📚 Gravitational Field Strength g | 引力场强度 g
Gravitational field strength is one of the central ideas in CIE A-Level Physics. It links the concept of a force field to the acceleration experienced by any mass placed in that field. Understanding g as a field quantity, rather than simply as 'gravity', is essential for solving problems involving orbits, satellites, and planetary motion.
引力场强度是 CIE A-Level 物理的核心概念之一。它把力场的概念与置于场中的任何质量所经历的加速度联系起来。将 g 理解为场量，而不仅仅是“重力”，对于解决轨道、卫星和行星运动问题至关重要。
1. Defining Gravitational Field Strength | 引力场强度的定义
Gravitational field strength at a point is defined as the gravitational force exerted per unit mass on a small test mass placed at that point. The test mass is assumed to be small enough that it does not significantly disturb the field being measured.
引力场强度定义为施加在置于该点的小检验质量上的每单位质量的引力。检验质量被认为足够小，以至于不会显著干扰正在测量的场。
g = F / m
In this equation, F is the gravitational force experienced by the test mass m. The SI unit of gravitational field strength is newton per kilogram, which is written as N kg⁻¹. This unit is equivalent to m s⁻², so g can also be interpreted as the acceleration due to gravity at that point.
在该方程中，F 是检验质量 m 所受到的引力。引力场强度的 SI 单位是牛顿每千克，写作 N kg⁻¹。该单位等价于 m s⁻²，因此 g 也可以解释为该点的重力加速度。
At the Earth's surface, the average value of g is approximately 9.81 N kg⁻¹. However, this value is not a universal constant; it depends on location, altitude, and the local distribution of mass.
在地球表面，g 的平均值约为 9.81 N kg⁻¹。但这个值并不是普适常数；它取决于位置、高度和当地的质量分布。
2. Scalar or Vector? Direction of g | 标量还是矢量？g 的方向
Although many exam questions quote g as a magnitude, gravitational field strength is a vector quantity. Its direction at any point is the direction of the gravitational force that would act on a small test mass placed at that point.
尽管许多考试题目引用 g 的大小，但引力场强度是一个矢量。它在任意点的方向是放置在该点的小检验质量所受引力的方向。
For an isolated point mass or a spherical planet, the gravitational field is directed radially inward toward the centre of the mass. This means the field vectors point along lines converging on the source mass.
对于孤立的点质量或球形行星，引力场沿径向指向质量中心。这意味着场矢量沿着汇聚于源质量的直线方向。
g = - (GM / r²) r̂
Here the negative sign shows that the field is opposite to the unit vector r̂, which points away from the source mass. In many CIE problems, only the magnitude GM / r² is needed, but you must state the direction separately when the question asks for a vector answer.
这里的负号表示场与单位矢量 r̂ 相反，r̂ 指向远离源质量的方向。在许多 CIE 问题中，只需要大小 GM / r²，但当题目要求矢量答案时，必须单独说明方向。
3. Field Strength for a Point Mass | 点质量的场强公式
ravitational field strength at a distance r from the centre of the mass is given by Newton' substituting F = GMm / r² into g = F / m, the test mass cancels, giving the useful result:
对于点质量 M，在距离其中心 r 处的引力场强度由牛顿万有引力定律给出。将 F = GMm / r² 代入 g = F / m，检验质量被消去，得到有用的结果：
g = GM / r²
In this expression, G is the universal gravitational constant, M is the source mass, and r is the distance from the centre of the source mass to the point where g is measured. The field strength follows an inverse square law, so doubling the distance reduces g by a factor of four.
在该表达式中，G 是万有引力常量，M 是源质量，r 是从源质量中心到测量 g 的点的距离。场强遵循平方反比定律，因此距离加倍会使 g 减小为原来的四分之一。
This formula assumes the source mass is either a point mass or a sphere with a radially symmetric mass distribution. For points outside the sphere, the entire mass can be treated as if it were concentrated at the centre.
该公式假设源质量是点质量或具有径向对称质量分布的球体。对于球体外部的点，整个质量可以视为集中在球心。
4. Gravitational Field Lines | 引力场线
Gravitational fields can be represented visually by field lines. These are arrows drawn in the direction of the gravitational force on a small test mass. For a point mass or a planet, the field lines are radial and point inward.
引力场可以用场线直观表示。这些箭头指向小检验质量所受引力的方向。对于点质量或行星，场线呈径向并指向内部。
The spacing of field lines indicates the strength of the field: closer lines mean a stronger field. Near the surface of a planet, the field lines are almost parallel and equally spaced, so the field can be approximated as uniform over small distances.
场线的间距表示场的强弱：线越密，场越强。在行星表面附近，场线几乎平行且等距，因此在较小距离内可以将场近似为均匀场。
on everywhere. This approximation is valid for small vertical displacements near the Earth's surface, where g ≈ 9.81 N kg⁻¹.
在均匀引力场中，g 的大小和方向处处相同。这一近似适用于地球表面附近较小的垂直位移，此时 g ≈ 9.81 N kg⁻¹。
5. Relation Between g and Gravitational Potential | g 与引力势的关系
Gravitational field strength is closely linked to gravitational potential V. In a radial field, the field strength is the negative gradient of the potential with respect to distance:
引力场强度与引力势 V 密切相关。在径向场中，场强是势对距离的负梯度：
g = - dV / dr
ng r direction. The negative sign shows that the field points toward decreasing potential.
对于点质量，引力势为 V = - GM / r。对 r 求导得到 dV / dr = GM / r²，因此沿 r 增大方向 g = - GM / r²。负号表明场指向势减小的方向。
n potential is ΔV = g Δh, which connects gravitational potential energy to field strength.
在均匀场中，该关系可以更简单地写作 g = - ΔV / Δr。如果质量在均匀场中移动垂直距离 Δh，势的变化为 ΔV = g Δh，这把引力势能与场强联系起来。
6. Variation of g with Height Above Earth | 地球表面上方 g 随高度的变化
At a height h above the Earth's surface, the distance from the centre of the Earth is r = R + h, where R is the Earth's radius. The gravitational field strength at that height is:
在地球表面上方高度 h 处，到地心的距离为 r = R + h，其中 R 是地球半径。该高度处的引力场强度为：
gh = GME / (R + h)²
This can also be expressed in terms of the surface value g₀ by taking the ratio:
这也可以通过取比值用表面值 g₀ 表示：
gh = g₀ × (R / (R + h))²
For small heights compared with the Earth' l approximation is Δg ≈ - 2g₀ h / R, but this is only valid when h is much smaller than R.
对于远小于地球半径的高度，g 的变化近似为线性。一个有用的近似是 Δg ≈ - 2g₀ h / R，但仅当 h 远小于 R 时才成立。
At an altitude of 100 km, g is only about 3% less than its surface value. This is why the uniform field approximation works well for laboratory-scale and many everyday projectile problems.
在 100 km 的高度，g 仅比表面值小约 3%。这就是为什么均匀场近似在实验室尺度和许多日常抛体问题中效果良好。
7. Variation of g Inside the Earth (Uniform Density Model) | 地球内部 g 的变化（均匀密度模型）
d at that point. For a planet of uniform density ρ, the enclosed mass is M(r) = (4/3)πρr³.
在均匀球形地球内部，只有半径 r 内包围的质量对该点的引力场有贡献。对于密度为 ρ 的均匀行星，包围的质量为 M(r) = (4/3)πρr³。
g = (4/3) π G ρ r
, r = 0, so g = 0. At the surface, r = R, and the formula gives the maximum surface value.
这表明在均匀行星内部，g 随距中心的距离线性增加。在中心处 r = 0，因此 g = 0。在表面处 r = R，公式给出最大的表面值。
The real Earth is not uniform, so the actual variation of g with depth is more complex. However, the uniform density model is a standard CIE approximation for understanding the principle.
真实地球并不是均匀的，因此 g 随深度的实际变化更为复杂。然而，均匀密度模型是 CIE 中理解这一原理的标准近似。
8. Apparent Weight and Free Fall | 视重与自由落体
. Apparent weight, however, is the normal contact force measured by a supporting surface or a spring balance. In free fall, the normal contact force becomes zero, so the apparent weight is zero even though the gravitational force mg is still acting.
重量是作用在物体上的引力，由 W = mg 计算。然而，视重是支撑面或弹簧秤测得的法向接触力。在自由落体中，法向接触力为零，因此视重为零，尽管引力 mg 仍然作用。
This explains why astronauts in orbit experience weightlessness. They are not beyond the reach of gravity; rather, they and their spacecraft are in continuous free fall around the Earth. The local g is still significant, but there is no contact force to provide a sensation of weight.
这解释了为什么轨道上的宇航员会体验到失重。他们并非不受重力影响；相反，他们和航天器一起围绕地球持续自由落体。当地的 g 仍然很大，但没有接触力提供重量的感觉。
9. Comparing g on Different Planets | 不同行星表面 g 的比较
The surface gravitational field strength of a planet depends on its mass and radius. A planet with a larger mass tends to have a larger g, but a larger radius reduces g because the surface is farther from the centre.
行星表面的引力场强度取决于其质量和半径。质量较大的行星往往具有较大的 g，但半径较大会使 g 减小，因为表面离中心更远。
| Body | 天体 | Mass / kg | 质量 / kg | Radius / m | 半径 / m | Surface g / N kg⁻¹ | 表面 g / N kg⁻¹ |
|---|---|---|---|
| Earth | 地球 | 5.97 × 10²⁴ | 6.37 × 10⁶ | 9.81 |
| Moon | 月球 | 7.35 × 10²² | 1.74 × 10⁶ | 1.62 |
| Mars | 火星 | 6.42 × 10²³ | 3.39 × 10⁶ | 3.73 |
/ g₂ = (M₁ / M₂) × (R₂ / R₁)². This avoids the need to calculate large powers separately.
要比较两颗行星，可以使用比值。由 g = GM / R²，表面场强之比为 g₁ / g₂ = (M₁ / M₂) × (R₂ / R₁)²。这避免了分别计算大数的幂。
10. Experimental Determination of g | g 的实验测定
One common CIE experiment to determine g uses free fall. An object is released from rest, and the time t taken to fall a known vertical distance s is measured. Since the acceleration is uniform, the motion follows:
CIE 中测定 g 的一个常见实验使用自由落体。物体从静止释放，测量下落已知垂直距离 s 所需的时间 t。由于加速度均匀，运动遵循：
s = ½ g t²
ice, repeated timings are taken for different heights, and a graph of s against t² is plotted. The gradient of the straight line is ½ g, so g can be found from the gradient.
整理后得到 g = 2s / t²。实际操作中，对不同高度进行多次计时，并绘制 s 对 t² 的图像。直线的斜率是 ½ g，因此可以从斜率求出 g。
her method uses a simple pendulum, where the period T is related to the length L by T = 2π√L / g This method is less dependent on reaction time but requires careful measurement of length.
另一种方法使用单摆，其周期 T 与摆长 L 的关系为 T = 2π√L / g。整理后得到 g = 4π²L / T²。该方法对反应时间的依赖较小，但需要仔细测量长度。
11. Common Misconceptions and Exam Tips | 常见误区与应试提示
the object placed in the field. The force on a mass m is then F = mg, which depends on m.
一个非常常见的错误是混淆引力场强度 g 与引力 F。场强是每单位质量的力，因此它与置于场中的物体质量无关。质量 m 所受的力则为 F = mg，这与 m 有关。
r² rather than the surface value. Check whether the point is near the surface or far away.
另一个常见错误是假设 g 总是 9.81 N kg⁻¹。在涉及卫星或远处点的 CIE 题目中，必须使用 g = GM / r²，而不是表面值。要检查该点是在表面附近还是远处。
When drawing field lines, always include arrows pointing toward the mass. Missing arrow directions is a common cause of lost marks. Also remember that g is a vector, so a full answer to a vector question must include both magnitude and direction.
画场线时，一定要画指向质量的箭头。缺少箭头方向是失分的常见原因。还要记住 g 是矢量，因此矢量题的完整答案必须同时包含大小和方向。
12. Worked Example | 例题
A spacecraft is at an altitude of 200 km above the Earth' of the Earth is 6.37 × 10⁶ m, calculate the gravitational field strength at the spacecraft's position.
一艘航天器位于地球表面上方 200 km 的高度。已知地球质量为 5.97 × 10²⁴ kg，地球半径为 6.37 × 10⁶ m，计算航天器所在位置的引力场强度。
r = R + h = 6.37 × 10⁶ m + 0.20 × 10⁶ m = 6.57 × 10⁶ m
Using g = GM / r² and substituting values gives:
使用 g = GM / r² 并代入数值，得到：
g = (6.67 × 10⁻¹¹ × 5.97 × 10²⁴) / (6.57 × 10⁶)²
This evaluates to approximately 9.22 N kg⁻¹. The field is directed toward the centre of the Earth. Notice that even at 200 km altitude, g has decreased only slightly from the surface value of 9.81 N kg⁻¹.
计算结果约为 9.22 N kg⁻¹。场的方向指向地心。注意，即使在 200 km 高度，g 仅比表面值 9.81 N kg⁻¹ 略有减小。
at this height, you would then multiply by the mass: F = mg = 1200 × 9.22 ≈ 1.11 × 10⁴ N.
如果题目问的是这个高度上质量为 1200 kg 的航天器所受的引力，你应乘以质量：F = mg = 1200 × 9.22 ≈ 1.11 × 10⁴ N。
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