Year 13 OCR Engineering: Formula and Theorem Quick Reference Handbook | Year 13 OCR 工程：公式定理速查手册

📚 Year 13 OCR Engineering: Formula and Theorem Quick Reference Handbook | Year 13 OCR 工程：公式定理速查手册

This quick reference handbook compiles essential formulas and key theorems for the Year 13 OCR Engineering specification. It consolidates the core principles from mechanics, materials, thermodynamics, fluid dynamics, electrical circuits, and control systems to support revision, problem-solving, and exam preparation.

本速查手册汇集了适用于 Year 13 OCR 工程课程的基本公式与核心定理。它整合了力学、材料、热力学、流体动力学、电路以及控制系统中的关键原理，为复习、解题和备考提供有力支持。

1. Linear Kinematics and Dynamics | 直线运动学与动力学

For an object moving with constant acceleration a, the following SUVAT equations relate displacement s, initial velocity u, final velocity v, and time t.

对于以恒定加速度 a 运动的物体，以下 SUVAT 方程关联位移 s、初速度 u、末速度 v 和时间 t。

v = u + at — final velocity equals initial velocity plus acceleration multiplied by time.

v = u + at — 末速度等于初速度加加速度乘以时间。

v = u + at

s = ut + ½at² — cement is initial velocity times time plus half the acceleration times the square of time.

s = ut + ½at² — 位移等于初速度乘以时间加上加速度与时间平方乘积的一半。

s = ut + ½at²

v² = u² + 2as — ty squared equals initial velocity squared plus twice the acceleration times displacement.

v² = u² + 2as — 末速度的平方等于初速度的平方加上加速度与位移乘积的两倍。

v² = u² + 2as

s = ½(u + v)t — displacement is the average velocity multiplied by time.

s = ½(u + v)t — 位移等于平均速度乘以时间。

s = ½(u + v)t

Newton's Second Law: the net force F acting on an object is equal to the product of its mass m and acceleration a.

牛顿第二定律：作用于物体的净力 F 等于其质量 m 与加速度 a 的乘积。

F = ma

Momentum p is the product of mass and velocity; impulse J equals the change in momentum.

动量 p 是质量与速度的乘积；冲量 J 等于动量的变化。

p = mv, J = F t = Δp

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2. Work, Energy and Power | 功、能与功率

Work done W by a constant force F acting over a displacement s at an angle θ to the direction of motion.

恒力 F 作用在位移 s 上且与运动方向夹角为 θ 时所做的功 W。

W = F s cos θ

Kinetic energy Ek is the energy an object possesses due to its motion, and gravitational potential energy Ep depends on height h above a reference level.

动能 Ek 是物体因运动而具有的能量，重力势能 Ep 取决于相对于参考水平面的高度 h。

Ek = ½ m v², Ep = m g h

Power P is the rate of doing work, equal to work divided by time or, for a constant force, the product of force and velocity.

功率 P 是做功的速率，等于功除以时间，或对于恒力等于力与速度的乘积。

P = W/t = F v

Efficiency η is the ratio of useful output energy or power to total input, often expressed as a percentage.

效率 η 是有用输出能量或功率与总输入的比值，通常以百分比表示。

η = (useful output / input) × 100%

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3. Stress, Strain and Material Constants | 应力、应变与材料常数

Tensile or compressive stress σ is the applied force F per unit cross-sectional area A.

拉伸或压缩应力 σ 是施加的力 F 除以横截面积 A。

σ = F / A

Normal strain ε is the change in length ΔL divided by the original length L₀. Shear stress τ and shear strain γ follow analogous definitions.

正应变 ε 是长度变化 ΔL 除以原始长度 L₀。切应力 τ 和切应变 γ 具有类似的定义。

ε = ΔL / L₀, τ = F / A, γ = Δx / L

Young's modulus E measures stiffness in tension or compression, and the shear modulus G measures rigidity under shear. Poisson's ratio ν relates lateral to axial strain.

杨氏模量 E 衡量拉伸或压缩时的刚度，剪切模量 G 衡量抗剪切的刚性。泊松比 ν 关联横向应变与轴向应变。

E = σ / ε, G = τ / γ, ν = - lateral strain / axial strain

Hooke's Law for a spring or elastic member: force F is proportional to extension x.

弹簧或弹性构件的胡克定律：力 F 与伸长量 x 成正比。

F = k x

The factor of safety is the ratio of the ultimate (or yield) stress to the allowable working stress.

安全系数是极限（或屈服）应力与许用工作应力的比值。

Factor of safety = ultimate stress / allowable stress

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4. Bending and Torsion of Beams | 梁的弯曲与扭转

For a beam under pure bending, the bending stress σ varies linearly with distance y from the neutral axis. The flexure formula relates bending moment M, second moment of area I, stress, and the radius of curvature R.

对于纯弯曲的梁，弯曲应力 σ 随到中性轴的距离 y 线性变化。弯曲公式关联弯矩 M、截面惯性矩 I、应力和曲率半径 R。

σ = M y / I, M / I = σ / y = E / R

ximum bending stress occurs at the greatest distance from the neutral axis, so σ_max = M y_max / I.

最大弯曲应力发生在离中性轴最远处，因此 σ_max = M y_max / I。

σ_max = M y_max / I

For a circular shaft under torsion, shear stress τ at radius ρ is given by the torsion formula, where T is the applied torque and J is the polar moment of inertia. The maximum shear stress is at the outer radius r.

对于受扭转的圆轴，在半径 ρ 处的切应力 τ 由扭转公式给出，其中 T 为施加的扭矩，J 为极惯性矩。最大切应力在外半径 r 处。

τ = T ρ / J, τ_max = T r / J

with a central point load F, length L, the maximum deflection δ_max occurs at the centre.

对于中点受集中载荷 F 的简支梁，长度 L，最大挠度 δ_max 出现在中间。

δ_max = F L³ / (48 E I)

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5. Fluid Mechanics | 流体力学

ydrostatic pressure at a depth h in a stationary fluid of density ρ is given by p = ρ g h.

密度 ρ 是单位体积的质量。在密度为 ρ 的静止流体中，深度 h 处的静水压力由 p = ρ g h 给出。

ρ = m / V, p = ρ g h

The continuity equation for an incompressible fluid states that the volume flow rate is constant: area A times velocity v remains the same along a streamline.

不可压缩流体的连续性方程表明体积流量恒定：面积 A 乘以速度 v 沿流线保持不变。

A₁ v₁ = A₂ v₂

Bernoulli's equation expresses the conservation of energy per unit volume along a streamline. The sum of pressure energy, kinetic energy per unit volume, and potential energy per unit volume is constant for steady, inviscid, incompressible flow.

伯努利方程表达了沿流线单位体积能量的守恒。对于稳定、无黏、不可压缩流体，压力能、动能和势能之和为常数。

p₁ + ½ ρ v₁² + ρ g h₁ = p₂ + ½ ρ v₂² + ρ g h₂

The Reynolds number Re characterises the flow regime; values below approximately 2000 indicate laminar flow, while values above 4000 usually indicate turbulent flow.

雷诺数 Re 表征流态；低于约 2000 为层流，高于约 4000 通常为湍流。

Re = ρ v d / μ

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6. Thermodynamics and Ideal Gases | 热力学与理想气体

The ideal gas law links pressure p, volume V, amount of substance n, and absolute temperature T. The universal gas constant R is 8.314 J mol⁻¹ K⁻¹.

理想气体定律将压力 p、体积 V、物质的量 n 和绝对温度 T 联系起来。通用气体常数 R 为 8.314 J mol⁻¹ K⁻¹。

p V = n R T

Boyle's law (constant T): p

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