📚 Fundamental Concepts of Measurement and Uncertainty | 测量与不确定度的基本概念
Physics is an empirical science. Every law, every formula, and every prediction ultimately rests on measurements. But no measurement is perfect. Understanding how to quantify the quality of a measurement and how to carry uncertainty through calculations is a core skill in IB Physics.
物理学是一门实验科学。每一条定律、每一个公式、每一个预测最终都建立在测量之上。但没有任何测量是完美的。理解如何量化测量的质量，以及如何在计算中传递不确定度，是IB物理的核心技能。
1. Physical Quantities and Units | 物理量与单位
A physical quantity is a property that can be measured and expressed by a numerical value and a unit. For example, the height of a student could be 1.75 m, where 1.75 is the numerical value and m (metre) is the unit. The value of a physical quantity has no meaning without its unit.
物理量是一种可以通过数值和单位来测量和表示的性质。例如，一位学生的身高可以是1.75 m，其中1.75是数值，m（米）是单位。没有单位，物理量的数值就没有意义。
In IB Physics, you must distinguish between base quantities and derived quantities. Base quantities are defined independently, while derived quantities are formed from base quantities through algebraic combinations.
在IB物理中，你必须区分基本量和导出量。基本量是独立定义的，而导出量是通过基本量的代数组合形成的。
- Base quantities: mass (kg), length (m), time (s), electric current (A), temperature (K), amount of substance (mol), luminous intensity (cd).
- 基本量：质量（kg）、长度（m）、时间（s）、电流（A）、温度（K）、物质的量（mol）、发光强度（cd）。
- Derived quantities: speed (m s⁻¹), force (kg m s⁻²), energy (kg m² s⁻²), pressure (kg m⁻¹ s⁻²).
- 导出量：速度（m s⁻¹）、力（kg m s⁻²）、能量（kg m² s⁻²）、压强（kg m⁻¹ s⁻²）。
2. The International System of Units (SI) | 国际单位制（SI）
The SI system provides standard definitions for all base units, ensuring that measurements are consistent worldwide. In IB Physics, you are expected to know the definitions of the SI base units and be able to convert between multiples and submultiples using prefixes.
国际单位制为所有基本单位提供了标准定义，确保世界各地的测量保持一致。在IB物理中，你需要知道SI基本单位的定义，并能够使用词头在倍数和分数单位之间进行换算。
Common prefixes range from pico (p, 10⁻¹²) to tera (T, 10¹²). A table of important prefixes is often needed when handling very large or very small quantities, such as the radius of an atom (10⁻¹⁰ m) or the age of the universe (10¹⁸ s).
常用词头范围从皮（p，10⁻¹²）到太（T，10¹²）。处理非常大或非常小的量时，例如原子半径（10⁻¹⁰ m）或宇宙年龄（10¹⁸ s），通常需要一张重要的词头表。
| Prefix 词头 | Symbol 符号 | Factor 因子 |
| giga 吉 | G | 10⁹ |
| mega 兆 | M | 10⁶ |
| kilo 千 | k | 10³ |
| centi 厘 | c | 10⁻² |
| milli 毫 | m | 10⁻³ |
| micro 微 | µ | 10⁻⁶ |
| nano 纳 | n | 10⁻⁹ |
| pico 皮 | p | 10⁻¹² |
3. Measurement Error and Uncertainty | 测量误差与不确定度
In IB Physics, the term "uncertainty" represents the range within which the true value of a measured quantity is expected to lie. It is not the same as a mistake; it is an inherent feature of any measurement.
在IB物理中，"不确定度"表示被测量的真值预期所在的区间。它不等同于错误；它是任何测量固有的一种特征。
There are two main sources of uncertainty: random errors and systematic errors. Random errors cause unpredictable fluctuations in readings, while systematic errors shift all readings consistently in one direction. A measurement can be precise but inaccurate if systematic errors are present.
不确定度有两个主要来源：随机误差和系统误差。随机误差导致读数发生不可预测的波动，而系统误差使所有读数一致地向一个方向偏移。如果存在系统误差，测量可能精确但不准确。
When you use a ruler with millimetre markings, the uncertainty is typically taken as ±0.5 mm, half the smallest division. For a digital instrument, the uncertainty is often the last displayed digit, for example ±0.01 g on a digital balance.
当你使用毫米刻度的直尺时，不确定度通常取最小刻度的一半，即±0.5 mm。对于数字仪器，不确定度通常是最末显示位，例如数字天平的±0.01 g。
4. Expressing Uncertainty: Absolute, Fractional, and Percentage | 不确定度的表示：绝对、分数和百分比
An absolute uncertainty Δx has the same unit as the measured quantity x. For example, a length measured as 25.0 cm with an absolute uncertainty of ±0.2 cm is written as 25.0 ± 0.2 cm.
绝对不确定度Δx与被测量x具有相同的单位。例如，测得长度为25.0 cm，绝对不确定度为±0.2 cm，记作25.0 ± 0.2 cm。
The fractional uncertainty is the ratio of the absolute uncertainty to the measured value: Δx / x. For 25.0 ± 0.2 cm, the fractional uncertainty is 0.2 / 25.0 = 0.008.
分数不确定度是绝对不确定度与测量值的比值：Δx / x。对于25.0 ± 0.2 cm，分数不确定度为0.2 / 25.0 = 0.008。
Multiplying the fractional uncertainty by 100% gives the percentage uncertainty. In this example, it is 0.8%. Percentage uncertainties are widely used in laboratory reports and exam answers.
将分数不确定度乘以100%得到百分比不确定度。在此例中为0.8%。百分比不确定度在实验报告和考试答案中被广泛使用。
Fractional uncertainty = Δx / x | Percentage uncertainty = (Δx / x) × 100%
5. Propagation of Uncertainties | 不确定度的传播
When a result is calculated from several measured quantities, the uncertainties of the inputs must be combined. IB Physics requires specific rules for addition/subtraction and multiplication/division.
当一个结果由几个测量量计算得出时，必须组合输入量的不确定度。IB物理要求掌握加减法和乘除法的特定规则。
For addition and subtraction, add the absolute uncertainties. For example, if A = 10.0 ± 0.1 and B = 4.0 ± 0.2, then A + B = 14.0 ± 0.3, and A - B = 6.0 ± 0.3. The absolute uncertainty is always the sum of the individual absolute uncertainties.
对于加减法，将绝对不确定度相加。例如，若A = 10.0 ± 0.1，B = 4.0 ± 0.2，则A + B = 14.0 ± 0.3，A - B = 6.0 ± 0.3。绝对不确定度始终是各绝对不确定度之和。
For multiplication and division, add the fractional or percentage uncertainties. If P = I²R, and I has a percentage uncertainty of 2% while R has 3%, then P has a percentage uncertainty of 2×2% + 3% = 7%. Note that the exponent of I multiplies its percentage uncertainty.
对于乘除法，将分数或百分比不确定度相加。若P = I²R，I的百分比不确定度为2%，R为3%，则P的百分比不确定度为2×2% + 3% = 7%。注意I的指数会乘以其百分比不确定度。
Addition/Subtraction: ΔZ = ΔA + ΔB | Multiplication/Division: ΔZ / Z = ΔA / A + ΔB / B
6. Best Estimate and Significant Figures | 最佳估计值与有效数字
The best estimate of a measured quantity is usually the mean (average) of repeated readings. The uncertainty of the mean can be estimated as half the range, especially when only a few readings are taken.
被测量的最佳估计值通常是多次读数的平均值。平均值的uncertainty可以估计为极差的一半，尤其是在读数次数较少时。
For example, if three readings are 12.1, 12.3, and 12.2, the mean is 12.2. The range is 12.3 - 12.1 = 0.2, so the uncertainty is 0.1. The result is written as 12.2 ± 0.1.
例如，三次读数为12.1、12.3和12.2，平均值为12.2。极差为12.3 - 12.1 = 0.2，所以不确定度为0.1。结果写为12.2 ± 0.1。
Significant figures must be consistent with the uncertainty. If a measurement is 12.2 ± 0.1, then reporting 12.21 ± 0.1 would be misleading because the extra digit is not meaningful. Usually, the uncertainty is rounded to one significant figure, and the measured value is rounded to the same decimal place.
有效数字必须与不确定度一致。如果测量值为12.2 ± 0.1，那么写成12.21 ± 0.1会误导，因为多余的数位没有意义。通常，不确定度四舍五入到一位有效数字，测量值四舍五入到相同的小数位数。
7. Graphing Data and Linearization | 数据作图与线性化
Graphs are essential tools for visualising relationships and determining physical constants. In IB Physics, a good graph has labelled axes with units, an appropriate scale, and data points with error bars where required.
图形是可视化关系和确定物理常数的重要工具。在IB物理中，一张好的图形应有带单位的坐标轴标签、合适的标度，以及必要时的误差棒数据点。
Many non-linear relationships can be turned into straight lines by choosing suitable variables. For example, for uniform acceleration, s = ½at² can be plotted as s versus t², giving a straight line with slope ½a.
许多非线性关系可以通过选择合适的变量转化为直线。例如，对于匀加速运动，s = ½at²可以画成s对t²的图，得到斜率为½a的直线。
For an exponential decay N = N₀e⁻λt, plotting ln N versus t gives a straight line with slope -λ and intercept ln N₀. Linearisation allows you to use linear regression and easily read off uncertainties.
对于指数衰减N = N₀e⁻λt，绘制ln N对t的图得到斜率为-λ、截距为ln N₀的直线。线性化使你可以使用线性回归并轻松读出不确定度。
8. Uncertainty in Slope and Intercept | 斜率与截距的不确定度
When a best-fit line is drawn, the uncertainty in its slope and intercept can be estimated by drawing the steepest and least steep lines that still pass through the error bars. These lines give the maximum and minimum possible values of the slope and intercept.
画出最佳拟合直线后，其斜率和截距的不确定度可以通过绘制仍然穿过误差棒的最陡和最平缓直线来估计。这两条线给出斜率和截距的最大和最小可能值。
Suppose the best-fit slope is 2.50 m s⁻², the steepest slope is 2.65 m s⁻², and the least steep is 2.35 m s⁻². The uncertainty is (2.65 - 2.35) / 2 = 0.15, so the slope is reported as 2.50 ± 0.15 m s⁻².
假设最佳拟合斜率为2.50 m s⁻²，最陡斜率为2.65 m s⁻²，最平缓斜率为2.35 m s⁻²。不确定度为(2.65 - 2.35) / 2 = 0.15，因此斜率报告为2.50 ± 0.15 m s⁻²。
When using digital software, the uncertainty can be obtained from the standard error of the fit, but for exams you should be able to perform the graphical method manually.
使用数字软件时，可以从拟合的标准误差获得不确定度，但在考试中你应该能够手动执行图形方法。
9. Systematic vs. Random Errors | 系统误差与随机误差
Random errors cause measurements to scatter randomly around the true value. They can be reduced by taking more readings and calculating the mean. The precision of a measurement is affected by random errors.
随机误差使测量值在真值周围随机散布。可以通过增加读数并计算平均值来减小。测量的精密度受随机误差影响。
Systematic errors cause all measurements to be consistently too high or too low. They often arise from poorly calibrated instruments or faulty experimental design. Systematic errors affect the accuracy of a measurement and cannot be reduced by repeating readings.
系统误差使所有测量值一致地偏高或偏低。它们通常源于校准不良的仪器或错误的实验设计。系统误差影响测量的准确度，并且不能通过重复读数来减小。
An example of a systematic error is a zero error in a stopwatch: if the stopwatch starts at 0.02 s, every timing will be too long by 0.02 s. A random error example is reaction time when starting and stopping the watch, which varies unpredictably.
系统误差的一个例子是秒表的零位误差：如果秒表从0.02 s开始，每次计时都会多0.02 s。随机误差的例子是启动和停止秒表时的反应时间，它会不可预测地变化。
10. Precision and Accuracy | 精密度与准确度
Precision describes how closely repeated measurements agree with each other. It is related to the size of the random uncertainty: smaller scatter means higher precision.
精密度描述重复测量的相互一致程度。它与随机不确定度的大小有关：散布越小，精密度越高。
Accuracy describes how close a measurement is to the true value. A measurement can be precise but inaccurate (e.g., a balance that is zeroed incorrectly gives consistent but wrong readings).
准确度描述测量值与真值的接近程度。一个测量可以精密但不准确（例如，未调零的天平给出一致但错误的读数）。
In IB Physics, you should use the terms "precision" and "accuracy" carefully. "Uncertainty" is used to express the reliability of a measurement quantitatively, while "error" refers to the deviation from the true value.
在IB物理中，你应该谨慎使用"精密度"和"准确度"这两个术语。"不确定度"用于定量表示测量的可靠性，而"误差"指与真值的偏差。
11. Reporting Results and Conclusion | 报告结果与结论
Every final result should be expressed in the form (value ± uncertainty) with the correct unit. The number of significant figures must match the uncertainty. For example, g = 9.81 ± 0.05 m s⁻², not 9.812 ± 0.05 m s⁻².
每个最终结果应以（数值 ± 不确定度）的形式表示，并带有正确的单位。有效数字的位数必须与不确定度匹配。例如，g = 9.81 ± 0.05 m s⁻²，而不是9.812 ± 0.05 m s⁻²。
When drawing conclusions, you must compare your measured value with the accepted value. The result is consistent with the accepted value if the accepted value lies within the uncertainty range. For example, if your measured value is 9.81 ± 0.05 m s⁻² and the accepted value is 9.81 m s⁻², the result is consistent.
下结论时，你必须将测量值与公认值进行比较。如果公认值位于不确定度区间内，则结果与公认值一致。例如，如果你的测量值为9.81 ± 0.05 m s⁻²，公认值为9.81 m s⁻²，结果就是一致的。
If the accepted value lies outside the uncertainty range, identify possible systematic errors that might have caused the discrepancy. Suggest specific improvements, such as using a more precise instrument or improving the method to reduce parallax error.
如果公认值位于不确定度区间之外，找出可能导致差异的系统误差。提出具体的改进措施，例如使用更精密的仪器或改进方法以减少视差误差。
Published by TutorHao | Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图