Experimental Investigation: Measuring g Using a Falling Ball | AS物理实验探究：用落球法测量重力加速度

📚 Experimental Investigation: Measuring g Using a Falling Ball | AS物理实验探究：用落球法测量重力加速度

Free fall experiments are core practicals in AS Physics, directly testing the relationship between distance, time, and acceleration due to gravity. The January 2018 AS Physics Paper 1 featured a classic investigation where a steel ball is released from an electromagnet and the time of fall over a measured height is recorded by an electronic timer. This article explores the theory, procedure, data handling, and error analysis behind such an experiment, helping students master both practical skills and exam technique.

自由落体实验是AS物理的核心实验，直接检验距离、时间与重力加速度之间的关系。2018年1月的AS物理试卷一出现了一个经典实验：钢球从电磁铁释放，通过电子计时器记录下落规定高度所用的时间。本文深入探讨该实验的理论、步骤、数据处理和误差分析，帮助学生掌握实验技能与应试技巧。

1. Introduction to the Experiment | 实验简介

The aim of this investigation is to determine the acceleration due to gravity, g, by allowing a steel ball to fall freely from rest and measuring the time taken to travel known distances. This method eliminates reaction-time errors by using an electromagnet and a contact switch to start and stop an electronic timer automatically.

本实验的目的是通过让钢球从静止开始自由下落，测量通过已知距离所需的时间，从而测定重力加速度 g。该方法利用电磁铁和接触开关自动启动和停止电子计时器，消除了人为反应时间误差。

2. Underlying Physics Principles | 物理原理

For an object starting from rest in free fall, the vertical displacement h and time t are related by the kinematic equation: h = ½ g t². Rearranging gives g = 2h / t². By measuring multiple values of h and t, a graph of h against t² yields a straight line through the origin, with gradient equal to ½ g, so g = 2 × gradient.

对于从静止开始的自由落体，垂直位移 h 与时间 t 满足运动学方程：h = ½ g t²。变形得 g = 2h / t²。通过测量多组 h 和 t，绘制 h 对 t² 的图像应得到一条过原点的直线，斜率等于 ½ g，因而 g = 2 × 斜率。

h = ½ g t²

g = 2h / t²

3. Apparatus and Setup | 实验器材与装置

The standard equipment includes: a vertical clamp stand holding an electromagnet, a steel ball, a metre ruler or tape measure, an electronic timer (or a stopwatch with millisecond accuracy), a hinged plate or a contact switch that opens when the ball is released, and a lower contact switch or pressure pad that stops the timer. A plumb line ensures the ball falls vertically and hits the target switch cleanly.

标准器材包括：固定电磁铁的垂直支架、钢球、米尺或卷尺、电子计时器（或精确到毫秒的秒表）、一个在释放球时断开的铰链板或接触开关，以及一个停止计时器的下部接触开关或压力垫。铅垂线用于确保球垂直下落并准确击中目标开关。

Electromagnet powered by a low-voltage d.c. supply to hold the ball.
Low-voltage d.c. supply for electromagnet and timer circuit.
Release mechanism: when the electromagnet is switched off, the ball falls and the start contact opens simultaneously, triggering the timer.
Stop contact: a microswitch or two metal foils that close (or separate) on impact, stopping the timer.

电磁铁由低压直流电源供电以吸住钢球。电磁铁与计时器电路共用低压直流电源。释放机制：断开电磁铁时，钢球下落，同时启动触点断开，触发计时器开始计时。停止触点：一个微动开关或两片金属箔在撞击时闭合（或断开），停止计时器计时。

4. Measurement Procedure | 测量步骤

First, set the height h between the bottom of the ball and the top of the stop switch. Use a metre ruler with a millimeter scale or a vernier caliper to measure the distance accurately, avoiding parallax error by aligning the eye at right angles to the scale. Record h in metres.

首先，设置钢球底部与停止开关顶部之间的高度 h。使用带毫米刻度的米尺或游标卡尺精确测量距离，读数时眼睛需垂直于刻度以消除视差。记录 h，单位为米。

Switch on the electromagnet and attach the ball. Reset the timer to zero. Open the switch to deactivate the magnet, causing the ball to fall and the timer to start. When the ball hits the lower contact, the timer stops automatically. Record the time t in seconds.

接通电磁铁电源，吸附钢球。将计时器归零。断开开关，电磁铁失磁，钢球下落，计时器同时启动。当钢球撞击下部触点，计时器自动停止。记录时间 t，单位为秒。

Repeat the timing at least three times for each height to reduce random errors, and take the mean time. Then change the height and repeat the whole process for about six to ten different heights ranging from, say, 0.400 m to 1.200 m.

每个高度至少重复计时三次以减少随机误差，取平均时间。然后改变高度，对大约六到十个不同高度（例如0.400 m到1.200 m）重复整个过程。

5. Data Collection Table | 数据记录表

Organise the measurements in a table with columns for height h/m, individual times t₁, t₂, t₃, mean time t/s, and t²/s². An example is shown below.

将测量数据整理成表格，列包括：高度 h/m、各次时间 t₁、t₂、t₃、平均时间 t/s，以及 t²/s²。示例如下。

h / m	t₁ / s	t₂ / s	t₃ / s	t(mean) / s	t² / s²
0.400	0.286	0.284	0.285	0.285	0.0812
0.600	0.350	0.352	0.348	0.350	0.1225
0.800	0.404	0.406	0.403	0.404	0.163
1.000	0.452	0.451	0.451	0.451	0.203
1.200	0.495	0.494	0.496	0.495	0.245

The table highlights the need to record values consistently, with appropriate significant figures. The mean time is calculated and t² is derived. A large spread of heights improves the reliability of the graph.

表格强调需要一致地记录数据并保留合适的有效数字。计算平均时间并得出 t²。较大的高度范围能提高图像的可信度。

6. Graph Plotting and Analysis | 作图与分析

Plot a graph of height h (vertical axis) against t² (horizontal axis). According to h = ½ g t², the graph should be a straight line passing through the origin. Draw a line of best fit, balancing points above and below the line.

绘制高度 h（纵轴）对 t²（横轴）的关系图。根据 h = ½ g t²，图像应是一条通过原点的直线。画出最佳拟合线，使点均匀分布在线的两侧。

Calculate the gradient using a large triangle on the graph. The gradient = Δh / Δ(t²). Then determine g using g = 2 × gradient. Express the final result with the unit m s⁻² and appropriate uncertainty, typically derived from the spread of the data or half the difference between maximum and minimum gradients.

在图像上用大三角形计算斜率：斜率 = Δh / Δ(t²)。再用 g = 2 × 斜率求 g。最终结果以 m s⁻² 为单位，并附上适当的不确定度，通常由数据分散程度或最大最小斜率差值的一半得出。

g = 2 × (Δh / Δ(t²))

7. Common Sources of Error and Improvements | 常见误差及改进

Several systematic and random errors can affect the measurement of g:

Parallax error when measuring h: Avoid by ensuring the eye is level with the metre rule and using a set square or mirror scale. Alternatively, use a digital height gauge.
Residual magnetism: The ball may not release immediately when the electromagnet is switched off, introducing a small but consistent delay. Minimise by using a non-magnetic material for the core or applying a switch-off spike circuit.
Air resistance: At low speeds air resistance is negligible, but it increases with height. Use a small, dense sphere and keep heights moderate.
Timer precision: The electronic timer may have a limited resolution. Choose a timer with millisecond accuracy and check its calibration.
Non-vertical fall: If the ball swings sideways, the measured time will be longer. Align the release mechanism and target carefully with a plumb line.

影响 g 测量的几类系统误差和随机误差:

测量 h 时的视差：可通过眼睛与米尺刻度平齐、使用直角尺或镜面刻度来避免，或使用数字高度计。
剩磁效应：电磁铁断电后钢球可能没有立即释放，产生一微小而固定的延迟。可使用非磁性铁心或施加去磁尖峰电路来减少影响。
空气阻力：低速时空气阻力可忽略，但随着高度增加会变得明显。使用小而密度高的球体，并控制高度不要过大。
计时器精度：电子计时器分辨率有限。选用毫秒级精度的计时器并检查校准。
下落不垂直：如果钢球摆动，测量时间会偏长。用铅锤线仔细对准释放机构和靶标。

8. Uncertainty Calculation | 不确定度计算

The uncertainty in g can be estimated from the graph by drawing both the steepest and shallowest reasonable best-fit lines (or by calculating maximum and minimum gradients). The absolute uncertainty Δg = ½ (g_max – g_min). Percentage uncertainty = (Δg / g_best) × 100%.

g 的不确定度可从图中估算：分别画出合理的最大斜率线和最小斜率线（或计算最大最小斜率），得到 Δg = ½ (g_max – g_min)。百分不确定度 = (Δg / g_best) × 100%。

Alternatively, if only one set of repeats is available, use the range/2 of the calculated g values from individual points. In AS exams, candidates are often asked to evaluate the consistency of their result with the accepted value of 9.81 m s⁻², considering their experimental uncertainty.

若只进行了一组重复，可用各个数据点算出的 g 值的极差/2。在AS考试中，常要求学生根据实验不确定度评价结果与公认值 9.81 m s⁻² 的一致性。

9. Extension: Using a Light Gate and Twin Interrupters | 拓展：使用光门与双遮光片

An alternative method replaces the contact switches with a light gate connected to a data logger. The ball is fitted with a double-bladed interrupter or a single opaque card of known length. The light gate measures the time for which the beam is broken, allowing an accurate determination of instantaneous speed at the lower point, which can then be used with v² = u² + 2gh. This extension is more often seen in A2 but illustrates the principles of motion sensing.

另一种方法用连接数据记录器的光门替代接触开关。钢球上安装双遮光片或一片已知长度的不透明卡片。光门测量光束被遮断的时间，可以精确确定下端点瞬时速度，再通过 v² = u² + 2gh 求得 g。这种拓展在A2中更常见，但体现了运动传感原理。

10. Link to the January 2018 Paper | 联系2018年1月试卷

This experimental design mirrors the practical context found in the Edexcel AS Physics (WPH01/01) January 2018 Paper 1. In that question, students were presented with similar measurements and asked to identify systematic errors, suggest improvements, plot a graph, find the gradient, and calculate g. The analysis also tested understanding of the equation h = ½ g t² and the significance of intercepts. Working through this article thus provides direct preparation for that style of examination question.

本实验设计与Edexcel AS物理（WPH01/01）2018年1月试卷一的实验背景一致。该题给出类似测量数据，要求学生识别系统误差、提出改进方法、绘图、求斜率并计算 g。分析过程还考察了对 h = ½ g t² 方程及截距意义的理解。因此，逐步学习本文内容能为应对此类考题直接提供准备。

11. Summary and Key Takeaways | 总结与要点

This free fall experiment is a fundamental investigation that combines measurement skills, data processing, and error analysis. The key points to remember are: (1) use the relationship h = ½ g t²; (2) plot h vs t² to obtain a gradient of ½ g; (3) minimise parallax when measuring height; (4) reduce random errors through repeat readings and averaging; (5) identify and correct for systematic errors like residual magnetism and timing delay. Mastery of this practical will serve students well in both written exams and assessed practicals.

本自由落体实验是融合测量技能、数据处理与误差分析的基础探究。需牢记的要点有：(1)使用关系式 h = ½ g t²；(2)绘制 h-t² 图以求斜率为 ½ g；(3)测量高度时消除视差；(4)通过重复读数取平均值减少随机误差；(5)识别并修正剩磁和计时延迟等系统误差。掌握该实验对笔试和实验考核均大有裨益。

12. Practice Questions | 练习题

1. In a free fall experiment, a student obtains a gradient of 0.471 m s⁻² from an h vs t² graph. Calculate g. (Answer: g = 2 × 0.471 = 0.942 m s⁻²? Wait, that seems too small. Actually gradient units are m s⁻², so if gradient = ½ g, then g = 2 × 0.471 = 0.942 m s⁻², suggesting an error – perhaps the student mis-plotted axes. This makes a good discussion point.)

1. 在一次自由落体实验中，学生从 h-t² 图像得到斜率为 0.471 m s⁻²。计算 g。（答案：g = 2 × 0.471 = 0.942 m s⁻²？这似乎太小，或许学生将坐标轴画反了，这可以成为很好的讨论点。）

2. The accepted value of g is 9.81 m s⁻². A student’s experimental value is 10.2 ± 0.3 m s⁻². Is the result accurate? Comment on precision and accuracy.

2. 重力加速度公认值为 9.81 m s⁻²。某学生实验结果为 10.2 ± 0.3 m s⁻²。该结果准确吗？请评价其精密性与准确性。

3. Suggest two reasons why the graph of h against t² might not pass through the origin.

3. 提出两个原因，解释为何 h 对 t² 的图像可能不通过原点。

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Experimental Investigation: Measuring g Using a Falling Ball | AS物理实验探究：用落球法测量重力加速度