📚 A-Level Physics: Gravitation Key Points | A-Level 物理：万有引力 考点精讲
Gravitation is a fundamental force that governs the motion of planets, stars, and satellites. In A-Level Physics, you are expected to understand Newton’s law of gravitation, gravitational fields, potentials, orbital mechanics, and related concepts. This revision guide summarises the key points, formulas, and common exam questions.
万有引力是支配行星、恒星和人造卫星运动的基本力。在 A-Level 物理中，你需要掌握牛顿万有引力定律、引力场、引力势、轨道力学及相关概念。本复习指南总结了核心考点、公式和常见考题。
1. Newton’s Law of Universal Gravitation | 牛顿万有引力定律
Newton’s law states that every particle 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.
牛顿定律指出：任何两个质点之间都存在相互吸引力，该力的大小与两质点的质量乘积成正比，与它们中心之间距离的平方成反比。
The gravitational force between two point masses (or spherical objects) is given by:
两个点质量（或球形物体）之间的引力由下式给出：
F = G m₁ m₂ / r²
where F is the force (N), G is the universal gravitational constant (6.67×10⁻¹¹ N m² kg⁻²), m₁ and m₂ are the masses (kg), and r is the separation (m). The force is always attractive and acts along the line joining the centres of mass.
其中 F 是力（N），G 是万有引力常量（6.67×10⁻¹¹ N m² kg⁻²），m₁ 和 m₂ 是质量（kg），r 是间距（m）。该力总是引力，方向沿两物体质心连线。
Strictly, this equation applies to point masses. For homogeneous spheres, we treat the mass as if it were concentrated at the centre. This is crucial for calculating forces between planets and satellites.
严格地说，该方程适用于质点。对于均匀球体，可将其质量视为集中于球心进行处理。这对于计算行星与卫星之间的引力至关重要。
2. Gravitational Field Strength | 引力场强度
The gravitational field strength g at a point is defined as the force per unit mass acting on a small test mass placed at that point: g = F / m. Its unit is N kg⁻¹, which is equivalent to m s⁻².
引力场强度 g 定义为放置在该点的小检验质量所受的力与其质量之比： g = F / m。其单位为 N kg⁻¹，等价于 m s⁻²。
For a point mass M (or outside a spherical mass), the field strength at distance r is:
对于点质量 M（或球体外部），距离 r 处的场强为：
g = G M / r²
This is an inverse square law. Near the Earth’s surface, g is approximately 9.81 N kg⁻¹. In calculations, we often denote this standard value as g₀. The field strength is a vector directed towards the centre of the mass.
这遵循平方反比定律。在地球表面附近，g 约为 9.81 N kg⁻¹。计算中，我们常以 g₀ 表示这一标准值。场强是一个矢量，方向指向质量中心。
Multiple masses produce a resultant field, found by vector addition of the individual field strengths. This is important for understanding neutral points where the net field is zero.
多个质量产生的合场强可通过各个场强的矢量叠加求得。这对于理解合场强为零的中性点很重要。
3. Gravitational Potential | 引力势
Gravitational potential V at a point is the work done per unit mass in bringing a small test mass from infinity to that point. Infinity is chosen as the zero of potential. The potential at distance r from a point mass M is:
引力势 V 是单位质量从无穷远移至该点外力所做的功。选取无穷远处势能为零。距离点质量 M 为 r 处的引力势为：
V = – G M / r
The potential is negative because work must be done against the gravitational field to move a mass from infinity to a point in the field. Its unit is J kg⁻¹.
势为负值，因为从无穷远将质量移到场中某点需要克服引力做功。它的单位是 J kg⁻¹。
Gravitational potential is a scalar. For a system of masses, the total potential at a point is the algebraic sum of the potentials due to each mass. Equipotential surfaces are surfaces of constant potential; no work is done when moving a mass along an equipotential surface.
引力势是标量。对于质量系统，某点的总势是各质量产生的势的代数和。等势面是势保持不变的曲面；沿等势面移动质量不做功。
4. Gravitational Potential Energy | 引力势能
The gravitational potential energy U of a system of two point masses m₁ and m₂ separated by distance r is defined as the work done to assemble them from infinite separation:
两个相距 r 的点质量 m₁ 和 m₂ 组成的系统的引力势能 U 定义为将它们从无穷远移至该距离所需做的功：
U = – G m₁ m₂ / r
For a mass m in the field of a larger mass M, this is often written U = m V = – G M m / r. The negative sign indicates a bound system; energy must be supplied to separate the masses to infinity (where U = 0).
对于在较大质量 M 的场中的质量 m，常写作 U = m V = – G M m / r。负号表示束缚系统；需要提供能量才能将物体分离到无穷远（此时 U = 0）。
The change in gravitational potential energy when a mass moves from r₁ to r₂ is ΔU = U₂ – U₁. In uniform fields (near Earth’s surface), we can use ΔU = mgΔh, but this is an approximation valid only for small height changes.
当质量从 r₁ 移动到 r₂ 时，引力势能的变化为 ΔU = U₂ – U₁。在均匀场（近地表面）中，我们可以使用 ΔU = mgΔh，但这只适用于高度变化很小的情况。
5. Orbital Motion: Velocity and Period | 轨道运动：速度与周期
For a satellite in a circular orbit around a central body of mass M, the gravitational force provides the necessary centripetal force:
对于绕中心质量 M 做圆周运动的卫星，引力提供所需向心力：
G M m / r² = m v² / r
Hence the orbital speed v and period T are:
因此，轨道速度 v 和周期 T 分别为：
v = √(G M / r)
T = 2π r / v = 2π √(r³ / G M)
Notice that the orbital speed depends only on the radius of the orbit and the mass of the central body; the satellite’s mass cancels out. The period squared is proportional to r³, which is Kepler’s third law.
注意，轨道速度仅取决于轨道半径和中心天体质量；卫星质量被约掉了。周期的平方与 r³ 成正比，这正是开普勒第三定律。
For an elliptical orbit, the total energy is still constant and given by E = – G M m / (2a) where a
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