📚 Gravitational Field Strength g | 引力场强度g
The concept of gravitational field strength is fundamental to understanding gravitational interactions in physics. It bridges Newton's law of universal gravitation and the local acceleration due to gravity, g, on Earth and other celestial bodies. For A-Level Physics students, mastering the definition, formula, and applications of g is essential for explaining satellite motion, weight, and gravitational potential.
引力场强度的概念是理解物理学中引力相互作用的基础。它连接了牛顿万有引力定律与地球及其他天体上由重力引起的局部加速度 g。对于 A-Level 物理学生而言，掌握 g 的定义、公式及其应用，对于解释卫星运动、重量和引力势至关重要。
1. Definition of Gravitational Field Strength | 引力场强度的定义
Gravitational field strength at a point is defined as the gravitational force per unit mass acting on a small test mass placed at that point. It is given by the equation g = F / m, where F is the gravitational force and m is the mass of the test mass. The SI unit is newtons per kilogram (N/kg). Because the force on a mass of 1 kg is numerically equal to the field strength, g is often expressed as an acceleration in m/s², though the two quantities are conceptually different but equivalent in value at the same location.
引力场强度定义为放置在该点的一个小测试质量所受的单位质量引力。它由公式 g = F / m 给出，其中 F 是引力，m 是测试质量的质量。国际单位是牛顿每千克（N/kg）。由于 1 kg 质量所受的力在数值上等于场强，g 也常用加速度单位 m/s² 表示，尽管这两个量在概念上不同，但在同一位置上数值是等价的。
2. Gravitational Field as a Vector | 作为矢量的引力场
Gravitational field strength is a vector quantity, meaning it has both magnitude and direction. The direction of the field at any point is defined as the direction of the gravitational force on a small positive test mass, which is always towards the centre of the source mass. Field lines are drawn as arrows pointing radially inwards for an isolated mass, and the spacing of the lines indicates the strength of the field − closer lines mean a stronger field.
引力场强度是矢量，既有大小也有方向。某点场的方向定义为作用在一个小的正测试质量上的引力方向，该方向始终指向源质量的中心。对于一个孤立的质量，场线用指向径向向内的箭头表示，而场线的间距表示场的强弱 − 线越密，场越强。
When drawing field lines around the Earth, they are directed downwards towards the ground and are approximately parallel and equally spaced near the surface, showing a uniform field. However, on a larger scale, the lines converge towards the Earth's centre, revealing the true radial nature of the field.
在绘制地球周围的场线时，它们指向地面且向下，在地表附近大致平行且等间距，显示出均匀场。但在更大尺度上，场线会汇聚于地心，揭示出场真正的径向特性。
3. Field due to a Point Mass | 点质量的引力场
For a point mass M (or a spherical mass outside its surface), the gravitational field strength at a distance r from its centre is given by g = GM / r² ance reduces the field strength to one quarter. The formula is derived by combining Newton's law of gravitation, F = GMm / r², with the definition g = F/m.
对于点质量 M（或球对称质量在其表面之外），距离其中心 r 处的引力场强度由 g = GM / r² 给出，其中 G 是万有引力常数（6.67 × 10⁻¹¹ N m² kg⁻²）。这一平方反比律表明，距离加倍会使场强降至原来的四分之一。该公式由牛顿引力定律 F = GMm / r² 与定义 g = F/m 结合推导而来。
g = GM / r²
value matches the measured free-fall acceleration, demonstrating the consistency of Newtonian gravity.
在地球表面，代入 M = 5.97 × 10²⁴ kg 和 R = 6.37 × 10⁶ m，得到 g ≈ 9.81 N kg⁻¹。该计算值与实测的自由落体加速度相符，证明了牛顿引力理论的一致性。
4. Uniform Gravitational Field Approximation | 均匀引力场的近似
Near the Earth's surface, the change in distance from the Earth's centre is tiny compared to the radius, so g is treated as a constant vector of magnitude 9.81 N kg⁻¹ directed vertically downwards. This is the uniform gravitational field approximation, which greatly simplifies problems involving projectiles, falling objects, and mechanical systems on Earth.
在地表附近，与地球半径相比，到地心距离的变化极小，因此 g 被视为一个恒定的矢量，大小为 9.81 N kg⁻¹，方向竖直向下。这就是均匀引力场近似，它极大地简化了涉及抛体、落体及地球上力学系统的问题。
In this model, field lines are represented by parallel, equally spaced arrows pointing downwards. The acceleration of free fall, g, is assumed to be constant regardless of height differences of a few metres or even hundreds of metres, giving an excellent approximation for most laboratory and everyday calculations.
在该模型中，场线表示为平行、等距且向下的箭头。自由落体加速度 g 被认为在几米甚至几百米的高度差内保持不变，为大多数实验室和日常计算提供了极好的近似。
5. Relation between g and G | g与万有引力常数G的关系
The universal gravitational constant G relates the gravitational force between two masses to their masses and separation, while g is the local manifestation of this universal law at a particular point in space. For a planet of mass M and radius R, surface gravitational field strength is g = GM / R². This shows that g depends on the planet's mass and size, not on the test mass itself.
万有引力常数 G 将两质量间的引力与它们的质量和间距联系起来，而 g 是这一普遍定律在空间特定点上的局部体现。对于质量为 M、半径为 R 的行星，表面引力场强度为 g = GM / R²。这表明 g 取决于行星的质量和大小，而与测试质量本身无关。
10⁻¹¹ N m² kg⁻²), a large mass such as a planet is required to produce a noticeable gravitational field strength. This relationship is fundamental in determining the mass of the Earth and other celestial bodies by measuring g and their radii.
由于 G 极其微小（约 6.67 × 10⁻¹¹ N m² kg⁻²），需要像行星这样的大质量才能产生可察觉的引力场强度。这一关系对于通过测量 g 和半径来确定地球及其他天体的质量至关重要。
6. Weight and Gravitational Force | 重量与引力
The weight W of an object is the gravitational force it experiences due to the gravitational field in which it is located. For an object of mass m, weight is given by W = mg, where g is the gravitational field strength at that location. Weight is a vector directed towards the centre of the gravitational source, while mass is a scalar quantity measured in kilograms.
物体的重量 W 是它所处引力场施加给它的引力。对于质量为 m 的物体，重量由 W = mg 给出，其中 g 是该位置的引力场强度。重量是一个矢量，指向引力源中心，而质量是以千克为单位的标量。
This equation is a specific case of Newton' rtant to distinguish between weight (a force) and mass (an intrinsic property); on the Moon, mass remains the same but weight decreases because g is smaller.
该方程是牛顿第二定律 F = ma 的一个特例，当唯一的作用力是重力且 a = g 时成立。然而，区分重量（力）和质量（内禀属性）很重要；在月球上，质量不变但重量会因 g 较小而减小。
7. Variation of g with Height | g随高度的变化
Above the Earth's surface, the gravitational field strength decreases with the square of the distance from the centre. If g₀ is the field strength at the surface (radius R), then at a height h above the surface the field strength becomes:
在地表上方，引力场强度随到中心距离的平方递减。若 g₀ 为表面（半径 R）处的场强，则在表面以上高度 h 处的场强为：
g = g₀ × (R / (R + h))²
4 ≈ 2.45 N kg⁻¹. At typical aircraft altitudes (h ≈ 10 km), the decrease is only about 0.3%, which is why the uniform field approximation holds well for near-surface motions.
例如，若 h = R（高度约 6 370 km），则 g = g₀ × (1/2)² = g₀/4 ≈ 2.45 N kg⁻¹。在典型飞机高度（h ≈ 10 km），减小幅度仅为约 0.3%，这就是均匀场近似对近地表运动依然适用的原因。
8. Variation of g with Latitude and Rotation | g随纬度与地球自转的变化
The measured value of g varies slightly with latitude due to two main factors: the Earth's rotation and its equatorial bulge. The centrifugal effect from Earth' ausing g to be lower at the equator (≈ 9.78 N kg⁻¹) compared to the poles (≈ 9.83 N kg⁻¹).
测得的 g 值随纬度略有变化，主要由两个因素导致：地球的自转及其赤道隆起。地球自转产生的离心效应减小了赤道处物体所受的净引力，且赤道比两极离地心更远，这两者都使得赤道的 g（约 9.78 N kg⁻¹）小于两极的 g（约 9.83 N kg⁻¹）。
The apparent weight of an object on a rotating planet is given by Wapp at the equator λ = 0°, the reduction is maximum, contributing to the latitude dependence.
在旋转行星上，物体的视重由 Wapp = mg − mω²R cos²λ 给出，其中 λ 是纬度，ω 是角速度。在极点 λ = 90°，修正项消失；而在赤道 λ = 0°，减小量最大，这便形成了 g 对纬度的依赖。
9. Gravitational Potential and Field Strength | 引力势与场强
s, V = −GM / r. The gravitational field strength g is related to the potential gradient by g = − dV / dr. The minus sign indicates that g points in the direction of decreasing potential, i.e. towards the mass.
某点的引力势 V 是将单位质量从无穷远移至该点所做的功。对于点质量，V = −GM / r。引力场强度 g 通过 g = − dV / dr 与势的梯度相关联。负号表明 g 指向势减小的方向，即指向质量。
g = − dV / dr
s the known formula but now explicitly shows the radial direction towards the centre. In regions of uniform field, V decreases linearly with height, giving a constant g.
对 V = −GM / r 求导得 dV/dr = +GM / r²，因此 g = −GM / r²，这与已知公式吻合，且明确显示了径向指向中心的方向。在均匀场区域，V 随高度线性下降，从而得到恒定的 g。
10. Measuring g: Free Fall Experiment | 测量g：自由落体实验
One common laboratory method to determine g involves dropping a steel ball from rest and measuring the time t it takes
Published by TutorHao | A-Level Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图