Measurements and Uncertainties | 测量与不确定度

📚 Measurements and Uncertainties | 测量与不确定度

In physics, making reliable measurements and understanding their limitations are essential for validating theories and models. This tutorial covers the IB Physics topic of measurements and uncertainties, including SI units, errors, uncertainty analysis, and graphical methods. By mastering these concepts, you will be able to design experiments, process data, and critically evaluate results in line with the IB internal assessment criteria.

在物理学中，进行可靠的测量并理解其局限性对于验证理论和模型至关重要。本教程涵盖 IB 物理中测量与不确定度这一主题，包括国际单位制、误差、不确定度分析以及图解方法。掌握这些概念后，你将能够根据 IB 内部评估标准设计实验、处理数据并批判性地评估结果。

1. Fundamental and Derived SI Units | 基本单位与导出单位

The International System of Units (SI) defines seven base units: metre (m) for length, kilogram (kg) for mass, second (s) for time, ampere (A) for electric current, kelvin (K) for thermodynamic temperature, mole (mol) for amount of substance, and candela (cd) for luminous intensity. All other units are derived from these, such as the newton (N = kg m s⁻²) or the joule (J = kg m² s⁻²).

国际单位制（SI）定义了七个基本单位：米（m）用于长度，千克（kg）用于质量，秒（s）用于时间，安培（A）用于电流，开尔文（K）用于热力学温度，摩尔（mol）用于物质的量，坎德拉（cd）用于发光强度。所有其他单位均可从这些基本单位导出，例如牛顿（N = kg m s⁻²）或焦耳（J = kg m² s⁻²）。

Checking the homogeneity of physical equations using dimensional analysis ensures that both sides have the same base units. This is a powerful tool for detecting errors in derived formulas.

使用量纲分析检验物理方程的一致性，可确保等式两边具有相同的基本单位。这是发现导出公式中错误的有力工具。

2. Scientific Notation and Metric Multipliers | 科学记数法与公制倍数

Scientific notation expresses numbers as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. This format simplifies working with very large or very small quantities, such as the speed of light 3.00 × 10⁸ m s⁻¹ or the Planck constant 6.63 × 10⁻³⁴ J s.

科学记数法将数字表示为 a × 10ⁿ 的形式，其中 1 ≤ a < 10，n 为整数。这种格式简化了极大或极小量的处理，例如光速 3.00 × 10⁸ m s⁻¹ 或普朗克常数 6.63 × 10⁻³⁴ J s。

Metric multipliers (prefixes) like kilo (k, 10³), mega (M, 10⁶), milli (m, 10⁻³), and micro (µ, 10⁻⁶) are used to avoid writing many zeros. Below is a summary of common SI prefixes.

公制倍数（前缀）如千（k, 10³）、兆（M, 10⁶）、毫（m, 10⁻³）、微（µ, 10⁻⁶）用于避免书写大量零。以下是常见 SI 前缀的总结。

Prefix	Symbol	Factor	Example
tera	T	10¹²	1 THz (terahertz) / 太赫兹
giga	G	10⁹	1 GW (gigawatt) / 吉瓦
mega	M	10⁶	1 MeV (megaelectronvolt) / 兆电子伏
kilo	k	10³	1 kg (kilogram) / 千克
centi	c	10⁻²	1 cm (centimetre) / 厘米
milli	m	10⁻³	1 mA (milliampere) / 毫安
micro	µ	10⁻⁶	1 µF (microfarad) / 微法
nano	n	10⁻⁹	1 nm (nanometre) / 纳米

3. Significant Figures and Orders of Magnitude | 有效数字与数量级

The number of significant figures in a measured value reflects the precision of the measurement. Rules: all non‑zero digits are significant; zeros between non‑zero digits are significant; leading zeros are not; trailing zeros in a number with a decimal point are significant. When multiplying or dividing, the result should have the same number of significant figures as the least precise input.

测量值中有效数字的位数反映了测量的精度。规则是：所有非零数字均为有效数字；非零数字之间的零为有效数字；前导零无效；带小数点数字的末尾零为有效数字。在乘除运算中，结果的有效数字位数应与最少精度的输入量相同。

The order of magnitude of a number is the power of ten when expressed in scientific notation; it provides a rough comparison of sizes. For example, the diameter of an atom (~10⁻¹⁰ m) is 10 orders of magnitude smaller than the diameter of a sand grain (~10⁻³ m).

一个数的数量级是其科学记数法中 10 的幂次；它提供了一种粗略的尺寸比较。例如，原子的直径（约 10⁻¹⁰ m）比沙粒的直径（约 10⁻³ m）小十个数量级。

4. Types of Errors: Random and Systematic | 误差类型：随机误差与系统误差

Random errors cause readings to be scattered about the true value; they arise from unpredictable fluctuations in readings, such as reaction time or electronic noise. They can be reduced by averaging multiple measurements. Systematic errors cause a consistent deviation in one direction and cannot be reduced by averaging; they often arise from faulty apparatus, incorrect calibration, or zero offset.

随机误差导致读数在真实值周围散开；它们源于读数中不可预测的波动，例如反应时间或电子噪声。可以通过多次测量取平均值来减小。系统误差导致读数朝一个方向一致偏离，无法通过取平均值减小；通常源于仪器故障、校准错误或零点偏移。

An example of a systematic error is a mass balance that reads 0.2 g when nothing is on it; all masses will be overestimated by 0.2 g. A random error could be parallax error when reading a meniscus, which may vary from reading to reading.

系统误差的一个例子是空载时读数为 0.2 g 的天平；所有质量都会被高估 0.2 g。随机误差的例子可能是读取弯月面时的视差，这种误差每次读数都可能不同。

5. Accuracy and Precision | 准确度与精确度

Accuracy refers to how close a measured value is to the true or accepted value. Precision refers to the degree of agreement among repeated measurements. A set of measurements can be precise but not accurate, if a systematic error is present. Conversely, measurements can be accurate on average but imprecise if there is large random scatter.

准确度指测量值与真实值或公认值的接近程度。精确度指多次测量结果之间的一致程度。当存在系统误差时，一组测量可能精确但不准确。相反，如果存在较大的随机散差，测量平均值可能准确但不够精确。

Think of a dartboard: precise throws cluster tightly, while accurate throws cluster around the bullseye. In experimental physics, we aim for both high precision and high accuracy.

以飞镖靶为例：精确的投掷紧密聚集，准确的投掷则围绕靶心。在实验物理中，我们追求高精确度和高准确度的兼得。

6. Absolute, Fractional, and Percentage Uncertainties | 绝对不确定度、分数不确定度和百分比不确定度

Every measurement has an uncertainty, often estimated as ± half the smallest scale division, or from the range of repeated readings. The absolute uncertainty Δx has the same units as the measurement. The fractional uncertainty is Δx / x, and the percentage uncertainty is (Δx / x) × 100%.

每个测量都有不确定度，通常估计为最小刻度值的一半，或由重复读数的范围确定。绝对不确定度 Δx 与测量值单位相同。分数不确定度为 Δx / x，百分比不确定度为 (Δx / x) × 100%。

absolute uncertainty = Δx, fractional uncertainty = Δx/x, percentage uncertainty = (Δx/x) × 100%

绝对不确定度 = Δx，分数不确定度 = Δx/x，百分比不确定度 = (Δx/x) × 100%

For a digital voltmeter reading of 2.34 mV, the absolute uncertainty might be ±0.01 mV (half the last digit). So the percentage uncertainty = (0.01/2.34)×100% ≈ 0.43%.

对于一个数字电压表读数 2.34 mV，绝对不确定度可能为 ±0.01 mV（最后一位的一半）。因此百分比不确定度 = (0.01/2.34)×100% ≈ 0.43%。

7. Combining Uncertainties: Addition and Subtraction | 不确定度的合成：加减运算

When quantities are added or subtracted, their absolute uncertainties add. If a = b + c or a = b − c, then the uncertainty in a is Δa = Δb + Δc. Note that the uncertainties are always added, never subtracted, because the worst-case combination must account for the maximum possible deviation.

当物理量相加或相减时，它们的绝对不确定度相加。若 a = b + c 或 a = b − c，则 a 的不确定度为 Δa = Δb + Δc。注意不确定度总是相加，绝不减去，因为最坏情况的组合必须考虑最大可能的偏离。

Example: Two lengths L₁ = (20.0 ± 0.2) cm and L₂ = (15.0 ± 0.3) cm give a total length L = 35.0 cm with absolute uncertainty ΔL = 0.2 + 0.3 = 0.5 cm. The result is expressed as L = (35.0 ± 0.5) cm.

示例：两个长度 L₁ = (20.0 ± 0.2) cm 和 L₂ = (15.0 ± 0.3) cm，总长 L = 35.0 cm，绝对不确定度 ΔL = 0.2 + 0.3 = 0.5 cm。结果表示为 L = (35.0 ± 0.5) cm。

8. Combining Uncertainties: Multiplication, Division, and Powers | 不确定度的合成：乘除与幂运算

For multiplication and division, fractional uncertainties add. If a = b × c or a = b / c, then:

Δa/a = Δb/b + Δc/c

接着，对于乘法和除法，分数不确定度相加。若 a = b × c 或 a = b / c，则：

Δa/a = Δb/b + Δc/c

If a quantity is raised to a power, a = bⁿ, the fractional uncertainty is multiplied by the magnitude of the exponent:

Δa/a = |n| × (Δb/b)

若某物理量被乘方，a = bⁿ，则分数不确定度乘以指数的绝对值：

Δa/a = |n| × (Δb/b)

Example: The speed v is calculated from distance d = (100.0 ± 1.0) m and time t = (10.0 ± 0.2) s. Then v = 10.0 m s⁻¹. Fractional uncertainties: Δd/d = 0.010, Δt/t = 0.020. Therefore, Δv/v = 0.010 + 0.020 = 0.030. Absolute uncertainty Δv = 0.030 × 10.0 = 0.3 m s⁻¹, so v = (10.0 ± 0.3) m s⁻¹.

示例：速度 v 由距离 d = (100.0 ± 1.0) m 和时间 t = (10.0 ± 0.2) s 计算得出。v = 10.0 m s⁻¹。分数不确定度：Δd/d = 0.010，Δt/t = 0.020。因此，Δv/v = 0.010 + 0.020 = 0.030。绝对不确定度 Δv = 0.030 × 10.0 = 0.3 m s⁻¹，故 v = (10.0 ± 0.3) m s⁻¹。

9. Graphing Physical Data and Error Bars | 物理数据作图与误差棒

Error bars on a graph indicate the uncertainty range of each data point. The length of the bar represents the absolute uncertainty in that variable. For linear graphs, it is common to plot vertical error bars for the dependent variable when the independent variable’s uncertainties are negligible. If both variables have significant uncertainties, both vertical and horizontal error bars are used.

图中的误差棒表示每个数据点的不确定度范围。误差棒的长度代表该变量的绝对不确定度。对于线性图，当自变量的不确定度可忽略时，通常绘制因变量的垂直误差棒。如果两个变量都有显著的不确定度，则同时使用垂直和水平误差棒。

When drawing a best-fit line, the line should pass as close as possible to all points, staying within the error bars where possible. The line does not necessarily have to pass through every point, especially outliers that lie far from the trend.

在绘制最佳拟合线时，应使直线尽可能靠近所有点，并尽可能留在误差棒范围内。该直线不必通过每个点，尤其是不必通过远离趋势的离群点。

10. Determining Uncertainty in Slope and Intercept | 斜率与截距的不确定度

To find the uncertainty in the gradient, draw maximum and minimum acceptable lines of fit (the steepest and shallowest lines that still pass through most error bars). Calculate their gradients m_max and m_min. The uncertainty in the gradient is Δm = (m_max − m_min) / 2. The best gradient is m_best = (m_max + m_min) / 2, or obtained from the best-fit line.

要确定斜率的不确定度，绘制最大和最小可接受拟合线（仍通过大多数误差棒的最陡和最浅直线）。计算它们的斜率 m_max 和 m_min。斜率的不确定度为 Δm = (m_max − m_min) / 2。最佳斜率 m_best = (m_max + m_min) / 2，或由最佳拟合线得到。

Similarly, the uncertainty in the intercept can be found by reading the intercepts of these extreme lines and applying the same subtraction method: Δc = (c_max − c_min) / 2. Always express the final gradient and intercept with their absolute uncertainties.

类似地，截距的不确定度可通过读取这两条极端线的截距并应用相同的减法获得：Δc = (c_max − c_min) / 2。最终斜率和截距必须附带其绝对不确定度来表示。

11. Worked Example: Resistivity of a Metal Wire | 示例分析：金属丝的电阻率

A student measures the length L = (0.845 ± 0.005) m, diameter d = (0.42 ± 0.01) mm, and resistance R = (8.2 ± 0.2) Ω of a metal wire. The cross-sectional area A = πd²/4, and resistivity ρ = RA/L. Calculate ρ and its uncertainty.

某学生测量了一根金属丝的长度 L = (0.845 ± 0.005) m，直径 d = (0.42 ± 0.01) mm，电阻 R = (8.2 ± 0.2) Ω。横截面积 A = πd²/4，电阻率 ρ = RA/L。计算 ρ 及其不确定度。

First, convert d to metres: d = 0.42 × 10⁻³ m, Δd = 0.01 × 10⁻³ m. Area A = π × (0.42×10⁻³)² / 4 ≈ 1.385 × 10⁻⁷ m². The fractional uncertainty in d

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Measurements and Uncertainties | 测量与不确定度