📚 Year 13 SQA Engineering: Formula and Theorem Quick Reference Handbook | Year 13 SQA 工程：公式定理速查手册
Welcome to the ultimate quick reference guide for Year 13 SQA Engineering. This handbook consolidates the essential formulas, theorems, and principles you need to master mechanics, materials, thermodynamics, electrical systems, and more. It is designed for rapid revision and easy memorisation, covering topics from kinematics to digital logic as specified in the Advanced Higher Engineering Science syllabus.
欢迎使用 Year 13 SQA 工程的终极速查手册。本手册整合了力学、材料学、热力学、电气系统等领域必须掌握的核心公式、定理和原理，旨在帮助快速复习和记忆。内容根据 Advanced Higher Engineering Science 课程大纲，覆盖运动学、数字逻辑等关键专题。
1. Kinematics and Dynamics | 运动学与动力学
Uniformly accelerated motion describes an object moving in a straight line with constant acceleration. The four fundamental SUVAT equations link initial velocity u, final velocity v, acceleration a, time t and displacement s.
匀加速运动描述物体在恒定加速度下的直线运动。四个基本 SUVAT 方程关联初速度 u、末速度 v、加速度 a、时间 t 和位移 s。
v = u + a t
s = u t + ½ a t²
v² = u² + 2 a s
s = ½ (u + v) t
Newton’s second law relates net force to acceleration: F = m a, where F is force (N), m is mass (kg) and a is acceleration (m/s²).
牛顿第二定律将合外力与加速度关联：F = m a，其中 F 为力（N），m 为质量（kg），a 为加速度（m/s²）。
F = m a
Momentum p is defined as the product of mass and velocity. The impulse-momentum principle states that impulse equals change in momentum.
动量 p 定义为质量与速度的乘积。冲量-动量原理指出冲量等于动量的变化。
p = m v
F Δt = Δp = m v – m u
2. Work, Energy and Power | 功、能和功率
Work done W by a constant force acting in the direction of displacement is W = F s. When a force is applied at an angle θ to the displacement, only the parallel component does work.
恒力沿位移方向所做的功 W = F s。当力与位移成角度 θ 时，只有平行分量做功。
W = F s cos θ
Kinetic energy Ek is the energy of a moving body. Gravitational potential energy Ep depends on height h above a reference level.
动能 Ek 是运动物体的能量。重力势能 Ep 取决于相对于参考面的高度 h。
Ek = ½ m v²
Ep = m g h
The work–energy principle states that the net work done on an object equals its change in kinetic energy. The principle of conservation of energy is fundamental in all engineering systems.
功能原理指出，对物体所做的净功等于其动能的变化量。能量守恒定律是所有工程系统的基础。
Wnet = ΔEk
Power P is the rate of doing work or transferring energy. For a constant force moving an object at velocity v, instantaneous power can be expressed as the product of force and velocity.
功率 P 是做功或能量传递的速率。对于以速度 v 运动的物体，瞬时功率可表示为力与速度的乘积。
P = W / t = F v
3. Statics and Moments | 静力学与力矩
For a body in static equilibrium, the vector sum of all forces and the sum of moments about any point must be zero.
处于静力平衡的物体，所有力的矢量和以及对任意点的力矩总和都必须为零。
ΣF = 0
ΣM = 0
The moment (torque) of a force about a pivot is the product of the force and the perpendicular distance from the line of action to the pivot. It causes rotation.
力对支点的力矩（扭矩）是力与力作用线到支点的垂直距离的乘积，引起转动。
M = F × d
A couple is a pair of equal, opposite parallel forces that produce pure rotation. The moment of a couple is the product of one force and the perpendicular distance between the forces.
力偶是一对大小相等、方向相反的平行力，产生纯转动。力偶矩等于其中一个力与两力作用线间垂直距离的乘积。
Mcouple = F × s
When analysing beams or structures, it is common to resolve forces into horizontal and vertical components. Typical support reactions and uniformly distributed loads (UDLs) can be treated as equivalent point loads.
分析梁或结构时，常将力分解为水平与垂直分量。典型的支座反力和均布载荷可转化为等效集中载荷。
4. Stress, Strain and Material Properties | 应力、应变与材料性质
Direct (normal) stress σ is the internal force per unit area within a material. Direct strain ε is the fractional change in length when a load is applied.
正应力 σ 是材料内部单位面积上的内力。正应变 ε 是施加载荷时长度的相对变化量。
σ = F / A
ε = ΔL / L₀
Young’s modulus E is a measure of the stiffness of a material, defined only in the linear elastic region of the stress-strain curve.
杨氏模量 E 衡量材料的刚度，仅在应力-应变曲线的线弹性区域内定义。
E = σ / ε
Shear stress τ and shear strain γ are analogous quantities for forces acting parallel to a surface. The shear modulus G relates them.
剪切应力 τ 和剪切应变 γ 是平行于表面作用的力的类似量。剪切模量 G 将它们关联起来。
τ = Fs / A
G = τ / γ
The factor of safety (FoS) is used to ensure a component can withstand loads greater than expected. It is the ratio of ultimate stress to allowable working stress.
安全系数用于确保部件能够承受超出预期的载荷，是极限应力与许用工作应力的比值。
FoS = σult / σallowable
5. Thermodynamics and Energy Systems | 热力学与能量系统
The First Law of Thermodynamics applied to a closed system states that the change in internal energy ΔU equals the net heat supplied Q minus the net work done W by the system (engineering sign convention).
热力学第一定律应用于闭口系统时指出，内能变化 ΔU 等于净输入热量 Q 减去系统对外所做的净功 W（工程符号惯例）。
ΔU = Q – W
For many engineering cycles, the thermal efficiency η is the ratio of useful work output to the heat input. The Carnot efficiency represents the theoretical maximum efficiency between two thermal reservoirs.
对于许多工程循环，热效率 η 是有用功输出与热量输入的比值。卡诺效率代表两个热源间的理论最大效率。
ηthermal = Wnet / Qin
ηCarnot = 1 – (Tcold / Thot)
Heat transfer by conduction through a flat slab follows Fourier’s law. The rate of heat transfer Qcond depends on the material’s thermal conductivity k, area A, temperature difference ΔT and thickness L.
通过平板的导热遵循傅里叶定律。导热速率 Qcond 取决于材料的导热系数 k、面积 A、温差 ΔT 和厚度 L。
Qcond = k A (ΔT / L)
The ideal gas law relates pressure P, volume V, amount of substance n and absolute temperature T. The specific gas constant Rspecific can also be used with mass m.
理想气体定律关联压力 P、体积 V、物质的量 n 和绝对温度 T。也可用比气体常数 Rspecific 和质量 m 表示。
P V = n R T = m Rspecific T
6. Electrical Principles | 电学原理
Ohm’s law states that the current I through a conductor is directly proportional to the potential difference V across it, provided temperature remains constant. Resistance R is measured in ohms (Ω).
欧姆定律指出，在温度不变的情况下，通过导体的电流 I 与导体两端的电势差 V 成正比。电阻 R 的单位为欧姆 (Ω)。
V = I R
Electric power P dissipated in a resistive component can be expressed in three equivalent forms.
电阻元件消耗的电功率 P 可用三种等价形式表示。
P = I V = I² R = V² / R
Resistance of a uniform conductor depends on its resistivity ρ, length L and cross-sectional area A.
均匀导体的电阻取决于其电阻率 ρ、长度 L 和横截面积 A。
R = ρ L / A
When resistors are connected in series, the total resistance is the sum of individual resistances. The same current flows through all components.
电阻串联时，总电阻为各电阻之和。通过每个元件的电流相同。
Rtotal = R₁ + R₂ + R₃ + …
For resistors in parallel, the reciprocal of the total resistance equals the sum of the reciprocals of individual resistances. The voltage across each branch is identical.
电阻并联时，总电阻的倒数等于各电阻倒数之和。各支路两端电压相同。
1/Rtotal = 1/R₁ + 1/R₂ + 1/R₃ + …
Kirchhoff’s current law (KCL): the sum of currents entering a node equals the sum of currents leaving. Kirchhoff’s voltage law (KVL): the algebraic sum of voltages around any closed loop is zero.
基尔霍夫电流定律 (KCL)：流入节点的电流之和等于流出节点的电流之和。基尔霍夫电压定律 (KVL)：任意闭合回路中电压的代数和为零。
7. Fluid Mechanics | 流体力学
The continuity equation for an incompressible fluid expresses conservation of mass. The volumetric flow rate Q remains constant along a streamline, relating cross-sectional area A and flow speed v.
不可压缩流体的连续性方程体现了质量守恒。体积流量 Q 沿流线保持恒定，将横截面积 A 与流速 v 联系起来。
A₁ v₁ = A₂ v₂
Q = A v
Bernoulli’s equation for steady, incompressible, inviscid flow along a streamline represents conservation of energy per unit volume. It relates pressure P, fluid density ρ, velocity v and elevation z.
伯努利方程适用于稳定、不可压缩、无黏性流沿流线流动，代表单位体积的能量守恒。它将压力 P、流体密度 ρ、流速 v 和高度 z 关联起来。
P₁ + ½ ρ v₁² + ρ g z₁ = P₂ + ½ ρ v₂² + ρ g z₂
For a fluid at rest, hydrostatic pressure increases linearly with depth h. The pressure at a point is given by the product of fluid density, gravitational acceleration and depth.
对于静止流体，静水压力随深度 h 线性增加。某点处的压力等于流体密度、重力加速度和深度的乘积。
P = ρ g h
The Reynolds number Re is a dimensionless quantity used to predict flow regime (laminar or turbulent). It depends on fluid density, velocity, a characteristic length D and dynamic viscosity μ.
雷诺数 Re 是用于预测流态（层流或湍流）的无量纲量。它取决于流体密度、流速、特征长度 D 和动力粘度 μ。
Re = ρ v D / μ
8. Simple Machines and Mechanical Advantage | 简单机械与机械效益
A simple machine is a device that changes the magnitude or direction of a force. The mechanical advantage (MA) is the ratio of load force to effort force.
简单机械是改变力的大小或方向的装置。机械效益 (MA) 是负载力与施力（输入力）的比值。
MA = Load / Effort
The velocity ratio (VR) or movement ratio is the ratio of distance moved by the effort to distance moved by the load in the same time. It is a purely geometric property.
速度比 (VR) 或位移比是相同时间内施力作用点移动的距离与负载移动距离的比值，是一个纯粹的几何特性。
VR = distance moved by effort / distance moved by load
The efficiency η of a machine is the percentage ratio of useful work output to total work input. It can also be expressed as the ratio of MA to VR.
机械效率 η 是有用功输出与总功输入的百分比比值，也可表示为 MA 与 VR 的比值。
η = (Work output / Work input) × 100%
η = (MA / VR) × 100%
Common simple machines and their typical ideal VR: a lever (VR = effort arm / load arm), a pulley system (VR = number of rope sections supporting the load), an inclined plane (VR = length / height) and a screw jack (VR = 2πr / pitch).
常见简单机械及其理想速度比：杠杆 (VR = 施力臂 / 负载臂)、滑轮组 (VR = 支持负载的绳索段数)、斜面 (VR = 长度 / 高度) 和螺旋千斤顶 (VR = 2πr / 螺距)。
9. Gear Systems and Transmission Ratios | 齿轮系统与传动比
Gears transmit rotary motion and torque between shafts. The speed (angular velocity ω) of meshing gears is inversely proportional to the number of teeth N. The gear ratio GR can be expressed in terms of teeth or speeds.
齿轮在轴间传递转动和扭矩。啮合齿轮的转速（角速度 ω）与齿数 N 成反比。齿轮比 GR 可用齿数或转速表示。
GR = Ndriven / Ndriver = ωdriver / ωdriven
For a simple gear train, the overall velocity ratio is the product of the ratios of consecutive meshing pairs. Idler gears change direction but do not affect the overall ratio.
对于简单轮系，总速度比是各对啮合齿轮比的乘积。惰轮改变转向但不影响总比值。
VRtotal = VR₁ × VR₂ × VR₃ × …
When friction
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