📚 Edexcel A-Level Physics: Combining Uncertainties (1.90) | 组合不确定度
In Edexcel A-Level Physics, the skill of combining uncertainties is central to experimental work and appears across every practical-based question. When you measure a length, time, mass or potential difference, each value carries an uncertainty, and any calculated result must express how those individual uncertainties combine.
在 Edexcel A-Level 物理中，组合不确定度是实验工作的核心技能，并出现在所有以实验为基础的考题中。当你测量长度、时间、质量或电势差时，每个数据都带有不确定度，任何计算结果都必须体现出这些单个不确定度是如何组合的。
1. Why Uncertainty Matters | 为什么不确定度重要
No measurement in physics is perfectly exact. The instruments we use have limited resolution, the experimenter may misread a scale, and the quantity itself may fluctuate. Uncertainty quantifies the range within which the true value is expected to lie, so a result quoted without an uncertainty is of little scientific value.
物理学中没有任何测量是完全精确的。我们使用的仪器分辨率有限，实验者可能读错刻度，被测量本身也可能波动。不确定度量化了真值预计所在的范围，因此没有给出不确定度的结果几乎没有科学价值。
In Edexcel practical questions, examiners expect you to distinguish between resolution, repeatability and reproducibility. For example, a digital voltmeter reading of 2.34 V has a resolution uncertainty of ±0.01 V, but the repeated readings may show a spread of ±0.06 V if the circuit is noisy.
在 Edexcel 实验题中，考官希望你区分分辨率、重复性和再现性。例如，数字电压表读数 2.34 V 的分辨率不确定度为 ±0.01 V，但如果电路噪声较大，重复读数可能显示 ±0.06 V 的波动范围。
2. Types of Error: Random vs Systematic | 随机误差与系统误差
Errors are grouped into two main types. Random errors cause readings to scatter on both sides of the true value and can be reduced by taking repeat measurements. Systematic errors shift all readings in one direction and are not reduced by repetition.
误差主要分为两类。随机误差使读数在真值两侧分散，可以通过多次重复测量来减小。系统误差使所有读数向同一个方向偏移，重复测量不能减小此类误差。
A zero error on a micrometer is a classic systematic error: if the micrometer reads 0.02 mm when fully closed, every length will be 0.02 mm too large until the zero correction is applied. A random error, by contrast, might come from human reaction time in starting and stopping a stopwatch.
千分尺的零误差是典型的系统误差：如果千分尺完全闭合时读数为 0.02 mm，那么在未进行零点修正前，每个长度都会偏大 0.02 mm。相比之下，随机误差可能来自启动和停止秒表时人的反应时间。
3. Key Terms: Accuracy, Precision, Resolution | 准确度、精密度与分辨率
Accuracy describes how close a measured value is to the accepted true value. Precision describes how closely repeated measurements agree with one another. Resolution is the smallest change that the measuring instrument can detect, such as 0.1 mm on a metre ruler or 0.01 s on a digital timer.
准确度描述测量值与公认真值的接近程度。精密度描述重复测量结果之间的彼此一致程度。分辨率是测量仪器能够检测到的最小变化，例如米尺上的 0.1 mm 或数字计时器上的 0.01 s。
It is possible to be precise but inaccurate: if a balance has a systematic offset of +0.05 g, five readings might all be 10.05 g, giving high precision but poor accuracy. Edexcel questions often ask you to identify which term applies to a described situation.
精密度高但准确度差是可能的：如果天平有 +0.05 g 的系统偏移，五次读数可能都是 10.05 g，精密度很高但准确度很差。Edexcel 考题经常要求你判断哪个术语适用于给定的情境。
4. Absolute and Percentage Uncertainty | 绝对不确定度与百分不确定度
Absolute uncertainty has the same unit as the measurement. It may be the instrument resolution, half the range of repeat readings, or a stated value. Percentage uncertainty expresses the absolute uncertainty as a fraction of the measured value multiplied by 100%:
绝对不确定度与测量值具有相同的单位。它可以是仪器分辨率、重复读数范围的一半，或给定的数值。百分不确定度将绝对不确定度表示为测量值的分数再乘以 100%：
percentage uncertainty = (absolute uncertainty ÷ measured value) × 100%
百分不确定度 = （绝对不确定度 ÷ 测量值） × 100%
For example, a current of 2.50 A with an absolute uncertainty of ±0.05 A has a percentage uncertainty of (0.05 ÷ 2.50) × 100% = 2.0%. This relative form is essential when combining uncertainties in products and quotients.
例如，电流为 2.50 A，绝对不确定度为 ±0.05 A，则百分不确定度为 (0.05 ÷ 2.50) × 100% = 2.0%。在乘除运算中组合不确定度时，这种相对形式至关重要。
5. Combining Uncertainties: Addition and Subtraction | 加减法中的不确定度合成
When two or more quantities are added or subtracted, the combined absolute uncertainty is the sum of the individual absolute uncertainties. This is the worst-case rule: even if one error is positive and another negative, the sum gives the maximum possible absolute uncertainty.
当两个或多个量相加或相减时，合成绝对不确定度等于各单个绝对不确定度之和。这是最坏情况规则：即使一个误差为正、另一个为负，求和仍给出最大可能的绝对不确定度。
If R = a + b − c, then ΔR = Δa + Δb + Δc
若 R = a + b − c，则 ΔR = Δa + Δb + Δc
-
Example: a = 8.0 ± 0.2 cm, b = 4.0 ± 0.1 cm, c = 3.0 ± 0.2 cm. Then R = 9.0 cm and ΔR = 0.2 + 0.1 + 0.2 = 0.5 cm.
-
示例：a = 8.0 ± 0.2 cm，b = 4.0 ± 0.1 cm，c = 3.0 ± 0.2 cm。则 R = 9.0 cm，ΔR = 0.2 + 0.1 + 0.2 = 0.5 cm。
Notice that the result should be written as 9.0 ± 0.5 cm, with the absolute uncertainty rounded to one significant figure and the value rounded to the same decimal place.
注意结果应写为 9.0 ± 0.5 cm，绝对不确定度通常保留一位有效数字，测量值四舍五入到相同的小数位。
6. Combining Uncertainties: Multiplication and Division | 乘除法中的不确定度合成
For multiplication and division, percentage uncertainties are combined by addition. Do not add absolute uncertainties when the quantities have different units or scales; convert each to a percentage first, then add the percentages.
对于乘法和除法，百分不确定度通过相加来合成。当各量单位或尺度不同时，不要直接相加绝对不确定度；应先将每个不确定度转换为百分数，再将百分数相加。
If R = a × b ÷ c, then %U(R) = %U(a) + %U(b) + %U(c)
若 R = a × b ÷ c，则 %U(R) = %U(a) + %U(b) + %U(c)
Worked example: a force F = 20.0 ± 0.4 N acts over a distance d = 0.50 ± 0.02 m. The work done W = Fd = 10.0 J. The percentage uncertainties are 2.0% for F and 4.0% for d, so the combined percentage uncertainty is 6.0%. The absolute uncertainty in W is 10.0 × 0.06 = 0.6 J, giving W = 10.0 ± 0.6 J.
计算示例：力 F = 20.0 ± 0.4 N 作用在距离 d = 0.50 ± 0.02 m 上。做功 W = Fd = 10.0 J。F 的百分不确定度为 2.0%，d 为 4.0%，因此合成百分不确定度为 6.0%。W 的绝对不确定度为 10.0 × 0.06 = 0.6 J，因此 W = 10.0 ± 0.6 J。
7. Power Rules and Scaling | 幂次法则与比例缩放
When a quantity is raised to a power, multiply the percentage uncertainty by that power. This arises because the quantity appears multiple times in the product. For a square, the percentage uncertainty doubles; for a cube, it triples.
当一个量被乘方时，需要将百分不确定度乘以该指数。这是因为该量在乘积中出现了多次。对于平方，百分不确定度加倍；对于立方，则变为三倍。
If R = aⁿ, then %U(R) = n × %U(a)
若 R = aⁿ，则 %U(R) = n × %U(a)
For example, the volume V of a cube with side length x = 5.0 ± 0.1 cm is x³ = 125 cm³. The percentage uncertainty in x is 2.0%, so the percentage uncertainty in V is 3 × 2.0% = 6.0%. The absolute uncertainty is 125 × 0.06 = 7.5 cm³, quoted as V = 125 ± 8 cm³ after rounding.
例如，边长为 x = 5.0 ± 0.1 cm 的立方体体积为 x³ = 125 cm³。x 的百分不确定度为 2.0%，因此 V 的百分不确定度为 3 × 2.0% = 6.0%。绝对不确定度为 125 × 0.06 = 7.5 cm³，四舍五入后可表示为 V = 125 ± 8 cm³。
Common scaling mistakes include forgetting to convert absolute to percentage uncertainty before applying the power, or applying the power to the absolute uncertainty directly. Always use percentage form for powers and products.
常见的缩放错误包括在应用幂次前忘记将绝对不确定度转换为百分不确定度，或将幂次直接应用于绝对不确定度。对于幂次和乘积，务必使用百分数形式。
8. Graphical Methods: Error Bars and Best/Worst Lines | 误差棒与最佳/最差拟合线
In Edexcel practical work, you often plot a graph and use its gradient or intercept. Error bars represent the absolute uncertainty in each plotted value. A best-fit line passes through the points, while worst-fit lines are drawn to give the maximum and minimum plausible gradients.
在 Edexcel 实验工作中，你经常需要绘制图线并利用其斜率或截距。误差棒表示每个绘图点的绝对不确定度。最佳拟合线穿过这些点，而最差拟合线用于给出最大和最小合理斜率。
To find the uncertainty in a gradient, draw the steepest and shallowest lines that still pass through the error bars. If the best gradient is m and the worst gradients are m₁ and m₂, then:
要确定斜率的不确定度，可画出仍然穿过误差棒的最陡和最浅线。如果最佳斜率为 m，两个最差斜率为 m₁ 和 m₂，则：
Δm = (m₁ − m₂) ÷ 2
Δm = （m₁ − m₂） ÷ 2
This method is especially useful for pendulum experiments, resistor networks and radioactive decay analysis, where the gradient has a physical meaning such as g or a decay constant.
该方法特别适用于单摆实验、电阻网络和放射性衰变分析，这些情况下斜率具有物理意义，例如 g 或衰变常数。
9. Experimental Design: Reducing Uncertainty | 实验设计：减小不确定度
A well-designed A-Level experiment reduces random uncertainty without introducing new systematic errors. Common techniques include measuring multiple periods instead of one, using larger distances or masses, and using instruments with better resolution.
设计良好的 A-Level 实验可以在不引入新系统误差的情况下减小随机不确定度。常用方法包括测量多个周期而不是单个周期、使用更大的距离或质量，以及使用分辨率更高的仪器。
-
Measure 20 oscillations of a pendulum and divide by 20: this reduces the timing uncertainty per oscillation by a factor of 20.
-
测量单摆的 20 次振动并除以 20：这将每次振动的时间不确定度降低到原来的 1/20。
-
Use a longer length in a free-fall experiment: the same absolute distance uncertainty becomes a smaller percentage uncertainty.
-
在
Published by TutorHao | A-Level 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