Conversion Factors | 单位换算因子

📚 Conversion Factors | 单位换算因子

In CIE A-Level Physics, a conversion factor is a ratio that expresses the same physical quantity in two different units. It is the tool you use whenever a calculation mixes cm with m, km h⁻¹ with m s⁻¹, or g cm⁻³ with kg m⁻³. Handling these factors confidently is essential for both multiple-choice and structured exam questions.

在 CIE A-Level 物理中，换算因子是表示同一物理量在两种不同单位下的比值。每当计算中混用 cm 与 m、km h⁻¹ 与 m s⁻¹、或 g cm⁻³ 与 kg m⁻³ 时，都需要使用换算因子。熟练处理这些因子对选择题和结构化考题都至关重要。

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1. Why Conversion Factors Matter in CIE Physics | 为什么换算因子在 CIE 物理中重要

CIE physics questions frequently present data in non-SI units such as km h⁻¹, g cm⁻³ or kWh. You must convert these into coherent SI units before substituting into equations. If the units are not consistent, even a correct formula will give a wrong numerical answer.

CIE 物理题经常给出非国际单位制的数据，例如 km h⁻¹、g cm⁻³ 或 kWh。你必须先将其转换为一致的 SI 单位，再代入公式。如果单位不一致，即使公式正确，数值答案也会出错。

A conversion factor is therefore not an extra step: it is part of the physics calculation. Writing units next to every number reduces careless mistakes and helps you check that an answer is physically reasonable.

因此，换算因子并不是额外步骤，而是物理计算的一部分。在每个数字旁写出单位可以减少粗心错误，也有助于你判断答案在物理上是否合理。

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2. SI Base Units and Prefixes | SI 基本单位与词头

All conversion factors start from the SI base units and the standard prefixes. The seven SI base units are the metre, kilogram, second, ampere, kelvin, mole and candela.

所有换算因子都源于 SI 基本单位和标准词头。七个 SI 基本单位是米、千克、秒、安培、开尔文、摩尔和坎德拉。

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

上表列出了 CIE 物理中最常用的词头。你应能快速写出例如 1 nm = 10⁻⁹ m、1 MJ = 10⁶ J、1 pF = 10⁻¹² F 等关系。

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3. Writing Conversion Factors as Fractions | 将换算因子写成分数

The safest method is to write a conversion factor as a fraction equal to 1. For example, since 1 km = 1000 m, the ratio 1000 m / 1 km is equal to 1, so multiplying by it changes the units but not the physical quantity.

最稳妥的方法是把换算因子写成分数，其值等于 1。例如，由于 1 km = 1000 m，比值 1000 m / 1 km 等于 1，因此乘以它只会改变单位，不会改变物理量本身。

4.5 km × (1000 m / 1 km) = 4500 m

This fraction method makes it easy to decide which unit goes on top. You want the old unit to cancel and the target unit to remain. Always include the units in the working, not just in the final answer.

这种分数方法让你容易判断哪个单位放在分子上。你希望旧单位被消去，目标单位保留下来。始终在计算过程中写出单位，而不仅仅在最终答案里写。

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4. Length, Area and Volume Conversions | 长度、面积与体积换算

Length conversions use exact powers of ten in SI. For area and volume, the factor must be squared or cubed. A common error is to write 1 cm² = 10⁻² m², but the correct relation is 1 cm² = (10⁻² m)² = 10⁻⁴ m².

长度换算在 SI 中使用 10 的整数次幂。面积和体积的换算因子必须平方或立方。常见错误是把 1 cm² 写成 10⁻² m²，但正确关系是 1 cm² = (10⁻² m)² = 10⁻⁴ m²。

• 1 cm = 10⁻² m, 1 mm = 10⁻³ m, 1 μm = 10⁻⁶ m
• 1 cm² = 10⁻⁴ m², 1 mm² = 10⁻⁶ m²
• 1 cm³ = 10⁻⁶ m³, 1 mm³ = 10⁻⁹ m³
• 1 L = 1 dm³ = 10⁻³ m³, 1 mL = 1 cm³ = 10⁻⁶ m³

这些关系表明：从 cm 到 m 是乘以 10⁻²，但从 cm² 到 m² 要乘以 10⁻⁴，从 cm³ 到 m³ 要乘以 10⁻⁶。体积换算时尤其要注意升与立方分米的关系。

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5. Speed and Acceleration Units | 速度与加速度单位

Speed conversions are extremely common. Since 1 km = 1000 m and 1 h = 3600 s, the conversion factor is 1000/3600 = 1/3.6. Therefore, 1 km h⁻¹ = (1/3.6) m s⁻¹ ≈ 0.278 m s⁻¹.

速度的换算非常常见。因为 1 km = 1000 m，1 h = 3600 s，换算因子为 1000/3600 = 1/3.6。因此，1 km h⁻¹ = (1/3.6) m s⁻¹ ≈ 0.278 m s⁻¹。

v (m s⁻¹) = v (km h⁻¹) ÷ 3.6

Acceleration is normally expressed in m s⁻². If a question gives a speed in km h⁻¹ and a time in seconds, convert the speed to m s⁻¹ first, then divide by the time in seconds.

加速度通常以 m s⁻² 表示。如果题目给出的速度单位是 km h⁻¹，而时间单位是秒，应先将速度换算为 m s⁻¹，再除以以秒为单位的时间。

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6. Density and Mass Conversions | 密度与质量换算

To convert 1 g cm⁻³ into kg m⁻³, write 1 g = 10⁻³ kg and 1 cm³ = 10⁻⁶ m³. Dividing gives 10⁻³ / 10⁻⁶ = 1000, so 1 g cm⁻³ = 1000 kg m⁻³.

要将 1 g cm⁻³ 换算为 kg m⁻³，写出 1 g = 10⁻³ kg、1 cm³ = 10⁻⁶ m³。相除得到 10⁻³ / 10⁻⁶ = 1000，因此 1 g cm⁻³ = 1000 kg m⁻³。

ρ (kg m⁻³) = 1000 × ρ (g cm⁻³)

Mass conversions are simpler: 1 tonne = 1000 kg, 1 g = 10⁻³ kg, and 1 mg = 10⁻⁶ kg. Pay attention when a material density is given in g cm⁻³ but a formula such as pressure p = ρgh expects kg m⁻³.

质量换算较简单：1 吨 = 1000 kg，1 g = 10⁻³ kg，1 mg = 10⁻⁶ kg。当材料密度以 g cm⁻³ 给出，而公式 p = ρgh 需要 kg m⁻³ 时，要特别注意换算。

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7. Force, Pressure and Energy | 力、压强与能量

The newton is already an SI derived unit: 1 N = 1 kg m s⁻². Pressure uses the pascal, 1 Pa = 1 N m⁻². Energy uses the joule, 1 J = 1 N m = 1 kg m² s⁻².

牛顿本身就是 SI 导出单位：1 N = 1 kg m s⁻²。压强使用帕斯卡，1 Pa = 1 N m⁻²。能量使用焦耳，1 J = 1 N m = 1 kg m² s⁻²。

Quantity	Common unit	Equivalent
pressure	1 atm	101325 Pa ≈ 1.01 × 10⁵ Pa
pressure	1 bar	1.00 × 10⁵ Pa
energy	1 eV	1.60 × 10⁻¹⁹ J
energy	1 kWh	3.60 × 10⁶ J

上表给出了常见非 SI 单位与 SI 单位之间的精确换算关系。计算压强或能量时，先统一为 Pa 和 J，再代入公式。

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8. Power and Electrical Units | 功率与电学单位

Power is measured in watts: 1 W = 1 J s⁻¹ = 1 kg m² s⁻³. Electrical units are built from the ampere. For example, 1 coulomb = 1 A s, 1 volt = 1 J C⁻¹, and 1 ohm = 1 V A⁻¹.

功率以瓦特为单位：1 W = 1 J s⁻¹ = 1 kg m² s⁻³。电学单位建立在安培基础上。例如，1 库仑 = 1 A s，1 伏特 = 1 J C⁻¹，1 欧姆 = 1 V A⁻¹。

In base SI units, the volt is kg m² s⁻³ A⁻¹ and the ohm is kg m² s⁻³ A⁻². These base-unit forms are useful for dimensional questions and for checking final units.

用 SI 基本单位表示，伏特为 kg m² s⁻³ A⁻¹，欧姆为 kg m² s⁻³ A⁻²。这些基本单位形式对于量纲题和检查最终单位非常有用。

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9. Converting Between Non-SI Common Units | 常见非国际单位换算

CIE papers sometimes require conversions involving inches, feet, miles, tonnes or temperature units. You should memorise the exact definitions where applicable: 1 inch = 2.54 cm exactly, 1 mile = 1609 m, 1 tonne = 1000 kg.

CIE 试卷有时会涉及英寸、英尺、英里、吨或温度单位的换算。你应记住适用的精确定义：1 英寸 = 2.54 cm（精确值），1 英里 = 1609 m，1 吨 = 1000 kg。

• 1 in = 2.54 cm = 0.0254 m
• 1 ft = 12 in = 0.3048 m
• 1 mile = 1609.344 m ≈ 1.609 km
• 1 tonne = 1000 kg
• T(K) = θ(°C) + 273.15

温度换算要区分温差和绝对温度。温差 1 °C 等于温差 1 K，但绝对温度必须加上 273.15。不要把温度差公式写成加 273.15。

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10. Significant Figures and Rounding After Conversion | 换算后的有效数字与修约

Conversion factors that are definitions, such as 1000 m km⁻¹ or 2.54 cm in⁻¹, are exact and do not limit significant figures. The final answer should keep the same number of significant figures as the least precise measured value.

定义为精确值的换算因子（如 1000 m km⁻¹ 或 2.54 cm in⁻¹）不计入有效数字限制。最终答案应保留与测量值中有效数字最少者相同的位数。

For example, if a distance is measured as 0.650 km and you need it in metres, 0.650 km = 650 m. The three significant figures are preserved because the conversion factor is exact.

例如，若测得距离为 0.650 km，需要以米表示，则 0.650 km = 650 m。由于换算因子是精确值，三个有效数字得以保留。

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11. Dimensional Checking Using Conversion Factors | 使用换算因子进行量纲检查

You can use conversion factors to check whether an expression has the correct SI base units. For example, if a formula is supposed to give energy, its units must reduce to kg m² s⁻².

你可以利用换算因子检查表达式是否具有正确的 SI 基本单位。例如，若某公式应

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