📚 Formula & Theorem Quick Reference Handbook | 公式定理速查手册
This handbook brings together the essential formulas, theorems and key equations required for the CIE GCSE Engineering course. It is designed as a fast revision aid, with each concept presented in clear paired statements and supported by correctly formatted mathematical expressions.
本手册汇总了 CIE GCSE 工程课程所必需的基本公式、定理和关键方程。它旨在作为快速复习工具，每个概念均以清晰的中英对照陈述呈现，并配有格式正确的数学表达式。
1. Stress and Strain | 应力与应变
Stress is the internal force per unit area within a material when an external load is applied. The basic formula is shown below.
应力是施加外部载荷时材料内部单位面积上所受的内力。基本公式如下。
σ = F / A
Here σ represents stress measured in pascals (Pa), F is the force in newtons (N), and A is the cross-sectional area in square metres (m²).
其中 σ 表示应力，单位为帕斯卡 (Pa)；F 是力，单位为牛顿 (N)；A 是横截面积，单位为平方米 (m²)。
Strain describes the deformation of the material and is defined as the ratio of extension to original length.
应变描述材料变形，定义为延伸长度与原长之比。
ε = ΔL / L₀
ΔL is the change in length and L₀ is the original gauge length. Strain is a dimensionless quantity.
ΔL 是长度变化量，L₀ 是原始标距长度。应变为无量纲量。
Direct stress can be tensile (pulling) or compressive (pushing), and both use the same formula.
直接应力可以是拉伸（拉）或压缩（压）两种形式，两者使用相同的公式。
2. Young’s Modulus and Hooke’s Law | 杨氏模量与胡克定律
Young’s modulus E is a measure of the stiffness of an elastic material and connects stress to strain in the linear region.
杨氏模量 E 是衡量弹性材料刚度的量，在弹性区域内将应力与应变联系起来。
E = σ / ε
Its unit is pascal (Pa), the same as stress, because strain has no units.
其单位与应力相同，为帕斯卡 (Pa)，因为应变无量纲。
Hooke’s Law states that, up to the limit of proportionality, extension is directly proportional to the applied force.
胡克定律指出，在比例极限范围内，延伸量与施加的力成正比。
F = k x
In this equation, F is the force, k is the spring constant (stiffness), and x is the extension from the equilibrium position.
方程中 F 表示力，k 为弹簧常数（刚度），x 为离开平衡位置的延伸量。
3. Factor of Safety | 安全系数
The factor of safety (FoS) is used to ensure that structures can support loads far beyond the maximum expected working load.
安全系数 (FoS) 用于确保结构能够承受远超最大预期工作载荷的负载。
FoS = Ultimate Stress / Working Stress
Ultimate stress is the maximum stress a material can withstand before failure, while working stress is the safe stress allowed in design.
极限应力是材料在失效前所能承受的最大应力，而工作应力是设计中允许的安全应力。
A higher factor of safety indicates a more conservative design, often used in critical structural applications.
较高的安全系数意味着设计更为保守，常用于关键结构的应用中。
4. Moments and Levers | 力矩与杠杆
The moment of a force about a pivot is the turning effect and is calculated by multiplying the force by the perpendicular distance.
力绕支点的力矩是转动效应，通过力乘以垂直距离来计算。
M = F × d
M represents moment in newton-metres (Nm), F is the force, and d is the perpendicular distance from the pivot to the line of action of the force.
M 表示力矩，单位为牛顿米 (Nm)；F 是力；d 是从支点到力作用线的垂直距离。
For a lever in equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments, known as the principle of moments.
对于处于平衡状态的杠杆，顺时针力矩之和等于逆时针力矩之和，这就是力矩原理。
For a simple first-order lever, the relationship reduces to: Load × Load arm = Effort × Effort arm.
对于简单一级杠杆，关系式简化为：载荷 × 载荷臂 = 动力 × 动力臂。
5. Mechanical Advantage (MA) and Velocity Ratio (VR) | 机械利益与速度比
Mechanical advantage measures how much a machine multiplies the input effort to lift a load.
机械利益衡量机器将输入动力放大以举起负载的倍数。
MA = Load / Effort
In an ideal frictionless system, MA would equal VR, but real machines always have some energy loss.
在理想无摩擦系统中，MA 将等于 VR，但实际机器总是存在一些能量损失。
The velocity ratio is determined solely by the geometry of the machine and indicates the ratio of distances moved.
速度比完全由机器的几何结构决定，表示移动距离的比值。
VR = Distance moved by effort / Distance moved by load
For many simple machines, VR can also be expressed in terms of number of teeth (gears) or rope sections (pulleys).
对于许多简单机械，VR 也可通过齿数（齿轮）或绳索段数（滑轮）来表示。
6. Efficiency of Machines | 机械效率
The efficiency of a machine tells us how well it converts the input work into useful output work, expressed as a percentage.
机器的效率表示它将输入功转化为有用输出功的程度，以百分比表示。
Efficiency (η) = (MA / VR) × 100%
Because MA is always less than VR due to friction, efficiency is always below 100% in real systems.
由于摩擦的存在，MA 总是小于 VR，因此实际系统的效率始终低于 100%。
Efficiency can also be calculated directly from work: η = (Work output / Work input) × 100%.
效率也可直接通过功来计算：η = (输出功 / 输入功) × 100%。
7. Pulleys and Gears | 滑轮与齿轮
For a simple pulley system with n load-bearing rope sections, the velocity ratio is equal to n.
对于具有 n 段承载绳索的简单滑轮系统，速度比等于 n。
VR = n (number of supporting rope strands)
The effort needed in an ideal system is Load / n, but friction increases the actual effort required.
理想系统中所需的动力为 载荷/n，但摩擦会增加实际所需的动力。
For meshing gears, VR is given by the tooth ratio: VR = Number of teeth on driven gear / Number of teeth on driver gear.
对于啮合齿轮，VR 由齿数比给出：VR = 从动齿轮齿数 / 主动齿轮齿数。
The torque ratio is inversely proportional to the speed ratio; a larger gear produces greater torque at lower speed.
扭矩比与转速比成反比；较大的齿轮以较低转速输出更大的扭矩。
8. Electrical Quantities | 电学量
Ohm’s Law relates potential difference, current and resistance in a circuit. It is the foundation of circuit analysis.
欧姆定律关联了电路中的电位差、电流和电阻，是电路分析的基础。
V = I × R
V is voltage in volts (V), I is current in amperes (A), and R is resistance in ohms (Ω).
V 表示电压，单位为伏特 (V)；I 表示电流，单位为安培 (A)；R 表示电阻，单位为欧姆 (Ω)。
Electric charge Q transferred by a steady current over time is given by:
稳定电流在一段时间内传递的电荷量 Q 由下式给出：
Q = I × t
where t is time in seconds and Q is in coulombs (C).
其中 t 为时间，单位为秒 (s)；Q 的单位为库仑 (C)。
9. Electrical Power and Energy | 电功率与电能
Electrical power consumed by a component is the product of voltage and current.
元器件消耗的电功率是电压与电流的乘积。
P = I × V
Using combinations of Ohm’s Law, power can also be expressed as P = I²R or P = V²/R.
结合欧姆定律，功率还可以表示为 P = I²R 或 P = V²/R。
Energy transferred electrically is power multiplied by time, often calculated in joules or kilowatt-hours.
电能传递的能量等于功率乘以时间，通常以焦耳或千瓦时计算。
E = P × t
In devices, energy can also be calculated directly from charge and voltage: E = Q × V.
在设备中，能量也可以直接通过电荷量和电压计算：E = Q × V。
10. Heat and Temperature | 热与温度
The quantity of heat energy required to raise the temperature of a substance is given by the specific heat capacity formula.
提高物质温度所需的热量由比热容公式给出。
Q = m × c × Δθ
Q is thermal energy in joules (J), m is mass in kilograms (kg), c is specific heat capacity in J/(kg °C), and Δθ is the temperature change in degrees Celsius.
Q 为热能，单位为焦耳 (J)；m 为质量，单位为千克 (kg)；c 为比热容，单位为 J/(kg °C)；Δθ 为温度变化，单位为摄氏度。
Linear thermal expansion (useful for bimetallic strips and engine components) can be estimated using:
线性热膨胀（对双金属片和发动机零件很重要）可用下式估算：
ΔL = α × L₀ × Δθ
where α is the coefficient of linear expansion for the material, L₀ is original length, and Δθ is temperature change.
其中 α 为材料的线膨胀系数，L₀ 为原始长度，Δθ 为温度变化量。
11. Work, Energy and Friction | 功、能与摩擦
Mechanical work is done when a force moves an object in the direction of the force.
当一个力使物体沿力的方向移动时，就做了机械功。
W = F × d
Gravitational potential energy gained by an object lifted to a height h is Ep = m × g × h, where g is 9.8 m/s² (often taken as 10 m/s² in exam problems).
物体被举升高度 h 所获得的引力势能为 Ep = m × g × h，其中 g 取 9.8 m/s²（考题中常取 10 m/s²）。
Kinetic energy of a moving object depends on mass and speed squared:
运动物体的动能取决于质量和速度的平方：
Ek = ½ × m × v²
The magnitude of friction force f between two dry surfaces is proportional to the normal reaction force R:
两干燥表面间摩擦力 f 的大小与法向反作用力 R 成正比：
f = μ × R
where μ is the coefficient of friction, a dimensionless number depending on the materials in contact.
其中 μ 为摩擦系数，是一个取决于接触材料的无量纲数。
12. Geometry and Conversions | 几何与换算
Engineering calculations frequently require areas and volumes of regular shapes. Area of a circle: A = π r²; circumference: C = 2 π r.
工程计算中经常需要规则形状的面积与体积。圆的面积：A = π r²；圆周长：C = 2 π r。
Rectangle area: A = l × w; triangle area: A = ½ b × h; volume of a cylinder: V = π r² h.
矩形面积：A = l × w；三角形面积：A = ½ b × h；圆柱体积：V = π r² h。
Common unit conversions useful in GCSE Engineering include: 1 m² = 10⁴ cm², 1 m³ = 10⁶ cm³, 1 kN = 1000 N, 1 MPa = 10⁶ Pa.
GCSE 工程中有用的常见单位换算包括：1 m² = 10⁴ cm², 1 m³ = 10⁶ cm³, 1 kN = 1000 N, 1 MPa = 10⁶ Pa。
When working with compound units, always check that quantities are expressed in consistent SI units before substituting into formulas.
在用复合单位解题时，务必在代入公式前检查各量是否使用了一致的国际单位制 (SI) 单位。
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