📚 AS Unit 1 Insert Jan 19: Simple Pendulum Experiment and Determination of g | AS物理单元1 2019年1月插页：单摆实验与g的测定
The January 2019 AS Physics Unit 1 insert provided a real-life experimental context for candidates to demonstrate their understanding of practical physics. It centred around a simple pendulum investigation, asking students to determine the acceleration of free fall, g, by analysing the relationship between the period of a pendulum and its length. This article revisits that investigation in depth, highlighting the key principles, measurement techniques, data analysis, and the error considerations expected of an AS‑Level physicist.
2019年1月AS物理单元1的插页为考生提供了一个真实的实验情境，考查他们运用实验物理知识的能力。该插页以单摆实验为核心，要求学生通过分析单摆周期与摆长的关系来测定自由落体加速度 g。本文深入回顾这一探究过程，重点阐述核心原理、测量技巧、数据分析方法以及一名AS阶段物理学者需要关注的误差考量。
1. Aim of the Investigation | 探究目的
The primary aim of this practical investigation was to determine the local acceleration of free fall, g, using a simple pendulum. By measuring the periodic time for small‑amplitude oscillations at different lengths, one can plot a linearised graph and extract g from its gradient. This classic experiment also develops essential skills in controlling variables, estimating uncertainties, and evaluating experimental procedures.
本次实验探究的主要目的是用一个单摆测定当地的重力加速度 g。通过测量不同摆长下小角度摆动的周期，可以绘制一条线性化图线，并从其斜率中求出 g 值。这个经典实验还能培养控制变量、估算不确定度以及评估实验步骤的关键能力。
2. Background Theory: Simple Harmonic Motion and the Pendulum | 背景理论：简谐运动与单摆
For a simple pendulum whose angular amplitude is small (typically less than about 10°), the motion approximates simple harmonic motion. The theoretical period T is given by the equation: T = 2π √(L/g), where L is the length from the point of suspension to the centre of mass of the bob, and g is the acceleration due to gravity. Squaring both sides yields a linear form: T² = (4π²/g) × L. A graph of T² against L therefore has a gradient equal to 4π²/g, from which g can be calculated.
当单摆的角振幅很小（通常小于约10°）时，其运动可近似为简谐运动。理论周期 T 由公式 T = 2π √(L/g) 给出，其中 L 是从悬挂点到摆球质心的长度，g 是重力加速度。两边平方后得到一个线性关系：T² = (4π²/g) × L。因此 T² 对 L 的图线斜率等于 4π²/g，由此可以计算出 g 值。
3. Equipment Described in the Insert | 插页中所描述的设备
The January 2019 insert depicted a typical laboratory arrangement. It included a small metal bob suspended from a clamp stand by a light, inextensible string. The vertical scale (a metre rule or similar height scale) allowed the pendulum length to be measured directly. A protractor was also provided to guide the release angle, ensuring the amplitude remained small enough to minimise the dependence of period on amplitude. A stopwatch was not shown in the insert but was assumed to be available for timing oscillations.
2019年1月的插页展示了一个典型的实验室装置。它包括一个通过轻质且不可伸长的细绳悬挂在支架上的小金属摆球。插页中的竖直标尺（类似于米尺或高度标尺）可以直接测量摆长。同时提供了一个量角器，用于指导释放角度，确保振幅足够小，从而尽量减小周期对振幅的依赖。插页中虽未显示秒表，但假定它是可供计时的。
4. Setting Up the Experiment | 实验装置的搭建
First, the string was securely attached to the clamp stand so that the suspension point was fixed. The length L was measured as the distance from the point of suspension to the centre of the bob. Since the insert ruler had a scale starting from the bench top, the actual length was obtained by subtracting the reading at the top of the string from the reading at the centre of the bob. The protractor was positioned near the point of release to confirm the small initial angle, typically 5°–10°.
首先，细绳被牢固地系在支架上，确保悬挂点固定不变。摆长 L 定义为从悬挂点到摆球质心的距离。由于插页中的标尺从桌面开始读数，实际长度可通过将摆球中心处的读数减去细绳顶端处的读数获得。量角器放置于释放点附近，用于确认初始角度较小，通常为5°至10°。
5. Carrying Out the Measurements | 进行测量
The bob was displaced to the chosen small angle and released smoothly without imparting any sideways force. The timing commenced as the bob passed its equilibrium position at the centre of the swing, to reduce reaction‑time errors. Instead of measuring a single oscillation, the time for a set of complete swings – often 10 or 20 – was recorded. Dividing the total time by the number of oscillations gave a more accurate value for the period T, greatly reducing the effect of human reaction time.
将摆球拉至选定的小角度，并平稳释放，不施加任何侧向力。计时从摆球经过平衡位置（摆动中心）时开始，以减小反应时间误差。实验中不是测量单次摆动，而是记录若干次（通常是10次或20次）完整摆动所用的时间。将总时间除以摆动次数便得到更为准确的周期 T，从而大大降低人为反应时间的影响。
6. Recording Data Table | 数据记录表
A sample data table from such an investigation might appear as shown below. The length L was varied systematically, and the corresponding timing for 10 oscillations was recorded twice to check consistency. The period T and its square T² were then calculated.
此类探究的一份样表可能如下所示。实验中系统性地改变摆长 L，并记录10次摆动的对应时间，重复两次以检验一致性，然后计算出周期 T 及其平方 T²。
| L / m | t₁ (10T) / s | t₂ (10T) / s | Mean t / s | T / s | T² / s² |
|---|---|---|---|---|---|
| 0.200 | 9.03 | 8.97 | 9.00 | 0.900 | 0.810 |
| 0.400 | 12.70 | 12.66 | 12.68 | 1.268 | 1.608 |
| 0.600 | 15.55 | 15.53 | 15.54 | 1.554 | 2.415 |
| 0.800 | 17.94 | 17.98 | 17.96 | 1.796 | 3.226 |
| 1.000 | 20.04 | 20.08 | 20.06 | 2.006 | 4.024 |
The small differences between t₁ and t₂ indicate good repeatability. Each mean period was derived by first averaging the two time readings and then dividing by 10.
t₁ 与 t₂ 之间微小的差异表明实验具有良好的重复性。每个平均周期是先求两次读数的平均值，再除以10得出的。
7. Plotting T² against L | 绘制 T² 与 L 的关系图
Using the processed data, a graph of T² (vertical axis) against L (horizontal axis) was plotted on millimetre‑square graph paper or with appropriate software. The points should lie close to a straight line passing through the origin, as predicted by the theory T² = (4π²/g) L. The line of best fit should be drawn so that the points are evenly distributed above and below it. Any obviously anomalous point – perhaps from a miscounted oscillation or a mistaken release – can be identified and excluded from the line.
利用处理后的数据，在毫米方格纸或适当的软件上绘制 T²（纵轴）与 L（横轴）的关系图。理论上，数据点应靠近一条通过原点的直线，因为 T² = (4π²/g) L。最佳拟合线应使各点均匀分布在直线两侧。任何明显的异常点——可能由计数错误或释放不当导致——可以识别出来，并在拟合直线时予以剔除。
8. Calculating g from the Gradient | 由斜率计算 g
A large right‑angled triangle was constructed on the best‑fit line to determine its gradient m. Using two widely separated points on the line (not data points), the slope is: m = Δy / Δx = Δ(T²) / ΔL. According to the theory, m = 4π² / g, therefore g = 4π² / m. For the sample data above, the points lie almost perfectly along a line with gradient ≈ 4.03 s²/m. Substituting: g = 4π² / 4.03 ≈ 9.79 m/s². The value can then be compared with the accepted local value (around 9.81 m/s²) to judge the experiment’s accuracy.
在最佳拟合线上构建一个较大的直角三角形以确定斜率 m。选取线上间隔较宽的两个点（不是原始数据点），斜率计算为：m = Δy / Δx = Δ(T²) / ΔL。根据理论关系，m = 4π² / g，因此 g = 4π² / m。就上述样例数据而言，各点几乎完美地落在一条斜率约为 4.03 s²/m 的直线上。代入计算：g = 4π² / 4.03 ≈ 9.79 m/s²。可将此值与当地公认值（约9.81 m/s²）进行比较，以评判实验的准确度。
9. Sources of Error and Uncertainty | 误差与不确定度来源
Several factors limit the precision of this investigation. The dominant uncertainty usually comes from the measurement of the pendulum length, because effective length is the distance to the centre of mass of the bob, which is difficult to locate exactly. The string may also stretch slightly under the weight of the bob. Reaction time in starting and stopping the stopwatch introduces random timing errors, though these are reduced by timing multiple oscillations. Friction at the point of suspension and air resistance cause the amplitude to slowly decay, but this has little effect on the period for small oscillations. Finally, an amplitude that is not small enough makes the period slightly dependent on angular displacement, violating the simple harmonic motion condition.
多个因素限制了本次实验的精确度。最主要的测量不确定度通常来自摆长的测量，因为有效长度是到摆球质心的距离，而质心难以精确定位。细绳在摆球的重力作用下也可能会略微伸长。启动和停止秒表时的反应时间会引入随机计时误差，但通过对多次摆动计时已有所减少。悬挂点处的摩擦和空气阻力会使振幅缓慢衰减，但在小摆动下对周期影响不大。最后，若振幅不够小，周期便会轻微依赖于角位移，从而违背简谐运动的条件。
10. Improving the Experiment | 改进实验的方法
To reduce length uncertainty, a larger fiducial marker, such as a horizontal pointer at the centre of the bob, can be used alongside a travelling microscope or a digital calliper to measure L more precisely. The bob can be replaced with a metal sphere of known radius so that L = distance to top of sphere + radius. A light gate linked to a digital timer could eliminate human reaction time entirely. Changing the string to a fine wire reduces stretch. To keep the amplitude consistently small, a fixed marker on the protractor can be used for every release. Repeating the entire experiment with a second independent set of lengths and averaging the obtained g values improves reliability.
为了减小长度测量不确定度，可使用一个更大的参照标记（例如摆球中心处的水平指针），并配合移测显微镜或数字游标卡尺来更精确地测量 L。用已知半径的金属球作摆球，则 L = 悬挂点到球顶的距离 + 半径。连接至数字计时器的光闸可完全消除人为反应时间。将细绳换成细金属丝可以减轻伸长量。为保持振幅恒定且足够小，每次释放时可将摆线对准量角器上的固定标记。用另一组独立的长度重复整个实验，并对所得 g 值取平均，也能提高结果的可靠性。
11. Conclusion | 结论
The simple pendulum experiment presented in the AS Unit 1 insert of January 2019 exemplifies how a straightforward set‑up can yield a precise determination of g when careful technique and thorough data analysis are applied. It illustrates core AS concepts such as linearisation of equations, minimising uncertainties through repeated measurements, and evaluating the validity of assumptions like small‑angle motion. Reflecting on the sources of error and considering possible improvements deepens one’s understanding of experimental physics and prepares students for the practical assessment demands of the course.
2019年1月AS单元1插页所展示的单摆实验，很好地体现了如何通过简单装置、细致操作和全面的数据分析来精确测定 g。它阐释了方程线性化、通过多次测量减小不确定度以及评估小角度运动等假设的有效性这些AS阶段的核心概念。反思误差来源并思考可能的改进措施，可以加深对实验物理的理解，并有助于学生应对课程中对实验评估能力的要求。
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