Understanding SI Units | 理解国际单位制 (SI)

📚 Understanding SI Units | 理解国际单位制 (SI)

In A-Level Physics, mastering the International System of Units (SI) is the foundation for all quantitative work. Precise measurement and clear communication of physical quantities rely on a standardized set of base units, derived combinations, prefixes, and conventions. This article explores the structure of the SI system, how to build and convert units, check the homogeneity of equations, and use dimensional analysis to verify physical relationships — essential skills for every CIE candidate.

在 A-Level 物理中，掌握国际单位制 (SI) 是所有定量分析的基础。精确测量和物理量的清晰表达依赖于一套标准化的基本单位、导出组合、前缀和书写惯例。本文探讨 SI 体系的结构、如何构建和换算单位、如何检查方程齐次性以及如何运用量纲分析验证物理关系——这些是每位 CIE 考生必备的核心技能。

1. Introduction to the SI System | SI 体系简介

The SI (Système International d'Unités) is the modern metric system used universally in science and engineering. It is maintained by the International Bureau of Weights and Measures (BIPM) and provides a coherent framework where all units are defined in terms of seven base quantities. This coherence means derived units can be expressed as products of powers of base units without conversion factors.

SI 是现代科学技术中通用的公制体系，由国际计量局 (BIPM) 维护。它提供了一个自洽的框架，所有单位均由七个基本量来定义。这种自洽性意味着导出单位可以直接表示为基本单位的幂的乘积，而无需换算因子。

For CIE A-Level Physics, you must be able to recall base units, express derived units, apply prefixes, and check equations using base units. Understanding SI goes beyond memorisation — it helps you grasp the very nature of physical quantities and their relationships.

对于 CIE A-Level 物理，你不仅需要记住基本单位、表达导出单位、使用前缀，还要能利用基本单位检查方程。理解 SI 不光是记忆，更能帮助你掌握物理量本身的性质及其相互关系。

---

2. The Seven Base Quantities and Units | 七个基本量和单位

The SI system is built upon seven independent base quantities, each with a precisely defined unit. No other base quantities are required to express all physical measurements. Memorising these is essential, including both the name and the symbol (case-sensitive).

SI 体系建立在七个独立的基本量之上，每个量都有精确定义的单位。任何物理测量都可以用这七个基本量表达。熟记这些基本量的名称和符号（区分大小写）至关重要。

Quantity	Unit Name	Symbol
length	metre	m
mass	kilogram	kg
time	second	s
electric current	ampere	A
thermodynamic temperature	kelvin	K
amount of substance	mole	mol
luminous intensity	candela	cd

Note that the kilogram is the only base unit that still has a prefix (kilo-) in its name for historical reasons. In CIE exams, you must always write units with correct capitalisation: e.g., 'A' for ampere, not 'a'; 'K' for kelvin, not 'k'.

注意，千克是唯一一个因历史原因名称中仍带有词头（kilo-）的基本单位。在 CIE 考试中，必须始终使用正确的字母大小写单位符号：例如安培用 'A' 而不是 'a'；开尔文用 'K' 而不是 'k'。

All other physical quantities are derived from combinations of these seven. For example, velocity is length divided by time, expressed in m s⁻¹. Recognising base units helps when simplifying complex expressions.

所有其他物理量均由这七个基本量组合导出。例如，速度是长度除以时间，用 m s⁻¹ 表示。识别基本单位有助于简化复杂表达式。

---

3. Derived Units | 导出单位

Derived units are formed by multiplying and dividing base units. Some have special names (e.g., newton, joule), but you should always be able to break them down into base SI units. This skill is frequently tested in Paper 2 and Paper 4.

导出单位通过对基本单位进行乘除运算得到。其中一些有专门的名称（如牛顿、焦耳），但你必须能随时将它们拆解成基本 SI 单位。这项技能经常在 Paper 2 和 Paper 4 中考查。

Common derived units include:

常见导出单位有：

Force: newton (N) = kg m s⁻²
Energy: joule (J) = N m = kg m² s⁻²
Power: watt (W) = J s⁻¹ = kg m² s⁻³
Electric charge: coulomb (C) = A s
Potential difference: volt (V) = J C⁻¹ = kg m² s⁻³ A⁻¹
Resistance: ohm (Ω) = V A⁻¹ = kg m² s⁻³ A⁻²

nt, the units on both sides must be N m = (kg m s⁻²)(m) = kg m² s⁻², which matches the joule.

当导出单位用基本单位表示时，方程中的任何不一致都会立刻显现。例如，如果你说功 = 力 × 位移，等式两边的单位必须为 N m = (kg m s⁻²)(m) = kg m² s⁻²，这与焦耳一致。

s⁻¹). Practice converting between special names and base units until it becomes automatic.

你还会遇到帕斯卡 (Pa = N m⁻² = kg m⁻¹ s⁻²) 或赫兹 (Hz = s⁻¹) 等单位。反复练习在专门名称和基本单位之间转换，直到熟练自如。

---

4. SI Prefixes and Scientific Notation | SI 词头与科学记数法

SI prefixes allow us to handle very large or very small numbers neatly. Each prefix represents a power of ten. You must know the standard prefixes from tera (10¹²) down to pico (10⁻¹²), as they are used in exam questions without further explanation.

SI 词头使我们能够整洁地处理极大或极小的数值。每个词头代表 10 的幂。你必须熟记从太 (10¹²) 到皮 (10⁻¹²) 的标准词头，因为考题中会直接使用它们而不再解释。

Key prefixes: tera (T, 10¹²), giga (G, 10⁹), mega (M, 10⁶), kilo (k, 10³), deci (d, 10⁻¹), centi (c, 10⁻²), milli (m, 10⁻³), micro (μ, 10⁻⁶), nano (n, 10⁻⁹), pico (p, 10⁻¹²). Remember that the symbol for micro is the Greek letter mu, often written as 'μ'.

关键词头：太 (T, 10¹²)、吉 (G, 10⁹)、兆 (M, 10⁶)、千 (k, 10³)、分 (d, 10⁻¹)、厘 (c, 10⁻²)、毫 (m, 10⁻³)、微 (μ, 10⁻⁶)、纳 (n, 10⁻⁹)、皮 (p, 10⁻¹²)。记住微的符号是希腊字母 mu，常写作 'μ'。

then combine with the unit. For example, 5.0 cm = 5.0 × 10⁻² m. Compound prefixes such as 'μμm' are not allowed; instead use pico (10⁻¹²) or an appropriate prefix.

换算时，始终先将词头替换为 10 的幂，再与单位结合。例如 5.0 cm = 5.0 × 10⁻² m。不允许使用诸如 'μμm' 的复合词头，而应使用皮 (10⁻¹²) 或其他合适的词头。

In calculations, express quantities in base SI units (e.g., metres, kilograms, seconds) before substituting into formulae, unless the question specifies otherwise. This avoids power-of-ten mistakes.

在计算时，除非题目另有要求，应先将物理量用基本 SI 单位（例如米、千克、秒）表示，再代入公式。这可以避免 10 的幂次出错。

---

5. Unit Conversions | 单位换算

Many CIE problems require converting between non-SI units (like km h⁻¹ to m s⁻¹) or between different prefixed units. A systematic approach is to write the given quantity, substitute conversion factors, and cancel units.

许多 CIE 题目要求在不同单位之间转换（如 km h⁻¹ 转 m s⁻¹），或在带不同词头的单位之间转换。系统的方法是先写出给定数值，然后代入换算因子并消去单位。

Example: convert 72 km h⁻¹ to m s⁻¹.
72 km h⁻¹ = 72 × (1000 m) / (3600 s) = 72 × (1/3.6) m s⁻¹ = 20 m s⁻¹.

示例：将 72 km h⁻¹ 换算为 m s⁻¹。
72 km h⁻¹ = 72 × (1000 m) / (3600 s) = 72 × (1/3.6) m s⁻¹ = 20 m s⁻¹。

= 10⁻⁶ m³. This is a common source of errors, so write out the square or cube explicitly.

涉及面积或体积的换算需要格外小心：1 cm² = (10⁻² m)² = 10⁻⁴ m²；1 cm³ = (10⁻² m)³ = 10⁻⁶ m³。这是常见的出错点，因此要明确地将平方或立方写出来。

1 g cm⁻³ = (10⁻³ kg) / (10⁻⁶ m³) = 1000 kg m⁻³. Doing this step-by-step reduces mistakes.

当换算密度单位时，例如从 g cm⁻³ 到 kg m⁻³，使用：1 g cm⁻³ = (10⁻³ kg) / (10⁻⁶ m³) = 1000 kg m⁻³。逐步进行可以减少错误。

---

广告 / Sponsored

6. Homogeneity of Equations | 方程的齐次性

An equation is homogeneous if the base units on each side are identical. Checking homogeneity is a quick way to spot algebraic errors or to eliminate multiple-choice options. Every physical equation must be dimensionally consistent.

如果一个方程两边的基本单位完全相同，它就是齐次的。检查齐次性是快速发现代数错误或排除选择题选项的有效方法。每一个物理方程都必须量纲一致。

² = m² s⁻²; RHS: (m s⁻¹)² + (m s⁻²)(m) = m² s⁻² + m² s⁻². Both terms are m² s⁻², so the equation is homogeneous.

步骤：将每个物理量替换成它的基本 SI 单位，化简并比较。例如，验证 v² = u² + 2as 是否齐次：左边：(m s⁻¹)² = m² s⁻²；右边：(m s⁻¹)² + (m s⁻²)(m) = m² s⁻² + m² s⁻²。两边均为 m² s⁻²，因此方程齐次。

Constants such as 2, π, or pure numbers have no units and do not affect homogeneity. If an equation fails the test, it cannot be correct. However, homogeneity alone does not guarantee correctness — a wrong coefficient can still produce the same base units.

像 2、π 这样的纯数字没有单位，不影响齐次性。如果一个方程未通过齐次性检验，则它必定是错误的。然而，仅仅齐次并不能保证正确——错误的系数仍可能产生相同的基本单位。

---

7. Dimensional Analysis | 量纲分析

Dimensional analysis extends the idea of units by representing physical quantities in terms of fundamental dimensions: length [L], mass [M], time [T], electric current [I], temperature [θ], amount of substance [N], and luminous intensity [J]. It is a powerful tool for deriving relationships when the underlying physics is partially known.

量纲分析将单位的理念加以推广，用基本量纲表示物理量：长度 [L]、质量 [M]、时间 [T]、电流 [I]、温度 [θ]、物质的量 [N] 和发光强度 [J]。当对背后的物理规律不完全了解时，量纲分析是推导关系的有力工具。

set up equations where exponents of [M], [L], and [T] must match on both sides to determine unknown powers.

想要求出一个量的量纲，使用其定义式。例如压强 P = F/A → [P] = [M][L][T]⁻² / [L]² = [M][L]⁻¹[T]⁻²。然后可以建立方程，令两边 [M]、[L]、[T] 的指数分别相等，以确定未知的幂次。

and gravitational field strength g: T = k lᵅ gᵝ. In dimensions: [T] = [L]ᵅ ([L][T]⁻²)ᵝ = [L]ᵅ⁺ᵝ [T]⁻²ᵝ. Equating gives: for [T]: 1 = -2β → β = -1/2; for [L]: 0 = α+β → α = 1/2. Thus T = k √l/g.

在 CIE 考试中，量纲分析用于要求推导表达式，如单摆的周期。假设 T 取决于摆长 l 和重力场强 g：T = k lᵅ gᵝ。量纲式为：[T] = [L]ᵅ ([L][T]⁻²)ᵝ = [L]ᵅ⁺ᵝ [T]⁻²ᵝ。令等式成立：对 [T] 有 1 = -2β → β = -1/2；对 [L] 有 0 = α+β → α = 1/2。因此 T = k √l/g。

It is important to note that dimensionless constants (like k) cannot be found this way. Dimensional analysis gives the correct form, but not the numerical coefficient.

需要注意的是，无量纲常数（如 k）无法用此方法求出。量纲分析只能给出正确的形式，无法给出数字系数。

---

8. Scientific Notation and Significant Figures | 科学记数法与有效数字

geable and clearly indicates significant figures. CIE expects you to round final answers to the least number of significant figures in the given data.

使用 SI 单位时，你必须以适当的精度给出数值答案。科学记数法（如 3.0 × 10⁸ m s⁻¹）让数值易于处理，并清晰地展示有效数字。CIE 要求你将最终答案四舍五入到与所给数据中最少的有效数字位数一致。

The number of significant figures depends on the measuring instrument. For example, a metre rule marked in mm can read to 0.001 m (1 mm), so 0.520 m has three significant figures. Digital multi-meters usually give three or four digits.

有效数字的位数取决于测量仪器。例如，一把最小刻度为 mm 的米尺可以读到 0.001 m (1 mm)，因此 0.520 m 有三个有效数字。数字万用表通常给出三位或四位数。

In multi-step calculations, avoid rounding intermediate results. Only round the final answer to the correct number of significant figures. Also, when adding or subtracting, the answer should have the same number of decimal places as the least precise measurement.

在多步运算中，避免对中间结果进行四舍五入。只在最终答案舍入到正确的有效数字位数。另外，进行加减运算时，结果的小数位数应与最不精确的测量值一致。

---

9. Estimation and Orders of Magnitude | 估算与数量级

order of magnitude is the power of ten when a value is expressed in scientific notation. For example, the mass of an electron (9.11 × 10⁻³¹ kg) has an order of magnitude of 10⁻³⁰ kg.

数量级让你能够快速估算物理量，这对检查计算结果是否合理非常有用。数量级是用科学记数法表示数值时的 10 的指数。例如，电子的质量（9.11 × 10⁻³¹ kg）的数量级为 10⁻³⁰ kg。

You should be able to estimate familiar quantities such as the height of a human (~1.7 m), the mass of a car (~1000 kg), the wavelength of visible light (~5 × 10⁻⁷ m), or the pressure of the atmosphere (~ 10⁵ Pa). These benchmarks help in understanding graph intercepts or spotting unit errors.

你应该能够估算常见的物理量，如人的身高（~1.7 m）、汽车的质量（~1000 kg）、可见光的波长（~5 × 10⁻⁷ m）或大气压强（~1 × 10⁵ Pa）。这些基准有助于理解图线截距或发现单位错误。

In the CIE syllabus, you may be asked to choose which of several wildly different values is plausible for a given physical context. Having a mental library of benchmark magnitudes can save time in the multiple-choice paper.

在 CIE 考试大纲中，你可能会被要求从几个相去甚远的数值中选出在给定物理情景下合理的一个。在心中建立一个基准数量级库可以节省选择题部分的答题时间。

---

10. Conventions for Writing SI Units | SI 单位的书写惯例

Strict conventions exist for writing unit symbols to avoid ambiguity. Unit symbols are printed in roman (upright) type, not italics, and remain unchanged in plural. The symbols are case-sensitive: 's' for second, 'S' for siemens (unit of conductance).

书写单位符号有严格的惯例以避免歧义。单位符号用正体而非斜体印刷，并且复数形式不变。符号区分大小写：'s' 表示秒，'S' 表示西门子（电导单位）。

There is always a space between the numerical value and the unit: write '10 m', not '10m'. The product of units is indicated by a space or a centred dot: N m or N·m. Do not use 'N × m' as it can be confused with vector cross product.

数值与单位之间始终保留一个空格：应写作 '10 m'，而不是 '10m'。单位的乘积用空格或半高圆点表示：N m 或 N·m。不要使用 'N × m'，因为这容易与向量叉积混淆。

Division of units is shown with a solidus (/) or negative exponents: m/s or m s⁻¹. Only one solidus may be used; for complex cases use brackets or negative powers: write J/(kg K) or J kg⁻¹ K⁻¹, never J/kg/K.

单位的相除用斜线 (/) 或负指数表示：m/s 或 m s⁻¹。只允许使用一条斜线；对于复杂情况应使用括号或负幂次：应写作 J/(kg K) 或 J kg⁻¹ K⁻¹，绝不可写作 J/kg/K。

Prefix symbols are attached directly to unit symbols without a space: km, ms, μA. Remember that kilogram (kg) is an exception because the prefix is already part of the base unit name.

词头符号应紧贴单位符号，中间不留空格：km、ms、μA。记住千克 (kg) 是个例外，因为其词头已是基本单位名称的一部分。

---

11. Summary and Exam Tips | 总结与考试技巧

Mastering SI units is not just about memorisation; it is about developing a reflex to check that every equation you write has consistent units. In CIE exams, marks are often allocated for showing correct units alongside final numerical answers.

掌握 SI 单位不只是记忆，而是要养成习惯，检查你写下的每一个方程是否单位一致。在 CIE 考试中，经常对在最终数值答案旁写上正确的单位有给分。

Always convert given data into base SI units before substituting into formulae. Use dimensional analysis to test whether a derived expression makes sense. If you finish a calculation and the unit is obviously wrong (e.g., joules for speed), revisit your working.

在代入公式前，始终将给定数据转换成基本 SI 单位。用量纲分析检验导出的表达式是否有意义。如果计算完毕后所得单位明显不对（如速度得到了焦耳），就应回头检查计算过程。

Practice by writing the base SI equivalent for complex derived units, converting prefixed units, and checking the homogeneity of textbook equations. Over time, the language of SI units will become second nature, giving you confidence in tackling A-Level Physics problems.

通过写出复杂导出单位的基本 SI 等价形式、换算带词头的单位、检查教材方程的齐次性来进行练习。渐渐地，SI 单位的语言将成为你的第二本能，使你在应对 A-Level 物理问题时有充分自信。

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）