📚 Combining Uncertainties (Spec 1.4.4) | 不确定度的合成（考点 1.4.4）
When you take a measurement in A-level physics, the result is never absolutely exact. Every instrument has a limited resolution, and every experiment has random and systematic errors. This article explains how to combine uncertainties when you use measured values in calculations, following the Edexcel specification point 1.4.4.
在 A-level 物理中，每次测量都不可能绝对精确。每种仪器都有有限的分辨率，每个实验都存在随机误差和系统误差。本文按照 Edexcel 考纲 1.4.4 的要求，解释如何在计算中合成测量值的不确定度。
1. Understanding Uncertainty | 理解不确定度
Uncertainty is the interval around a measured value that is expected to contain the true value. It is not the same as a mistake; it reflects the limitations of the measuring instrument and the method.
不确定度是测量值周围的一个区间，真值预期落在该区间内。它不同于错误，它反映的是测量仪器和测量方法的局限性。
For a single reading, the absolute uncertainty is usually taken as plus or minus the smallest scale division or half of it, depending on the instrument. For a digital instrument, it is often the smallest digit.
对于单次读数，绝对不确定度通常取最小分度值或其一半，具体取决于仪器。对于数字仪器，通常取最后一位数字的一个单位。
absolute uncertainty = ± smallest scale division (or half)
绝对不确定度 = ± 最小分度值（或一半）
2. Absolute, Fractional and Percentage Uncertainty | 绝对、相对和百分比不确定度
Absolute uncertainty is the uncertainty expressed in the same unit as the measurement. Fractional uncertainty is the ratio of absolute uncertainty to the measured value, and percentage uncertainty is this ratio multiplied by 100%.
绝对不确定度是用与测量值相同单位表示的不确定度。相对不确定度是绝对不确定度与测量值的比值，百分比不确定度是这个比值乘以 100%。
% uncertainty = (absolute uncertainty ÷ measured value) × 100%
百分比不确定度 = （绝对不确定度 ÷ 测量值） × 100%
For example, a length measured as 20.0 cm ± 0.5 cm has an absolute uncertainty of 0.5 cm and a percentage uncertainty of 2.5%.
例如，长度测量为 20.0 cm ± 0.5 cm，绝对不确定度为 0.5 cm，百分比不确定度为 2.5%。
3. Why Combine Uncertainties? | 为什么要合成不确定度？
Most physical quantities are not measured directly. You measure two or more variables and then calculate a result, such as speed from distance and time. Each measured variable carries its own uncertainty, so the final calculated value also has an uncertainty. Combining uncertainties allows you to state a final result with a realistic range.
大多数物理量不是直接测量的。你需要测量两个或多个变量，然后计算出结果，例如由距离和时间算出速度。每个被测量的变量都有自己的不确定度，因此最终计算值也有不确定度。合成不确定度可以让你给出一个具有现实范围的最终结果。
Edexcel exams often ask you to combine uncertainties using a small set of rules. These rules depend on whether the quantities are added, subtracted, multiplied, divided or raised to a power.
Edexcel 考试经常要求你使用一组简单的规则来合成不确定度。这些规则取决于量是相加、相减、相乘、相除还是乘方。
4. Adding and Subtracting Quantities | 加减量的合成规则
When two measured quantities are added or subtracted, their absolute uncertainties add together. This gives the largest possible absolute uncertainty in the result.
当两个测量量相加或相减时，它们的绝对不确定度相加。这给出结果中可能的最大绝对不确定度。
If R = A + B or R = A – B, then:
如果 R = A + B 或 R = A – B，则：
ΔR = ΔA + ΔB
ΔR = ΔA + ΔB
Here ΔR, ΔA and ΔB are the absolute uncertainties in R, A and B. Note that even when you subtract quantities, the uncertainties still add, never cancel.
这里 ΔR、ΔA 和 ΔB 分别是 R、A、B 的绝对不确定度。注意，即使量之间是相减，不确定度仍然相加，永远不会相消。
- Example: A = 10.0 cm ± 0.1 cm, B = 5.0 cm ± 0.2 cm, so R = A + B = 15.0 cm ± 0.3 cm.
- 示例：A = 10.0 cm ± 0.1 cm，B = 5.0 cm ± 0.2 cm，所以 R = A + B = 15.0 cm ± 0.3 cm。
5. Multiplying and Dividing Quantities | 乘除量的合成规则
When two measured quantities are multiplied or divided, their percentage uncertainties add together. This rule applies because relative errors combine in the same way for products and quotients.
当两个测量量相乘或相除时，它们的百分比不确定度相加。这个规则适用，因为相对误差在乘法和除法中以相同的方式合成。
If R = A × B or R = A ÷ B, then:
如果 R = A × B 或 R = A ÷ B，则：
%U_R = %U_A + %U_B
百分比U_R = 百分比U_A + 百分比U_B
Equivalently, you can write this in fractional form:
等价地，你可以用分数形式表示：
ΔR/R = ΔA/A + ΔB/B
ΔR/R = ΔA/A + ΔB/B
After finding the percentage uncertainty in R, convert it back to an absolute uncertainty by multiplying by the calculated value of R.
求出 R 的百分比不确定度后，将其乘以 R 的计算值，即可转换回绝对不确定度。
- Example: A = 20.0 ± 0.4, B = 5.0 ± 0.1, so %U_A = 2%, %U_B = 2%, therefore %U_R = 4%.
- 示例：A = 20.0 ± 0.4，B = 5.0 ± 0.1，所以百分比U_A = 2%，百分比U_B = 2%，因此百分比U_R = 4%。
6. Raising to a Power | 幂次合成规则
When a measured quantity is raised to a power, the percentage uncertainty in the result is the percentage uncertainty in the quantity multiplied by the absolute value of the power.
当测量量被乘方时，结果的百分比不确定度等于该量的百分比不确定度乘以指数的绝对值。
If R = Aⁿ, then:
如果 R = Aⁿ，则：
%U_R = |n| × %U_A
百分比U_R = |n| × 百分比U_A
This can be extended to formulas with several variables. For R = k Aᵐ Bⁿ, where k is a constant with no uncertainty:
这可以推广到含有多个变量的公式。对于 R = k Aᵐ Bⁿ，其中 k 是无不确定度的常数：
%U_R = |m| × %U_A + |n| × %U_B
百分比U_R = |m| × 百分比U_A + |n| × 百分比U_B
For example, the period of a simple pendulum depends on the square root of its length, so the power on L is ½. The percentage uncertainty in the period is therefore half the percentage uncertainty in the length.
例如，单摆的周期与摆长的平方根相关，因此 L 的指数是 ½。所以周期的百分比不确定度是摆长百分比不确定度的一半。
7. Worked Example: Ohm’s Law | 例题：欧姆定律
A resistor is tested by measuring the potential difference V = 2.50 V ± 0.05 V and the current I = 0.50 A ± 0.02 A. Calculate the resistance R = V ÷ I and the absolute uncertainty in R.
通过测量电势差 V = 2.50 V ± 0.05 V 和电流 I = 0.50 A ± 0.02 A 来测试一个电阻。计算电阻 R = V ÷ I 以及 R 的绝对不确定度。
Step 1: Find the percentage uncertainties.
步骤 1：求百分比不确定度。
%U_V = (0.05 ÷ 2.50) × 100% = 2%
%U_I = (0.02 ÷ 0.50) × 100% = 4%
Step 2: Since R is found by division, add the percentage uncertainties.
步骤 2：因为 R 由除法得到，所以将百分比不确定度相加。
%U_R = 2% + 4% = 6%
Step 3: Calculate R and then its absolute uncertainty.
步骤 3：计算 R，再求绝对不确定度。
R = 2.50 V ÷ 0.50 A = 5.00 Ω
ΔR = 6% × 5.00 Ω = 0.30 Ω
Final result: R = 5.00 Ω ± 0.30 Ω.
最终结果：R = 5.00 Ω ± 0.30 Ω。
8. Uncertainties from Graphs | 从图像获得不确定度
Many A-level experiments involve plotting a graph to find a gradient or intercept. The uncertainty in a gradient can be estimated by drawing a best-fit line and a worst-fit line, then comparing their gradients.
许多 A-level 实验涉及绘制图像来求斜率或截距。斜率的不确定度可以通过绘制最佳拟合线和最差拟合线，然后比较它们的斜率来估计。
Error bars on points show the uncertainty in each plotted value. A typical method is to draw the steepest and shallowest lines that still pass through all the error bars. The gradient uncertainty is half the difference between the maximum and minimum gradients.
数据点上的误差棒表示每个绘制值的不确定度。典型方法是画出仍然通过所有误差棒的最陡和最平缓的直线。斜率不确定度是最大斜率与最小斜率差值的一半。
This graphical method is especially useful when the relationship is linear, such as in a force-extension experiment for Hooke’s law.
当关系为线性时，这种图像方法特别有用，例如在胡克定律的力-伸长实验中。
9. Common Pitfalls and Exam Tips | 常见错误与考试技巧
Many students lose marks by forgetting to convert percentage uncertainty back to absolute uncertainty at the end of a calculation. Always check the unit and the number of significant figures in your final uncertainty.
许多学生因为在计算结束时忘记将百分比不确定度转换回绝对不确定度而丢分。务必检查最终不确定度的单位和有效数字位数。
Another common error is using the wrong rule. Remember: add absolute uncertainties for addition and subtraction, but add percentage uncertainties for multiplication and division.
另一个常见错误是使用错误的规则。记住：加减时合成为绝对不确定度，乘除时合成为百分比不确定度。
- Use half the range when you repeat a measurement several times.
- 重复测量多次时，使用极差的一半作为不确定度。
- State uncertainties to one or two significant figures, and match the decimal places of the measured value.
- 不确定度保留一位或两位有效数字，并与测量值的小数位匹配。
- Always include units for absolute uncertainties.
- 绝对不确定度始终包含单位。
10. Practice Questions | 练习题
1. A student measures mass m = 200 g ± 4 g and volume V = 50 cm³ ± 2 cm³. Calculate the density ρ = m ÷ V and its percentage uncertainty.
1. 某学生测量质量 m = 200 g ± 4 g 和体积 V = 50 cm³ ± 2 cm³。计算密度 ρ = m ÷ V 及其百分比不确定度。
2. A distance s = 40.0 m ± 0.5 m and time t = 5.0 s ± 0.2 s are used to find average speed v = s ÷ t. Determine the absolute uncertainty in v.
2. 使用距离 s = 40.0 m ± 0.5 m 和时间 t = 5.0 s ± 0.2 s 求平均速度 v = s ÷ t。确定 v 的绝对不确定度。
3. The power P dissipated in a resistor is given by P = I²R, where I = 2.00 A ± 0.05 A and R = 10.0 Ω ± 0.2 Ω. Find the percentage uncertainty in P.
3. 电阻上消耗的功率由 P = I²R 给出，其中 I = 2.00 A ± 0.05 A，R = 10.0 Ω ± 0.2 Ω。求 P 的百分比不确定度。
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