Measurements and their Errors | 测量与误差

📚 Measurements and their Errors | 测量与误差

In AS Level Physics, measurements are the foundation of every experiment. OxfordAQA International AS Level Physics requires you to understand not only how to take readings but also how to quantify how reliable those readings are. This topic covers SI units, prefixes, errors, uncertainties, and graph-analysis skills that are examined across both Paper 1 and Paper 2.

在AS物理中，测量是所有实验工作的基础。牛津AQA国际AS物理不仅要求你掌握如何读数，还要求你能够量化这些读数的可靠性。本专题涵盖SI单位、词头、误差、不确定度以及图像分析技能，这些内容在Paper 1和Paper 2中都会被考查。

1. Physical Quantities and SI Units | 物理量与SI单位

Every physical quantity consists of a numerical value and a unit. The International System of Units (SI) defines seven base quantities. In AS physics, the most commonly used base quantities are mass (kg), length (m), time (s), electric current (A) and temperature (K). Amount of substance (mol) is also part of the SI system, but you will rarely need it in mechanics or electricity questions.

每个物理量都由一个数值和一个单位组成。国际单位制（SI）定义了七个基本量。在AS物理中，最常用的基本量是质量（kg）、长度（m）、时间（s）、电流（A）和温度（K）。物质的量（mol）也属于SI体系，但在力学或电学问题中你很少会用到它。

Base quantity 基本量	Base unit 基本单位	Symbol 符号
Mass 质量	kilogram 千克	kg
Length 长度	metre 米	m
Time 时间	second 秒	s
Electric current 电流	ampere 安培	A
Temperature 温度	kelvin 开尔文	K

A derived unit is written in terms of base units. For example, force: 1 N = 1 kg m s⁻²; energy: 1 J = 1 kg m² s⁻²; pressure: 1 Pa = 1 kg m⁻¹ s⁻². A common exam task is to express a unit such as the volt in base units. The method is always the same: write down a defining equation (V = W / Q), then replace each quantity with its base-unit expression.

导出单位用基本单位表示。例如，力：1 N = 1 kg m s⁻²；能量：1 J = 1 kg m² s⁻²；压强：1 Pa = 1 kg m⁻¹ s⁻²。常见考题是要求你用基本单位表示伏特这样的单位。方法始终相同：写出一个定义式（V = W / Q），然后用基本单位表达式替换每个物理量。

You should also check the homogeneity of equations: substitute the base units of every quantity into both sides and confirm they match. A homogeneous equation may still be physically wrong, but an equation whose units do not match must be wrong.

你还应该检验方程的量纲一致性：把每个量的基本单位代入方程两边，确认它们一致。量纲一致的方程仍可能在物理上错误，但两边单位不一致的方程必定错误。

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2. Prefixes and Unit Conversions | 词头与单位换算

SI prefixes allow you to write very large and very small quantities conveniently. You must recall the following prefixes and their powers of ten.

SI词头让你能够方便地书写非常大和非常小的量。你必须记住以下词头及其10的幂次。

Prefix 词头	Symbol 符号	Power of 10 幂次
pico 皮	p	10⁻¹²
nano 纳	n	10⁻⁹
micro 微	μ	10⁻⁶
milli 毫	m	10⁻³
centi 厘	c	10⁻²
kilo 千	k	10³
mega 兆	M	10⁶
giga 吉	G	10⁹

When converting, always replace the prefix with its power of ten. For example, 25 μm = 25 × 10⁻⁶ m = 2.5 × 10⁻⁵ m; 4.2 MW = 4.2 × 10⁶ W. Take particular care with squared and cubed units: 1 cm² = (10⁻² m)² = 10⁻⁴ m², and 1 cm³ = (10⁻² m)³ = 10⁻⁶ m³.

换算时，始终用10的幂次替换词头。例如，25 μm = 25 × 10⁻⁶ m = 2.5 × 10⁻⁵ m；4.2 MW = 4.2 × 10⁶ W。要特别注意平方和立方单位：1 cm² = (10⁻² m)² = 10⁻⁴ m²，1 cm³ = (10⁻² m)³ = 10⁻⁶ m³。

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3. Random Errors and Systematic Errors | 随机误差与系统误差

Random errors are unpredictable fluctuations in the measurement that vary in magnitude and sign. They make repeated readings spread around the true value. Examples include noise in an electrical circuit, vibration of a bench, or judging the position of a pointer between scale divisions. Random errors can be reduced by repeating the measurement and calculating the mean, and by using more precise instruments.

随机误差是测量中不可预测的波动，其大小和方向随机变化。它们使重复读数围绕真值分散分布。例如电路中的噪声、实验台的振动，或估读指针在刻度之间的位置。随机误差可以通过重复测量并计算平均值、以及使用更精密的仪器来减小。

Systematic errors are consistent biases that shift every reading in the same direction. Causes include an incorrectly calibrated instrument, a zero error, or parallax error when reading a scale. Systematic errors cannot be reduced by averaging; they must be removed by calibrating the instrument, checking the zero reading, or changing the technique.

系统误差是使每次读数都朝同一方向偏移的一致偏差。原因包括仪器校准不当、零位误差，或读数时的视差。系统误差不能通过求平均来减小；必须通过校准仪器、检查零位读数或改变方法加以消除。

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4. Accuracy, Precision and Resolution | 准确度、精密度与分辨率

Accuracy describes how close a measurement is to the true value. Precision describes how close repeated measurements are to each other, that is, the spread of readings. A measurement can be precise but inaccurate if a systematic error shifts all readings away from the true value.

准确度描述测量值与真值的接近程度。精密度描述重复测量值彼此之间的接近程度，即读数的离散程度。如果系统误差使所有读数偏离真值，测量可能精密但不准确。

Resolution is the smallest change in the quantity being measured that the instrument can display. A metre ruler has a resolution of 1 mm; a digital stopwatch has a resolution of 0.01 s, although human reaction time means the practical uncertainty is much larger. Do not confuse resolution with accuracy: a high-resolution instrument is not automatically accurate if it has a zero error.

分辨率是仪器能够显示的被测量的最小变化。米尺的分辨率为1 mm；数字秒表的分辨率为0.01 s，但人的反应时间意味着实际不确定度要大得多。不要把分辨率与准确度混淆：如果存在零位误差，高分辨率仪器并不自动准确。

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5. Absolute, Fractional and Percentage Uncertainty | 绝对、分数与百分比不确定度

The absolute uncertainty Δx is the actual range of doubt in a measured value and carries the same unit as the measurement. It can be estimated from the resolution of the instrument, from the spread of repeated readings, or from the manufacturer’s specification. For an analogue scale, the uncertainty is usually taken as half the smallest division; for a digital instrument, it is usually taken as one resolution step (the smallest digit).

绝对不确定度Δx是测量值中怀疑范围内的实际大小，与测量值具有相同单位。它可以由仪器分辨率、重复读数的离散程度或厂家规格来估计。对于模拟刻度，不确定度通常取最小分度的一半；对于数字仪器，通常取一个分辨率步长（最小数字）。

You will often need to quote uncertainty in three equivalent forms:

你常常需要以三种等价形式表述不确定度：

Fractional uncertainty = Δx / x

分数不确定度 = Δx / x

Percentage uncertainty = (Δx / x) × 100%

百分比不确定度 = (Δx / x) × 100%

For example, a length measured as 5.0 cm with an absolute uncertainty of 0.1 cm has a fractional uncertainty of 0.1 / 5.0 = 0.02 and a percentage uncertainty of 2%.

例如，测得长度5.0 cm，绝对不确定度为0.1 cm，则分数不确定度为0.1 / 5.0 = 0.02，百分比不确定度为2%。

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6. Combining Uncertainties | 不确定度的合成

When a final result is calculated from several measured quantities, the uncertainties must be combined. The rules below are the ones OxfordAQA expects you to apply.

当最终结果由多个测量量计算得出时，必须合成不确定度。以下是牛津AQA要求你运用的规则。

• For addition and subtraction: add the absolute uncertainties.

  加减法：绝对不确定度相加。

• For multiplication and division: add the fractional or percentage uncertainties.

  乘除法：分数或百分比不确定度相加。

• For a power, such as y = xⁿ: multiply the percentage uncertainty by the power.

  幂次关系，如 y = xⁿ：百分比不确定度乘以幂指数。

If y = a + b, then Δy = Δa + Δb

若 y = a + b，则 Δy = Δa + Δb

If y = a × b, then Δy / y = Δa / a + Δb / b

若 y = a × b，则 Δy / y = Δa / a + Δb / b

Worked example: a current I = 2.0 ± 0.1 A flows through a resistor R = 10 ± 0.5 Ω. The power is P = I²R. The percentage uncertainty in I is 0.1 / 2.0 = 5%, so in I² it is 2 × 5% = 10%. The percentage uncertainty in R is 0.5

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