📚 Year 13 OCR Engineering: Formula & Theorem Quick Reference Handbook | 公式定理速查手册
This quick reference handbook brings together the essential formulas, theorems, and constants required for the Year 13 OCR Engineering course. It is designed as a revision aid to help you recall key relationships in mechanics, materials, thermodynamics, fluids, electronics, and control systems. Use it alongside your textbook and practice papers to build speed and confidence before the exam.
本速查手册汇集了 Year 13 OCR 工程课程所需的核心公式、定理与常数。它是为帮助你快速回顾力学、材料、热力学、流体、电子及控制系统中的关键关系而设计的复习工具。请将其与教材和练习试卷结合使用，在考前提升答题速度与信心。
1. Stress and Strain | 应力与应变
Direct stress is the internal force per unit area within a material. Tensile stress is positive; compressive stress is negative. Strain is the ratio of change in length to the original length.
正应力是材料内部单位面积上的内力。拉应力为正，压应力为负。应变是长度变化量与原始长度的比值。
Tensile stress: σ = F / A, where F is the applied force (N) and A is the original cross‑sectional area (m²).
拉应力：σ = F / A，其中 F 为施加的力（N），A 为原始横截面积（m²）。
Direct strain: ε = ΔL / L0, with ΔL = extension or contraction (m) and L0 the original length (m).
正应变：ε = ΔL / L0，ΔL 为伸长或缩短量（m），L0 为原始长度（m）。
Shear stress: τ = F / A, where F is the shear force parallel to the area A. Shear strain γ is the angular deformation in radians.
剪应力：τ = F / A，F 为平行于面积 A 的剪切力。剪应变 γ 是以弧度为单位的角变形。
Poisson’s ratio: ν = −εlateral / εaxial. For most metals ν ≈ 0.3.
泊松比：ν = −ε横向 / ε轴向。大多数金属的 ν ≈ 0.3。
2. Elastic Constants and Material Properties | 弹性常数与材料性质
Young’s modulus describes the stiffness of a material in the linear‑elastic region. Shear and bulk moduli relate shear and volumetric strains to stress.
杨氏模量描述材料在线弹性区域的刚度。剪切模量和体积模量分别将剪应变和体积应变与应力联系起来。
Young’s modulus: E = σ / ε (slope of the elastic portion of the stress–strain curve). Units are Pa.
杨氏模量：E = σ / ε（应力–应变曲线弹性段的斜率）。单位为 Pa。
Shear modulus (modulus of rigidity): G = τ / γ. Relation: G = E / [2(1 + ν)].
剪切模量（刚性模量）：G = τ / γ。关系式：G = E / [2(1 + ν)]。
Bulk modulus: K = −V (Δp / ΔV). For isotropic materials, K = E / [3(1 − 2ν)].
体积模量：K = −V (Δp / ΔV)。对于各向同性材料，K = E / [3(1 − 2ν)]。
Factor of safety (FoS): FoS = ultimate tensile strength / allowable working stress.
安全系数：安全系数 = 极限抗拉强度 / 许用工作应力。
Toughness is the total energy absorbed per unit volume up to fracture (area under the stress–strain curve). Hardness is the resistance to indentation.
韧性是断裂前单位体积吸收的总能量（应力–应变曲线下的面积）。硬度是对压入的抵抗能力。
3. Beams and Bending | 梁与弯曲
Bending moments cause internal stresses that vary linearly through the beam’s depth. The neutral axis experiences zero stress.
弯矩引起沿梁高度线性分布的内应力。中性轴处的应力为零。
Bending formula: σ / y = M / I = E / R, where M is the bending moment (Nm), I is the second moment of area (m⁴), y is the distance from the neutral axis (m), and R is the radius of curvature (m).
弯曲公式：σ / y = M / I = E / R，M 为弯矩（Nm），I 为截面惯性矩（m⁴），y 为到中性轴的距离（m），R 为曲率半径（m）。
Maximum bending stress: σmax = M ymax / I.
最大弯曲应力：σmax = M ymax / I。
Second moment of area (common shapes): Rectangle about centroidal axis: I = b h³ / 12. Solid circular section: I = π d⁴ / 64.
截面惯性矩（常见形状）：矩形对形心轴：I = b h³ / 12。实心圆截面：I = π d⁴ / 64。
Deflection of a cantilever beam (end load F): δ = F L³ / (3 E I). For a simply supported beam with central load: δ = F L³ / (48 E I).
悬臂梁（端部载荷 F）的挠度：δ = F L³ / (3 E I)。简支梁中心受载：δ = F L³ / (48 E I)。
4. Dynamics and Kinematics | 动力学与运动学
The SUVAT equations describe uniformly accelerated linear motion. Newton’s laws link force, mass, and acceleration.
SUVAT 方程描述匀加速直线运动。牛顿定律将力、质量和加速度联系起来。
SUVAT equations (constant acceleration a): v = u + at ; s = ut + ½ a t² ; v² = u² + 2 a s ; s = (u + v) t / 2.
SUVAT 方程（匀加速度 a）：v = u + at ；s = ut + ½ a t² ；v² = u² + 2 a s ；s = (u + v) t / 2。
Newton’s second law: F = m a, where F is resultant force (N), m is mass (kg), a is acceleration (m s⁻²).
牛顿第二定律：F = m a，F 为合外力（N），m 为质量（kg），a 为加速度（m s⁻²）。
Momentum: p = m v. Impulse = Δp = F Δt.
动量：p = m v。冲量 = Δp = F Δt。
Centripetal acceleration for circular motion at radius r: a = v² / r = ω² r. Centripetal force F = m v² / r.
向心加速度（圆周运动半径 r）：a = v² / r = ω² r。向心力 F = m v² / r。
Angular motion: ω = Δθ / Δt ; α = Δω / Δt. Tangential velocity v = ω r.
角运动：ω = Δθ / Δt ；α = Δω / Δt。切向速度 v = ω r。
5. Work, Energy and Power | 功、能与功率
Work is done when a force moves its point of application. Energy is stored or transferred; power is the rate of doing work.
当力使其作用点移动时即做功。能量被储存或转移；功率是做功的速率。
Work done: W = F s cos θ, where θ is the angle between force and displacement.
做功：W = F s cos θ，θ 为力与位移之间的夹角。
Kinetic energy: KE = ½ m v². Gravitational potential energy: GPE = m g h.
动能：KE = ½ m v²。重力势能：GPE = m g h。
Elastic potential energy stored in a spring: E = ½ k x² = ½ F ΔL (Hooke’s law region).
弹簧储存的弹性势能：E = ½ k x² = ½ F ΔL（胡克定律区域）。
Power: P = W / t ; for constant force and velocity, P = F v. Units: watt (W).
功率：P = W / t ；恒力恒速时，P = F v。单位：瓦特（W）。
Efficiency: η = (useful power output / total power input) × 100%.
效率：η =（有用输出功率 / 总输入功率）× 100%。
6. Fluid Mechanics and Bernoulli | 流体力学与伯努利
Fluid pressure increases with depth. Bernoulli’s equation expresses conservation of energy along a streamline for an incompressible, inviscid flow.
流体压强随深度增加。伯努利方程表达了不可压缩、无粘性流体沿流线的能量守恒。
Hydrostatic pressure: p = ρ g h, where ρ is fluid density (kg m⁻³), g = 9.81 m s⁻², h is depth (m).
流体静压：p = ρ g h，ρ 为流体密度（kg m⁻³），g = 9.81 m s⁻²，h 为深度（m）。
Continuity equation (mass conservation): A1 v1 = A2 v2 for an incompressible fluid.
连续性方程（质量守恒）：对于不可压缩流体，A1 v1 = A2 v2。
Bernoulli’s equation (steady, incompressible, inviscid flow along a streamline): p + ½ ρ v² + ρ g h = constant.
伯努利方程（稳定、不可压缩、无粘性沿流线）：p + ½ ρ v² + ρ g h = 常数。
Reynolds number indicates flow regime: Re = ρ v L / μ, where L is characteristic length and μ dynamic viscosity. Re < 2000 laminar, > 4000 turbulent.
雷诺数指示流态：Re = ρ v L / μ，L 为特征长度，μ 为动力粘度。Re < 2000 层流，> 4000 湍流。
Drag force: FD = ½ CD ρ A v², where CD is the drag coefficient.
阻力：FD = ½ CD ρ A v²，CD 为阻力系数。
7. Thermodynamics and Heat | 热力学与热量
The first law of thermodynamics is an energy balance. Ideal gas behaviour and heat transfer mechanisms are central to thermal engineering.
热力学第一定律是能量平衡。理想气体行为和传热机制是热力工程的核心。
First law of thermodynamics: ΔU = Q − W, where ΔU is the change in internal energy of the system, Q is heat added to the system, W is work done BY the system. (Sign convention may vary; OCR uses ΔU = Q − W.)
热力学第一定律：ΔU = Q − W，ΔU 为系统内能变化，Q 为加入系统的热量，W 为系统对外做的功。（OCR 使用此符号规定。）
Ideal gas equation: p V = n R T, with p in Pa, V in m³, n in mol, R = 8.31 J mol⁻¹ K⁻¹, T in K.
理想气体方程：p V = n R T，p 为 Pa，V 为 m³，n 为 mol，R = 8.31 J mol⁻¹ K⁻¹，T 为 K。
Work done by an expanding gas at constant pressure: W = p ΔV.
恒压下气体膨胀做功：W = p ΔV。
Sensible heat: Q = m c Δθ, where c is specific heat capacity (J kg⁻¹ K⁻¹). Latent heat: Q = m L.
显热：Q = m c Δθ，c 为比热容（J kg⁻¹ K⁻¹）。潜热：Q = m L。
Thermal efficiency of a heat engine: η = Wnet / Qin = 1 − Qout / Qin. Carnot efficiency: ηCarnot = 1 − Tcold / Thot (temperatures in kelvin).
热机热效率：η = W净 / Q入 = 1 − Q出 / Q入。卡诺效率：ηCarnot = 1 − T冷 / T热（温度用开尔文）。
Conduction rate (Fourier’s law): P = k A (ΔT / L), where k is thermal conductivity (W m⁻¹ K⁻¹).
传导速率（傅里叶定律）：P = k A (ΔT / L)，k 为导热系数（W m⁻¹ K⁻¹）。
8. Electrical Circuit Analysis | 电路分析
Ohm’s law and Kirchhoff’s rules are fundamental for analysing DC circuits. Power dissipation and RC time constants are also essential.
欧姆定律和基尔霍夫定则是分析直流电路的基础。功率耗散和 RC 时间常数同样必不可少。
Ohm’s law: V = I R, where V is voltage (V), I is current (A), R is resistance (Ω).
欧姆定律：V = I R，V 为电压（V），I 为电流（A），R 为电阻（Ω）。
Resistors in series: Rtotal = R1 + R2 + … . Resistors in parallel: 1 / Rtotal = 1 / R1 + 1 / R2 + … .
电阻串联：R总 = R1 + R2 + … 。电阻并联：1 / R总 = 1 / R1 + 1 / R2 + … 。
Power in a resistor: P = I V = I² R = V² / R.
电阻功率：P = I V = I² R = V² / R。
Kirchhoff’s current law (KCL): at any node, Σ Iin = Σ Iout.
基尔霍夫电流定律：任一节点处，Σ I入 = Σ I出。
Kirchhoff’s voltage law (KVL): around any closed loop, the algebraic sum of e.m.f.s equals the sum of voltage drops (Σ V = 0).
基尔霍夫电压定律：任一闭合回路中，电动势代数和等于电压降之和（Σ V = 0）。
Capacitance: C = Q / V. Time constant for an RC circuit: τ = R C (seconds).
电容：C = Q / V。RC 电路的时间常数：τ = R C（秒）。
9. Logic Gates and Boolean Algebra | 逻辑门
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