📚 Year 11 CCEA Engineering: Quick Reference Handbook of Formulas and Theorems | Year 11 CCEA 工程：公式定理速查手册
This handbook gathers the key formulas and principles required for CCEA GCSE Engineering in Year 11. It covers kinematics, mechanics, materials, electricity, simple machines and thermal effects, helping you recall and apply the relationships quickly during design, analysis and problem solving.
本手册汇集了 CCEA Year 11 工程课程所需的核心公式与定理，涵盖运动学、力学、材料、电学、简单机械及热效应等内容，帮助你在设计、分析和解题中快速回顾与运用这些关系。
1. Kinematic Equations | 运动学公式
For an object moving with uniform acceleration a, initial velocity u, final velocity v, displacement s and time t, the following equations apply.
对于以匀加速度 a 运动的物体，初速度 u、末速度 v、位移 s 和时间 t 满足以下方程。
Velocity–time relation: v = u + a t
速度–时间关系：v = u + a t
Displacement using average velocity: s = ½ (u + v) t
用平均速度求位移：s = ½ (u + v) t
Displacement with acceleration: s = u t + ½ a t2
含加速度的位移：s = u t + ½ a t2
Velocity–displacement relation: v2 = u2 + 2 a s
速度–位移关系：v2 = u2 + 2 a s
2. Newton’s Laws of Motion | 牛顿运动定律
First law: An object remains at rest or in uniform motion unless acted upon by a resultant force.
第一定律：除非受到合外力作用，物体将保持静止或匀速直线运动状态。
Second law: The acceleration of an object is directly proportional to the resultant force and inversely proportional to its mass. F = m a
第二定律：物体的加速度与合外力成正比，与质量成反比。F = m a
Third law: When two objects interact, they exert equal and opposite forces on each other.
第三定律：两物体相互作用时，彼此施加的力大小相等、方向相反。
3. Forces, Moments and Equilibrium | 力、力矩与平衡
Weight is the force due to gravity: W = m g, where g = 9.81 m s-2 on Earth.
重力是由于引力产生的力：W = m g，地球表面 g = 9.81 m s-2。
The moment of a force about a pivot is the product of the force and the perpendicular distance from the pivot: M = F × d
力对支点的力矩等于力与支点到力作用线垂直距离的乘积：M = F × d
Principle of moments: For a body in rotational equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments.
力矩原理：处于转动平衡的物体，顺时针力矩之和等于逆时针力矩之和。
Conditions for static equilibrium: resultant force = 0 and resultant moment = 0.
静力平衡条件：合外力为零且合力矩为零。
4. Stress, Strain and Young’s Modulus | 应力、应变与杨氏模量
Tensile or compressive stress is force per unit area: σ = F / A (unit: Pa or N m-2).
拉伸或压缩应力是单位面积上的力：σ = F / A（单位：Pa 或 N m-2）。
Strain is the extension per original length, a dimensionless ratio: ε = ΔL / L0
应变是伸长量与原长之比，无量纲：ε = ΔL / L0
Young’s modulus E measures stiffness: E = σ / ε (unit: Pa). It applies within the linear elastic region.
杨氏模量 E 衡量材料刚度：E = σ / ε（单位：Pa），适用于线弹性区域。
5. Work, Energy and Power | 功、能量和功率
Work done when a force moves an object: W = F d cos θ, where θ is the angle between force and displacement. If force and motion are parallel, W = F d.
力做功的定义：W = F d cos θ，其中 θ 为力与位移的夹角。当力与运动方向平行时，W = F d。
Gravitational potential energy: Ep = m g h
重力势能：Ep = m g h
Kinetic energy: Ek = ½ m v2
动能：Ek = ½ m v2
Power is the rate of doing work or transferring energy: P = W / t. For a constant force and velocity, P = F v.
功率是做功或能量转换的速率：P = W / t。当力恒定且速度不变时，P = F v。
Conservation of energy: Energy cannot be created or destroyed, only converted from one form to another.
能量守恒：能量不能被创造或消灭，只能从一种形式转换为另一种形式。
6. Ohm’s Law and Electrical Power | 欧姆定律与电功率
Ohm’s law: V = I R, where V is voltage (V), I is current (A) and R is resistance (Ω).
欧姆定律：V = I R，其中 V 为电压 (V)，I 为电流 (A)，R 为电阻 (Ω)。
Resistors in series: Rtotal = R1 + R2 + …
电阻器串联：Rtotal = R1 + R2 + …
Resistors in parallel: 1 / Rtotal = 1 / R1 + 1 / R2 + …
电阻器并联：1 / Rtotal = 1 / R1 + 1 / R2 + …
Electrical power: P = V I = I2 R = V2 / R
电功率：P = V I = I2 R = V2 / R
7. Mechanical Advantage, Velocity Ratio and Efficiency | 机械效益、速度比与效率
Mechanical advantage (MA) is the ratio of load to effort: MA = Load / Effort
机械效益（MA）是负载与施加力之比：MA = 负载 / 施加力
Velocity ratio (VR) is the ratio of distance moved by effort to distance moved by load: VR = deffort / dload
速度比（VR）是施力端移动距离与负载端移动距离之比：VR = d施力 / d负载
Efficiency (η) measures how well a machine converts input work to output work: η = (MA / VR) × 100% = (Work output / Work input) × 100%
效率（η）衡量机器将输入功转化为输出功的程度：η = (MA / VR) × 100% = (输出功 / 输入功) × 100%
8. Gears and Pulleys | 齿轮与滑轮
For two meshing gears, the velocity ratio is given by the tooth count: VR = Number of teeth on driven gear / Number of teeth on driver gear
对于两个咬合齿轮，速度比由齿数决定：VR = 从动轮齿数 / 主动轮齿数
The speed ratio: Speed of driver / Speed of driven = Teeth on driven / Teeth on driver
转速比：主动轮转速 / 从动轮转速 = 从动轮齿数 / 主动轮齿数
For a pulley system, VR equals the number of rope sections supporting the load (for ideal fixed and movable pulleys).
滑轮系统中，VR 等于支持负载的绳段数（理想定滑轮和动滑轮组合）。
9. Fluid Pressure | 流体压强
Pressure due to a static liquid column: p = ρ g h, where ρ is fluid density (kg m-3), g is gravitational field strength and h is depth (m).
静止液柱产生的压强：p = ρ g h，其中 ρ 为液体密度 (kg m-3)，g 为重力场强度，h 为深度 (m)。
Pressure in a hydraulic system is transmitted equally: p = F1 / A1 = F2 / A2. This allows force multiplication.
液压系统中的压强处处相等：p = F1 / A1 = F2 / A2，从而实现力的放大。
10. Thermal Expansion | 热膨胀
Linear expansion of a solid: ΔL = α L0 ΔT, where α is the coefficient of linear expansion (K-1 or °C-1), L0 is original length and ΔT is temperature change.
固体线膨胀：ΔL = α L0 ΔT，其中 α 为线膨胀系数 (K-1 或 °C-1)，L0 为原长，ΔT 为温度变化。
Area expansion: ΔA ≈ 2α A0 ΔT; volume expansion: ΔV ≈ 3α V0 ΔT for isotropic materials.
面积膨胀：ΔA ≈ 2α A0 ΔT；体积膨胀：对于各向同性材料 ΔV ≈ 3α V0 ΔT。
Thermal stress can arise if expansion is restricted, and engineers use expansion joints or selection of materials to accommodate dimensional changes.
若膨胀受到约束会产生热应力，工程师通过使用伸缩缝或材料选择来适应尺寸变化。
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