📚 3: Quantities and Units | 3：物理量与单位
In physics, every measurement and theoretical expression relies on clearly defined quantities and units. Whether you are calculating the acceleration of a car or the wavelength of light, you must be able to express results using the International System of Units (SI) and work confidently with base units, derived units, prefixes and dimensions. This article covers the key knowledge required for the Edexcel A‑Level Physics course, from the seven SI base quantities to homogeneity checks and vector operations.
在物理学中，每一次测量和理论表达都依赖于明确定义的物理量和单位。无论是计算汽车的加速度还是光的波长，你都必须能够用国际单位制 (SI) 表达结果，并熟练使用基本单位、导出单位、词头和量纲。本文涵盖了 Edexcel A‑Level 物理课程所要求的核心知识，从七个 SI 基本量到方程的同质性检验和向量运算。
1. What Are Physical Quantities? | 什么是物理量？
A physical quantity is a property of a material or system that can be quantified by measurement. It always consists of a numerical value and a unit. For example, stating that a table’s length is 1.85 metres gives both the magnitude (1.85) and the standard of comparison (metre). Quantities fall into two broad groups: base quantities, which are defined independently, and derived quantities, which are formed by combining base quantities.
物理量是可以通过测量来量化的物质或系统的属性。它总是由数值和单位组成。例如，说一张桌子长 1.85 米，既给出了大小 (1.85) 也给出了比较标准 (米)。物理量分为两大类：独立定义的基本量和由基本量组合而成的导出量。
2. SI Base Quantities and Units | SI 基本量和单位
The International System of Units (SI) is built upon seven base quantities. Every A‑Level physicist must memorise them, together with their symbols and units:
国际单位制 (SI) 建立在七个基本量之上。每一位 A‑Level 物理学习者都必须记住它们及其符号和单位：
| Base Quantity | Symbol | SI Base Unit | Unit Symbol |
|---|---|---|---|
| Mass | m | kilogram | kg |
| Length | l, x, r, etc. | metre | m |
| Time | t | second | s |
| Electric current | I | ampere | A |
| Thermodynamic temperature | T | kelvin | K |
| Amount of substance | n | mole | mol |
| Luminous intensity | Iv | candela | cd |
These seven base units are defined with extraordinary precision—for instance, the metre is defined by fixing the speed of light in vacuum at exactly 299 792 458 m s⁻¹. Every other mechanical unit can be expressed as a combination of kg, m and s.
这七个基本单位被极其精确地定义——例如，米是通过将真空中光速固定为精确的 299 792 458 m s⁻¹ 来定义的。所有其他力学单位都可以表示为 kg、m 和 s 的组合。
3. Derived Quantities and Units | 导出量和单位
Derived quantities are formed by multiplying or dividing base quantities. Their units are correspondingly combinations of base units. Velocity, acceleration, force, energy and pressure are all examples. A systematic way to check a derived unit is to write the defining equation and substitute base units.
导出量由基本量相乘或相除构成。它们的单位相应地是基本单位的组合。速度、加速度、力、能量和压强都是例子。检验导出单位的一个系统方法是写出定义方程并代入基本单位。
velocity = displacement / time → unit: m s⁻¹
加速度 = 位移 / 时间 → 单位: m s⁻¹
force = mass × acceleration → unit: kg m s⁻² (newton, N)
力 = 质量 × 加速度 → 单位: kg m s⁻² (牛顿, N)
energy = force × distance → unit: kg m² s⁻² (joule, J)
能量 = 力 × 距离 → 单位: kg m² s⁻² (焦耳, J)
In mechanics, all quantities can ultimately be expressed in terms of just three base quantities: mass (M), length (L) and time (T). This dimensional representation is a powerful tool for verifying equations.
在力学中，所有物理量最终都可以只用三个基本量来表示：质量 (M)、长度 (L) 和时间 (T)。这种量纲表示法是检验方程的有力工具。
4. Homogeneity of Physical Equations | 物理方程的同质性
A valid physical equation must be homogeneous: each term on both sides of the equation must have the same base‑unit combination. Checking homogeneity does not guarantee that an equation is correct, but it quickly exposes many mistakes. For the familiar kinematic equation
一个有效的物理方程必须是同质的：方程两边每一项的基本单位组合必须相同。检验同质性并不能保证方程正确，但能迅速揭露许多错误。对于熟悉的运动学方程
s = u t + ½ a t²
the same unit, metre, so the equation is homogeneous. Homogeneity tests also help when converting between unit systems or when deriving new relationships.
我们检查量纲。s 是位移，单位为 m。u t 的单位是 (m s⁻¹) × s = m。½ 是纯数，a t² 的单位是 (m s⁻²) × s² = m。所有三项具有相同的单位米，因此方程是同质的。同质性检验在单位制转换或推导新关系时也非常有用。
5. Metric Prefixes and Unit Conversions | 公制前缀与单位转换
To handle the enormous range of magnitudes encountered in physics—from the radius of an atomic nucleus to the distance between galaxies—we use SI prefixes. You are expected to know the standard set from pico (10⁻¹²) to tera (10¹²) and be able to convert freely between them.
为了处理物理学中遇到的巨大数量级范围——从原子核半径到星系间的距离——我们使用 SI 前缀。你需要掌握从皮可 (10⁻¹²) 到太拉 (10¹²) 的标准前缀集，并能在它们之间自由转换。
| Prefix | Symbol | Factor |
|---|---|---|
| pico | p | 10⁻¹² |
| nano | n | 10⁻⁹ |
| micro | μ | 10⁻⁶ |
| milli | m | 10⁻³ |
| centi | c | 10⁻² |
| deci | d | 10⁻¹ |
| kilo | k | 10³ |
| mega | M | 10⁶ |
| giga | G | 10⁹ |
| tera | T | 10¹² |
× 10⁻⁹ m = 5.0 × 10⁻⁸ m. Mastering prefix manipulation reduces arithmetic errors in exams.
进行单位转换时，务必显式写出转换因子。例如，将 50 nm 转换为米：50 nm = 50 × 10⁻⁹ m = 5.0 × 10⁻⁸ m。熟练掌握前缀换算可减少考试中的算术错误。
6. Scientific Notation | 科学记数法
arge or very small quantities clearly. For instance, the charge on an electron is better written as 1.60 × 10⁻¹⁹ C than as 0.000000000000000000160 C. In physics examinations, answers are often expected in standard form unless the question specifies otherwise.
科学记数法将一个数表示为尾数 (介于 1 和 10 之间的数) 与 10 的幂的乘积。它对于清晰书写极大或极小的量不可或缺。例如，电子的电荷写成 1.60 × 10⁻¹⁹ C 要比写成 0.000000000000000000160 C 好得多。在物理考试中，除非题目另有规定，答案通常要求用标准形式表达。
10⁻³. When combining numbers in scientific notation, handle the mantissas and the exponents separately.
始终调整 10 的幂使尾数落在 1 到 10 之间。因此，3450 = 3.45 × 10³，而 0.0027 = 2.7 × 10⁻³。当用科学记数法组合数字时，分别处理尾数和指数。
7. Significant Figures | 有效数字
The number of significant figures (sf) indicates the precision of a measured or calculated value. In Edexcel exams, final answers are often required to be given to an appropriate number of significant figures—usually the same as the least precise piece of data used. The rules for counting significant figures are straightforward:
有效数字的位数表示一个测量值或计算值的精密度。在 Edexcel 考试中，最终答案通常要求给出适当位数的有效数字——一般与所使用的最不精确的数据的一致。计算有效数字的规则很直接：
- Non‑zero digits are always significant. (123 has 3 sf)
- 非零数字总是有效的。(123 有 3 位有效数字)
- Zeros between non‑zero digits are significant. (1002 has 4 sf)
- 非零数字之间的零是有效的。(1002 有 4 位有效数字)
- Leading zeros are not significant. (0.0045 has 2 sf)
- 前导零不是有效数字。(0.0045 有 2 位有效数字)
- Trailing zeros in a number with a decimal point are significant. (2.300 has 4 sf)
- 带小数点的数中末尾的零是有效的。(2.300 有 4 位有效数字)
When rounding, remember that a result should never appear more precise than the original data. For a multiplication, the answer should have no more significant figures than the least precise factor.
进行舍入时，切记结果不应表现得比原始数据更精确。对于乘法，答案的有效数字位数不应超过最不精确因子的位数。
8. Scalars and Vectors | 标量和向量
Quantities in physics are classified as scalars or vectors. A scalar has magnitude only; examples include mass, temperature, energy, distance and speed. A vector has both magnitude and direction; displacement, velocity, acceleration, force and momentum are vectors. A clear distinction is essential because vectors cannot be added by simple arithmetic—direction must be taken into account.
物理量分为标量和向量。标量只有大小；例子包括质量、温度、能量、距离和速率。向量既有大小又有方向；位移、速度、加速度、力和动量都是向量。区分二者至关重要，因为向量不能通过简单算术相加——必须考虑方向。
Vectors are often represented by bold type or by an arrow above the symbol, for example F s an arrow whose length is proportional to the magnitude and whose orientation shows the direction.
向量常用粗体或在符号上方加箭头表示，例如 F 或 F→。在纸上，你可以将向量画成一个箭头，其长度与大小成正比，其指向表示方向。
9. Vector Addition and Resolution | 向量相加与分解
To find the resultant of two vectors, you can use the tip‑to‑tail method or, for perpendicular vectors, Pythagoras' theorem. If two forces of 3.0 N and 4.0 N act at right angles, the magnitude of the resultant is √3.0² + 4.0² = 5.0 N. The direction relative to the 3.0 N force is given by tan⁻¹(4.0/3.0).
求两个向量的合向量，你可以使用三角形法，对于互相垂直的向量可使用勾股定理。如果两个力 3.0 N 和 4.0 N 成直角作用，合力的大小为 √3.0² + 4.0² = 5.0 N。相对于 3.0 N 力的方向由 tan⁻¹(4.0/3.0) 给出。
Resolving a vector into perpendicular components—usually horizontal and vertical—is an equally important skill. A vector of magnitude R at an angle θ to the horizontal has horizontal component R cos θ and vertical component R sin θ. Resolution is the basis for analysing projectiles, forces on slopes and equilibrium problems.
将向量沿互相垂直的方向分解——通常是水平和竖直方向——是一项同样重要的技能。一个大小为 R、与水平方向成 θ 角的向量，其水平分量为 R cos θ，竖直分量为 R sin θ。分解是分析抛体运动、斜面上的力和平衡问题的基础。
You must be comfortable switching between a vector's magnitude‑direction form and its component form, and vice versa. Mastering this skill will make many mechanics topics far more manageable.
你必须能在向量的大小‑方向形式与分量形式之间自如切换。掌握这一技能将使许多力学专题变得容易得多。
10. Orders of Magnitude and Estimation | 数量级与估算
An order‑of‑magnitude calculation rounds a quantity to the nearest power of ten. Biologists might estimate the number of cells in a leaf; physicists estimate the wavelength of visible light (~10⁻⁷ m) or the mass of a proton (~10⁻²⁷ kg). Such estimates are useful for designing experiments and for checking whether a calculated answer is physically plausible.
数量级计算将物理量舍入到最接近的 10 的幂。生物学家可能估算一片叶子的细胞数目；物理学家估算可见光的波长 (~10⁻⁷ m) 或质子的质量 (~10⁻²⁷ kg)。这类估算对设计实验和检验计算答案在物理上是否合理都很有用。
ombine powers of ten. For instance, to estimate the height of a stack of 500 sheets of paper, you might approximate the thickness of one sheet as 0.1 mm (= 10⁻⁴ m). The total height is 500 × 10⁻⁴ m = 5 × 10⁻² m, or about 5 cm.
在考试中，你可能被要求不借助计算器估算一个数值。关键在于将每个输入近似到一位有效数字，然后组合 10 的幂。例如，若要估算 500 张纸叠起来的高度，你可将一张纸的厚度近似为 0.1 mm (= 10⁻⁴ m)。总高度为 500 × 10⁻⁴ m = 5 × 10⁻² m，大约 5 cm。
Orders of magnitude also help you internalise the physical world’s scale. Knowing that the Earth's radius is around 10⁷ m, while an atom is about 10⁻¹⁰ m, gives you a 17‑order‑of‑magnitude perspective that is essential for any physicist.
数量级还能帮你内化物理世界的尺度。知道地球半径约为 10⁷ m，而原子约为 10⁻¹⁰ m，让你获得跨越 17 个数量级的视角，这对任何物理学家都是必需的。
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