Determining g | 测量重力加速度 g

📚 Determining g | 测量重力加速度 g

In A-Level Physics, determining the acceleration of free fall, g, is a core practical skill. It links kinematics, experimental design, graphing and uncertainty analysis. CIE candidates are expected to describe methods, process data graphically, and evaluate sources of error.

在 A-Level 物理中，测量自由落体加速度 g 是一项核心实验技能。它联系运动学、实验设计、绘图和不确定度分析。CIE 考生需能描述方法、用图像处理数据并评估误差来源。

The accepted standard value of g near the Earth’s surface is about 9.81 m s⁻². However, the local value may differ slightly because of altitude, latitude and the presence of large underground rock formations. The aim of a good experiment is not only to obtain a value close to the accepted one, but also to identify why remaining differences occur.

地球表面附近公认的 g 标准值约为 9.81 m s⁻²。但当地值可能因海拔、纬度和大型地下岩层的存在而略有不同。一个好的实验不仅是为了得到接近公认值的数值，还要找出剩余差异产生的原因。

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1. What Is g and Why Measure It? | 什么是 g，为何要测量？

The acceleration of free fall, g, is the acceleration experienced by an object when gravity is the only force acting on it. In the absence of air resistance, all objects near the same location fall with the same acceleration regardless of their mass.

自由落体加速度 g 是物体仅受重力作用时的加速度。在没有空气阻力的情况下，同一地点附近的所有物体无论质量大小，都以相同的加速度下落。

Determining g is a practical test of your understanding of uniformly accelerated motion. It also develops skills in measuring length and time, reducing systematic errors, plotting linear graphs and propagating uncertainties. These are exactly the skills assessed in CIE Paper 3 and Paper 5.

测量 g 是对你理解匀加速运动的实践检验。它还能培养测量长度和时间、减少系统误差、绘制线性图像和传递不确定度的能力。这些正是 CIE 卷三和卷五所考查的技能。

In CIE practical papers, you may be asked to choose suitable apparatus, explain how to reduce parallax error, justify whether the graph should pass through the origin, and comment on the reliability of the final result.

在 CIE 实验卷中，你可能需要选择合适的仪器、解释如何减少视差、判断图像是否应通过原点，并评价最终结果的可靠性。

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2. Core Kinematic Equations for Free Fall | 自由落体的核心运动学方程

When an object is released from rest and falls a vertical distance h under gravity, the relevant equation of motion is:

物体从静止释放，在重力作用下下落竖直距离 h，所用运动方程为：

h = ½ g t²

This assumes initial velocity u = 0, acceleration a = g downwards, and negligible air resistance. Rearranging gives g = 2h / t², which is the basis of the free fall method.

该式假设初速度 u = 0，向下加速度 a = g，空气阻力可忽略。整理得 g = 2h / t²，这就是自由落体法的依据。

If the object already has an initial downward velocity u, the full equation s = ut + ½at² is used. However, in most school experiments the object is deliberately released from rest to simplify the analysis.

如果物体已有向下的初速度 u，则应使用完整公式 s = ut + ½at²。但在大多数学校实验中，物体被有意从静止释放，以简化分析。

For a falling object passing through two light gates separated by a distance s, the speeds v₁ and v₂ at the two positions are related by:

对于穿过相距 s 的两个光门的落体，两个位置处的速度 v₁ 和 v₂ 满足：

v₂² = v₁² + 2gs

This equation is useful when timing is replaced by direct speed measurement using light gates and a data logger.

当用光门和数据记录器直接测量速度取代计时时，这个方程非常有用。

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3. Method 1: Electromagnet and Trapdoor | 方法一：电磁铁与落板开关

This is one of the most common CIE practical methods. A steel ball is held by an electromagnet. When the current is switched off, the ball falls and a timer starts. The ball hits a trapdoor switch at a measured distance h below, and the timer stops, giving the fall time t.

这是 CIE 实验中最常见的方法之一。钢球被电磁铁吸住。切断电流后，球下落，计时器启动。球击中下方已知距离 h 处的落板开关，计时器停止，给出下落时间 t。

The distance h should be measured from the bottom of the ball to the top of the trapdoor switch. This reference point must be kept the same for every trial, otherwise a systematic error will be introduced.

距离 h 应从球的底部量到落板开关的顶部。每次试验必须保持参考点一致，否则会引入系统误差。

For each height h, the time t should be measured at least three times and averaged. Changing h in regular steps, for example 0.20 m, 0.40 m, 0.60 m, 0.80 m and 1.00 m, gives enough data for a reliable graph.

每个高度 h 的时间 t 应至少测量三次并取平均值。按规律改变 h，例如 0.20 m、0.40 m、0.60 m、0.80 m 和 1.00 m，可获得足够的数据绘制可靠的图像。

Using h = ½ g t², a graph of h against t² should be a straight line through the origin with gradient equal to g/2. Therefore g = 2 × gradient.

由 h = ½ g t²，以 h 为纵轴、t² 为横轴作图，应得到过原点的直线，斜率等于 g/2。因此 g = 2 × 斜率。

If the best-fit line does not pass through the origin, it may indicate a systematic error such as a timing delay or an incorrect zero setting for h. This is a valuable discussion point for CIE evaluation questions.

如果最佳拟合直线不通过原点，可能表明存在系统误差，例如计时延迟或 h 的零点设置不正确。这是 CIE 评价题中一个有价值的讨论点。

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4. Method 2: Light Gates and Data Logging | 方法二：光门与数据记录

A more modern method uses light gates connected to a data logger or timer. A card of known length L is attached to the falling object. As the card passes through a light gate, the timer records the interruption time Δt. The speed v is calculated from v = L / Δt.

更现代的方法使用连接数据记录器或计时器的光门。落体上附有已知长度 L 的挡光片。当挡光片通过光门时，计时器记录遮光时间 Δt。速度 v 由 v = L / Δt 计算。

To determine g, two light gates are placed a vertical distance s apart. The speeds v₁ and v₂ at the two gates are measured. Using v₂² = v₁² + 2gs gives:

为测定 g，将两个光门

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