📚 Mastering Combining Uncertainties in Edexcel A-Level Physics | 掌握爱德思A-Level物理中的不确定度合成
Every measurement in Edexcel A-Level Physics has a limit to its precision. Combining uncertainties is the tool that lets you report how reliable a calculated result is, whether you are finding density, acceleration, or the gradient of a graph.
在爱德思A-Level物理中，每一次测量都有其精度限制。合成不确定度正是用来判断计算结果有多可靠的工具，无论你是在求密度、加速度，还是图像斜率。
1. Why Uncertainties Matter | 为什么不确定度重要
Experimental results are never absolutely exact. An uncertainty tells another scientist the range within which the true value probably lies, and it allows a fair comparison between two measured values.
实验结果从来都不是绝对精确的。不确定度告诉其他科学家真实值可能所在的范围，并且可以对两个测量值进行公平比较。
Examiners reward clear uncertainty calculations because they show you understand the limits of the data, not just the final number.
考官会奖励清晰的不确定度计算，因为这表明你理解数据的局限性，而不只是得到一个最终数字。
2. Absolute, Fractional and Percentage Uncertainties | 绝对、相对和百分比不确定度
An absolute uncertainty has the same units as the quantity itself. Fractional uncertainty is absolute uncertainty divided by the measured value, and percentage uncertainty multiplies this fraction by 100%.
绝对不确定度与物理量本身具有相同的单位。相对不确定度是绝对不确定度除以测量值，百分比不确定度则是将这个比值乘以100%。
fractional uncertainty = Δx ÷ x
percentage uncertainty = (Δx ÷ x) × 100%
For example, a length measured as L = 25.0 ± 0.2 cm has a fractional uncertainty of 0.2 ÷ 25.0 = 0.008 and a percentage uncertainty of 0.8%.
例如，长度测量为 L = 25.0 ± 0.2 cm 时，其相对不确定度为 0.2 ÷ 25.0 = 0.008，百分比不确定度为 0.8%。
3. Reading Instruments and Resolution | 仪器读数与分辨率
The resolution of an instrument is the smallest change it can display. For an analogue scale such as a ruler or a moving-coil meter, the absolute uncertainty is usually taken as half of the smallest scale division.
仪器的分辨率是它能显示的最小变化量。对于诸如直尺或动圈式电表之类的模拟刻度，绝对不确定度通常取最小刻度分度的一半。
For a digital instrument, the absolute uncertainty is normally the smallest digit displayed, unless the manufacturer states otherwise.
对于数字仪器，绝对不确定度通常为其显示的最小位数，除非制造商另有说明。
| Instrument | Typical absolute uncertainty |
|---|---|
| Metre ruler | ±0.5 mm or ±1 mm |
| Vernier calliper | ±0.01 cm |
| Micrometer screw gauge | ±0.01 mm |
| Digital ammeter | ± last displayed digit |
4. Combining Uncertainties: Addition and Subtraction | 不确定度合成：加减法
When two quantities are added or subtracted, their absolute uncertainties are added directly. This gives the maximum likely uncertainty in the result.
当两个量相加或相减时，它们的绝对不确定度直接相加。这样得到结果中可能出现的最大不确定度。
If y = a + b or y = a − b, then Δy = Δa + Δb
For example, if two lengths are 12.0 ± 0.1 cm and 8.0 ± 0.1 cm, their difference is 4.0 ± 0.2 cm because 0.1 + 0.1 = 0.2.
例如，两个长度分别为 12.0 ± 0.1 cm 和 8.0 ± 0.1 cm，它们的差值为 4.0 ± 0.2 cm，因为 0.1 + 0.1 = 0.2。
5. Combining Uncertainties: Multiplication and Division | 不确定度合成：乘除法
For multiplication or division, you add percentage uncertainties rather than absolute uncertainties. This works because each variable contributes proportionally to the final result.
对于乘法或除法，你需要将百分比不确定度相加，而不是绝对不确定度相加。这是因为每个变量按比例影响最终结果。
If y = (a × b) ÷ c, then %Δy = %Δa + %Δb + %Δc
Suppose speed v = distance ÷ time, with d = 20.0 ± 0.2 m and t = 4.0 ± 0.1 s. The percentage uncertainties are 1.0% and 2.5%, so the total percentage uncertainty is 3.5%. The speed is 5.0 m s⁻¹, so the absolute uncertainty is 0.035 × 5.0 = 0.175 ≈ 0.18 m s⁻¹.
假设速度 v = 距离 ÷ 时间，其中 d = 20.0 ± 0.2 m，t = 4.0 ± 0.1 s。百分比不确定度分别为 1.0% 和 2.5%，因此总百分比不确定度为 3.5%。速度为 5.0 m s⁻¹，所以绝对不确定度为 0.035 × 5.0 = 0.175 ≈ 0.18 m s⁻¹。
6. Power Rules and Mixed Calculations | 幂次规则与混合运算
When a quantity is raised to a power, multiply the percentage uncertainty by the modulus of that power. A square root is the same as a power of one-half, so its percentage uncertainty is half the original.
当一个量被乘方时，需要将百分比不确定度乘以该幂次的绝对值。平方根相当于二分之一次幂，因此其百分比不确定度是原来的一半。
If y = aⁿ, then %Δy = |n| × %Δa
For example, the volume of a sphere is V = (4/3)πr³, so the percentage uncertainty in V is three times the percentage uncertainty in r. If r = 2.00 ± 0.01 cm, then %Δr = 0.5%, and %ΔV = 3 × 0.5% = 1.5%.
例如，球体体积为 V = (4/3)πr³，因此 V 的百分比不确定度是 r 的百分比不确定度的三倍。如果 r = 2.00 ± 0.01 cm，则 %Δr = 0.5%，%ΔV = 3 × 0.5% = 1.5%。
7. Repeated Readings and Mean Uncertainty | 重复读数与平均值的不足
Taking repeated readings reduces random uncertainty and gives a more reliable mean value. A simple A-level estimate of the uncertainty in the mean is half the range of the repeated values.
进行多次重复读数可以减少随机不确定度，并给出更可靠的平均值。A-level 中一个简单的平均值不确定度估计是重复值范围的一半。
mean x̄ = Σx ÷ N, uncertainty in mean = range ÷ 2
If three readings are 10.1 s, 10.2 s and 10.0 s, the mean is 10.1 s and the range is 0.2 s, so the uncertainty is ±0.1 s. The result is reported as 10.1 ± 0.1 s.
如果三次读数分别为 10.1 s、10.2 s 和 10.0 s，平均值为 10.1 s，范围为 0.2 s，因此不确定度为 ±0.1 s。结果应报告为 10.1 ± 0.1 s。
8. Graphical Analysis: Error Bars | 图解法：误差棒
When plotting experimental data, each point should have an error bar representing its absolute uncertainty. The length of the bar is twice the absolute uncertainty, centred on the plotted point.
在绘制实验数据时，每个点都应带有表示其绝对不确定度的误差棒。误差棒的长度是绝对不确定度的两倍，并以所绘点为中心。
Error bars allow you to judge whether a straight line or a curve is genuinely supported by the data, and they show which points are less reliable.
误差棒可以帮助你判断数据是否真正支持直线或曲线关系，并显示出哪些点不太可靠。
9. Uncertainty in Gradients and Intercepts | 斜率和截距的不确定度
To find the uncertainty in a gradient, draw the best-fit line and then the steepest and shallowest reasonable lines that still pass through the error bars. The gradient uncertainty is half the difference between the maximum and minimum slopes.
为了求出斜率的不确定度，先画出最佳拟合线，然后画出仍然能穿过误差棒的最陡和最缓合理直线。斜率的不确定度就是最大斜率与最小斜率差值的一半。
Δgradient = (slope_max − slope_min) ÷ 2
The same approach can be applied to the intercept by reading the maximum and minimum intercept values from the alternative lines.
同样的方法也可以应用于截距，即从两条替代直线中读取最大截距和最小截距。
10. Practical Worked Example | 实际解题示例
A cylinder has mass m = 20.0 ± 0.1 g, diameter d = 1.00 ± 0.01 cm and height h = 5.00 ± 0.05 cm. Calculate its density and the uncertainty in the density.
一个圆柱体质量 m = 20.0 ± 0.1 g，直径 d = 1.00 ± 0.01 cm，高度 h = 5.00 ± 0.05 cm。计算其密度以及密度的不确定度。
Radius r = d ÷ 2 = 0.50 cm, so its uncertainty is 0.005 cm. The volume is V = πr²h = π × 0.25 × 5.00 = 3.927 cm³. Density ρ = m ÷ V = 20.0 ÷ 3.927 = 5.09 g cm⁻³.
半径 r = d ÷ 2 = 0.50 cm，因此其不确定度为 0.005 cm。体积为 V = πr²h = π × 0.25 × 5.00 = 3.927 cm³。密度 ρ = m ÷ V = 20.0 ÷ 3.927 = 5.09 g cm⁻³。
%Δρ = %Δm + 2 × %Δr + %Δh
Percentage uncertainties are m: 0.5%, r: 1.0% so 2 × r: 2.0%, h: 1.0%. Total is 3.5%. Absolute uncertainty = 5.09 × 0.035 = 0.178 ≈ 0.18 g cm⁻³. The density is 5.09 ± 0.18 g cm⁻³.
百分比不确定度分别为 m：0.5%，r：1.0%（故 2 × r 为 2.0%），h：1.0%。总和为 3.5%。绝对不确定度 = 5.09 × 0.035 = 0.178 ≈ 0.18 g cm⁻³。密度为 5.09 ± 0.18 g cm⁻³。
11. Common Pitfalls and Exam Tips | 常见错误与考试技巧
Do not mix absolute and percentage uncertainties in the same rule. Addition and subtraction require absolute uncertainties, while multiplication and division require percentage uncertainties.
不要在同一条规则中混用绝对不确定度和百分比不确定度。加法和减法使用绝对不确定度，而乘法和除法使用百分比不确定度。
Always round an uncertainty to one significant figure unless the question states otherwise, and match the final calculated value to the same number of decimal places as the uncertainty.
除非题目另有说明，否则始终将不确定度四舍五入到一位有效数字，并将最终计算值的位数与不确定度的小数位数保持一致。
When a formula contains both addition and multiplication, combine the addition step first using absolute uncertainties, then
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Cancel reply