbits) of the orbit.
第三定律指出，轨道周期T的平方与轨道半长轴（对于圆轨道为半径r）的立方成正比。
T² = (4π² / GM) r³
s can be derived by equating gravitational force to centripetal force and using v = 2πr/T:
该公式可以通过将引力等于向心力并使用v = 2πr/T推导而来：
T² / r³ = 4π² / GM
The ratio T²/r³ is the same for all objects orbiting the same central mass. This allows astronomers to determine the mass of a planet by observing the orbital period of its moons.
对于绕同一中心质量运行的所有物体，T²/r³的比值是相同的。这使天文学家能够通过观测卫星的轨道周期来确定行星的质量。
7. Geostationary Satellites | 地球同步卫星
A geostationary satellite is one that remains fixed above a particular point on the Earth's equator. It has an orbital period equal to the Earth's rotational period (24 hours) and orbits in the same direction as the Earth's rotation.
地球同步卫星是固定在地球赤道某一点上方的卫星。它的轨道周期等于地球自转周期（24小时），并且绕行方向与地球自转方向相同。
To find the geostationary orbital radius, we use Kepler's third law with T = 24 hours = 86,400 seconds:
为求地球同步轨道半径，我们使用开普勒第三定律，取T = 24小时 = 86,400秒：
r³ = GMT² / 4π²
Using M = 5.97 × 10²⁴ kg for the Earth, this gives:
取地球质量M = 5.97 × 10²⁴ kg，可得：
r ≈ 4.23 × 10⁷ m ≈ 35,800 km above the surface
Geostationary satellites must orbit in the equatorial plane — they cannot be placed above non-equatorial latitudes.
They are used for telecommunications, broadcasting, and weather monitoring because ground antennas do not need to track them.
are geostationary (a geosynchronous orbit has T = 24 h but may be inclined or elliptical).
地球同步卫星必须在赤道平面内运行——它们不能放置在非赤道纬度上方。
它们用于通信、广播和气象监测，因为地面天线无需跟踪它们。
地球同步与地球静止的区别：所有地球静止轨道都是地球同步的，但并非所有地球同步轨道都是地球静止的（地球同步轨道周期T = 24小时，但可以是倾斜或椭圆形的）。
8. Escape Velocity | 逃逸速度
Escape velocity is the minimum speed an object must have at a given point in a gravitational field in order to escape from it completely — that is, to reach infinity with zero kinetic energy remaining.
逃逸速度是物体在引力场中某一点所需的最小速度，以便完全逃脱该引力场——即以剩余动能为零到达无穷远处。
Using conservation of energy, the initial kinetic energy must equal the work done against gravity to reach infinity:
利用能量守恒，初始动能必须等于克服引力到达无穷远处所做的功：
½mv² = GMm / r
vesc = √2GM / r
At the Earth's surface, this gives approximately 11.2 km s⁻¹. Note that escape velocity is independent of the mass of the escaping object.
在地球表面，这大约为11.2 km s⁻¹。注意，逃逸速度与逃逸物体的质量无关。
Escape velocity is a scalar speed, not a direction-dependent vector — the object can escape in any direction not intersecting the planet.
In practice, rockets do not need to reach full escape velocity at the surface because they can burn fuel continuously while ascending.
The concept of escape velocity is closely linked to black holes: if the escape velocity at the surface of an object exceeds the speed of light, not even light can escape.
逃逸速度是标量速度，而非依赖方向的矢量——物体可以沿任何不与行星相交的方向逃逸。
实际上，火箭不需要在地表就达到完整的逃逸速度，因为它们在上升过程中可以持续燃烧燃料。
逃逸速度的概念与黑洞密切相关：如果物体表面的逃逸速度超过光速，那么连光也无法逃脱。
9. Energy of Orbiting Satellites | 轨道卫星的能量
A satellite in a stable circular orbit has both kinetic energy and gravitational potential energy. The total mechanical energy of the satellite is constant as it orbits.
在稳定圆轨道上的卫星同时具有动能和引力势能。卫星的总机械能在绕行过程中保持不变。
The kinetic energy of the satellite is:
卫星的动能为：
KE = ½mv² = GMm / 2r
This follows from substituting v² = GM/r into ½mv².
这是将v² = GM/r代入½mv²得到的。
The gravitational potential energy is:
引力势能为：
PE = −GMm / r
Therefore the total energy is:
因此总能量为：
Etotal = KE + PE = −GMm / 2r
The total energy is negative, confirming that the satellite is bound to the Earth — it cannot escape without additional energy input.
To move a satellite to a higher orbit, energy must be supplied: the satellite gains gravitational potential energy, and although its speed decreases, the total energy increases (becomes less negative).
When a satellite moves to a lower orbit (e.g., due to atmospheric drag), it loses potential energy, gains kinetic energy, and total energy decreases.
总能量为负值，确认卫星被束缚在地球周围——若没有额外能量输入，它无法逃逸。
要将卫星移到更高的轨道，必须提供能量：卫星获得引力势能，虽然其速度减小，但总能量增加（负值变小）。
当卫星移动到更低的轨道时（例如由于大气阻力），它失去势能、获得动能，且总能量减小。
10. Worked Examples and Exam Tips | 例题与考试技巧
Example 1: Orbital speed of the ISS — The International Space Station orbits at a height of approximately 408 km above the Earth's surface. Calculate its orbital speed. (RE = 6.37 × 10⁶ m, ME = 5.97 × 10²⁴ kg, G = 6.67 × 10⁻¹¹ N m² kg⁻²)
例题1：国际空间站的轨道速度——国际空间站轨道距地球表面约408 km。计算其轨道速度。（RE = 6.37 × 10⁶ m, ME = 5.97 × 10²⁴ kg, G = 6.67 × 10⁻¹¹ N m² kg⁻²）
r = 6.37 × 10⁶ + 0.408 × 10⁶ = 6.778 × 10⁶ m
v = √GM/r = √((6.67 × 10⁻¹¹)(5.97 × 10²⁴) / 6.778 × 10⁶) ≈ 7.66 × 10³ m s⁻¹
The ISS travels at approximately 7.66 km s⁻¹, completing one orbit in about 92 minutes.
国际空间站以约7.66 km s⁻¹的速度运行，大约92分钟绕地球一圈。
Example 2: Field strength at height h — Calculate the gravitational field strength at a height of 100 km above the Earth's surface.
例题2：高度h处的场强——计算距地球表面100 km高度处的引力场强度。
g = GM / (RE + h)² = (6.67 × 10⁻¹¹)(5.97 × 10²⁴) / (6.47 × 10⁶)² ≈ 9.51 N kg⁻¹
Even at 100 km altitude, g is only slightly less than its surface value of 9.81 N kg⁻¹, which is why astronauts in low Earth orbit still experience a strong gravitational pull — they are in free fall, not in a gravity-free environment.
即使在100 km的高空，g也只比地表值9.81 N kg⁻¹略小。这就是为什么近地轨道上的宇航员仍感受到强大的引力——他们处于自由落体状态，而非无重力环境。
Exam tips for AQA/OxfordAQA:
针对AQA/牛津AQA的考试技巧：
Always define your symbols clearly before substituting values into equations.
Remember that r in all gravitational equations is measured from the centre of mass, not from the surface.
Check whether a question asks for field strength g or potential V — they have different units and signs.
Practise deriving equations from F = GMm/r² — examiners frequently award method marks for derivations.
For satellite questions, always state that the gravitational force provides the centripetal force before writing equations.
Use the correct number of significant figures (usually 2 or 3 in exam answers).
在代入数值之前，务必先明确每个符号的定义。
记住所有引力方程中的r是从质心而非表面开始测量的。
检查题目问的是场强g还是势V——它们的单位和正负号不同。
练习从F = GMm/r²推导方程——考官经常为推导过程提供方法分。
对于卫星问题，在写方程之前先说明引力提供了向心力。
使用正确的有效数字位数（考试答案通常为2或3位）。
Key equations summary at a glance:
| Quantity | Equation | Key points |
| Gravitational force | F = Gm₁m₂ / r² | Always attractive; inverse square |
| Field strength | g = GM / r² | Vector; units N kg⁻¹ |
| Potential | V = −GM / r | Scalar; negative; zero at infinity |
| Orbital speed | v = √GM/r | Independent of satellite mass |
| Kepler's third law | T² = (4π²/GM)r³ | Same central M → same ratio |
| Escape speed | vesc = √2GM/r | Independent of object's mass |
Gravitational fields and satellite motion are among the most rewarding topics in A Level Physics because the equations are elegant, the concepts are intuitive once mastered, and exam questions follow predictable patterns. Master the relationships between F, g, V, and orbital motion, practise converting between different representations, and you will be well prepared for any question AQA or OxfordAQA can throw at you.
引力场与卫星运动是A Level物理中最具回报价值的专题之一：方程优雅、概念一旦掌握便非常直观，且考题模式可预测。掌握F、g、V与轨道运动之间的关系，练习不同表示方式之间的转换，你就能从容应对AQA或牛津AQA的任何考题。
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