📚 Combining Uncertainties | A-Level物理中的不确定度合成
In Edexcel A-Level Physics, Topic 1.3 expects you to handle measured quantities and their uncertainties. Combining uncertainties correctly is essential for practical work, data analysis, and written Paper 3 questions.
在Edexcel A-Level物理中，Topic 1.3要求你掌握测量量及其不确定度。正确合成不确定度对实验操作、数据分析以及笔试Paper 3都至关重要。
1. Why Uncertainties Matter | 为什么不确定度很重要
Every measurement has a limit of precision. If you calculate a quantity from several measurements, the final answer cannot be more precise than the least precise input. Combining uncertainties propagates these limits into the result.
每次测量都有精度极限。若由多个测量量计算某个量，最终答案不可能比最不精确的输入更精确。合成不确定度就是把这些极限传递到结果中。
Edexcel questions often ask you to state an uncertainty, estimate a percentage uncertainty, or compare results using uncertainties. Clear propagation is a key skill.
Edexcel题目常要求写出不确定度、估计百分不确定度，或用不确定度比较实验结果。清晰的传递方法是关键技能。
2. Absolute and Percentage Uncertainty | 绝对不确定度与百分不确定度
Absolute uncertainty Δx is the interval within which the true value lies, for example x = 25.0 ± 0.2 cm. Percentage uncertainty is (Δx / x) × 100%.
绝对不确定度 Δx 是真值所在的区间，例如 x = 25.0 ± 0.2 cm。百分不确定度为 (Δx / x) × 100%。
For a 25.0 cm measurement with Δx = 0.2 cm, percentage uncertainty = (0.2 / 25.0) × 100% = 0.80%.
对于25.0 cm、Δx = 0.2 cm的测量，百分不确定度 = (0.2 / 25.0) × 100% = 0.80%。
| Instrument / 仪器 | Typical absolute uncertainty / 典型绝对不确定度 |
| Metre rule / 米尺 | ±1 mm |
| Micrometer screw gauge / 螺旋测微器 | ±0.01 mm |
| Digital voltmeter / 数字电压表 | ±1 in last digit unless stated / 未注明时末位±1 |
| Stopwatch / 秒表 | ±0.2 s for human reaction / 人手反应±0.2 s |
3. Combining Uncertainties: Addition and Subtraction | 加减运算中的不确定度合成
When quantities are added or subtracted, add the absolute uncertainties. This is the worst-case combination and is accepted at A-Level.
当量做加减运算时，将绝对不确定度相加。这是最坏情况的合成方式，A-Level考试中接受。
R = A + B − C ⇒ ΔR = ΔA + ΔB + ΔC
If x = 12.0 ± 0.1 cm and y = 4.0 ± 0.2 cm, then L = x − y = 8.0 cm and ΔL = 0.1 + 0.2 = 0.3 cm, giving L = 8.0 ± 0.3 cm.
若 x = 12.0 ± 0.1 cm、y = 4.0 ± 0.2 cm，则 L = x − y = 8.0 cm，ΔL = 0.1 + 0.2 = 0.3 cm，所以 L = 8.0 ± 0.3 cm。
Do not add percentage uncertainties here. For subtraction especially, the percentage uncertainty can become very large if the result is small, but the rule still uses absolute uncertainties.
这里不要加百分不确定度；尤其做减法时结果很小，百分不确定度会显得很大，但规则仍然使用绝对不确定度。
4. Combining Uncertainties: Multiplication and Division | 乘除运算中的不确定度合成
When quantities are multiplied or divided, add the percentage uncertainties. This applies to both simple products and quotients.
当量做乘除运算时，将百分不确定度相加。这适用于简单的乘积和商。
R = AB / C ⇒ %ΔR = %ΔA + %ΔB + %ΔC
Suppose F = 20.0 ± 0.2 N and d = 5.0 ± 0.1 m. Work done W = Fd = 100 J. Here %ΔF = 1.0%, %Δd = 2.0%, so %ΔW = 3.0%. Then ΔW = 0.030 × 100 = 3 J, so W = 100 ± 3 J.
示例：F = 20.0 ± 0.2 N，d = 5.0 ± 0.1 m。功 W = Fd = 100 J。其中 %ΔF = 1.0%，%Δd = 2.0%，所以 %ΔW = 3.0%。于是 ΔW = 0.030 × 100 = 3 J，即 W = 100 ± 3 J。
Always convert to percentage uncertainties before adding in multiplication or division. Never add absolute uncertainties directly in these cases.
乘除运算前必须先转换为百分不确定度再相加。这类情况下绝不能直接相加绝对不确定度。
5. Combining Uncertainties: Powers and Roots | 幂和开方中的不确定度合成
If a quantity is raised to a power n, multiply the percentage uncertainty by n. A root counts as a fractional power: √x = x^(1/2), so n = 1/2.
若某量取 n 次幂，则百分不确定度乘以 n。开方按分数幂处理：√x = x^(1/2)，故 n = 1/2。
R = Aⁿ ⇒ %ΔR = |n| × %ΔA
For example, the volume of a sphere is V = (4/3)πr³, so %ΔV = 3 × %Δr. If r = 10.0 ± 0.1 cm, then %Δr = 1.0%, so %ΔV = 3.0%.
例如球的体积 V = (4/3)πr³，所以 %ΔV = 3 × %Δr。若 r = 10.0 ± 0.1 cm，则 %Δr = 1.0%，因此 %ΔV = 3.0%。
For a root such as T = 2π√(L/g), the power on L is 1/2, so %ΔT from L is 0.5 × %ΔL. Factors like 2π and g-as-constant carry no uncertainty.
对于如 T = 2π√(L/g) 的开方关系，L 的幂为 1/2，所以由 L 引起的 %ΔT 为 0.5 × %ΔL。2π 和作为常数的 g 不引入不确定度。
6. Multiplying by a Constant | 乘以常数时的处理
A constant has no uncertainty. For R = kA, the absolute uncertainty becomes ΔR = kΔA; the percentage uncertainty stays the same as A.
常数没有不确定度。对于 R = kA，绝对不确定度变为 ΔR = kΔA；百分不确定度与 A 保持不变。
R = kA ⇒ ΔR = kΔA and %ΔR = %ΔA
If d = 5.0 ± 0.2 cm, then the circumference C = πd = 15.7 cm gives ΔC = π × 0.2 = 0.63 cm ≈ 0.6 cm. So C = 15.7 ± 0.6 cm.
若 d = 5.0 ± 0.2 cm，则圆周长 C = πd = 15.7 cm，ΔC = π × 0.2 = 0.63 cm ≈ 0.6 cm。因此 C = 15.7 ± 0.6 cm。
7. Mixed Operations and Step-by-Step Propagation | 混合运算与逐步传递
For a formula with several operation types, propagate uncertainties in exactly the same order as the arithmetic. Keep intermediate values with one extra significant figure, then round at the end.
对包含多种运算的公式，按照算术顺序逐步传递不确定度。中间值保留一位多余的有效数字，最后再修约。
For example, in Ek = ½mv², first handle v²: %ΔEk = %Δm + 2 × %Δv. Then convert the final percentage uncertainty back to absolute uncertainty.
例如 Ek = ½mv²，先处理 v²：%ΔEk = %Δm + 2 × %Δv，最后再把最终百分不确定度转回绝对不确定度。
- Step 1: Compute the nominal result.
- Step 2: Combine percentage or absolute uncertainties as the rules dictate.
- Step 3: Round final uncertainty to 1 significant figure, or 2 if the first digit is 1.
- 第一步：计算标称结果。
- 第二步：根据规则合成百分或绝对不确定度。
- 第三步：最终不确定度保留1位有效数字；若首位是1可保留2位。
8. Uncertainty Bars and Graphical Analysis | 误差棒与图像分析
Error bars show absolute uncertainty on a graph. For a straight line fitted to data, draw the best-fit line and at least one worst-fit line — steepest or shallowest — that passes near the error bars.
误差棒在图中表示绝对不确定度。对数据拟合直线时，画最佳拟合线和至少一条最坏拟合线——最陡或最浅——并使其通过误差棒附近。
The gradient uncertainty can be estimated as: Δm = |m_best − m_worst|. A large gap between the extreme gradients means a large uncertainty in the gradient.
斜率不确定度可估算为：Δm = |m_best − m_worst|。极端斜率之间的差距大，意味着斜率不确定度大。
For a quantity obtained from the vertical intercept, use the same max-min method: take the largest and smallest plausible intercepts from your extreme lines.
对于由纵截距得到的量，使用相同的最大-最小法：从极端线中取最大和最小的合理截距。
9. Significant Figures in Final Answers | 最终答案的有效数字
Quote final uncertainties
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