📚 Measurements and Uncertainties for IB Physics HL and OCR Physics A | IB物理HL与OCR物理A的测量与不确定度
In both IB Physics HL and OCR A Level Physics, the first topic lays the foundation for all experimental work: measurements, units, and uncertainties. A measurement is only useful if we know how reliable it is. This article covers the concepts shared by the two specifications, including SI units, significant figures, systematic and random errors, absolute and percentage uncertainties, propagation of uncertainties, and graphical analysis using error bars. Mastering these ideas will improve your practical write-ups and your exam performance.
在IB物理HL和OCR A-Level物理中，第一个主题为所有实验工作打下基础：测量、单位与不确定度。只有知道测量有多可靠，它才有用。本文涵盖两个大纲共有的概念，包括国际单位制、有效数字、系统误差与随机误差、绝对和百分比不确定度、不确定度的传播以及使用误差棒的图像分析。掌握这些内容将提升你的实验报告水平和考试成绩。
1. SI Units and Fundamental Quantities | 国际单位制与基本物理量
The International System of Units (SI) defines seven base quantities: length (metre, m), mass (kilogram, kg), time (second, s), electric current (ampere, A), temperature (kelvin, K), amount of substance (mole, mol), and luminous intensity (candela, cd). IB Physics HL and OCR both require you to recall these base units and use them in dimensional analysis.
国际单位制定义了七个基本量：长度（米，m）、质量（千克，kg）、时间（秒，s）、电流（安培，A）、温度（开尔文，K）、物质的量（摩尔，mol）和发光强度（坎德拉，cd）。IB物理HL和OCR都要求你记住这些基本单位，并能在量纲分析中使用它们。
| Base quantity | Unit name | Symbol |
|---|---|---|
| length | metre | m |
| mass | kilogram | kg |
| time | second | s |
| electric current | ampere | A |
| temperature | kelvin | K |
| amount of substance | mole | mol |
| luminous intensity | candela | cd |
All other units are derived from these seven base units. For example, force is measured in newtons, where 1 N = 1 kg m s⁻². Being able to express a derived unit in base units is a common exam skill in both specifications.
所有其他单位都由这七个基本单位导出。例如，力的单位是牛顿，1 N = 1 kg m s⁻²。能够用基本单位表示导出单位是两个大纲中常见的考试技能。
2. Derived Units and Homogeneity | 导出单位与量纲一致性
A derived unit can be written as a product or quotient of base units. For instance, the unit of pressure, the pascal, is N m⁻² = kg m⁻¹ s⁻². Checking that both sides of an equation have the same base units is called checking for homogeneity. This is a powerful tool for spotting mistakes in algebra or recalled formulas.
导出单位可以写成基本单位的乘积或商。例如，压强单位帕斯卡表示为 N m⁻² = kg m⁻¹ s⁻²。检查方程两边是否具有相同的基本单位称为检查量纲一致性。这是发现代数或记错公式错误的强大工具。
[pressure] = kg m⁻¹ s⁻²
For kinetic energy E = ½ mv², the unit is kg × (m s⁻¹)² = kg m² s⁻². If both sides of an energy equation do not reduce to kg m² s⁻², the equation cannot be correct.
对于动能 E = ½ mv²，其单位为 kg × (m s⁻¹)² = kg m² s⁻²。如果能量方程的两边不能化为 kg m² s⁻²，该方程就不可能是正确的。
3. Significant Figures and Rounding | 有效数字与修约
Significant figures (s.f.) show the precision of a measured or calculated value. Leading zeros are not significant, trailing zeros after a decimal point are significant, and zeros between non-zero digits are significant. For example, 0.00350 has three significant figures.
有效数字显示测量值或计算值的精密度。前导零不计入有效数字，小数点后的尾随零计入，非零数字之间的零也计入。例如，0.00350 有三位有效数字。
When multiplying or dividing, quote your answer to the same number of significant figures as the least precise input. When adding or subtracting, use the least number of decimal places. Avoid rounding at intermediate steps; keep extra digits and round only the final answer.
乘除运算时，答案的有效数字位数应与精度最低的输入值相同。加减运算时，应使用最少的小数位数。避免在中间步骤修约；保留额外位数，只对最终答案进行修约。
- 0.0045 has 2 s.f.
- 1.0045 has 5 s.f.
- 10.0 has 3 s.f.
中文对应：0.0045 有 2 位有效数字；1.0045 有 5 位有效数字；10.0 有 3 位有效数字。
4. Accuracy vs Precision | 准确度与精确度
Accuracy describes how close a measured value is to the true or accepted value. Precision describes how closely repeated measurements agree with one another, regardless of whether they are correct. A set of readings can be precise but inaccurate if a systematic error is present. In exam questions, you must distinguish the two terms clearly.
准确度描述测量值与真值或公认值的接近程度。精确度描述重复测量之间的一致程度，而不论它们是否正确。如果存在系统误差，一组读数可能精确但不准确。在考试题目中，你必须清楚地区分这两个术语。
Think of darts thrown at a target: tight grouping far from the bullseye is precise but not accurate; scattered around the bullseye is accurate on average but less precise. Similar reasoning applies to laboratory measurements.
想象向靶心掷飞镖：集中在远离靶心的一处是精确但不准确；散布在靶心周围平均而言是准确但不够精确。类似的推理也适用于实验室测量。
5. Systematic and Random Errors | 系统误差与随机误差
Systematic errors shift every measurement in the same direction. Common sources include a zero error on a balance, a wrongly calibrated instrument, or a parallax error from not viewing a scale squarely. They affect accuracy, not precision.
系统误差使每次测量都朝同一方向偏移。常见来源包括天平未归零、仪器校准错误或未正视标尺造成的视差。系统误差影响准确度而非精确度。
Random errors cause readings to scatter unpredictably. Sources include reaction time, vibrations, or noise in electronic equipment. Repeating measurements and taking an average reduces random error but does not remove systematic error.
随机误差导致读数不可预测地分散。来源包括反应时间、振动或电子设备中的噪声。重复测量并取平均值可减少随机误差，但不能消除系统误差。
6. Absolute, Fractional and Percentage Uncertainty | 绝对、相对与百分比不确定度
Every measurement has an absolute uncertainty, often equal to half the smallest scale division for an analogue instrument or the smallest digit for a digital instrument. The absolute uncertainty is written as ±Δx. Fractional uncertainty is Δx / x, and percentage uncertainty is (Δx / x) × 100%.
每个测量都有绝对不确定度，对于模拟仪器通常等于最小刻度的一半，对于数字仪器则等于最小位数。绝对不确定度写作 ±Δx。相对不确定度为 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%
A length measured as 24.0 cm ± 0.1 cm has percentage uncertainty (0.1 / 24.0) × 100% ≈ 0.42%. Percentage uncertainties are used when combining measurements by multiplication, division, or powers.
某长度测得 24.0 cm ± 0.1 cm，其百分比不确定度为 (0.1 / 24.0) × 100% ≈ 0.42%。在乘除或幂次运算中合成测量值时，要使用百分比不确定度。
7. Reading Scales and Digital Instruments | 读数标尺与数字仪器
For an analogue scale such as a ruler or voltmeter, the absolute uncertainty is usually taken as half the smallest division. For example, a ruler with 1 mm divisions gives ±0.5 mm for a single reading, but if two readings are needed to find a length, the total uncertainty is ±1 mm.
对于尺子或电压表等模拟标尺，绝对不确定度通常取最小刻度的一半。例如，刻度为 1 mm 的尺子单次读数不确定度为 ±0.5 mm，但如果需要两次读数来求
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