📚 Testing a Relationship in Physics Experiments | 物理实验中验证关系
In A-Level Physics, many practical investigations are designed to test whether a predicted mathematical relationship between two physical quantities holds true. You will need to design an experiment, collect reliable data, and then analyse that data using graphical methods to confirm, refute, or refine the proposed relationship. This article covers the key techniques you must master – from identifying the form of a relationship, through linearising data, to assessing the uncertainties in your final conclusion.
在A-Level物理中，许多实验旨在检验两个物理量之间的预测数学关系是否成立。你需要设计实验、收集可靠数据，然后运用图表方法分析数据，以确认、否定或修正所提出的关系。本文将涵盖你必须掌握的关键技术——从识别关系的形式，到数据的线性化处理，再到评估最终结论中的不确定度。
1. Understanding Experimental Relationships | 理解实验关系
riable, x) is systematically altered. A valid relationship is often expressed as an equation, such as y = kx for direct proportion or y = kx² for a quadratic dependence. In your practical work, you are typically asked to test a specific hypothesis, e.g. 'The period of a pendulum squared is directly proportional to its length'. Your task is to gather evidence that either supports or contradicts that statement.
实验关系描述了一个物理量（因变量y）随另一个物理量（自变量x）的系统性变化而变化的方式。有效的关系通常用一个方程表达，例如正比关系y = kx，或平方依赖关系y = kx²。在你的实验工作中，通常要求检验一个特定的假设，例如“单摆周期的平方与其摆长成正比”。你的任务是收集支持或否定该陈述的证据。
2. Common Types of Relationships in Physics | 物理中常见关系类型
Several standard mathematical forms appear repeatedly in A-Level experiments:
A-Level实验中反复出现以下几种标准数学形式：
ht line through the origin. Example: V = IR for an ohmic resistor at constant temperature.
正比关系：y = mx，图像为过原点的直线。例如：恒温下欧姆导体满足V = IR。
rce E = V + Ir, giving terminal voltage V = E − Ir (graph of V against I has intercept E).
带截距的线性关系：y = mx + c。例如：电动势E = V + Ir，给出路端电压V = E − Ir（V对I的图线截距为E）。
Inverse proportion: y = k/x or y = k/x². Example: Coulomb's law F ∝ 1/r².
反比关系：y = k/x或y = k/x²。例如：库仑定律F ∝ 1/r²。
r law: y = kxⁿ, where n can be any number. Example: the period of a simple pendulum T = 2π√L/g can be squared to give T² = (4π²/g)L, showing T² ∝ L.
幂律关系：y = kxⁿ，其中n可以是任意数。例如：单摆的周期T = 2π√L/g可两边平方得到T² = (4π²/g)L，显示T² ∝ L。
Exponential decay: y = y₀ e−λx. Example: capacitor discharge: V = V₀ e−t/RC.
指数衰减：y = y₀ e−λx。例如：电容器放电：V = V₀ e−t/RC。
Recognising these patterns helps you decide how to process data to achieve a straight-line graph.
认识这些模式有助于你决定如何处理数据以获得一条直线图形。
3. Planning the Experiment | 计划实验
stance, when investigating T² versus L for a pendulum, length L is the independent variable, period T (and hence T²) is the dependent variable, and the amplitude (small angles) and mass of the bob must be kept constant. Select a suitable range of the independent variable – ments; a metre rule (resolution ±1 mm) is adequate for lengths from about 20 cm to 100 cm.
在进行任何测量之前，应明确你的自变量、因变量以及必须控制的变量。例如，研究单摆的T²与L的关系时，摆长L是自变量，周期T（进而T²）是因变量，而振幅（小角度）和摆球质量必须保持不变。选择自变量的合适范围——理想情况下至少覆盖五到七个值，均匀分布在一个能产生可测量变化的范围内。考虑仪器的分辨率；对于约20 cm到100 cm的长度，米尺（分辨率±1 mm）是足够的。
Also decide how many repeats you will take for each measurement to reduce random error. Typically, three repeats and calculation of a mean is a minimum; for time-dependent quantities like pendulum period, measure the time for multiple oscillations (e.g. 10T) and then divide to improve precision.
还要决定每个测量值将重复多少次以减小随机误差。通常，至少进行三次重复并计算平均值；对于像单摆周期这样依赖时间的量，测量多次振荡的时间（例如10T）然后除以次数以提高精度。
4. Data Collection and Tabulation | 数据收集与表格制作
Record all raw data in a neatly designed table. The table must have a descriptive title and clearly labelled columns with units and uncertainties. The independent variable should be in the first column, followed by the raw dependent variable readings and any calculated quantities. A well-structured table for a pendulum experiment might look like this:
将所有原始数据记录在设计整齐的表格中。表格必须有说明性的标题，列标题应清晰标明单位和不确定度。自变量应放在第一列，然后是原始的因变量读数和任何计算量。一个单摆实验的结构良好的表格可能如下所示：
| Length, L / m (±0.001 m) | Time for 10T, t₁ / s | t₂ / s | t₃ / s | Mean t / s | Period T / s | T² / s² |
|---|---|---|---|---|---|---|
| 0.200 | 8.96 | 8.90 | 8.89 | 8.92 | 0.892 | 0.796 |
| 0.400 | 12.70 | 12.66 | 12.68 | 12.68 | 1.268 | 1.608 |
pwatch to ±0.01 s) and apply the correct number of significant figures in derived columns.
始终以仪器的精度记录读数（例如秒表为±0.01 s），并在导出列中应用正确的有效数字位数。
5. Plotting the Initial Graph | 绘制初步图表
Plot the raw dependent variable against the independent variable on graph paper or using software. Use appropriate scales that occupy at least half the graph area and label axes with quantity and unit (e.g. L / m). Plot data points as small crosses or dots with circles, and if possible add uncertainty bars showing the absolute uncertainty in each measurement. The initial graph reveals the overall trend: a straight line suggests direct proportion or a linear relationship; a curve that flattens suggests inverse or exponential behaviour. This visual inspection tells you whether further transformation is needed.
在坐标纸或使用软件将原始的因变量对自变量作图。选择合适的标度，至少占据图面的一半，并用物理量和单位标记坐标轴（例如L / m）。将数据点画成小十字或带圈的点，如果可能，加上显示每个测量绝对不确定度的误差棒。初始图揭示总体趋势：一条直线表明正比或线性关系；趋于平缓的曲线表明反比或指数行为。这种可视化检查告诉你是否需要进行进一步变换。
For our pendulum example, plotting T² directly against L immediately yields a straight line through the origin – no further linearisation is required because the relationship has already been converted to a linear form by squaring the period.
对于我们的单摆例子，直接将T²对L作图立即得到一条通过原点的直线——无需进一步线性化，因为通过将周期平方，关系已经转换为线性形式。
6. Linearising Non-Linear Relationships | 将非线性关系线性化
Many physical relationships produce curves when plotted raw. To test such relationships rigorously, you must transform the variables to produce a straight-line graph. Common strategies include:
许多物理关系在原始数据绘图时呈现曲线。为了严格检验这类关系，你必须变换变量以生成一条直线图。常见策略有：
- For y ∝ x²: plot y against x².
- y ∝ x²时：绘制y对x²的图。
- For y ∝ 1/x: plot y against 1/x.
- y ∝ 1/x时：绘制y对1/x的图。
- For y ∝ 1/x²: plot y against 1/x².
- y ∝ 1/x²时：绘制y对1/x²的图。
g x + log k. Plot log y against log x; the gradient gives n and the intercept gives log k.
- 对于幂律y = kxⁿ：取对数得log y = n log x + log k。绘制log y对log x的图；斜率给出n，截距给出log k。
- For exponential decay y = y₀ e−λx: take natural logs: ln y = ln y₀ − λx. Plot ln y against x; the gradient is −λ.
- 对于指数衰减y = y₀ e−λx：取自然对数：ln y = ln y₀ − λx。绘制ln y对x的图；斜率为−λ。
This transformation is critical because a straight line is the only graph form that can be reliably verified and used to extract constants.
这一变换至关重要，因为直线是唯一可以可靠验证并用于提取常数的图形形式。
7. Drawing the Line of Best Fit | 绘制最佳拟合线
Once you have a linearised graph, use a transparent ruler to draw a single straight line that passes as close as possible to all data points, balancing the number of points above and below the line. Do not force the line through the origin unless the theory demands it and you have verified it with a separate check. If error bars are plotted, the line should pass through the uncertainties of most points. For a more advanced analysis, draw two further lines: the steepest and shallowest plausible lines that still fit the data within errors. These worst-fit lines help determine the uncertainty in the gradient.
一旦得到线性化图形，用透明直尺绘制一条尽可能通过所有数据点的单一直线，使线的两侧点数大致平衡。除非理论要求且你已通过单独检查验证，否则不要强迫直线穿过原点。如果绘制了误差棒，直线应穿过大多数点的不确定区间。对于更深入的分析，再画两条辅助线：在误差范围内仍能拟合数据的最陡和最浅的合理直线。这些最差拟合线有助于确定斜率的不确定度。
Label the best-fit line with its equation if possible, and mark on the graph the coordinates of two widely separated points on the line (not data points) used to calculate the gradient.
如果可能，给最佳拟合线标上它的方程，并在图上标出线上相距较远的两个点（非数据点）的坐标，用于计算斜率。
8. Interpreting Gradient and Intercept | 解释斜率和截距
ulum, the gradient m = 4π²/g. Thus, from the measured gradient you can determine an experimental value for the acceleration due to gravity: g = 4π²/m. Compare this experimental value with the accepted value (9.81 m s⁻²) by calculating the percentage difference: |experimental − accepted| / accepted × 100%.
最佳拟合线的斜率和y轴截距根据理论方程具有物理意义。对于单摆的T²对L图，斜率m = 4π²/g。因此，由测得的斜率可以确定重力加速度的实验值：g = 4π²/m。通过计算百分差异将该实验值与公认值（9.81 m s⁻²）进行比较：|实验值 − 公认值| / 公认值 × 100%。
If the theory predicts a zero intercept but your graph shows a small non-zero value, comment on possible systematic errors (e.g. a mis-measured length, offset zero). A non-zero intercept can often reveal a constant offset in your measurements.
如果理论预测截距为零，但你的图线显示一个小的非零值，应评论可能的系统误差（例如长度测量错误，零点偏移）。非零截距常常可以揭示测量中的恒定偏移。
9. Uncertainties and Error Analysis | 不确定度与误差分析
Every experimental gradient must be quoted with an absolute uncertainty. To find the uncertainty in the gradient, calculate the gradient of the steepest (mmax) and shallowest (mmin) worst-fit lines. Then the uncertainty Δm = (mmax − mmin gate the uncertainty into any derived quantity, such as g. If g = 4π²/m, then Δg/g = Δm/m, so Δg = g × (Δm/m).
每个实验斜率必须附带一个绝对不确定度。要求出斜率的不确定度，计算最陡（mmax）和最浅（mmin）最差拟合线的斜率。然后不确定度Δm = (mmax − mmin) / 2。相对不确定度Δm / m给出了数据质量的度量。截距的不确定度也可类似求得。始终将不确定度传播到任何导出量中，例如g。若g = 4π²/m，则Δg/g = Δm/m，因此Δg = g × (Δm/m)。
unit. A small percentage uncertainty and a graph whose error bars are small compared with the scatter indicates a reliable experiment.
以（值 ± 不确定度）的形式连同正确单位陈述你的最终结果。较小的百分不确定度以及误差棒远小于数据离散度的图形表明实验是可靠的。
10. Evaluating the Experiment | 评估实验
A full evaluation identifies the main sources of error and suggests realistic improvements. Random errors, such as reaction time in starting a stopwatch, can be reduced by measuring longer time intervals (many oscillations) or using light gates. Systematic errors, like a zero error on a balance or a metre rule not being perfectly vertical, may cause a constant shift in results. Discuss whether your straight line passes through the origin as predicted; if not, pinpoint which systematic effect could account for the discrepancy.
完整的评估要找出主要的误差来源，并提出切实可行的改进建议。随机误差，例如启动秒表的反应时间，可通过测量较长时间间隔（多次振荡）或使用光门来减小。系统误差，如天平零点误差或米尺不严格垂直，可能导致结果发生恒定偏移。讨论你的直线是否按预测通过原点；若未通过，指出哪一种系统效应可能导致该偏差。
You should also comment on the limitations of the equipment and the range of data. For instance, did you use small enough amplitudes for the pendulum to satisfy the simple harmonic approximation? Could air resistance affect the motion? Suggest specific changes, such as using a clamp with a precise height adjustment or a digital timer with a precision of 0.001 s.
你还应评论设备的局限性和数据的范围。例如，单摆振幅是否足够小以满足简谐近似？空气阻力是否会影响运动？提出具体改进，例如使用具有精确高度调节的夹具，或精度为0.001 s的数字计时器。
11. Drawing a Valid Conclusion | 得出有效结论
statement with evidence: the straight line achieved after appropriate transformation, the gradient and its uncertainty, and the comparison with accepted theory. For example: ' .78 ± 0.24 m s⁻², which agrees with the accepted 9.81 m s⁻² within the experimental uncertainty.' Avoid overclaiming if the uncertainty is large or the graph shows systematic deviation.
你的结论必须直接回到最初的假设。说明数据是否支持所提出的关系，并用证据支持你的陈述：经过适当变换后得到的直线、斜率及其不确定度、以及与公认理论的比较。例如：“T²对L的图线是一条通过原点的直线，证实T²与L成正比。重力加速度的实验值为9.78 ± 0.24 m s⁻²，在实验不确定度范围内与公认值9.81 m s⁻²一致。”如果不确定度较大或图线显示系统性偏差，应避免过度断言。
Effectively testing a relationship in physics is not just about following a recipe; it requires critical thinking about the validity of data, the appropriateness of transformations, and a full appreciation of measurement uncertainty. Mastering these skills will serve you well in your A-Level practical assessments and beyond.
有效检验物理关系不仅仅是按部就班地操作；它需要对数据的有效性、变换的适当性以及测量不确定度的全面理解进行批判性思考。掌握这些技能将在你的A-Level实验考核及今后的学习中大有裨益。
Published by TutorHao | Physics Revision Series | aleveler.com
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