AS Physics: Measurements and Errors – Deriving Uncertainty Formulas | AS物理：测量与误差

📚 AS Physics: Measurements and Errors – Deriving Uncertainty Formulas | AS物理：测量与误差 – 不确定度公式推导

In AS Physics, the topic of Measurements and Their Errors is fundamental because no experimental result is meaningful without a quantified uncertainty. This article focuses on the step-by-step derivations of the key uncertainty propagation formulas that are required for OxfordAQA International AS Physics. We will build up from basic definitions to combination rules for sums, differences, products, quotients, and powers, and finally discuss a worked example using a pendulum to determine g.

在AS物理中，测量及其误差主题至关重要，因为任何实验结果如果缺乏量化的不确定度都没有意义。本文重点详细推导牛津AQA国际AS物理所要求的核心误差传播公式。我们将从基本定义开始，逐步建立和、差、积、商和幂的组合规则，最后通过单摆测定g的例子展示实际应用。

1. Absolute, Relative, and Percentage Uncertainties | 绝对不确定度、相对不确定度与百分比不确定度

Any measured quantity x is reported as x ± Δx, where Δx is the absolute uncertainty representing the range within which the true value is expected to lie. The absolute uncertainty has the same units as the quantity itself.

任何测量量x都表示为 x ± Δx，其中 Δx 是绝对不确定度，表示真值预期所在的范围。绝对不确定度与测量量具有相同的单位。

The fractional uncertainty (or relative uncertainty) is the ratio Δx/x, a dimensionless number. The percentage uncertainty is simply the fractional uncertainty multiplied by 100%. These relative forms are essential when combining uncertainties through multiplication or division.

相对不确定度（或称分数不确定度）是比值 Δx/x，是一个无量纲数。百分比不确定度就是将相对不确定度乘以 100%。在进行乘除运算时，这些相对形式非常重要。

2. Uncertainty in Direct Measurements | 直接测量中的不确定度

For a single reading taken from a digital instrument, the absolute uncertainty is taken as the smallest division of the display. For an analogue scale, it is half of the smallest scale division. When repeated readings are taken, the absolute uncertainty can be estimated by half the range (the spread) of the values, or by using the standard deviation where appropriate.

对于数字仪器的一次读数，绝对不确定度取为显示器的最小刻度。对于模拟量表，取最小刻度的一半。当进行多次读数时，绝对不确定度可以用测量值的半区间（极差的一半）来估计，或者在适当情况下使用标准偏差。

If a length L is measured five times giving readings of 1.23 m, 1.25 m, 1.24 m, 1.23 m, 1.25 m, the range is 0.02 m, so ΔL ≈ 0.01 m. The result is expressed as L = 1.24 ± 0.01 m.

若长度L 被测量五次，读数为 1.23 m、1.25 m、1.24 m、1.23 m、1.25 m，则极差为 0.02 m，故 ΔL ≈ 0.01 m。结果表示为 L = 1.24 ± 0.01 m。

3. Derivation for Sums and Differences | 和与差的误差公式推导

Consider two measured quantities A = a ± Δa and B = b ± Δb. Let Z = A + B. The maximum possible value of Z is Z_max = (a + Δa) + (b + Δb) = (a + b) + (Δa + Δb). The minimum possible value is Z_min = (a − Δa) + (b − Δb) = (a + b) − (Δa + Δb). Thus the absolute uncertainty in Z is ΔZ = Δa + Δb.

考虑两个测量量 A = a ± Δa 和 B = b ± Δb。设 Z = A + B。Z 的最大可能值为 Z_max = (a + Δa) + (b + Δb) = (a + b) + (Δa + Δb)，最小可能值为 Z_min = (a − Δa) + (b − Δb) = (a + b) − (Δa + Δb)。因此，Z 的绝对不确定度为 ΔZ = Δa + Δb。

For the difference Z = A − B, we apply the worst-case method. The maximum of Z occurs when A is as large as possible and B is as small as possible: Z_max = (a + Δa) − (b − Δb) = (a − b) + (Δa + Δb). The minimum is (a − Δa) − (b + Δb) = (a − b) − (Δa + Δb). Hence, once again ΔZ = Δa + Δb.

对于差 Z = A − B，我们采用最坏情况法。当 A 尽可能大而 B 尽可能小时，Z 取最大值：Z_max = (a + Δa) − (b − Δb) = (a − b) + (Δa + Δb)。最小值为 (a − Δa) − (b + Δb) = (a − b) − (Δa + Δb)。因此，同样 ΔZ = Δa + Δb。

Rule: For Z = A ± B, ΔZ = Δa + Δb.

规则：对于 Z = A ± B，ΔZ = Δa + Δb。

This simple additive rule tells us that uncertainties always add in absolute terms for addition and subtraction, making the overall uncertainty at most the sum of individual absolute uncertainties.

这一简单的加法规则告诉我们，在加法和减法中，不确定度总是以绝对值相加，使得总不确定度至多为各个绝对不确定度之和。

4. Derivation for Products | 乘积的误差公式推导

Let Z = A B. Using the measured values a and b and their uncertainties, we can compute the maximum expected value of Z: Z_max = (a + Δa)(b + Δb) = ab + aΔb + bΔa + ΔaΔb.

设 Z = A B。利用测量值 a 和 b 及其不确定度，可以计算 Z 的最大预期值：Z_max = (a + Δa)(b + Δb) = ab + aΔb + bΔa + ΔaΔb。

The term ΔaΔb is a product of two small uncertainties and is much smaller than the other terms; it can be neglected. Hence the absolute uncertainty in Z is approximately ΔZ ≈ aΔb + bΔa.

项 ΔaΔb 是两个小不确定度的乘积，远小于其他项；可以忽略。因此，Z 的绝对不确定度近似为 ΔZ ≈ aΔb + bΔa。

Dividing by the central value Z = ab, we obtain the fractional uncertainty:

除以中心值 Z = ab，得到相对不确定度：

ΔZ/Z ≈ (aΔb + bΔa) / ab = Δb/b + Δa/a.

ΔZ/Z ≈ (aΔb + bΔa) / ab = Δb/b + Δa/a。

Thus for multiplication, the fractional uncertainties add. The same result can be obtained by considering the natural logarithm: ln Z = ln A + ln B, leading to dZ/Z = dA/A + dB/B, and converting differentials to finite uncertainties gives the same additive combination. In terms of percentage uncertainties, the percentage uncertainty in Z is the sum of the percentage uncertainties in A and B.

因此，对于乘法，相对不确定度相加。通过考虑自然对数也可得到相同结果：ln Z = ln A + ln B，导出 dZ/Z = dA/A + dB/B，并将微分转换为有限不确定度，得到相同的加和组合。就百分比不确定度而言，Z 的百分比不确定度等于 A 和 B 的百分比不确定度之和。

5. Derivation for Quotients | 商的误差公式推导

For division, Z = A/B. Again we apply the worst-case scenario. The maximum value of Z occurs with the largest numerator and smallest denominator: Z_max = (a + Δa)/(b − Δb).

对于除法，Z = A/B。再次应用最坏情况。当分子最大而分母最小时，Z 取得最大值：Z_max = (a + Δa)/(b − Δb)。

We can rewrite the fraction using series expansion or algebraic manipulation: Z_max = (a(1+Δa/a))/(b(1−Δb/b)) ≈ (a/b) (1+Δa/a)(1+Δb/b + (Δb/b)² + …). Ignoring second-order terms, we get Z_max ≈ (a/b)(1 + Δa/a + Δb/b). Similarly, Z_min ≈ (a/b)(1 − Δa/a − Δb/b).

我们可以通过级数展开或代数操作改写分式：Z_max = (a(1+Δa/a))/(b(1−Δb/b)) ≈ (a/b) (1+Δa/a)(1+Δb/b + (Δb/b)² + …)。忽略二阶及更高阶项，得到 Z_max ≈ (a/b)(1 + Δa/a + Δb/b)。类似地，Z_min ≈ (a/b)(1 − Δa/a − Δb/b)。

Therefore the fractional uncertainty is ΔZ/Z ≈ Δa/a + Δb/b, exactly the same as for a product. The rule for multiplication and division is unified: fractional uncertainties add when quantities are multiplied or divided.

因此，相对不确定度为 ΔZ/Z ≈ Δa/a + Δb/b，与乘积的情形完全相同。乘法和除法的规则是统一的：当量被乘或除时，相对不确定度相加。

6. Derivation for Powers | 幂函数的误差公式推导

Let Z = Aⁿ, where n is a constant (may be positive, negative, or fractional). Using the binomial expansion for a small uncertainty Δa:

设 Z = Aⁿ，其中 n 为常数（可以是正数、负数或分数）。利用微小不确定度 Δa 的二项式展开：

(a + Δa)ⁿ = aⁿ (1 + Δa/a)ⁿ ≈ aⁿ [1 + n (Δa/a) + …].

(a + Δa)ⁿ = aⁿ (1 + Δa/a)ⁿ ≈ aⁿ [1 + n (Δa/a) + …]。

The first-order approximation gives the change in Z: ΔZ ≈ n aⁿ⁻¹ Δa. Dividing by Z = aⁿ, we get:

一阶近似给出 Z 的变化：ΔZ ≈ n aⁿ⁻¹ Δa。除以 Z = aⁿ，得出：

ΔZ/Z ≈ |n| (Δa/a).

ΔZ/Z ≈ |n| (Δa/a)。

The absolute value is used because an uncertainty in A contributes a proportional change irrespective of the sign of n. For example, if n = −1 (reciprocal), the relative uncertainty is the same as that in A. For n = ½ (square root), the fractional uncertainty is half that of A.

使用绝对值是因为无论 n 的符号如何，A 的不确定度都会贡献一个成比例的变化。例如，若 n = −1（倒数），相对不确定度与 A 的相对不确定度相同。若 n = ½（平方根），相对不确定度是 A 的一半。

7. General Approach Using Partial Derivatives | 使用偏导数的一般方法

When a quantity is a function of several variables, Z = f(A, B, …), the worst-case absolute uncertainty can be derived from the total differential. Treating uncertainties as small changes:

当某个量是多个变量的函数，Z = f(A, B, …)，最坏情况下的绝对不确定度可以从全微分推导。将不确定度视为微小变化：

ΔZ ≈ |∂f/∂A| Δa + |∂f/∂B| Δb + …

Here, the partial derivatives are evaluated at the measured values, and the absolute values ensure all contributions add constructively (worst-case). This general formula reduces to our earlier rules:

此处，偏导数在测量值处计算，绝对值确保所有贡献正向相加以获得最坏情况。此一般公式可归约到之前的规则：

For Z = A + B, ∂f/∂A = 1, ∂f/∂B = 1 ⇒ ΔZ = Δa + Δb.

对于 Z = A + B，∂f/∂A = 1，∂f/∂B = 1 ⇒ ΔZ = Δa + Δb。
For Z = A B, ∂f/∂A = B, ∂f/∂B = A ⇒ ΔZ = b Δa + a Δb, which divided by ab gives the fractional form.

对于 Z = A B，∂f/∂A = B，∂f/∂B = A ⇒ ΔZ = b Δa + a Δb，除以 ab 即得相对形式。
For Z = Aⁿ, ∂f/∂A = n Aⁿ⁻¹ ⇒ ΔZ = n aⁿ⁻¹ Δa ⇒ ΔZ/Z = n Δa/a (absolute value taken).

对于 Z = Aⁿ，∂f/∂A = n Aⁿ⁻¹ ⇒ ΔZ = n aⁿ⁻¹ Δa ⇒ ΔZ/Z = n Δa/a（取绝对值）。

This approach is extremely powerful and can be used for any formula, including trigonometric or logarithmic functions, although at AS level the focus is on algebraic combinations.

这种方法非常强大，可用于任何公式，包括三角函数或对数函数，尽管在AS阶段重点还是代数组合。

8. Combining Rules: Density of a Sphere | 组合规则实例：球体密度

Imagine measuring the mass m and radius r of a solid sphere to find its density ρ = m / ((4/3)πr³). The constant factor 4/3 π does not affect the fractional uncertainty. Treat ρ ∝ m r⁻³. Using the product and power rules:

假设测量一个实心球的质量m 和半径r 以求其密度 ρ = m / ((4/3)πr³)。常数因子 4/3 π 不影响相对不确定度。视 ρ ∝ m r⁻³。应用乘积和幂规则：

Δρ/ρ = Δm/m + 3 (Δr/r).

Δρ/ρ = Δm/m + 3 (Δr/r)。

Suppose m = 1.20 ± 0.01 kg and r = 0.050 ± 0.001 m. The fractional uncertainty in mass is 0.01/1.20 = 0.00833 (0.833%), and in radius it is 0.001/0.050 = 0.020 (2.0%). The fractional uncertainty in density is 0.00833 + 3 × 0.020 = 0.06833, or 6.8%. This direct addition illustrates how the cube on r amplifies the uncertainty.

设 m = 1.20 ± 0.01 kg，r = 0.050 ± 0.001 m。质量的相对不确定度为 0.01/1.20 = 0.00833（0.833%），半径的相对不确定度为 0.001/0.050 = 0.020（2.0%）。密度的相对不确定度为 0.00833 + 3 × 0.020 = 0.06833，即 6.8%。这一直接相加说明 r 的三次方放大了不确定度。

The absolute uncertainty in ρ can then be calculated by multiplying the

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AS Physics: Measurements and Errors – Deriving Uncertainty Formulas | AS物理：测量与误差 – 不确定度公式推导