📚 A-Level Physics Unit 5 Insert (Jan19): Experimental Investigation of a Simple Pendulum | A-Level物理单元5插入页(2019年1月): 单摆实验探究
The January 2019 insert for the Edexcel IAL Physics Unit 5 exam presented a classic experimental investigation that required students to determine a physical constant from first-hand data. The task centred on the simple pendulum, a core practical that links periodic motion, data logging and graphical analysis. Candidates were given a set of length and timing measurements and asked to calculate the acceleration due to gravity, g, together with its associated uncertainty.
2019年1月 Edexcel IAL 物理单元5考试所附的插入页提供了一项经典的实验探究，要求学生根据原始数据确定一个物理常数。该任务以单摆实验为核心，这是一个将周期运动、数据记录和图像分析紧密结合的核心实验。考生会得到一组摆长与计时数据，并被要求计算重力加速度 g 及其不确定度。
1. Overview of the Insert Task | 插入页任务概述
The insert typically contains a table of raw measurements, a labelled diagram of the apparatus and a series of guiding questions. For the January 2019 paper, candidates were expected to process the data independently, calculate mean periods, square them and plot a graph of T² against L. The aim was to deduce g from the slope of the best-fit line and to estimate the percentage uncertainty in the final result.
插入页通常包含一个原始测量数据表、一个标注清晰的装置示意图以及一系列引导性问题。在2019年1月的试卷中，考生需要独立处理数据，计算平均周期、平方值，并绘制 T² 对 L 的图像。其目标是根据最佳拟合直线的斜率推算出 g，并估算最终结果的百分不确定度。
Understanding the layout of the insert is crucial. The data table often includes columns for pendulum length L, time for 20 oscillations (t₂₀), and leave blanks for the period T and T². Students must fill these in carefully, paying attention to significant figures and units. The final part of the question typically asks for a justified conclusion on whether the determined value agrees with the accepted value of 9.81 m s⁻².
理解插入页的布局至关重要。数据表通常包括摆长 L、20 次振动的时间 (t₂₀)，并为周期 T 和 T² 留空。学生须仔细填写，注意有效数字和单位。题目的最后一部分通常会要求一个有理有据的结论，判断所测得的数值是否与公认值 9.81 m s⁻² 吻合。
2. Experimental Apparatus | 实验装置
The simple pendulum used in the investigation consists of a small metal bob suspended by a light, inextensible string from a rigid support. A metre rule is used to measure the length from the point of suspension to the centre of the bob, and a stopwatch records the time for multiple oscillations. The insert often shows a diagram where a fiducial marker, such as a vertical pin, indicates the equilibrium position to help reduce timing errors.
该探究实验使用的单摆由一个悬挂在轻质、不可伸长的细绳上的小金属球构成，绳的另一端固定在刚性支架上。用米尺测量从悬挂点到球心的长度，用秒表记录多次振动的时间。插入页通常会展示一幅示意图，图中有一个基准标记（例如一根垂直的针），标示平衡位置，以帮助减少计时误差。
A clamps and stand, a protractor for checking small angular displacements (typically <10°) and a set square to align the metre rule vertically are implied. Although not always listed, such details are essential for reproducing the experiment and for answering questions on precision and accuracy. The string should be thin compared with the bob to minimise the effect of air resistance on the period.
实验装置还隐含用到铁架台和夹子、用于检查小角度位移（通常 <10°）的量角器，以及保证米尺竖直的三角尺。尽管这些细节不一定被列出，对于重现实验和回答关于精密度与准确度的问题而言却必不可少。相比小球，细绳应足够细，以尽量减少空气阻力对周期的影响。
3. Recording Data and Reducing Errors | 数据记录与减少误差
The insert emphasises measuring the time for 20 complete swings rather than a single swing. This technique significantly reduces the percentage uncertainty arising from human reaction time. The period T is then found by dividing the total time by 20. Repeating the measurement and taking the average further improves reliability.
插入页强调测量 20 次完整摆动的时间而非单次摆动。这一方法能显著降低由人的反应时间引起的百分不确定度。随后用总时间除以 20 即可得到周期 T。重复测量并取平均值会进一步提高可靠性。
Length L must be measured from the point of suspension to the centre of the bob. This is often done by measuring to the top and bottom of the bob and averaging, or by using a caliper to find the bob’s diameter. The insert may give a value for the diameter, expecting candidates to add half of it to the string length. All raw data are recorded with consistent decimal places reflecting the instrument’s resolution, e.g. to ±0.1 cm for a metre rule.
摆长 L 必须从悬挂点测量到小球的中心。常通过分别测量到小球顶端和底端的距离并取平均值来实现，或者用游标卡尺测量小球直径。插入页可能会给出直径值，期望考生将其一半加到绳长上。所有原始数据都要记录为一致的小数位数，以反映仪器的分度值，例如米尺读到 ±0.1 cm。
4. Data Table and Processing | 数据表与处理
A typical data set from the insert might appear as follows (reconstructed for revision). The table includes the raw length and time for 20 oscillations, plus calculated columns for T and T².
插入页中的典型数据可能如下所示（为复习需要而重建）。该表包含原始摆长和 20 次振动的时间，以及计算出的 T 列与 T² 列。
| L / m (±0.001 m) | t₂₀ / s (±0.01 s) | T = t₂₀/20 / s | T² / s² |
|---|---|---|---|
| 0.200 | 17.92 | 0.896 | 0.803 |
| 0.400 | 25.40 | 1.270 | 1.613 |
| 0.600 | 31.04 | 1.552 | 2.409 |
| 0.800 | 35.84 | 1.792 | 3.211 |
| 1.000 | 40.10 | 2.005 | 4.020 |
When processing the data, students should retain an appropriate number of significant figures. T is usually given to 3 or 4 s.f. and T² to 3 s.f. because squaring propagates the uncertainty. It is good practice to calculate T² values before plotting and to check for any outliers before drawing the graph.
处理数据时，学生应保留合适的有效数字位数。T 通常保留 3 或 4 位有效数字，T² 保留 3 位，因为平方会传递不确定度。好的做法是先计算 T² 值再绘图，并在绘制之前检查是否有异常值。
5. Graph Plotting | 绘图
The standard approach is to plot T² on the y-axis against L on the x-axis. The expected relationship is linear and passes through the origin, provided the pendulum approximates a simple harmonic oscillator with small amplitude. The theory gives the equation:
标准方法是以 T² 为 y 轴、L 为 x 轴绘图。只要单摆在小振幅条件下近似为简谐振子，预期关系就是一条过原点的直线。理论给出的方程为：
T² = (4π²/g) L
Candidates must draw a best-fit straight line through the data points, being careful to maximise the number of points on or near the line. The line should be drawn with a sharp pencil and a transparent ruler. When the insert provides grid paper, the scale must be chosen to use at least half of the available space in both directions. The origin does not have to be included if the data range does not approach zero, but in this experiment a zero-zero intercept is expected theoretically, and the line can be forced through the origin if the question instructs.
考生必须通过数据点画一条最佳拟合直线，注意让尽可能多的点落在线上或附近。画线时应使用削尖的铅笔和透明直尺。如果插入页提供了方格纸，所选的坐标标度必须使两个方向至少利用可用空间的一半。如果数据范围不趋近于零，原点可以不包括在内；但本实验中理论上截距应为零，如果题目要求，可使直线通过原点。
6. Calculating g from the Slope | 由斜率计算 g
Once the graph is drawn, the slope m is determined using a large triangle on the best-fit line. Select two widely spaced points (x₁, y₁) and (x₂, y₂) on the line, not directly from the data table, and compute:
绘制好图像后，在最佳拟合直线上用一个大三角形来确定斜率 m。选择线上两个间隔较大的点 (x₁, y₁) 和 (x₂, y₂)（而非直接取自数据表），计算：
m = (T₂² – T₁²) / (L₂ – L₁)
From the theory, m = 4π² / g, so g = 4π² / m. Using typical data, if m = 4.00 s² m⁻¹, then g = 4π² / 4.00 = 9.87 m s⁻². This approximate value is slightly higher than the accepted 9.81 m s⁻², hinting at small systematic errors.
根据理论，m = 4π² / g，因此 g = 4π² / m。代入典型数据，若 m = 4.00 s² m⁻¹，则 g = 4π² / 4.00 = 9.87 m s⁻²。这个近似值略高于公认值 9.81 m s⁻²，暗示存在微小的系统误差。
It is vital to show the working on the graph paper explicitly, by marking the coordinates used for the slope. Candidates should also state the final value of g with its absolute uncertainty calculated in the next section and round it to an appropriate number of significant figures, typically 2 or 3 s.f.
在坐标纸上清晰地展示计算过程至关重要，要标记出用于计算斜率的坐标点。考生还应给出 g 的最终值，并附上下一节计算出的绝对不确定度，将其修约到适当的有效数字，通常为 2 或 3 位有效数字。
7. Uncertainty Analysis | 不确定度分析
To find the uncertainty in g, you must first determine the uncertainty in the slope. Draw the steepest and shallowest possible straight lines that still fit the error bars (worst-fit lines). From these, obtain the maximum gradient m_max and the minimum gradient m_min. The absolute uncertainty in the slope Δm is half the difference:
要计算 g 的不确定度，必须首先确定斜率的不确定度。画出仍能拟合误差线的最陡和最浅的“最差拟合”直线。从这两条线获得最大斜率 m_max 和最小斜率 m_min。斜率的不确定度 Δm 为两者差值的一半：
Δm = (m_max – m_min) / 2
Then the uncertainty in g is found by propagation. Since g = 4π² / m, the relative uncertainty in g equals the relative uncertainty in m: Δg/g = Δm/m. Hence Δg = g × (Δm/m). Express the final result as g ± Δg, and calculate the percentage uncertainty = (Δg/g) × 100%.
再通过不确定度传递求 g 的不确定度。因为 g = 4π² / m，g 的相对不确定度等于 m 的相对不确定度：Δg/g = Δm/m。因此 Δg = g × (Δm/m)。将最终结果表示为 g ± Δg，并计算百分不确定度 = (Δg/g) × 100%。
If error bars are not given, you may estimate Δm by considering the scatter of points about the best-fit line. The percentage uncertainty in L is often negligible compared to that in T², so the dominant source is usually the timing. A well-conducted experiment should yield a percentage uncertainty below 5%.
如果未给出误差线，可通过考察点相对于最佳拟合线的分散程度来估算 Δm。与 T² 的不确定度相比，L 的百分不确定度通常可忽略，因此主要来源通常是计时。一个操作良好的实验，其百分不确定度应低于 5%。
8. Comparing with the Accepted Value | 与公认值比较
The accepted value of g is 9.81 m s⁻². To assess the accuracy of the experiment, compute the percentage difference: |(experimental value – 9.81) / 9.81| × 100%. If this percentage difference is less than the estimated percentage uncertainty, the result is considered consistent with the accepted value. Otherwise, systematic errors are likely present.
重力加速度 g 的公认值为 9.81 m s⁻²。为评估实验的准确度，计算百分差异：|(实验值 – 9.81) / 9.81| × 100%。若该百分差异小于估算的百分不确定度，则结果被认为与公认值一致；否则很可能存在系统误差。
In the January 2019 insert scenario, many candidates found their g value to be slightly higher, around 9.9 m s⁻². This discrepancy could be due to the length measurement being systematically too short, perhaps because the effective length was slightly less than measured when the bob rotated, or because the amplitude was slightly large, violating the small-angle approximation.
在2019年1月插入页的情境中，许多考生发现自己的 g 值略高，约为 9.9 m s⁻²。这一差异可能源于摆长测量系统性地偏短，也可能因为小球旋转时有效长度比测量值略小，或者振幅略大而违背了小角度近似。
9. Sources of Systematic Error | 系统误差的来源
Several systematic errors can affect the pendulum experiment. If the string has significant mass or stretches during the swing, the simple model fails. Similarly, air resistance damps the motion, but it has a negligible effect on the period for small swings. The most likely systematic error is mis-measuring L, by failing to account for the bob’s radius or by measuring at an angle rather than vertically.
若干系统误差会影响单摆实验。如果绳子质量显著或在摆动过程中伸长，简单的模型就会失效。同理，空气阻力会阻尼运动，但在小角度摆动时对周期的影响可忽略。最有可能的系统误差是错误测量 L，例如没有计入小球的半径，或没有竖直测量而是倾斜着测量。
Timing errors can be systematic if the stopwatch is started or stopped consistently early or late. Using a light gate and a data logger would eliminate reaction time entirely, but the insert assumes a manual stopwatch. The amplitude should be kept below 10° to ensure the period is independent of amplitude; otherwise T increases slightly, leading to an overestimate of g if not corrected.
如果秒表总是按得过早或过晚，计时误差可成为系统误差。使用光门和数据记录仪可以完全消除反应时，但插入页假定使用手动秒表。振幅应保持在 10° 以下，以确保周期与振幅无关；否则 T 会略为增大，若不修正，会导致 g 被高估。
10. Reducing Uncertainties and Improving Precision | 降低不确定度与提高精密度
To improve precision, increase the number of oscillations timed (e.g., 50 swings) to further reduce the relative time uncertainty. Use a fiducial marker, such as a needle behind the string, and start and stop the stopwatch when the string passes the marker at the centre of the swing, because the bob moves fastest there and timing is more reproducible. Measure L with a vernier caliper or a travelling microscope for greater resolution.
为提高精密度，可以增加计时的振动次数（例如 50 次），以进一步降低相对时间不确定度。使用基准标记，如在绳子后方放一根针，当绳摆过标记中心位置时启动和停止秒表；因为小球在此处运动最快，计时更可重复。使用游标卡尺或移测显微镜测量 L 可获得更高的分辨率。
Repeating the entire set of measurements for each length and averaging periods, then plotting the mean T² against L, reduces random scatter. When plotting, choose scales that spread the data points over at least half the grid. If the line shows a non-zero intercept, comment on it but do not force the line through the origin unless instructed; an intercept may indicate a systematic error in length measurement.
对每个长度重复整套测量并求周期平均值，然后绘制平均 T² 对 L 的图像，可减少随机散射。绘图时，选择能使数据点散布于至少半个网格的标度。如果直线显示出非零截距，应加以评述，但除非题目要求，不要强行让直线通过原点；截距可能表明长度测量存在系统误差。
11. Exam Tips for Unit 5 Insert Questions | 单元5插入页题型考试技巧
When tackling the insert question, read the rubric carefully to see whether raw times are given for single or multiple swings. Show all your working, including the division by 20 and the squaring step. On the graph, label axes with quantities and units, use a sensible scale, and draw the best-fit line as soon as you have plotted the points. Do not join the dots. Remember to record the two points used for the slope directly on the graph.
解答插入页题目时，要仔细阅读题干，看清所给的原始时间是单次摆动还是多次摆动。展示所有计算步骤，包括除以 20 以及平方的步骤。在图上，用物理量和单位标注坐标轴，使用合理的标度，数据点一画好就立即画出最佳拟合直线。不要用折线连接各点。记得将用来求斜率的两个点直接标注在图上。
For the uncertainty calculation, many marks are allocated to drawing the worst-fit lines on the same grid and clearly labelling them. State m_max and m_min explicitly, compute Δm and then Δg. Always give g to an appropriate number of significant figures, matching the precision of your measurements. Finally, write a short conclusion comparing your value with the accepted one and discussing whether the discrepancy lies within experimental error.
不确定度计算部分，许多分数都用于在同一个网格上画出最差拟合线并清晰标注。要明确写出 m_max 和 m_min，计算 Δm 然后求 Δg。始终将 g 给出与测量精密度相匹配的合适有效数字。最后，写一段简短的结论，将自己的值与公认值比较，并讨论差异是否落在实验误差范围内。
12. Worked Example: Applying the Skills | 范例：应用所学技能
Using the sample data above, suppose the best-fit line gives a slope m = 3.98 s² m⁻¹. Then g = 4π² / 3.98 = 9.92 m s⁻². By drawing worst-fit lines, one obtains m_max = 4.10 s² m⁻¹ and m_min = 3.86 s² m⁻¹. Then Δm = (4.10 – 3.86)/2 = 0.12 s² m⁻¹. The absolute uncertainty Δg = 9.92 × (0.12/3.98) ≈ 0.30 m s⁻². Thus g = 9.9 ± 0.3 m s⁻² to 2 significant figures in the uncertainty. The percentage difference from 9.81 is |(9.9 – 9.81)/9.81| × 100% ≈ 0.9%. The percentage uncertainty is (0.3/9.9) × 100% ≈ 3%, so the result is consistent with the accepted value.
以上述样本数据为例，假设最佳拟合直线的斜率 m = 3.98 s² m⁻¹，则 g = 4π² / 3.98 = 9.92 m s⁻²。通过绘制最差拟合线，得到 m_max = 4.10 s² m⁻¹，m_min = 3.86 s² m⁻¹。于是 Δm = (4.10 – 3.86)/2 = 0.12 s² m⁻¹。绝对不确定度 Δg = 9.92 × (0.12/3.98) ≈ 0.30 m s⁻²。因此 g = 9.9 ± 0.3 m s⁻²（不确定度保留 2 位有效数字）。与 9.81 的百分差异为 |(9.9 – 9.81)/9.81| × 100% ≈ 0.9%。百分不确定度为 (0.3/9.9) × 100% ≈ 3%，因此该结果与公认值一致。
This worked example mirrors exactly what examiners expect: a clear gradient calculation, explicit worst-fit lines, correct propagation of error, and a well-reasoned conclusion. Practice with different sets of data from past papers will build speed and confidence for the real Unit 5 written alternative to practical examination.
这个范例完全对应了考官的期望：清晰的斜率计算、明确的最差拟合线、正确的误差传递以及有理有据的结论。用往年真题的不同数据进行练习，将为真正的单元5 笔试实验替代考试建立速度和信心。
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