📚 9630-PH02 Specimen 2016 v2: Experimental Investigation – Simple Pendulum | 9630-PH02 样本卷2016 v2 实验探究：单摆
In the 9630-PH02 International AS Physics specimen paper, a typical experimental investigation involves analysing the motion of a simple pendulum to determine the acceleration due to gravity g. This article walks you through the key stages of such an investigation, from data collection to uncertainty evaluation, mirroring the skills assessed in the examination.
在 9630-PH02 国际 AS 物理样本卷中，典型的实验探究是利用单摆的运动来测定重力加速度 g。本文带你走完这样一项探究的关键步骤，从数据采集到不确定度评估，完整呈现考试中考查的实验技能。
1. Overview of the Experiment | 实验概述
The aim is to measure g by investigating how the period T of a simple pendulum depends on its length L. By keeping the amplitude small, the motion approximates simple harmonic motion, and the period is given by T = 2π √(L/g). A graphical approach using T² against L makes it possible to extract g from the gradient.
本实验的目标是通过探究单摆周期 T 与其长度 L 的关系来测量 g。在保持小振幅的条件下，运动近似为简谐运动，周期公式为 T = 2π √(L/g)。利用 T² 对 L 作图，从斜率中即可提取出 g。
The investigation typically requires you to identify independent, dependent and controlled variables, record data in a well-designed table, plot a graph, draw a best-fit line, calculate the gradient and its uncertainty, and finally determine a value for g with an estimate of the uncertainty.
探究过程通常要求你识别自变量、因变量和控制变量，在精心设计的表格中记录数据，绘制图像并画出最佳拟合线，计算斜率及其不确定度，最终给出 g 的数值及其不确定度估计。
2. Apparatus and Method | 实验装置与方法
You will need a small dense bob attached to a light, inextensible string, a clamp stand, a metre rule, a stopwatch, a protractor (optional) and a means of measuring the length accurately such as a vernier calliper for the bob’s diameter. The length L is measured from the point of suspension to the centre of the bob.
你需要一个质量较大、体积较小的摆球，系在轻质且不可伸长的细线上，还需要铁架台、米尺、秒表、量角器（可选）以及用于准确测量长度的工具，如测量摆球直径的游标卡尺。长度 L 是从悬挂点到摆球中心测得的距离。
A typical method involves measuring L for a range of values (e.g. from 0.300 m to 1.200 m in steps of about 0.100 m). For each length, set the pendulum swinging with a small angle (less than 10°), time at least 20 complete oscillations, and calculate the period T by dividing the total time by the number of oscillations. Repeat each timing to reduce random error.
典型方法是测量一系列长度 L 的值（如从 0.300 m 到 1.200 m，步长约为 0.100 m）。对于每个长度，使单摆以小角度（小于 10°）摆动，记录至少 20 次完整振荡的时间，并将总时间除以振荡次数算得周期 T。对每次计时重复测量以减小随机误差。
3. Measurements and Data Recording | 测量量与数据记录
Record your data in a table with columns for L / m, t₁ / s (for 20 oscillations), t₂ / s, mean t / s, period T / s and T² / s². Always include units in the column headings and ensure consistent significant figures based on the precision of your instruments.
将数据记录在表格中，表头包括 L / m、t₁ / s（20 次振荡）、t₂ / s、平均时间 t / s、周期 T / s 以及 T² / s²。务必在表头包含单位，并根据仪器精度保持有效数字的一致性。
| L / m | t₁ / s | t₂ / s | mean t / s | T / s | T² / s² |
|---|---|---|---|---|---|
| 0.400 | 25.98 | 26.02 | 26.00 | 1.300 | 1.690 |
| 0.600 | 31.20 | 31.24 | 31.22 | 1.561 | 2.437 |
| 0.800 | 36.04 | 36.08 | 36.06 | 1.803 | 3.251 |
| 1.000 | 40.42 | 40.38 | 40.40 | 2.020 | 4.080 |
The period is calculated to one more significant figure than the raw data to minimise rounding errors in later calculations. T² is then computed.
周期计算时比原始数据多保留一位有效数字，以尽量减少后续计算中的舍入误差。然后计算 T²。
4. Data Processing: The Period-Length Relationship | 数据处理：周期与长度的关系
From theory, for a simple pendulum undergoing small oscillations, T = 2π √(L/g). Squaring both sides gives T² = (4π²/g) L. This shows that T² is directly proportional to L, with the constant of proportionality being 4π²/g.
根据理论，对于做小角度摆动的单摆，有 T = 2π √(L/g)。两边平方得 T² = (4π²/g) L。这表明 T² 与 L 成正比，比例常数为 4π²/g。
Therefore, a graph of T² (y-axis) against L (x-axis) should yield a straight line passing through the origin. The gradient of this line, m, equals 4π²/g, so g can be found from g = 4π² / m.
因此，以 T² 为 y 轴、L 为 x 轴作图，将得到一条过原点的直线。该直线的斜率 m 等于 4π²/g，所以 g 可由 g = 4π² / m 求得。
5. Plotting a Best-Fit Line | 绘制最佳拟合线
Plot the data points on graph paper with appropriate scales. The x-axis and y-axis should be labelled with the quantity and unit, e.g. L / m and T² / s². Draw a single, straight best-fit line that passes as close to as many points as possible, with roughly equal numbers of points above and below the line.
在坐标纸上以合适的标度绘制数据点。x 轴和 y 轴应标出物理量及单位，例如 L / m 和 T² / s²。画出一条单一的、笔直的最佳拟合线，尽可能贴近所有点，使落在线上下两侧的点数量大致相等。
Do not force the line through the origin unless there is a convincing physical reason. In this experiment, theoretically the line should pass through the origin, but small systematic errors (like an incorrect measurement of L) may shift the intercept. In examination-style investigations, you are often asked to comment on whether the intercept agrees with the expected value.
除非有充分的理论依据，否则不要强制让直线穿过原点。在本实验中，理论上线应过原点，但小的系统误差（如长度 L 测量不准）可能会使截距偏离。在考卷风格的探究中，通常要求你对截距是否与预期值一致做出评论。
6. Calculating the Acceleration Due to Gravity g | 计算重力加速度 g
Choose two points far apart on the best-fit line – not experimental data points – to calculate the gradient m. Use a large triangle to minimise percentage uncertainty.
在最佳拟合线上选取相距较远的两个点（而非实验数据点）来计算斜率 m。使用一个大的三角形以减小百分比不确定度。
m = Δ(T²) / ΔL
Suppose the chosen points give m = 4.08 s²/m. Then:
g = 4π² / m = 4π² / 4.08 ≈ 9.67 m/s²
If the accepted value is 9.81 m/s², the percentage difference can be calculated. The specimen paper often asks you to compare your experimental value with the standard value and discuss reasons for any discrepancy.
如果标准值为 9.81 m/s²，可以计算百分差。样本卷通常会要求你将实验值与标准值进行比较，并讨论产生差异的原因。
7. Uncertainty Analysis | 不确定度分析
Uncertainties arise from the measurements of length and time. The absolute uncertainty in L is often taken as ±1 mm using a metre rule, or ±0.5 mm if a vernier calliper is used for the diameter and the length is obtained carefully. For the stopwatch, the uncertainty in the total time for 20 oscillations might be ±0.2 s due to reaction time, though human error can be larger.
不确定度来源于长度和时间的测量。使用米尺时，长度 L 的绝对不确定度通常取 ±1 mm；若用游标卡尺测量直径并仔细获得长度，可取 ±0.5 mm。对于秒表，由于反应时间，20 次振荡总时间的不确定度可能是 ±0.2 s，尽管人为误差可能更大。
The percentage uncertainty in T can be estimated from the spread of repeated timings. The uncertainty in T² is then twice the percentage uncertainty in T. When combined with the percentage uncertainty in L, you can find the overall percentage uncertainty in the calculated value of g.
T 的百分不确定度可以通过重复计时的偏差范围来估计。T² 的不确定度则是 T 的不确定度的两倍。将其与 L 的百分不确定度合成，便可求得算出的 g 值的总百分不确定度。
A common approach in the investigation is to also determine the uncertainty in the gradient by drawing the steepest and shallowest acceptable lines (worst-fit lines) and using the difference in gradient to find Δm. Then the uncertainty in g can be found from Δg = g × (Δm / m).
实验探究中常用的方法还包括绘制可接受的最陡和最缓直线（最差拟合线），利用斜率差求出 Δm。然后由 Δg = g × (Δm / m) 计算出 g 的不确定度。
8. Sources of Error and Improvements | 误差来源与改进
Random errors include human reaction time when starting and stopping the stopwatch. This can be minimised by timing many oscillations (at least 20) and repeating measurements. Using a light gate or photogate timer would reduce this error significantly, though such equipment may not always be available.
随机误差包括启动和停止秒表时的人为反应时间。通过记录多个振荡的次数（至少 20 次）并重复测量，可以使此误差最小化。使用光门或光电计时器可以显著降低这一误差，尽管此类设备并非总是可用。
A systematic error could arise if the length L is measured from the clamp to the top of the bob rather than to its centre. Always measure to the centre of mass, and add the radius of the bob to the string length. Failing to measure the bob’s radius introduces a constant offset that affects the intercept of the T²–L graph.
如果长度 L 是从夹具测到摆球的顶部而非球心，则会产生系统误差。务必测量到质心，将摆球半径加到线上长度中。漏测摆球半径会引入一个恒定的偏移量，从而影响 T²–L 图像的截距。
Another possible error is allowing the pendulum to swing with a large amplitude, which makes the motion depart from simple harmonic and causes T to increase slightly. Keeping the angle below 10° and using a protractor helps maintain validity.
另一个可能的误差是让单摆以较大振幅摆动，这将使运动偏离简谐，导致 T 略有增加。将角度保持在 10° 以下并使用量角器有助于保证有效性。
9. Conclusion | 结论
Using the experimental data from the pendulum investigation, the value of g was determined to be 9.67 ± 0.14 m/s² (example with 1.5% uncertainty). The result agrees with the accepted value within the experimental uncertainty, confirming the validity of the method and the underlying relationship T² ∝ L.
利用单摆实验数据，测得 g 的值为 9.67 ± 0.14 m/s²（举例，不确定度为 1.5%）。该结果在实验不确定度范围内与公认值一致，验证了该方法的有效性以及 T² ∝ L 的基本关系。
The investigation demonstrates key practical skills: planning, recording data with appropriate precision, plotting and interpreting a straight-line graph, using the gradient to compute a derived quantity, and critically evaluating the results.
本探究展示了关键实验技能：制定方案、以合适的精度记录数据、绘制并解读直线图像、利用斜率计算导出量，以及批判性地评估结果。
10. Common Examination Tips | 试卷常见问题提示
When facing a specimen 9630-PH02 experimental question, always read the stem carefully to identify which variables you must control and which you must measure. Raw data tables should include repeated readings where possible, and column headings must show units divided by the quantity, e.g. L/m or t/s.
面对 9630-PH02 样本卷的实验题时，务必仔细阅读题干，明确需要控制哪些变量、测量哪些变量。原始数据表格应尽可能包含重复读数，且表头必须用物理量/单位的形式表示，如 L/m 或 t/s。
In calculations, show the substitution clearly and quote your final value to the appropriate number of significant figures, usually the smallest number of significant figures from the measured data. When discussing uncertainties, always link your statements to the data you have collected.
在计算过程中，清晰展示代入过程，并按照测量数据中最小的有效数字位数给出最终结果。在讨论不确定度时，始终将你的陈述与你所收集的数据关联起来。
Practice plotting graphs with a sharp pencil and a clear ruler; examiners expect data points as small crosses or encircled dots, and a clean, thin best-fit line. Always state the gradient calculation points you used – not experimental data points – and show the coordinates.
练习用削尖的铅笔和清晰的直尺作图；考官期望数据点画为小十字或加圈的圆点，最佳拟合线应干净、细直。务必写明你用来计算斜率的点（不能是实验数据点），并标出坐标。
Published by TutorHao | Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导