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Unit 6: Experimental Physics and Measurement Uncertainty | 第6单元：实验物理与测量不确定度

2026-08-21 14:03 · 本文部分内容由AI辅助创作

📚 Unit 6: Experimental Physics and Measurement Uncertainty | 第6单元：实验物理与测量不确定度

Unit 6 is the practical and experimental paper in many A-level Physics specifications. It tests not only whether you can recall physical theory, but also whether you can take measurements, estimate uncertainties, plot graphs, and evaluate a procedure critically. In other words, it rewards candidates who behave like working physicists rather than candidates reciting facts. This article covers the core skills for Unit 6, from reading a single instrument correctly to writing a full, mark-scoring evaluation.

第6单元是许多 A-level 物理课程中的实验与操作考查部分。它不仅考察你是否能回忆物理理论，还考察你能否进行测量、估算不确定度、绘制图像并批判性地评价实验方案。换句话说，它奖励的是像真正的物理工作者那样思考的考生，而不是只会背诵事实的考生。本文涵盖第6单元的核心技能，从正确读取一件仪器，到写出一份能拿满分的完整评价。

1. Understanding the Structure of Unit 6 | 理解第6单元的结构

In Unit 6, exam questions are usually based on familiar practical tasks such as measuring g by free fall, investigating springs, building circuits, or observing standing waves. You need to describe apparatus, identify independent and dependent variables, state control variables, calculate results, and comment on reliability. The contexts are deliberately familiar, because the examiner is testing transferable practical skill rather than obscure physics.

在第6单元中，考试题通常基于熟悉的实验任务，例如通过自由落体测量 g、研究弹簧、搭建电路或观察驻波。你需要描述仪器、确定自变量和因变量、说明控制变量、计算结果并评价可靠性。题目背景刻意选用熟悉的情境，因为考官考查的是可迁移的实验技能，而不是冷僻的物理知识。

The paper often asks you to plan a method or improve an existing method. Therefore, you should learn a set of standard techniques rather than memorising individual experiments word for word. A candidate who understands why a technique works can apply it to an unfamiliar context, which is exactly what the extended-response questions demand.

试卷常要求你设计一种方法或改进现有方法。因此，你应该掌握一套标准实验技术，而不是逐字记住个别实验。理解某项技术为何有效的考生，才能把它迁移到陌生情境中，而这正是长答题所要求的。

Typical question stems include: 'Describe how you would measure...', 'State one significant source of uncertainty and how it could be reduced...', 'Plot a graph of... and use the gradient to determine...', and 'Comment on the reliability of your results.' Answering these well requires vocabulary, structure, and quantitative care in equal measure.

常见的题干包括："描述你将如何测量……"、"指出一个重要的不确定度来源及其减小方法……"、"画出……的图像并用斜率求出……"以及"评价你结果的可靠性"。要答好这些题，需要在术语、结构和定量严谨性三方面同样到位。

2. Instruments, Resolution and Single-Reading Uncertainty | 仪器、分辨率与单次读数不确定度

st change it can display. A metre rule has a resolution of 1 mm, while a digital voltmeter' is taken as half the resolution, but for a digital reading it is usually ± the last digit.

每种仪器都有一个分辨率：即它能显示的最小变化。米尺的分辨率为 1 mm，数字电压表的分辨率是最后一位数字所代表的值。在许多考试大纲中，单次读数的不确定度取分辨率的一半，但对于数字读数通常取最后一位数字的 ±1。

single-reading uncertainty = ± (resolution / 2)

For a stopwatch, the dominant uncertainty is usually your reaction time, not the display resolution. Reaction time is typically about 0.1 s to 0.3 s, so taking a single time measurement is much less reliable than timing several oscillations and dividing by the number of oscillations. Using n oscillations reduces the reaction-time uncertainty per oscillation by a factor of n.

对于秒表，主导的不确定度通常是你的反应时间，而不是显示分辨率。反应时间一般为 0.1 s 至 0.3 s 左右，因此测一次时间远不如测量多次振动再除以振动次数可靠。测 n 次振动可以把每次振动分摊到的反应时间不确定度减小为原来的 1/n。

InstrumentTypical resolutionTypical single-reading uncertainty
Metre rule1 mm± 0.5 mm (± 1 mm if the end is worn)
Vernier caliper0.1 mm± 0.05 mm
Micrometer screw gauge0.01 mm± 0.005 mm (plus zero error)
Digital ammeter0.01 A± 0.01 A
Digital stopwatch0.01 s± 0.1 s to ± 0.3 s (reaction time)
Thermometer0.5 °C± 0.25 °C
Protractor1°± 0.5°

gful; 25.0 cm ± 0.003 cm is almost certainly wrong because no metre rule can resolve that.

当你写下一个测量值时，一定要让不确定度与读数相匹配。"25.0 cm ± 0.1 cm"是有意义的，而"25.0 cm ± 0.003 cm"几乎肯定是错的，因为没有米尺能分辨到那种程度。

3. Random and Systematic Errors | 随机误差与系统误差

Random errors cause readings to scatter about a true value in an unpredictable direction. They arise from fluctuating conditions, imprecise judgement, and the limited resolution of instruments. Repeating measurements and averaging reduces their effect, but it never removes them completely. Random error limits precision.

随机误差使读数以不可预测的方向围绕真值散布。它们来源于条件波动、判断不精确以及仪器分辨率有限。重复测量并取平均可以减小其影响，但永远无法完全消除。随机误差限制的是精密度。

Systematic errors shift every reading in the same direction by the same amount or the same proportion. They arise from a zero error, an incorrectly calibrated scale, or a method that always biases the result, such as reading a meniscus from above or starting a stopwatch late by the same reaction lag every time. Systematic error limits accuracy, and averaging cannot remove it.

系统误差使每一个读数都朝同一方向偏移相同的量或相同的比例。它们来源于零点误差、刻度未校准，或某种总是使结果偏向一边的方法，例如从上方读弯月面，或每次都因为同样的反应延迟而晚按秒表。系统误差限制的是准确度，取平均无法消除它。

A useful distinction to remember is: accuracy describes how close a reading is to the true value, while precision describes how close repeated readings are to each other. A set of readings can be highly precise yet inaccurate if a systematic error is present, and this is precisely the situation examiners like to test.

一个值得记住的区分是：准确度描述读数与真值的接近程度，精密度描述重复读数彼此之间的接近程度。如果存在系统误差，一组读数可能精密度很高却不准确——这正是考官喜欢考查的情形。

In written answers, name the error, classify it, and say what it does. For example: 'The metre rule has a worn zero end, producing a systematic error that makes every measured length too large; this cannot be reduced by averaging but can be reduced by measuring from the 1.00 cm mark instead.'

在书面作答中，要指出误差名称、分类，并说明其后果。例如："米尺零端磨损，产生系统误差，使每个测量长度偏大；这无法通过取平均减小，但可以改从 1.00 cm 刻度处量起以减小影响。"

4. Absolute, Fractional and Percentage Uncertainty | 绝对、分数与百分比不确定度

tainty is Δx / x, a dimensionless number, and the percentage uncertainty is that fraction multiplied by 100 % forms are interchangeable, and exam questions frequently require you to move between them.

绝对不确定度 Δx 与被测量本身具有相同的单位，例如 ± 0.1 cm。分数不确定度是 Δx / x，它没有量纲；百分比不确定度则是该分数乘以 100 %。这三种形式可以互相转换，考试题经常要求你在它们之间来回换算。

percentage uncertainty = (Δx / x) × 100 %

cm has a fractional uncertainty of 0.1 / 25.0 = 0.004 and a percentage uncertainty of 0.4 %. The same absolute uncertainty applied to a smaller length, say 2.0 cm ± 0.1 cm, gives 5 %, which shows why measuring a small quantity directly is usually poor practice.

例如，25.0 cm ± 0.1 cm 的长度，其分数不确定度为 0.1 / 25.0 = 0.004，百分比不确定度为 0.4 %。同样的绝对不确定度若加在更小的长度上，比如 2.0 cm ± 0.1 cm，则为 5 %，这说明了为什么直接测量很小的量通常是糟糕的做法。

tage uncertainty in y is n times the percentage uncertainty in x. This rule is the single most useful shortcut in Unit 6 calculations.

如果一个量被提升到某个幂次，其百分比不确定度要乘以该幂次。对于 y = xⁿ，y 的百分比不确定度是 x 的百分比不确定度的 n 倍。这条规则是第6单元计算中最有用的捷径。

y = xⁿ → Δy / y = n (Δx / x)

5. Combining Uncertainties in Calculations | 计算中的不确定度合成

When quantities are multiplied or divided, you combine the percentage uncertainties. When they are added or subtracted, you combine the absolute uncertainties. Getting this the wrong way round is one of the most common and most costly mistakes in the paper.

当量之间相乘或相除时，你把它们的百分比不确定度合成；当量之间相加或相减时，你把绝对不确定度合成。把这两种情形弄反，是试卷中最常见、也最失分的错误之一。

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ulum equation g = 4π²L / T². Suppose L = 50.0 cm with an uncertainty of 0.1 cm, giving 0.2 %, and the period is measured as T = 1.42 s with an uncertainty of 0.01 s, giving 0.70 %. Because T is squared, its percentage uncertainty doubles to 1.4 %. A simple addition gives a total uncertainty of about 1.6 %; combining in quadrature, √0.2² + 1.4², gives about 1.4 %. Follow whichever convention your specification uses.

以单摆公式 g = 4π²L / T² 为例。设 L = 50.0 cm，不确定度 0.1 cm，即 0.2 %；周期测得 T = 1.42 s，不确定度 0.01 s，即 0.70 %。由于 T 被平方，其百分比不确定度加倍为 1.4 %。简单相加得总不确定度约 1.6 %；按平方和开方合成，√0.2² + 1.4²，约为 1.4 %。请按你所在考试大纲的惯例处理。

Substituting the values gives g = 4π² × 0.500 / 1.42² = 9.79 m s⁻². Taking 1.4 % esult is g = 9.79 ± 0.14 m s⁻², which is consistent with the accepted value of 9.81 m s⁻².

代入数值可得 g = 4π² × 0.500 / 1.42² = 9.79 m s⁻²。取 9.79 的 1.4 % 得 ± 0.14 m s⁻²，因此最终结果为 g = 9.79 ± 0.14 m s⁻²，与公认值 9.81 m s⁻² 相符。

6. Uncertainty in the Mean and Repeat Readings | 平均值的不确定度与重复读数

Repeating a measurement and taking the mean reduces the effect of random error. The uncertainty in the mean is often estimated as half the range of the readings, that is, (maximum − minimum) / 2. This simple estimate is acceptable in most A-level contexts and is easy to justify in an answer.

重复测量并取平均值可以减小随机误差的影响。平均值的不确定度常取读数极差的一半，即（最大值 − 最小值）/ 2。这种简单估计在多数 A-level 情境下都可接受，而且在作答时容易说明理由。

Suppose five timings of a pendulum' mean is quoted as 14.3 ± 0.2 s for ten oscillations, or 1.43 ± 0.02 s for one oscillation.

假设单摆十次振动的五次计时分别为 14.1 s、14.4 s、14.1 s、14.5 s 和 14.3 s。平均值为 14.28 s，极差的一半为 (14.5 − 14.1) / 2 = 0.2 s，因此十次振动的平均值写作 14.3 ± 0.2 s，单次振动则为 1.43 ± 0.02 s。

You should then compare this spread-based uncertainty with the instrument uncertainty. If the half range is smaller than the resolution-based uncertainty, quote the larger of the two, because the true uncertainty cannot be smaller than what the instrument can resolve. Examiners frequently look for this comparison.

随后你应把基于散布的不确定度与仪器不确定度相比较。如果极差的一半小于基于分辨率的不确定度，就取两者中较大的那个，因为真实不确定度不可能小于仪器所能分辨的程度。考官常常在寻找这一比较。

Statistically, the standard error of the mean decreases as 1 / √n, so quadrupling the number of readings halves the random contribution. In an evaluation, you can say that taking more repeats would reduce random error but would not affect a systematic error.

从统计上说，平均值的标准误按 1 / √n 减小，因此把读数数量增加到四倍，随机误差的贡献减半。在评价中你可以说，增加重复次数会减小随机误差，但对系统误差毫无影响。

7. Recording Data, Units and Significant Figures | 记录数据、单位与有效数字

Data tables should have a clear heading for every column, with the quantity and its unit, for example 'Time t / s' or 'Length L / cm'. Units are separated from the quantity with a slash, not written inside brackets after every entry. Raw readings and processed values should be in separate columns.

数据表应为每一列写清表头，包含物理量及其单位，例如"Time t / s"或"Length L / cm"。单位用斜杠与物理量分开，而不是在每个数据后面加括号写一遍。原始读数与处理后的数值应分列不同栏。

All raw readings in a column should be recorded to the same number of decimal places, matching the resolution of the instrument. If a stopwatch reads to 0.01 s, every raw time should be written with two decimal places, even if that means writing 14.10 s and 14.40 s.

同一列中的所有原始读数应记录到相同的小数位数，并与仪器的分辨率一致。如果秒表读到 0.01 s，每个原始时间都应写两位小数，即使这意味着写成 14.10 s 和 14.40 s。

4 m s⁻² rather than 9.789 ± 0.1372 m s⁻². Carrying extra digits through intermediate steps is fine, but the final answer must be rounded sensibly.

不确定度按惯例保留一位有效数字，测量值本身则四舍五入到相同的小数位。应写作 9.79 ± 0.14 m s⁻²，而不是 9.789 ± 0.1372 m s⁻²。在中间步骤多保留几位数字没问题，但最终答案必须合理取整。

8. Plotting Graphs, Error Bars and Lines of Best Fit | 绘图、误差棒与最佳拟合线

Graphs are the heart of experimental physics, because they let you test a relationship and extract a constant. Axes must be labelled with the quantity, symbol and unit, for example 'T² / s²' on the y-axis and 'L / m' on the x-axis. The scale should use at least half of the printed grid in both directions.

图像是实验物理的核心，因为它让你检验一个关系并提取一个常数。坐标轴必须标注物理量、符号和单位，例如 y 轴标"T² / s²"、x 轴标"L / m"。标度应在两个方向上都至少占满网格的一半。

rror bars make it visually obvious whether a point is genuinely anomalous or simply scattered within uncertainty. If a point lies more than about two error bars from the line, it can be treated as anomalous and should be identified as such.

数据点的不确定度用误差棒表示，即竖直方向 ± Δy、水平方向 ± Δx。误差棒能让你一眼看出某个点究竟是真正的异常点，还是仅在不确定度范围内散布。如果一个点偏离直线超过约两个误差棒，就可视为异常点，并应在作答中指出。

The line of best fit should pass through the error bars of as many points as possible while balancing points above and below it. Do not simply join the first and last points, and do not force the line through the origin unless theory requires it. If theory predicts a straight line through the origin, however, a non-zero intercept suggests a systematic error.

最佳拟合线应尽可能穿过更多点的误差棒，同时使其上方与下方的点数大致平衡。不要简单地把首尾两点连起来，也不要强行让直线过原点，除非理论上要求如此。不过，如果理论预测直线应过原点，那么非零截距就提示存在系统误差。

9. Finding Gradients and Intercepts with Uncertainties | 求斜率与截距及其不确定度

To measure a gradient, choose two points far apart on the line of best fit, not on the original data points, and form a large triangle. The gradient is the rise divided by the run, and its unit is the unit of the y-axis divided by the unit of the x-axis.

求斜率时，应在最佳拟合线上选取相距较远的两点（而不是原始数据点），构成一个大三角形。斜率等于纵向变化除以横向变化，其单位是 y 轴单位除以 x 轴单位。

gradient m = Δy / Δx

emes of the resulting range. Quote the gradient as the best value ± half the range: m = (mmax − mmin) / 2.

斜率的不确定度可通过画出仍然穿过所有误差棒的最陡与最平直线，再取所得范围的两端来求得。斜率写作最佳值 ± 范围的一半：m = (mmax − mmin) / 2。

dient. Writing the physical meaning of the gradient and intercept, with units, is usually worth an explicit mark.

记住要把斜率赋予物理意义。对于 T² 对 L 的图像，斜率为 4π² / g，因此 g = 4π² / 斜率。写出斜率与截距的物理含义并带上单位，通常可以拿到明确的一分。

10. Linearising Relationships with Logarithms | 用对数线性化关系

pulate the equation until it has the form y = mx + c, then plot the appropriate variables.

许多物理关系并非线性，因此必须先线性化才能画成直线。技巧是把方程变形为 y = mx + c 的形式，然后绘出相应的变量。