📚 Gravitational Field Strength g: Definition and Measurement | 引力场强度g：定义与测量
In A-Level physics, the gravitational field is one of the most important ideas in mechanics. The symbol g appears in equations for weight, projectile motion, circular motion and orbital motion. This article defines gravitational field strength and explores practical ways to measure it in the laboratory.
在 A-Level 物理中，引力场是力学中最重要的概念之一。符号 g 出现在重力、抛体运动、圆周运动和轨道运动的方程中。本文将定义引力场强度，并探讨在实验室中测量它的实际方法。
1. What Is a Gravitational Field? | 什么是引力场？
A gravitational field is a region of space in which a mass experiences a gravitational force. Every object with mass creates a gravitational field around itself.
引力场是指空间中任何有质量的物体都会受到引力作用的区域。每个有质量的物体都会在自身周围产生引力场。
The field is described using field lines. Around a point mass or a sphere, the lines point radially inwards towards the centre. Near the surface of the Earth, the field lines are very nearly parallel and equally spaced, so the field is treated as uniform over small distances.
引力场可以用场线来描述。在点质量或球体周围，场线沿径向指向中心。在地球表面附近，场线几乎平行且等距，因此在较小范围内可把引力场视为均匀场。
A field is a vector quantity. To describe it completely, we must state both its magnitude and its direction.
场是矢量，必须同时说明它的大小和方向，才能完整地描述它。
2. Defining Gravitational Field Strength | 引力场强度的定义
Gravitational field strength g at a point is defined as the force per unit mass acting on a small point mass placed at that point.
引力场强度 g 在一点的定义为：放在该点的一个小质点所受到的力与它的质量之比。
This definition can be written as:
g = F / m
Here, F is the gravitational force in newtons, m is the test mass in kilograms, and g is the gravitational field strength in newtons per kilogram.
其中，F 是引力，单位是牛顿；m 是试探质量，单位是千克；g 是引力场强度，单位是牛顿每千克。
The unit N kg⁻¹ is equivalent to m s⁻². This tells us that gravitational field strength has the same unit as acceleration. Indeed, for a freely falling object with no air resistance, the gravitational field strength is equal to the acceleration of free fall.
单位 N kg⁻¹ 与 m s⁻² 等价。这说明引力场强度与加速度具有相同的单位。事实上，对于没有空气阻力的自由落体物体，引力场强度等于自由落体加速度。
Near the Earth’s surface, the accepted value is approximately g = 9.81 N kg⁻¹ or 9.81 m s⁻². The direction of g is towards the centre of the Earth.
在地球表面附近，公认值约为 g = 9.81 N kg⁻¹，即 9.81 m s⁻²。g 的方向指向地球中心。
3. Point Masses and Spherical Bodies | 点质量与球形天体
Newton’s law of gravitation states that the force between two point masses m₁ and m₂ separated by distance r is:
牛顿万有引力定律指出，两个点质量 m₁ 和 m₂ 相距 r 时，它们之间的引力为：
F = G m₁ m₂ / r²
Here G is the gravitational constant, which is a universal constant with the value 6.674 × 10⁻¹¹ N m² kg⁻².
其中 G 是引力常量，是普适常量，数值为 6.674 × 10⁻¹¹ N m² kg⁻²。
For a mass M that creates the field, and a small test mass m at distance r from its centre:
对于产生引力场的质量 M 和距离其中心 r 处的小试探质量 m：
g = F / m = G M / r²
This equation applies exactly for a point mass. By a theorem first proved by Newton, it also applies outside a uniform spherical shell or a spherically symmetric sphere: the sphere behaves as if all its mass were concentrated at its centre.
这个方程严格适用于点质量。根据牛顿最早证明的定理，它也适用于均匀球壳或球对称球体的外部：球体可视为全部质量集中在球心。
Therefore, at the surface of a spherical planet of mass M and radius R, we can write:
因此，在质量为 M、半径为 R 的球形行星表面，可以写出：
g = G M / R²
This formula is extremely useful because it links the local measurable quantity g to the planet’s mass and radius.
这个公式非常有用，因为它把局地可测量的 g 与行星的质量和半径联系了起来。
4. Relationship with Weight and Mass | 与重力和质量的关系
The weight of an object is the gravitational force acting on it. For an object of mass m in a gravitational field of strength g:
物体的重力是作用在它上面的引力。对于质量为 m 的物体，处在引力场强度为 g 的场中时：
W = m g
Mass is an intrinsic property of an object; it does not change when the object is moved to another planet. Weight is a force; it changes when g changes.
质量是物体本身的性质，不随物体移动到另一颗行星而改变。重力是一种力，会随着 g 的变化而变化。
On the Moon, an astronaut with a mass of 80 kg still has a mass of 80 kg, but their weight is much smaller because the Moon’s gravitational field strength is only about 1.62 N kg⁻¹.
在月球上，一个质量为 80 kg 的宇航员质量仍然是 80 kg，但宇航员的重力要小得多，因为月球表面的引力场强度只有约 1.62 N kg⁻¹。
This is why balances that measure weight directly must be recalibrated if they are used in a different gravitational field.
这就是为什么直接测量重力的秤在不同引力场中使用时必须重新校准。
5. g and G: Not the Same | g 和 G 不是同一个量
Students often confuse g with G. They are completely different quantities.
学生经常把 g 和 G 混淆。它们是完全不同的物理量。
g is the gravitational field strength or free-fall acceleration. It depends on the mass of the source and the distance from the source. On Earth, g ≈ 9.81 N kg⁻¹.
g 是引力场强度或自由落体加速度。它取决于源物体的质量和到源物体的距离。在地球上，g ≈ 9.81 N kg⁻¹。
G is the universal gravitational constant. It is fundamental to Newton’s law of gravitation and has the same value everywhere in the universe.
G 是万有引力常量，是牛顿万有引力定律中的基本常量，在整个宇宙中处处相同。
From g = GM/r², we can see that g is derived from G together with a particular source mass and distance. In qualitative terms, G is “how strong gravity is between masses”, while g is “the acceleration produced by a specific mass at a specific location”.
从 g = GM/r² 可以看出，g 是由 G 与特定的源质量和距离共同决定的。通俗地说，G 描述的是质量之间引力的强弱，而 g 描述的是某个特定质量在特定位置所产生的加速度。
6. Measuring g by Free Fall | 自由落体法测量 g
One of the simplest practical methods is to measure the time taken for an object to fall a measured distance.
最简单的实验方法之一是测量物体下落一段已知距离所用的时间。
A steel ball is released from rest, and a timer is started at the same moment. When the ball passes through a light gate placed at distance s below the release point, the timer stops.
让钢球由静止释放，同时启动计时器。当钢球经过释放点下方距离 s 处的光电门时，计时器停止。
For acceleration g from rest, the distance fallen in time t is:
对于从静止开始、加速度为 g 的运动，时间 t 内下落距离为：
s = ½ g t²
Rearranging gives:
整理可得：
g = 2s / t²
So g is found by measuring s and t. Modern experiments usually use electromagnetic release and light gates to avoid reaction-time errors.
因此测量 s 和 t 即可求出 g。现代实验通常使用电磁铁释放和光电门来避免反应时间误差。
Alternatively, two light gates can measure the speed of the ball at two heights. From v² = u² + 2as, and using u = 0, the acceleration can be found from the difference of the squares of the speeds.
另一种方法是使用两个光电门测量钢球在两个高度处的速度。由 v² = u² + 2as，令 u = 0，可通过速度平方之差求出加速度。
7. Measuring g with a Simple Pendulum | 单摆法测量 g
The simple pendulum is a classic laboratory method for determining g. For small oscillations, the period T of a pendulum depends on the length l and the gravitational field strength g:
单摆是测定 g 的经典实验方法。在小角度摆动下，单摆周期 T 取决于摆长 l 和引力场强度 g：
T = 2π √(l / g)
This equation is valid only when the angular amplitude is small, usually less than 10 degrees.
该方程只在小振幅条件下成立，通常要求摆角小于 10 度。
In the experiment, the length l is measured from the point of suspension to the centre of the bob. The time for N oscillations is measured with a stopwatch, and the period is calculated as T = total time / N.
实验中，摆长 l 是从悬点到摆球中心的距离。用秒表测出 N 次全振动的时间，周期 T 等于总时间除以 N。
Rearranging the period equation gives:
将周期公式变形可得：
g = 4π² l / T²
For example, if l = 0.505 m and T = 1.42 s, then g is approximately 9.88 m s⁻². This agrees well with the standard value of 9.81 m s⁻².
例如，若 l = 0.505 m，T = 1.42 s，则 g 约为 9.88 m s⁻²，与标准值 9.81 m s⁻² 符合得很好。
8. Measuring g with a Spring Balance | 弹簧测力计法测量 g
A static method uses a calibrated spring balance. A known mass m is suspended from the spring and allowed to come to rest.
一种静力测量方法是使用校准过的弹簧测力计。把已知质量 m 挂在弹簧下，让系统静止。
In equilibrium, the upward spring force is equal to the downward weight. If the spring balance reads F, then:
在平衡状态下，向上的弹簧力等于向下的重力。如果弹簧测力计读数为 F，则：
F = m g
Therefore:
因此：
g = F / m
This method is straightforward but not as precise as timing methods. The reading of the spring balance must be taken when the system is completely stationary, and the spring must be accurately calibrated.
这种方法简单直接，但精度不如计时类方法。读数时系统必须完全静止，而且弹簧必须经过准确校准。
One important practical detail is that the spring itself has mass. This can cause a small systematic error because the upper parts of the spring stretch slightly differently from the lower parts.
一个重要的实验细节是弹簧本身有质量。这会带来较小的系统误差，因为弹簧上部的伸长与下部略有不同。
9. Experimental Errors and Improvements | 实验误差与改进
In timing methods such as the pendulum, the most common random error is human reaction time when starting and stopping the stopwatch.
在单摆等计时方法中，最常见的随机误差是启动和停止秒表时的人为反应时间。
This error is reduced by measuring the time for many oscillations, for example 20 or 30 oscillations, rather than timing a single swing.
可以通过测量多次全振动的时间来减小这种误差，例如连续测量 20 次或 30 次，而不是只测量单次摆动的时间。
In the free-fall method, air resistance can slow the ball. Using a dense steel ball reduces this systematic error because the acceleration due to air resistance is smaller for a larger mass.
在自由落体法中，空气阻力会使球减速。使用密度较大的钢球可以减小这种系统误差，因为质量越大，空气阻力引起的加速度影响越小。
Parallax errors can occur when reading a metre ruler. They can be reduced by placing the ruler
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