IB Physics: Newton's Law of Gravitation and Its Applications | IB物理：万有引力定律及应用

📚 IB Physics: Newton’s Law of Gravitation and Its Applications | IB物理：万有引力定律及应用

Gravitation is a fundamental interaction that governs the motion of planets, satellites, and falling apples alike. In the IB Physics syllabus, Newton’s law of universal gravitation serves as the cornerstone for understanding orbital mechanics, gravitational fields, and energy conservation in space.

万有引力是一种基本的相互作用，它同时支配着行星、卫星和苹果下落的运动。在IB物理课程中，牛顿万有引力定律是理解轨道力学、引力场和空间能量守恒的基石。

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

Newton’s law states that every point mass attracts every other point mass with a force that is 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的力，该力的大小与两质点质量的乘积成正比，与它们质心之间距离的平方成反比。

F = G m₁m₂ / r²

Here G is the gravitational constant, G = 6.67 × 10⁻¹¹ N m² kg⁻², m₁ and m₂ are the masses, and r is the centre-to-centre distance.

其中G为引力常量，G = 6.67 × 10⁻¹¹ N·m²·kg⁻²，m₁和m₂为质量，r为两质点质心间的距离。

The force is always attractive, acting along the line joining the two masses.
力始终为吸引力，作用方向沿两质量连线。
For extended spherical objects, r is measured from the centre of one sphere to the centre of the other.
对于球对称的物体，r从一个球心量到另一个球心。
The inverse-square relation means that doubling the distance reduces the force to one quarter.
平方反比关系意味着距离加倍时，力减小为原来的四分之一。

2. Gravitational Field Strength | 引力场强度

A gravitational field is a region of space where a mass experiences a gravitational force. The field strength g is defined as the force per unit mass placed in the field.

引力场是空间中质量会受到引力作用的区域。引力场强度g定义为置于场中单位质量所受到的引力。

g = F / m = GM / r²

This equation shows that the field strength at a distance r from the centre of a mass M depends only on M and r, not on the test mass. Near the Earth’s surface, g ≈ 9.8 N kg⁻¹.

该方程表明，距离质量M中心r处的场强只取决于M和r，与试探质量无关。在地球表面附近，g ≈ 9.8 N·kg⁻¹。

g is a vector quantity, directed toward the centre of the attracting mass.
g为矢量，方向指向吸引质量中心。
At the surface of a spherical mass, g = GM/R², where R is the radius of the mass.
在球形质量表面，g = GM/R²，其中R为质量半径。
Inside a uniform spherical shell, the field is zero (shell theorem).
在均匀球壳内部，场强为零（球壳定理）。

3. Gravitational Potential Energy | 引力势能

In a uniform gravitational field near the Earth’s surface, gravitational potential energy is given by U = mgh. However, for large distances, the field is not uniform, and a more general expression is required.

在地球表面附近的均匀引力场中，引力势能表示为U = mgh。但在距离很大时，场不再均匀，需要更一般的表达式。

U = -GMm / r

The negative sign indicates that the gravitational potential energy is zero at infinity and becomes more negative as the masses get closer. This means work must be done against the gravitational force to separate the masses.

负号表示引力势能以无穷远处为零，随着质量间距离减小而变得更负。这意味着要使两质量分离，必须克服引力做功。

The reference point for gravitational potential energy is taken at r → ∞.
引力势能的参考点取在r → ∞处。
Bounded systems have negative total energy, while unbounded systems have positive or zero energy.
束缚系统的总能量为负，非束缚系统的总能量为正或为零。

4. Gravitational Potential | 引力势

Gravitational potential V is the gravitational potential energy per unit mass at a point in a field.

引力势V是引力场中某一点处单位质量的引力势能。

V = -GM / r

The unit of gravitational potential is J kg⁻¹. It is a scalar quantity, and the potential difference between two points equals the work done per unit mass in moving between them.

引力势的单位为J·kg⁻¹。它是标量，两点之间的势差等于单位质量在两点间移动时所做的功。

Quantity	Expression	Unit
Gravitational potential V	V = -GM/r	J kg⁻¹
Gravitational potential energy U	U = mV = -GMm/r	J
Field strength g	g = -dV/dr	N kg⁻¹

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

Johannes Kepler derived three empirical laws describing planetary motion around the Sun.

开普勒在观测数据的基础上总结出了描述行星绕太阳运动的三条经验定律。

First law: Planets move in elliptical orbits with the Sun at one focus.
第一定律：行星沿椭圆轨道运行，太阳位于椭圆的一个焦点上。
Second law: A line joining a planet and the Sun sweeps out equal areas in equal intervals of time.
第二定律：行星与太阳的连线在相等时间内扫过相等的面积。
Third law: The square of the orbital period is proportional to the cube of the semi-major axis.
第三定律：轨道周期的平方与半长轴的立方成正比。

T² ∝ a³

For an approximately circular orbit, a ≈ r, and Kepler’s third law can be derived from Newtonian mechanics together with the law of gravitation.

对于近似圆形的轨道，a ≈ r，开普勒第三定律可由牛顿力学与万有引力定律共同推导出来。

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

A satellite in a circular orbit experiences a centripetal force provided by the gravitational attraction of the central body.

在圆形轨道上运行的卫星，其向心力由中心天体的引力提供。

GMm / r² = mv² / r

Cancelling m and solving for v gives the orbital speed at radius r.

消去m并解出v，可得半径为r处的轨道速度。

v = √(GM / r)

This shows that orbital speed decreases with increasing orbital radius. A higher orbit moves more slowly than a lower orbit.

这表明轨道速度随轨道半径增大而减小。高轨道的运动速度比低轨道更慢。

7. Orbital Speed and Period | 轨道速度与周期

The orbital period T is the time taken for one complete revolution. Combining v = 2πr/T with v = √(GM/r) yields a direct relation between T and r.

轨道周期T是完成一整圈公转所需的时间。将v = 2πr/T与v = √(GM/r)结合，可得到T与r的直接关系。

T² = (4π² / GM) r³

This is a powerful form of Kepler’s third law. It allows astronomers to determine the mass of a central object by measuring the orbit of a satellite around it.

这是开普勒第三定律的有力形式。天文学家可以通过测量周围卫星的轨道来确定中心天体的质量。

Orbit altitude	r from Earth’s centre	Orbital speed	Period
Low Earth orbit (200 km)	6.57 × 10⁶ m	7.8 km s⁻¹	≈ 88 minutes
Geostationary orbit (35,800 km)	4.22 × 10⁷ m	3.1 km s⁻¹	24 hours

8. Escape Velocity | 逃逸速度

Escape velocity is the minimum speed required for an object to escape a gravitational field and never return, reaching zero velocity at infinity.

逃逸速度是物体克服引力场永不返回所需的最小速度，它在无穷远处速度为零。

½mv² = GMm / R

Solving for v gives the escape speed from the surface of a mass M with radius R.

解出v即得到从质量为M、半径为R的质量表面逃逸的速度。

vₑₛ꜀ = √(2GM / R)

For the Earth, vₑₛ꜀ ≈ 11.2 km s⁻¹. Note that escape velocity is independent of the mass of the escaping object.

地球的逃逸速度约为11.2 km·s⁻¹。注意逃逸速度与逃逸物体的质量无关。

9. Energy in Orbits | 轨道能量

A satellite in a circular orbit has both kinetic energy K and gravitational potential energy U. The total mechanical energy is the sum of the two.

在圆轨道上运行的卫星同时具有动能K和引力势能U。机械能总量为两者之和。

E = K + U = ½mv² – GMm / r

Using v² = GM/r, we find:

利用v² = GM/r，可得：

E = -GMm / 2r

The total energy is negative, confirming that the satellite is bound to the central mass.
总能量为负，确认卫星被束缚在中心质量附近。
As r increases, E becomes less negative, meaning the total energy increases.
随着r增大，E的负值减小，即总能量增大。
To move to a higher orbit, a satellite must gain energy from its engines.
要进入更高的轨道，卫星必须从发动机获得能量。

10. Geostationary Satellites | 地球同步卫星

A geostationary satellite orbits above the equator with a period equal to the Earth’s rotational period (24 hours). It remains fixed relative to a point on the Earth’s surface.

地球同步卫星在赤道正上方运行，其周期等于地球自转周期（24小时）。它相对于地球表面上的某一点保持静止。

r³ = GMT² / 4π²

Orbital radius r ≈ 4.22 × 10⁷ m, about 35,800 km above the ground.
轨道半径r ≈ 4.22 × 10⁷ m，即距地面约35,800 km。
The orbit must lie in the equatorial plane.
轨道必须位于赤道平面内。
The orbital speed is approximately 3.1 km s⁻¹.
轨道速度约为3.1 km·s⁻¹。
Geostationary satellites are used for telecommunications, weather monitoring, and broadcasting.
同步卫星用于通信、气象监测和广播。

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

Astronauts in orbit experience apparent weightlessness, even though gravity is still significant at their altitude. This occurs because both the astronauts and their spacecraft are in free fall around the Earth.

轨道上的宇航员会经历表观失重状态，尽管他们所在高度引力仍然显著。这是因为宇航员和飞船都在围绕地球自由下落。

N = mg – mv² / r

In orbit, the centripetal acceleration v²/r equals g at that altitude, so the normal reaction force N becomes zero. The condition for apparent weightlessness is therefore:

在轨道上，向心加速度v²/r等于该高度处的g，因此支持力N变为零。表观失重的条件是：

v² / r = g

Gravity is still present; astronauts are not beyond the Earth’s gravitational field.
引力仍然存在；宇航员并未脱离地球引力场。
True weightlessness would occur only at infinite distance or in the absence of gravitational fields.
真正的完全失重只会在无穷远处或不存在引力场的区域出现。

12. Limitations and General Relativity | 局限性及广义相对论

Newton’s law of gravitation is remarkably accurate for most everyday and astronomical situations, but it fails under extreme conditions such as strong gravitational fields or motions near the speed of light.

牛顿万有引力定律在大多数日常和天文场景中非常精确，但在强引力场或接近光速运动的极端条件下会失效。

Einstein’s general relativity describes gravity as the curvature of spacetime, correctly explaining phenomena such as gravitational lensing, the precession of Mercury’s orbit, and black holes.

爱因斯坦的广义相对论把引力描述为时空的弯曲，能够正确解释引力透镜、水星近日点进动以及黑洞等现象。

For IB Physics, Newton’s formulation remains the required model for solving orbital and gravitational problems.
在IB物理中，牛顿表述仍是解决轨道和引力问题所要求的模型。
General relativity is a conceptual extension that appears in the optional relativity topic.
广义相对论在相对论选修专题中作为概念性扩展出现。

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IB Physics: Newton’s Law of Gravitation and Its Applications | IB物理：万有引力定律及应用