📚 Energy in a Gravitational Field | 引力场中的能量
Gravitational fields store energy whenever a mass is placed in them. In CIE A-Level Physics, you need to distinguish between gravitational potential energy, gravitational potential, and the work done when masses move through a field. These ideas explain satellite orbits, escape speeds, and energy changes between orbits.
只要将质量置于引力场中，引力场就会储存能量。在 CIE A-Level 物理中，你需要区分引力势能、引力势以及质量在引力场中移动时所做的功。这些概念可以解释卫星轨道、逃逸速度以及轨道之间的能量变化。
1. Gravitational Potential Energy Near Earth's Surface | 地球表面附近的引力势能
Close to the Earth' . The zero of potential energy is usually chosen at the surface, but only changes in potential energy are physically significant.
在地球表面附近，引力场近似为匀强场。当质量 m 被竖直提升 Δh 高度时，克服重力所做的功为 W = mgΔh。这部分功以引力势能的形式储存起来，因此 ΔU = mgΔh。势能零点通常选在地面，但只有势能的变化才具有物理意义。
2. 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 without acceleration. For a point mass M, the potential at a distance r from its centre is V = −GM/r. The unit is J kg⁻¹. The negative sign arises because gravitational force is attractive: an external agent does negative work when the test mass approaches from infinity.
引力势 V 是单位质量的小检验质量从无穷远处无加速地移到该点时所做的功。对于点质量 M，在距其中心 r 处的引力势为 V = −GM/r。单位是 J kg⁻¹。负号来自引力的吸引性：当检验质量从无穷远处靠近时，外界对该质量做负功。
3. Gravitational Potential Energy in a Radial Field | 径向场中的引力势能
Since potential is energy per unit mass, the gravitational potential energy U of a mass m at distance r from a point mass M is U = mV = −GMm/r. This equation applies to any separation r, not only near a planet's surface. As r increases, U becomes less negative, meaning the mass has gained potential energy relative to points closer to the source.
由于势是单位质量具有的能量，质量 m 在距点质量 M 为 r 处的引力势能为 U = mV = −GMm/r。该方程适用于任意距离 r，而不仅限于行星表面附近。当 r 增大时，U 的负值变小，这意味着相对于离源较近的点，该质量获得了势能。
4. Field Strength and Potential Gradient | 场强与势梯度
Gravitational field strength g is related to the rate at which gravitational potential changes with distance. In general, g = −dV/dr. For a radial field, differentiating V = −GM/r gives g = GM/r², matching Newton' e negative sign shows that the field points in the direction in which potential decreases.
引力场强 g 与引力势随距离变化的快慢有关。一般来说，g = −dV/dr。对于径向场，对 V = −GM/r 求导可得 g = GM/r²，这与牛顿的平方反比定律一致。负号表明场指向势降低的方向。
5. Equipotential Surfaces | 等势面
An equipotential surface is a surface on which the gravitational potential is constant. No work is done by the gravitational field when a mass moves along an equipotential surface because the potential difference is zero. In a radial field, equipotential surfaces are concentric spheres centred on the source mass. In a uniform field, they are parallel planes perpendicular to the field lines.
等势面是引力势保持不变的面。当质量沿等势面移动时，由于势差为零，引力场不做功。在径向场中，等势面是以源质量为中心的同心球面。在匀强场中，等势面是垂直于场线的平行平面。
6. Total Energy of an Orbiting Satellite | 在轨卫星的总能量
tic energy is K = ½mv² = GMm/(2r). The potential energy is U = −GMm/r. The total energy is E = K + U = −GMm/(2r). A negative total energy means the satellite is bound to the planet.
对于质量为 m、在半径 r 的圆轨道上绕质量 M 的行星运动的卫星，引力提供向心力：GMm/r² = mv²/r。因此动能为 K = ½mv² = GMm/(2r)。势能为 U = −GMm/r。总能量为 E = K + U = −GMm/(2r)。总能量为负表示卫星被行星束缚。
7. Escape Velocity | 逃逸速度
planet of mass M and radius R, the condition is K + U ≥ 0. This gives ½mv² − GMm/R ≥ 0, so vₑ = √2GM/R. For Earth, vₑ ≈ 11.2 km s⁻¹. The escape speed is independent of the object's mass.
逃逸速度是物体恰好能到达无穷远处且剩余动能为零所需的最小发射速度。在质量为 M、半径为 R 的行星表面，条件为 K + U ≥ 0。由此可得 ½mv² − GMm/R ≥ 0，所以 vₑ = √2GM/R。对于地球，vₑ ≈ 11.2 km s⁻¹。逃逸速度与物体质量无关。
8. Energy Changes Between Orbits | 轨道之间的能量变化
(1/r₁ − 1/r₂). Interestingly, the kinetic energy decreases because the orbital speed is lower in a higher orbit, but the potential energy increases by a larger amount.
当卫星从半径为 r₁ 的低轨道转移到半径为 r₂ 的高轨道时，必须由发动机做功。总能量从 E₁ = −GMm/(2r₁) 变为 E₂ = −GMm/(2r₂)。由于 r₂ > r₁，E₂ 的负值比 E₁ 小，因此总能量增加了。所需的外部功为 ΔE = GMm/2 (1/r₁ − 1/r₂)。有趣的是，由于高轨道的轨道速度较低，动能反而减小，但势能的增加量更大。
9. Work Done and Potential Difference | 做功与势差
If a mass m moves between two points where the gravitational potential changes from V₁ to V₂, the work done on the mass by an external force is W = m(V₂ − V₁). This is useful for problems involving launching rockets, transferring satellites, or moving masses within a planet's field. A mass moving freely under gravity loses potential energy and gains kinetic energy, keeping the total energy constant when no external work is done.
如果质量 m 在两点之间移动，且引力势从 V₁ 变化到 V₂，则外力对质量所做的功为 W = m(V₂ − V₁)。这一关系可用于解决火箭发射、卫星转移或质量在行星场中移动等问题。仅受引力作用的物体会失去势能并获得动能，在无外力做功时总能量保持不变。
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