Gravitational Potential Energy | 引力势能

📚 Gravitational Potential Energy | 引力势能

Moving beyond the approximate mgh, the Newtonian expression U = −GMm/r uncovers the fundamentally negative nature of binding energy and underpins everything from satellite orbits to interplanetary travel.

在 A-Level 物理中，引力势能描述的是物体因处于引力场中某位置而具有的做功能力。超越近似的 mgh 公式，牛顿引力理论给出的 U = −GMm/r 揭示了束缚能根本上的负值特性，并支撑着从卫星轨道到行星际航行的一切。

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1. Defining Gravitational Potential Energy | 引力势能的定义

For two point masses M and m separated by a distance r, the gravitational potential energy (U) of the system is defined as the work done by an external agent to bring the two masses from an infinite separation — where the potential energy is taken as zero — to the final separation r, without changing their kinetic energies.

对于相距 r 的两个质点 M 和 m，系统的引力势能 U 定义为：外界作用力将两质点从相距无穷远（势能取为零）移至最终间距 r 的过程中所做的功，且在此过程中不改变它们的动能。

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2. Work Done Against Gravity | 反抗引力做功

When mass m is moved towards M without a net change in speed, the applied force must be equal and opposite to the gravitational attraction. With displacement directed towards M and the external force pointing away, the external force does negative work. Integrating yields W_ext = −GMm/r. Hence the gravitational potential energy acquired is U = −GMm/r.

当质量 m 被移向 M 且净速度不变时，施加的外力必须与引力大小相等、方向相反。由于位移指向 M 而外力背离 M，外力做负功。通过积分可得外力做功 W_ext = −GMm/r，因此系统获得的引力势能为 U = −GMm/r。

U = −GMm / r

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3. Reference Point at Infinity | 无穷远处的参考零势能点

Setting the zero of potential energy at infinite separation is mathematically convenient: it makes U vanish when the masses have no gravitational interaction. Because gravity is always attractive, bringing masses closer together causes U to fall below zero and become increasingly negative.

将势能零点取在无穷远在数学上十分方便：当质点间完全没有引力相互作用时，U 自然为零。由于引力始终是吸引力，质点彼此靠近时 U 会降至零以下，并变得越来越负。

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4. The Negative Sign – Physical Meaning | 负号及其物理意义

ical energy E = K + U is negative, the mass m cannot escape — it remains trapped in a closed orbit or falls towards M. Zero total energy corresponds to the threshold of escape.

势能为负表明两质点构成一个束缚系统；必须提供能量才能将它们分离至无穷远。如果总机械能 E = K + U 为负，质量 m 就无法逃脱 — 它将保持在闭轨道上或落向 M。总能量为零则对应刚好能够逃逸的临界状态。

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5. Gravitational Potential vs Potential Energy | 引力势与引力势能

;GM/r. Working with potential simplifies the analysis of fields produced by multiple masses, as V is a scalar that obeys superposition.

引力势 V 是放在该点的单位质量具有的引力势能：V = U/m。对于点质量 M，V = −GM/r。使用引力势可以简化对多个质量产生的场的分析，因为势是服从叠加原理的标量。

• V = −GM/r   (gravitational potential)
• U = mV   (potential energy of mass m at that point)

引力势 V = −GM/r；质量 m 在该点的势能 U = mV。

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6. Potential Energy of a System of Point Masses | 多质点体系的引力势能

For a collection of point masses, the total gravitational potential energy is the sum of the pairwise contributions of every two-body interaction. Each pair (i, j) separated by r_ij adds a term −Gm_im_j/r_ij. This additive property follows directly from the superposition of potentials.

对于一组点质量，总引力势能等于所有两两相互作用项之和。每对质点 (i, j) 相距 r_ij 贡献一项 −Gm_im_j/r_ij。这一可加性源于势的叠加原理。

U_total = −G ∑_pairs (m_im_j / r_ij)

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7. Energy in Circular Orbits | 圆轨道中的能量

y, the kinetic energy is K = ½mv² = GMm/(2r), while the potential energy is U = −GMm/r. The total mechanical energy becomes

对于质量为 m 的卫星在半径 r 的圆轨道上绕中心质量 M 飞行，向心力条件给出 v² = GM/r。于是动能 K = ½mv² = GMm/(2r)，势能为 U = −GMm/r。总机械能为

E = K + U = −GMm / (2r)

This reveals that the total energy is negative, equal in magnitude to the kinetic energy but half the magnitude of the potential energy. A satellite in a lower (smaller r) orbit possesses a more negative total energy, meaning it is more tightly bound.

这表明总能量为负，其大小等于动能，却是势能大小的一半。在较低（r 较小）轨道上的卫星拥有更负的总能量，意味着它被束缚得更紧。

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8. Orbital Changes and Energy Transfers | 轨道变化与能量转移

Moving a satellite from a lower orbit to a higher one requires an input of energy: the total energy must increase (become less negative). This is typically achieved through engine burns that boost the velocity, temporarily creating an elliptical transfer orbit before circularising at the new radius. Throughout any conservative manoeuvre, the sum K + U remains constant along the trajectory.

将卫星从低轨道转移到高轨道需要输入能量：总能量必须增加（负值绝对值减小）。这通常通过发动机点火增速来实现，先临时形成椭圆转移轨道，再在新半径处圆化。在任何保守机动过程中，K + U 的总和沿轨道保持不变。

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9. Comparing U = −GMm/r with mgh | 与 mgh 公式的比较

Near the Earth’s surface, where the elevation h is much smaller than the Earth’s radius R_E, we can expand

在地球表面附近，当高度 h 远小于地球半径 R_E 时，可以展开

U(h) = −GMm/(R_E+h) ≈ −GMm/R_E + (GMm/R_E²)h

Recognising g = GM/R_E², the change in potential energy only is ΔU ≈ mgh. Hence mgh is a local linear approximation that ignores the additive constant, which is irrelevant when only differences in energy matter.

注意到 g = GM/R_E²，仅势能的改变量 ΔU ≈ mgh。因此 mgh 是一个局部线性近似，它忽略了一个常数项；当只关心能量差值时，这个常数项无关紧要。

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10. Escape Velocity – Energy Perspective | 从能量角度看逃逸速度

t the surface. Setting E = ½mv⊆²; − GMm/R = 0 gives the escape speed

一个物体从质量为 M、半径为 R 的行星表面发射，当它在表面的总机械能为零时，刚好能逃逸到无穷远。令 E = ½mv_e² − GMm/R = 0 可得逃逸速度

v_e = √(2GM/R) = √(2gR)

Escape velocity is independent of the mass of the escaping object and depends only on the source mass and distance from its centre.

逃逸速度与逃逸物体的质量无关，仅取决于源头质量以及距其中心的距离。

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11. Graphical Representations | 引力势能的图像

the horizontal axis. Overlaying horizontal lines of constant total energy E helps classify orbits: negative E yields elliptical (or circular) orbits, E = 0 gives a parabolic escape trajectory, and positive E corresponds to a hyperbolic fly-by.

U 随 r 变化的图像是一条双曲线：当 r → ∞ 时 U → 0，当 r → 0 时 U 变得越来越负，图形以横轴为渐近线。在此图上叠加代表恒定总能量 E 的水平线有助于轨道分类：E 为负给出椭圆（或圆）轨道，E = 0 对应抛物线逃逸轨道，E 为正对应双曲线飞掠轨道。

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12. Common Pitfalls and Worked Example | 常见错误与例题

Example: ltitude 400 km. (Earth’s mass M = 5.97×10²&sup4; kg, radius R = 6370 km).

例题：计算将质量 500 kg 的卫星从地球表面送入高度 400 km 的圆轨道所需的能量。（地球质量 M = 5.97×10²&sup4; kg，半径 R = 6370 km）

Initial energy at surface: E_i nus;GMm/R. Final orbital radius r = R + h = 6.77×10&sup6; m, total energy in orbit E_f = &

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