📚 Combined-215: Mastering Uncertainties in A-Level Physics Practicals | 综合-215：掌握A-Level物理实验中的不确定度
Every measurement in a physics experiment carries some doubt – this is called uncertainty. In Edexcel A-Level Physics, understanding how to quantify and combine uncertainties is essential for evaluating the reliability of your results. Whether you are adding lengths or dividing voltage by current, you need to know how the uncertainties propagate through your calculations. This tutorial breaks down the key rules for combining absolute, relative and percentage uncertainties, with worked examples that mirror the style of exam questions.
物理实验中的每一次测量都带有一定的疑问——这就是不确定度。在Edexcel A-Level物理中，理解如何量化和合成不确定度对评估结果的可靠性至关重要。无论是将长度相加还是用电压除以电流，你都需要知道不确定度如何在计算过程中传递。本教程将分解合成绝对、相对和百分比不确定度的关键规则，并配以与考试题型一致的详细例题。
1. What Are Uncertainties? | 什么是不确定度？
rument, the method and the observer. In A-Level physics, uncertainties are usually expressed as an absolute value (±Δx) or as a percentage (% nt must be recorded with its uncertainty to allow a proper analysis of experimental error.
不确定度是指测量真值预期落入的数值范围。它反映了测量仪器、方法和观察者的局限性。在A-Level物理中，不确定度通常以绝对值（±Δx）或百分比（%不确定度）表示。每次测量都必须记录其不确定度，以便合理分析实验误差。
2. Types of Uncertainties | 不确定度的类型
There are two main types: systematic and random. Systematic uncertainties bias all readings in one direction, often due to a faulty instrument or calibration error, and they affect accuracy. Random uncertainties cause readings to scatter randomly around the true value and affect precision. In practical work we usually deal with random uncertainties, assumed to follow a normal distribution about the mean.
不确定度主要有两种：系统不确定度和随机不确定度。系统不确定度会使所有读数偏向一个方向，通常由仪器故障或校准误差引起，影响准确度。随机不确定度使读数在真值周围随机散布，影响精密度。在实验操作中，我们通常处理的是随机不确定度，并假设它围绕均值呈正态分布。
3. Absolute Uncertainty and Fractional (Relative) Uncertainty | 绝对不确定度与相对不确定度
vision, while for a digital instrument it is the smallest readable digit. The relative uncertainty is Δx / x (no units), and the percentage uncertainty is (Δx / x) × 100%. Converting between these forms is a key skill.
绝对不确定度（Δx）是测量值范围的一半；对于模拟标尺的单次读数，通常取最小刻度的一半，而对于数字仪器则是末位可读数字。相对不确定度为 Δx / x（无单位），百分比不确定度为 (Δx / x) × 100%。熟练转换这些形式是一项关键技能。
4. Combining Uncertainties for Addition and Subtraction | 加减法的不确定度合成
est rule: the total absolute uncertainty is the sum of the individual absolute uncertainties, regardless of whether you are adding or subtracting. Always use the same units.
当两个测量量相加或相减时，绝对不确定度直接相加。若 Q = a + b 或 Q = a − b，则 ΔQ = Δa + Δb。这是最简单的规则：总绝对不确定度等于各绝对不确定度之和，无论加法还是减法都适用。务必使用相同的单位。
5. Combining Uncertainties for Multiplication and Division | 乘除法的不确定度合成
For multiplication or division, you add the relative e) uncertainties. If Q = a × b or Q = a / b, then ΔQ/Q = Δa/a + Δb/b. In percentage terms: % uncertainty in Q = % uncertainty in a + % uncertainty in b. Do not add absolute uncertainties here – tive uncertainty, multiply it by the calculated Q to obtain the absolute uncertainty in Q.
对于乘法或除法，你需要将相对（或百分比）不确定度相加。若 Q = a × b 或 Q = a / b，则 ΔQ/Q = Δa/a + Δb/b。用百分比表示为：Q 的百分比不确定度 = a 的百分比不确定度 + b 的百分比不确定度。切勿直接相加绝对不确定度，那样会高估误差。求出合成后的相对不确定度后，将其乘以计算出的 Q 值，即可得到 Q 的绝对不确定度。
6. Rules for Powers and Roots | 幂和方根的不确定度合成
When a quantity is raised to a power n, the relative uncertainty is multiplied by |n|. If Q = an, then ΔQ/Q = |n| × (Δa/a). For example, if Q = a2 a/a). This rule follows from the derivative method and is essential for quantities like kinetic energy (v2) or the period of a pendulum (T ∝ √L).
当一个量被乘方 n 次时，相对不确定度要乘以 |n|。若 Q = an，则 ΔQ/Q = |n| × (Δa/a)。例如，若 Q = a2，则 Q 的百分比不确定度是 a 的百分比不确定度的两倍。对于平方根，n = ½，因此 ΔQ/Q = ½ × (Δa/a)。这一规则源自求导方法，对动能（v2）或单摆周期（T ∝ √L）等量至关重要。
7. Propagating Uncertainties Through Complex Formulas | 通过复杂公式传递不确定度
tainties in intermediate values, then move to multiplication/division/powers using relative uncertainties. A common example is density ρ = m / V, where V = l1 × l2 × l3. The relative uncertainty in V is the sum of the relative uncertainties in each length, and then ρ’s relative uncertainty is the sum of those of m and V. Break the problem into sections to avoid mistakes.
大多数A-Level计算涉及多种运算的组合。应逐步处理：首先用加减规则求出中间值的绝对不确定度，然后使用相对不确定度处理乘除和乘方。一个常见例子是密度 ρ = m / V，其中 V = l1 × l2 × l3。V 的相对不确定度是各长度相对不确定度之和，然后 ρ 的相对不确定度是 m 和 V 的相对不确定度之和。将问题分解成若干部分可避免出错。
8. Worked Example – Resistivity of a Wire | 例题 – 金属丝的电阻率
eter d = 0.36 ± 0.02 mm, length L = 1.250 ± 0.005 m. Resistivity ρ = RA/L, with A = π(d/2)2. Step 1: calculate A = π × (0.18 mm)2 = 1.018 × 10−7 m2 (after converting mm2 to m2). Relative uncertainty in d: 0.02/0.36 = 0.0556 (5.56%). Since A depends on d2, relative uncertainty in A = 2 × 5.56% = 11.12%. R relative uncertainty = 0.2/4.8 = 0.0417 (4.17%), L = 0.005/1.250 = 0.004 (0.40%). The combined relative uncertainty for ρ = 4.17% + 11.12% + 0.40% = 15.69%. ρ = (4.8 × 1.018×10−7) / 1.250 = 3.91 × 10−7 Ω·m. Absolute uncertainty = 3.91×10−7 × 0.1569 ≈ 0.61×10−7 Ω·m. Final result: ρ = (3.9 ± 0.6) × 10−7 Ω·m, matching the precision of the data.
假设测得电阻 R = 4.8 ± 0.2 Ω，直径 d = 0.36 ± 0.02 mm，长度 L = 1.250 ± 0.005 m。电阻率 ρ = RA/L，其中 A = π(d/2)2。步骤1：计算 A = π × (0.18 mm)2 = 1.018 × 10−7 m2（单位换算后）。d 的相对不确定度：0.02/0.36 = 0.0556（5.56%）。由于 A 依赖于 d2，A 的相对不确定度 = 2 × 5.56% = 11.12%。R 相对不确定度 = 0.2/4.8 = 0.0417（4.17%），L 相对不确定度 = 0.005/1.250 = 0.004（0.40%）。ρ 的合成相对不确定度 = 4.17% + 11.12% + 0.40% = 15.69%。ρ = (4.8 × 1.018×10−7) / 1.250 = 3.91 × 10−7 Ω·m。绝对不确定度 = 3.91×10−7 × 0.1569 ≈ 0.61×10−7 Ω·m。最终结果：ρ = (3.9 ± 0.6) × 10−7 Ω·m，与数据精度匹配。
9. Presentation of Final Results | 最终结果的表示
Always quote the calculated value with its absolute uncertainty and round them consistently. The uncertainty is usually given to 1 significant figure (2 if the first digit is 1 or 2). The main value should be rounded to the same decimal place as the uncertainty. Also state the percentage uncertainty where helpful. For the example above, only 2 significant figures in 0.6 makes the value 3.9 sensible.
始终将计算值与绝对不确定度一同给出，并保持舍入一致性。不确定度通常保留1位有效数字（如果首位数是1或2则可保留2位）。主值应四舍五入至与不确定度相同的小数位。此外，有益时可给出百分比不确定度。上述例子中，不确定度0.6只有1位有效数字，因此主值3.9是合理的。
10. Practical Tips for Minimising Uncertainty | 减小不确定度的实验技巧
Increase the number of readings and take an average to reduce random uncertainty. Use instruments with a better resolution for the largest sources of uncertainty – for example, a micrometer rather than a ruler for measuring wire diameter. Measure multiple periods for oscillations to improve the precision of time. Always zero the instrument before use and avoid parallax errors by reading scales perpendicularly.
增加读数次数并取平均值可减小随机不确定度。对主要的不确定度来源，使用分辨率更高的仪器——例如用千分尺而非直尺测量金属丝直径。测量振荡的多个周期以提高时间精度。使用前务必将仪器调零，并通过垂直于标尺读取来避免视差误差。
11. Common Exam Pitfalls | 常见考试误区
Many students confuse absolute and relative rules, adding absolute uncertainties for multiplication. Another frequent error is forgetting to convert units before calculating relative uncertainties (e.g. mixing millimetres and metres). Also, when a measurement is taken as a difference (like a temperature rise), the uncertainty in the difference is the sum of the absolute uncertainties of the two readings, not a percentage. Be careful with powers: squaring a quantity doubles the percentage uncertainty, but many forget to multiply by the power.
许多学生会混淆绝对与相对规则，在乘法中直接相加绝对不确定度。另一个常见错误是在计算相对不确定度前忘记转换单位（如混淆毫米和米）。此外，当一个测量值是差值（如温升）时，差值的不确定度是两个读数绝对不确定度之和，而非百分比。对于乘方也要小心：平方会使百分比不确定度加倍，但很多人忘记乘以幂指数。
12. Summary – The Core Rules | 总结 – 核心规则
- Addition/subtraction: add absolute uncertainties
- Multiplication/division: add relative (or percentage) uncertainties
- Powers: multiply relative uncertainty by |n|
- Finally, always convert the combined relative uncertainty back to absolute form for the final answer
- 加减法：绝对不确定度相加
- 乘除法：相对（或百分比）不确定度相加
- 乘方：相对不确定度乘以 |n|
- 最后，始终将合成后的相对不确定度转换回绝对形式以便给出最终答案
Master these rules and practise with past paper data to become confident in handling uncertainties – an essential skill for top marks in Edexcel A-Level physics practical evaluations.
掌握这些规则并利用历年真题数据进行练习，你就能自信地处理不确定度——这是在Edexcel A-Level物理实验评估中获得高分的关键技能。
Published by TutorHao | Edexcel A-Level Physics Revision Series | aleveler.com
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