📚 Relating SI Units | 关联国际单位制单位
In CIE A Level Physics, relating SI units means expressing every derived unit in terms of the seven SI base units. This skill is tested directly in unit-conversion questions and indirectly when you check whether an equation is dimensionally consistent.
在 CIE A Level 物理中，关联国际单位制单位是指用七个基本单位表示每一个导出单位。这一技能不仅直接考查单位换算题，也会在检验公式是否量纲一致时间接考查。
1. SI Base Units and Base Quantities | SI基本单位与基本量
Memorise the seven SI base quantities and their base units. Every other unit in mechanics and electricity can be built from these seven, so they are the foundation of unit analysis.
熟记七个SI基本量及其基本单位。力学和电学中的所有其他单位都可以由这七个基本单位组合而成，因此它们是单位分析的基础。
| Quantity | Unit | 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 |
2. Derived Units as Combinations of Base Units | 导出单位作为基本单位的组合
Any quantity that is not one of the seven base quantities has a derived unit. To express a derived unit in base units, substitute the defining equation and replace each symbol with its own base unit.
任何不在七个基本量之列的物理量都具有导出单位。要用基本单位表示导出单位，只需代入定义式，并将每个物理量符号替换为其基本单位。
speed = distance ÷ time → m s⁻¹
acceleration = speed change ÷ time → m s⁻²
density = mass ÷ volume → kg m⁻³
momentum = mass × velocity → kg m s⁻¹
Always simplify the resulting product of powers so the unit is written as a single expression, for example kg m s⁻¹ rather than kg × m ÷ s.
始终要把得到的幂次积化简为一个单一表达式，例如写成 kg m s⁻¹，而不是 kg × m ÷ s。
3. Expressing the Newton in Base Units | 用基本单位表示牛顿
The newton is the SI unit of force, but it is not a base unit. It is defined by Newton’s second law: force equals mass times acceleration.
牛顿是力的SI单位，但它不是基本单位。它由牛顿第二定律定义：力等于质量乘以加速度。
F = ma
Substitute the base units of mass and acceleration into the equation.
将质量和加速度的基本单位代入该方程。
1 N = 1 kg × 1 m s⁻² = kg m s⁻²
This is one of the most frequently required unit relationships in CIE exam papers. You should be able to write it down instantly.
这是 CIE 试卷中要求最频繁的单位关系之一。你应该能够立即写出它。
4. Work, Energy and Power: Joules and Watts | 功、能量与功率：焦耳与瓦特
Work is force multiplied by displacement in the direction of the force. Since the newton is kg m s⁻², the joule can be expressed in base units as follows.
功等于力乘以沿力方向的位移。由于牛顿是 kg m s⁻²，焦耳可以用基本单位表示如下。
W = Fs → 1 J = 1 N m = kg m² s⁻²
Power is the rate of doing work, so the watt is one joule per second.
功率是做功的速率，因此瓦特等于每秒一焦耳。
P = W ÷ t → 1 W = 1 J s⁻¹ = kg m² s⁻³
Remember that all forms of energy, including kinetic energy and gravitational potential energy, must have the same base unit: kg m² s⁻².
记住所有形式的能量，包括动能和重力势能，必须具有相同的基本单位：kg m² s⁻²。
5. Pressure and Frequency | 压强与频率
Pressure is defined as force acting normally per unit area. Starting from the base units of force and area gives the base-unit expression for the pascal.
压强定义为垂直作用在单位面积上的力。从力和面积的基本单位出发，可以得到帕斯卡的基本单位表达式。
p = F ÷ A → 1 Pa = 1 N m⁻² = kg m⁻¹ s⁻²
Frequency is the number of complete cycles per second. Its SI unit, the hertz, is therefore equivalent to the reciprocal second.
频率是每秒完整周期的次数。它的SI单位赫兹因此等于每秒的倒数。
f = 1 ÷ T → 1 Hz = s⁻¹
Do not confuse the hertz with the radian per second; angular frequency has the unit rad s⁻¹, and the radian itself is dimensionless.
不要将赫兹与弧度每秒混淆；角频率的单位是 rad s⁻¹，而弧度本身是无量纲的。
6. Electrical Derived Units: Coulomb, Volt and Ohm | 电学导出单位：库仑、伏特和欧姆
Electrical units often look complicated, but they can all be expressed in the four base units kg, m, s and A. Start with charge from current and time.
电学单位往往看起来复杂，但它们都可以用 kg、m、s 和 A 这四个基本单位表示。首先从电流和时间得出电荷。
Q = It → 1 C = 1 A s
Potential difference is energy transferred per unit charge, so the volt is the joule per coulomb.
电势差是每单位电荷转移的能量，因此伏特是焦耳每库仑。
V = W ÷ Q → 1 V = 1 J C⁻¹ = kg m² s⁻³ A⁻¹
Resistance is potential difference divided by current, so the ohm is the volt per ampere.
电阻是电势差除以电流，因此欧姆是伏特每安培。
R = V ÷ I → 1 Ω = 1 V A⁻¹ = kg m² s⁻³ A⁻²
You should also be able to show that the farad is kg⁻¹ m⁻² s⁴ A² and the tesla is kg s⁻² A⁻¹ if your specification includes capacitance or magnetic flux density.
如果你的考纲涉及电容或磁通量密度，你还应能够证明法拉是 kg⁻¹ m⁻² s⁴ A²，特斯拉是 kg s⁻² A⁻¹。
7. Checking Equations by Unit Homogeneity | 通过单位一致性检验方程
A valid physical equation must be homogeneous: every term must have the same base units. This gives a powerful method for spotting errors in a formula.
一个有效的物理方程必须具有单位一致性：每一项必须具有相同的基本单位。这为发现公式错误提供了一种有力方法。
Consider the equation of motion for constant acceleration:
考虑匀加速运动的运动学方程：
s = ut + ½at²
Check each term separately.
分别检验每一项。
[s] = m
[ut] = (m s⁻¹)(s) = m
[½at²] = (m s⁻²)(s²) = m
All three terms reduce to metres, so the equation is homogeneous. Note that unit analysis cannot find incorrect dimensionless constants such as ½, but it can prove that an equation is definitely wrong if the units do not match.
三项都化简为米，因此该方程具有单位一致性。注意单位分析无法发现像 ½ 这样的无量纲常数错误，但如果单位不匹配，它就能证明一个方程一定是错误的。
8. Predicting Units of Constants | 推断常数的单位
When an equation contains a constant with unknown units, rearrange the equation to make the constant the subject, then substitute the known units of the other quantities.
当方程中的常数单位未知时，重新整理方程使该常数成为主项，然后代入其他量的已知单位。
For Newton’s law of gravitation:
对于牛顿万有引力定律：
F = Gm₁m₂ ÷ r²
Rearrange for the gravitational constant G:
重新整理求得引力常数 G：
G = Fr² ÷ (m₁m₂)
[G] = (kg m s⁻²)(m²) ÷ (kg × kg) = m³ kg⁻¹ s⁻²
This technique is often used in multiple-choice questions where you must identify the correct base units for a constant such as the Planck constant or the gravitational constant.
这种技巧常用于选择题中，要求你识别某个常数如普朗克常量或引力常数的正确基本单位。
9. Prefixes and Powers of Ten | 词头与10的幂
CIE questions frequently mix prefixed units with base units. You must convert every prefix to its power of ten before checking homogeneity or carrying out a calculation.
CIE 的考题经常将带词头的单位与基本单位混用。在检验单位一致性或进行计算之前，你必须先把每个词头转换为对应的10的幂。
| Prefix | Symbol | Power of ten |
|---|---|---|
| tera | T | 10¹² |
| giga | G | 10⁹ |
| mega | M | 10⁶ |
| kilo | k | 10³ |
| centi | c |
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