IB Physics: Gravitation Key Points | IB 物理：万有引力考点精讲

📚 IB Physics: Gravitation Key Points | IB 物理：万有引力考点精讲

Mastering gravitation is essential for the IB Physics syllabus, covering Newton’s law, gravitational fields, potential, orbital mechanics, and the application of Kepler’s laws. This article distills core concepts, formulas, and exam tips to help you confidently tackle multiple-choice and extended-response questions on universal gravitation.

掌握万有引力是 IB 物理考纲的核心要求，涵盖牛顿定律、引力场、引力势、轨道力学及开普勒定律的应用。本文精炼核心概念、公式与应试技巧，助你从容应对万有引力的选择题和长答题。

1. Newton’s Law of Universal Gravitation | 牛顿万有引力定律

Every point mass attracts every other point mass with a force directly proportional to the product of their masses and inversely proportional to the square of their separation.

任意两个质点都相互吸引，引力的大小与两质点的质量乘积成正比，与它们之间距离的平方成反比。

F = G m₁m₂ / r²

F is the gravitational force (N), G = 6.67 × 10⁻¹¹ N m² kg⁻² is the universal gravitational constant, m₁ and m₂ are the masses (kg), and r is the centre-to-centre distance (m).

F 为引力(牛顿)，G = 6.67 × 10⁻¹¹ N m² kg⁻² 为万有引力常量，m₁、m₂ 为两质点的质量(千克)，r 为它们质心之间的距离(米)。

The law is an inverse-square law: doubling the distance cuts the force to one quarter. It applies strictly only to point masses or spherically symmetric bodies, where r is the distance between centres.

该定律遵循平方反比规律：距离加倍，引力减小到原来的四分之一。它严格适用于质点或球对称天体，r 取球心间距。

2. Gravitational Field Strength | 引力场强度

The gravitational field strength at a point is the force per unit mass experienced by a small test mass placed at that point.

引力场强度定义为单位质量在该点所受的引力，即放在该点的小检验质量所受的力与其质量之比。

g = F / m

Field strength g is a vector pointing towards the source mass, with units N kg⁻¹. Near a spherical body of mass M, the magnitude is given by g = GM / r².

场强 g 是矢量，方向指向场源质量，单位为 N kg⁻¹。在质量为 M 的球体外，其大小为 g = GM / r²。

On the surface of Earth, g ≈ 9.81 N kg⁻¹. Note that g is equivalent to the acceleration due to gravity for a free-falling object, so it is also expressed in m s⁻².

在地球表面，g 约等于 9.81 N kg⁻¹。注意 g 与自由落体加速度等效，因此也可以用 m s⁻² 表示。

3. Gravitational Potential | 引力势

Gravitational potential at a point is the work done per unit mass to bring a small test mass from infinity to that point.

引力势是指将单位质量从无穷远处移至该点外力所做的功。

V = –GM / r

V is always negative for isolated masses, with zero potential defined at infinity. The negative sign indicates that gravity does work on the mass as it moves inward, reducing potential energy.

对于孤立质量，引力势恒为负值，无穷远处定义为零势能点。负号表示当物体向内移动时引力做正功，势能减小。

The gradient of gravitational potential with respect to displacement gives the gravitational field strength: g = –ΔV/Δr.

引力势随位移的变化率(梯度)的负值即为引力场强度：g = –ΔV/Δr。

4. Gravitational Potential Energy | 引力势能

The gravitational potential energy of a two-mass system is the energy associated with their gravitational interaction, taken as zero when they are infinitely separated.

两质点系统的引力势能是它们因引力相互作用而具有的能量，以无穷远处为零势能点。

Eₚ = –G M m / r

This is the work done to assemble the system by bringing m from infinity to a distance r from M. For a satellite of mass m orbiting at radius r, its potential energy is U = –GMm / r.

这对应将质量 m 从无穷远处移至距 M 为 r 处外力做功的总和。对于在轨道半径 r 上运行的质量为 m 的卫星，其势能为 U = –GMm / r。

The force is conservative, so the work done moving between two points depends only on the difference in potential energy: W = –ΔEₚ.

引力是保守力，因此两点间移动所做的功只取决于势能差：W = –ΔEₚ。

5. Kepler’s Laws of Planetary Motion | 开普勒行星运动定律

First Law (Law of Ellipses): Planets orbit the Sun in elliptical paths with the Sun at one focus.

第一定律(椭圆定律)：所有行星绕太阳运动的轨道都是椭圆，太阳位于椭圆的一个焦点上。

Second Law (Law of Equal Areas): A line joining a planet to the Sun sweeps out equal areas in equal times, meaning the planet moves faster when nearer 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，即行星在近日点附近运动更快。

Third Law (Law of Periods): The square of the orbital period T is proportional to the cube of the semi-major axis r of its orbit: T² ∝ r³.

第三定律(周期定律)：行星公转周期的平方与轨道半长轴的立方成正比：T² ∝ r³。

For circular orbits, this can be derived from Newton’s law: T² = (4π² / GM) × r³.

对于圆轨道，可由牛顿定律推导得出：T² = (4π² / GM) × r³。

6. Orbital Motion and Circular Orbits | 轨道运动与圆周轨道

For a satellite in a stable circular orbit, the centripetal force required is provided entirely by gravity.

对于稳定圆周轨道上的卫星，所需的向心力完全由引力提供。

GMm / r² = m v² / r

This leads to an expression for orbital speed v = √(GM / r). Note that v does not depend on the satellite’s own mass; only on the central mass M and orbital radius r.

由此可得轨道速率 v = √(GM / r)。注意 v 与卫星本身质量无关，只取决于中心天体质量 M 和轨道半径 r。

A smaller orbital radius results in a higher orbital speed, consistent with Kepler’s second law.

轨道半径越小，轨道速率越大，这与开普勒第二定律一致。

7. Orbital Velocity and Period | 轨道速度与周期

From the circular orbit condition, the period T is the circumference divided by speed: T = 2πr / v. Substituting v yields T² = (4π² / GM) r³.

由圆轨道条件可得到周期 T 等于周长除以速率：T = 2πr / v。代入 v 得到 T² = (4π² / GM) r³。

This is Kepler’s third law for circular orbits. It allows calculation of the mass of a central body if T and r are known for any orbiting object.

这就是圆轨道的开普勒第三定律。若已知任一绕行天体的周期 T 和轨道半径 r，即可推算中心天体的质量。

For geostationary satellites, T must equal Earth’s rotational period (24 hours), which fixes the orbital radius at approximately 4.23 × 10⁷ m from Earth’s centre.

对于地球同步卫星，T 必须等于地球自转周期(24小时)，由此可确定其轨道半径约为距地心 4.23 × 10⁷ m。

8. Energy in Orbits | 轨道中的能量

The total mechanical energy of a satellite in a circular orbit is the sum of its kinetic and potential energies.

在圆轨道上，卫星的总机械能为其动能与势能之和。

Kinetic energy: Eₖ = ½ m v² = GMm / (2r). Potential energy: Eₚ = –GMm / r.

动能：Eₖ = ½ m v² = GMm / (2r)。势能：Eₚ = –GMm / r。

Thus, total energy E_tot = Eₖ + Eₚ = –GMm / (2r). The magnitude of kinetic energy is half the magnitude of potential energy, and total energy is always negative for a bound orbit.

因此，总能量 E_tot = Eₖ + Eₚ = –GMm / (2r)。动能的大小是势能大小的一半，束缚轨道的总能量恒为负值。

To move a satellite to a higher orbit, energy must be supplied; the total energy becomes less negative, meaning more work must be done against gravity.

将卫星移向更高轨道需提供能量；总能量负值变小，表明必须克服引力做更多功。

9. Weightlessness and Apparent Weight | 失重与表观重量

Astronauts in orbit experience apparent weightlessness, not because gravity is absent, but because they are in free fall towards Earth together with their spacecraft.

轨道上的航天员经历表观失重，并非因为引力消失，而是因为他们与航天器一起相对地球自由下落。

The only force acting is gravity, which supplies the centripetal acceleration needed for circular motion. A scale would read zero because both the astronaut and the scale are accelerating at the same rate.

唯一作用的力是引力，它提供了圆周运动所需的向心加速度。如果将航天员放在秤上，读数将为零，因为航天员和秤都在以相同的加速度下落。

Apparent weight is the normal reaction force from a supporting surface; in orbit this equals zero. On Earth’s surface, apparent weight is mg only in an inertial frame; in an accelerating lift, apparent weight changes.

表观重量是支持面提供的法向反作用力；在轨道上它为零。在地球表面，惯性系中表观重量为 mg；在加速的电梯中，表观重量会发生改变。

10. Variation of g with Altitude and Depth | 重力加速度随高度和深度的变化

The value of g above Earth’s surface decreases with altitude: g(h) = GM / (R + h)², where R is Earth’s radius and h is height above surface.

地表上方的 g 值随高度增加而减小：g(h) = GM / (R + h)²，其中 R 为地球半径，h 为离地表高度。

Below the surface, assuming uniform density, the mass contributing to gravity is only the mass inside the radius r. This gives g(r) = (4/3) π G ρ r, so g decreases linearly to zero at the centre.

在地表以下，假设均匀球体密度，有效引力质量仅限于半径 r 以内的球体。此时 g(r) = (4/3) π G ρ r，g 值线性减小，在地心处为零。

Exam questions often ask you to sketch the graph of g against distance from Earth’s centre, showing a linear increase from centre to surface (if uniform density is assumed) and an inverse-square decay beyond.

考题常要求画出 g 随距地心距离变化的图像，显示从地心到地表线性增加(假设均匀密度)，地表之外按平方反比衰减。

11. Escape Velocity | 逃逸速度

Escape velocity is the minimum speed needed for an object to escape a planet’s gravitational field without further propulsion, reaching infinity with zero speed.

逃逸速度是物体无需进一步推进、恰好能在无穷远处速度减为零时脱离行星引力场所需的最小速率。

Setting total energy at surface to zero: ½ m v_esc² – GMm / R = 0, so v_esc = √(2GM / R).

令表面处总能量为零：½ m v_esc² – GMm / R = 0，得 v_esc = √(2GM / R)。

Escape velocity depends only on the mass and radius of the planet, not on the object’s mass. For Earth, v_esc ≈ 11.2 km s⁻¹.

逃逸速度只取决于行星的质量和半径，与物体质量无关。对于地球，v_esc 约等于 11.2 km s⁻¹。

A black hole is a body where the escape velocity exceeds the speed of light, so nothing, not even light, can escape from within the event horizon.

黑洞是逃逸速度超过光速的天体，因此在事件视界内任何物质甚至光都无法逃逸。

12. Satellites: Geostationary vs Polar | 卫星：地球同步轨道与极轨道

A geostationary satellite orbits in Earth’s equatorial plane with a period of 24 hours, appearing fixed in the sky to a ground observer. Its orbital radius is fixed by Kepler’s third law.

地球同步卫星在地球赤道平面内运行，周期为24小时，对地面观测者来说悬停在天空固定位置。其轨道半径由开普勒第三定律唯一确定。

Polar satellites orbit at low altitudes, passing over the poles, allowing Earth’s entire surface to be scanned due to Earth’s rotation underneath. Their period is typically about 90–100 minutes.

极轨卫星在低高度轨道运行，飞越两极，借助地球自转可扫描整个地表。周期通常约为90-100分钟。

Applications: Geostationary satellites are used for communication and weather monitoring; polar satellites are used for Earth observation, environmental monitoring, and spy satellites.

应用：地球同步卫星用于通信和气象监测；极轨卫星用于地球观测、环境监测和侦察卫星。

Calculating orbital radius for a geostationary orbit is a classic IB problem combining Newton’s law and circular motion.

计算地球同步轨道半径是结合牛顿定律与圆周运动的经典 IB 题目。

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IB Physics: Gravitation Key Points | IB 物理：万有引力 考点精讲