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

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

This handbook summarises the essential formulas, laws and theorems you must recall for the Year 13 WJEC Engineering examinations. Use it for quick revision and exam-paper practice, ensuring you can apply each relationship accurately and efficiently.

本手册汇总了 Year 13 WJEC 工程考试必须掌握的核心公式、定律与定理。可用作快速复习和试卷练习，确保你能准确高效地应用每一个关系式。

1. Statics & Equilibrium | 静力学与平衡

ΣF = 0

For a body in static equilibrium, the vector sum of all external forces acting on it must be zero. This condition must be satisfied in two perpendicular directions, typically horizontal and vertical.

对于处于静力平衡的物体，所有外力的矢量和必须为零。这个条件必须在两个互相垂直的方向上同时满足，通常取水平和竖直方向。

ΣM = 0

The sum of all moments about any chosen point is zero. A moment is the product of a force and the perpendicular distance from its line of action to the pivot: M = F × d.

对任意选定点的所有力矩之和为零。力矩是力与其作用线到支点的垂直距离的乘积：M = F × d。

F ≤ μR

The friction force between two dry surfaces is limited by the coefficient of static friction µ and the normal reaction R. When limiting equilibrium is reached, F = μR.

两个干燥表面之间的摩擦力受静摩擦系数 µ 和法向反力 R 的限制。达到极限平衡时，F = μR。

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2. Stress, Strain & Young’s Modulus | 应力、应变与杨氏模量

σ = F / A

Direct stress (σ) is the internal force per unit cross-sectional area. It is measured in pascals (Pa). Tensile stress is taken as positive.

正应力（σ）是单位横截面积上的内力，单位为帕斯卡（Pa）。拉应力取正值。

ε = ΔL / L₀

Axial strain (ε) is the dimensionless ratio of change in length to the original length.

轴向应变（ε）是长度变化量与原长之比，量纲为一。

E = σ / ε (within proportional limit)

Young’s modulus (E) describes the stiffness of a material in the linear elastic region. It is constant for a given material and has units of Pa.

杨氏模量（E）描述材料在线弹性范围内的刚度。对于给定材料为常数，单位为 Pa。

ν = – ε_lateral / ε_axial

Poisson’s ratio (ν) is the negative ratio of transverse strain to axial strain. For most engineering metals ν ≈ 0.3.

泊松比（ν）是横向应变与轴向应变之比的负值。大多数工程金属的 ν 约为 0.3。

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3. Beam Bending Theory | 梁弯曲理论

M / I = σ / y = E / R

The bending equation relates the bending moment M, second moment of area I, bending stress σ at distance y from the neutral axis, and the radius of curvature R. The neutral axis passes through the centroid of the cross-section.

弯曲公式关联了弯矩 M、截面二次矩 I、距中性轴 y 处的弯曲应力 σ 以及曲率半径 R。中性轴通过横截面的形心。

Maximum bending stress occurs at the furthest face from the neutral axis, where y = y_max:

最大弯曲应力出现在离中性轴最远的表面，此处 y = y_max：

σ_max = M y_max / I

Common standard second moments of area for a rectangle (breadth b, depth d) about its centroidal axis: I = bd³/12.

常见矩形截面（宽 b、高 d）对其形心轴的截面二次矩：I = bd³/12。

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4. Torsion of Circular Shafts | 圆轴扭转

T / J = τ / r = Gθ / L

The torsion equation for a circular shaft links the applied torque T, polar second moment of area J, shear stress τ at radius r, shear modulus G, angle of twist θ (in radians) and shaft length L.

圆轴扭转方程联系了施加的扭矩 T、极截面二次矩 J、半径 r 处的剪切应力 τ、剪切模量 G、扭转角 θ（弧度）和轴长 L。

For a solid circular shaft, J = πd⁴/32, and for a hollow shaft with inner diameter dᵢ and outer diameter dₒ, J = π(dₒ⁴ – dᵢ⁴)/32.

对于实心圆轴，J = πd⁴/32；对于内径 dᵢ、外径 dₒ 的空心轴，J = π(dₒ⁴ – dᵢ⁴)/32。

τ_max = T dₒ / (2J)

Maximum shear stress occurs at the outer surface of the shaft. Power transmitted by a rotating shaft: P = Tω, where ω is angular velocity in rad/s.

最大剪切应力发生在轴的外表面。旋转轴传递的功率：P = Tω，其中 ω 为角速度，单位为 rad/s。

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5. Kinematics & Dynamics | 运动学与动力学

v = u + a t

s = u t + ½ a t²

v² = u² + 2 a s

These equations of motion apply for constant linear acceleration a, where u is initial velocity, v final velocity, s displacement and t time.

这些运动学方程适用于恒定的线性加速度 a，其中 u 为初速度，v 为末速度，s 为位移，t 为时间。

F = m a

Newton’s second law: the resultant force acting on a body equals the product of its mass and acceleration.

牛顿第二定律：作用在物体上的合力等于质量与加速度的乘积。

Impulse = Δp = F Δt

Impulse equals the change in momentum. In a closed system, momentum is conserved: Σ m₁u₁ = Σ m₂v₂.

冲量等于动量的变化量。在封闭系统中，动量守恒：Σ m₁u₁ = Σ m₂v₂。

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6. Work, Energy & Power | 功、能与功率

W = F s cos θ

Work done by a constant force is the product of the force, displacement and the cosine of the angle between them.

恒力所做的功等于力、位移以及它们之间夹角余弦的乘积。

E_k = ½ m v²

E_p = m g h

Kinetic energy and gravitational potential energy. The principle of conservation of energy states that total mechanical energy remains constant if only conservative forces act.

动能和重力势能。能量守恒定律指出，如果只有保守力做功，则总机械能保持不变。

P = W / t = F v

Power is the rate of doing work. For a force moving at constant velocity v, instantaneous power is P = Fv.

功率是做功的速率。对于以恒定速度 v 运动的力，瞬时功率为 P = Fv。

η = (useful output / input) × 100%

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

效率是有用输出功率与输入功率之比，常用百分比表示。

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7. Electrical Principles | 电学原理

V = I R

Ohm’s law for a resistor at constant temperature. The potential difference across a component is proportional to the current flowing through it.

恒定温度下电阻的欧姆定律。元件两端的电位差与流过它的电流成正比。

P = I V = I² R = V² / R

Electrical power dissipated in a resistive component can be expressed in these three equivalent forms. The energy transferred E = P t.

电阻元件消耗的电功率可用这三种等价形式表示。转移的能量 E = P t。

ΣI_in = ΣI_out (KCL)

ΣV_loop = 0 (KVL)

Kirchhoff’s current law states that the algebraic sum of currents entering a node is zero. Kirchhoff’s voltage law states that the sum of e.m.f.s around any closed loop equals the sum of potential differences.

基尔霍夫电流定律：流入节点的电流代数和为零。基尔霍夫电压定律：任一闭合回路中，电动势之和等于电位差之和。

R_total = R₁ + R₂ + … (series)

1/R_total = 1/R₁ + 1/R₂ + … (parallel)

Equivalent resistance rules for series and parallel combinations.

串联与并联电阻的等效电阻规则。

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8. Thermodynamics | 热力学

p V = n R T

The ideal gas equation links pressure p, volume V, amount of substance n, molar gas constant R (8.31 J mol⁻¹ K⁻¹) and absolute temperature T (in kelvins).

理想气体状态方程联系了压力 p、体积 V、物质的量 n、摩尔气体常数 R（8.31 J mol⁻¹ K⁻¹）和绝对温度 T（开尔文）。

p V = constant (Boyle’s law, T constant)

V / T = constant (Charles’s law, p constant)

p / T = constant (Gay-Lussac’s law, V constant)

These relationships hold for a fixed mass of ideal gas.

以上关系适用于固定质量的理想气体。

Q = m c ΔT

Sensible heat transfer relates heat Q, mass m, specific heat capacity c and temperature change ΔT.

显热传递关联了热量 Q、质量 m、比热容 c 和温度变化 ΔT。

Q̇ = k A ΔT / d

Steady-state conduction through a slab of thickness d, area A and thermal conductivity k, with a temperature difference ΔT.

通过厚度为 d、面积为 A、导热系数为 k 的平板的稳态导热，温度差为 ΔT。

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9. Fluid Mechanics | 流体力学

p = ρ g h

Pressure at a depth h in a static fluid of density ρ, with g being gravitational field strength.

密度为 ρ 的静止流体中，深度 h 处的压力，g 为重力场强度。

F_buoyancy = ρ_fluid g V_displaced

Archimedes’ principle: the upthrust on a submerged or floating body equals the weight of fluid displaced.

阿基米德原理：浸没或漂浮物体所受的浮力等于排开流体的重量。

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

Bernoulli’s equation for steady, incompressible, inviscid flow along a streamline. It expresses the conservation of mechanical energy per unit volume.

沿流线的定常、不可压、无黏流动的伯努利方程。它表示单位体积机械能的守恒。

Q = A v

Volumetric flow rate Q equals the product of cross-sectional area A and average flow velocity v. For a liquid, A₁v₁ = A₂v₂ (continuity).

体积流量 Q 等于横截面积 A 与平均流速 v 的乘积。对于液体，A₁v₁ = A₂v₂（连续性方程）。

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10. Engineering Mathematics Essentials | 工程数学精要

sin θ = opposite / hypotenuse

cos θ = adjacent / hypotenuse

tan θ = sin θ / cos θ

Basic trigonometric ratios used for resolving forces and analysing vector components.

用于分解力和分析矢量分量的基本三角比。

c² = a² + b² – 2ab cos C (cosine rule)

a / sin A = b / sin B (sine rule)

These rules are often needed when constructing force polygons or analysing non-right-angled trusses.

构建力多边形或分析非直角桁架时经常需要用到这些规则。

d/dx (xⁿ) = n xⁿ⁻¹, ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C

Basic differentiation and integration are essential for finding slopes, rates of change, areas and centroids. The area between a curve and the x-axis is evaluated by definite integration.

基础的微分与积分对于求斜率、变化率、面积和形心至关重要。曲线与 x 轴之间的面积通过定积分求得。

θ = s / r (radians)

ω = Δθ / Δt, α = Δω / Δt

Circular measure and rotational kinematics. Angular velocity ω and angular acceleration α are central to analysing gears, pulleys and rotating machinery.

弧度制与旋转运动学。角速度 ω 和角加速度 α 是分析齿轮、带轮和旋转机械的核心。

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