📚 AS OCR Engineering: Quick Reference Handbook of Formulas and Theorems | AS OCR 工程：公式定理速查手册
This quick reference guide brings together all the essential formulas, principles and theorems you will meet in the AS OCR Engineering course. Each entry is presented with its standard notation, a short explanation and a paired Chinese version to reinforce understanding. Use it for revision, problem-solving and building confidence before the examination.
本速查手册汇集了 AS OCR 工程课程中所有重要的公式、原理和定理。每个条目均配有标准符号、简短说明以及对应的中文释义，以加深理解。可在复习、解题和考前使用，帮助建立信心。
1. Engineering Mathematics Essentials | 工程数学基础
Pythagoras’ theorem relates the sides of a right‑angled triangle: a² + b² = c², where c is the hypotenuse.
a² + b² = c²
勾股定理描述直角三角形边的关系：a² + b² = c²，其中 c 为斜边。
Trigonometric ratios in a right triangle: sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent.
sin θ = opp / hyp
cos θ = adj / hyp
tan θ = opp / adj
直角三角形中的三角函数：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。
Conversion between degrees and radians: θ (rad) = θ (°) × π / 180.
θ (rad) = θ (°) × (π / 180)
度与弧度的换算：θ (弧度) = θ (°) × π / 180。
Resolution of a force or vector: Fₓ = F cos θ, Fy = F sin θ. Magnitude: F = √(Fₓ² + Fy²).
Fₓ = F cos θ, Fy = F sin θ
F = √(Fₓ² + Fy²)
力或向量的分解：Fₓ = F cos θ，Fy = F sin θ。大小：F = √(Fₓ² + Fy²)。
2. Statics, Forces and Moments | 静力学、力和力矩
Newton’s second law: resultant force equals mass times acceleration.
F = m a
牛顿第二定律：合力等于质量乘以加速度。
For equilibrium of a body the vector sum of all forces and the sum of all moments about any point must be zero.
Σ F = 0, Σ M = 0
物体平衡的条件是：所有力的矢量和为零，且对任一点所有力矩之和为零。
Moment of a force about a pivot: product of the force and the perpendicular distance from the pivot.
M = F × d
力矩：力与力臂（力到支点的垂直距离）的乘积。
Principle of moments: for equilibrium, sum of clockwise moments equals sum of anticlockwise moments.
Σ M꜀꜀w = Σ Mₐ꜀w
力矩原理：平衡时，顺时针力矩之和等于逆时针力矩之和。
Coefficient of static friction: μₛ = F / N, where F is the limiting friction and N the normal reaction.
μₛ = F / N
静摩擦系数：μₛ = F / N，F 为最大静摩擦力，N 为法向反力。
3. Stress, Strain and Young’s Modulus | 应力、应变与杨氏模量
Direct stress (σ): force applied perpendicularly divided by the original cross‑sectional area.
σ = F / A
正应力 (σ)：垂直施加的力除以原始截面积。
Direct strain (ε): change in length divided by the original length.
ε = ΔL / L₀
正应变 (ε)：长度变化量除以原始长度。
Young’s modulus (E) measures the stiffness of a material within its elastic limit.
E = σ / ε
杨氏模量 (E) 表征材料在弹性限度内的刚度。
For many materials stress and strain are proportional up to the limit of proportionality, described by Hooke’s law: σ ∝ ε, or σ = E ε.
σ = E ε (Hooke’s law)
对多数材料，在比例极限内应力与应变成正比，即胡克定律：σ ∝ ε，或 σ = E ε。
Ultimate tensile strength (UTS) is the maximum stress a material can withstand while being stretched before necking.
极限抗拉强度 (UTS) 是材料在颈缩前可承受的最大应力。
4. Material Properties and Safety Factors | 材料性质与安全系数
Density (ρ) is mass per unit volume.
ρ = m / V
密度 (ρ)：单位体积的质量。
Factor of safety is the ratio of the material’s ultimate (or yield) stress to the allowable working stress.
Factor of safety = σᵤₗₜ / σwₒᵣₖ
安全系数：材料的极限应力（或屈服应力）与许用工作应力之比。
Hardness is resistance to indentation or scratching; common scales include Brinell, Vickers and Rockwell.
硬度是抵抗压入或划痕的能力；常用标度有布氏、维氏和洛氏。
Strain energy stored in an elastically deformed component: U = ½ F ΔL or U = ½ σ ε V.
U = ½ F ΔL = ½ σ ε V
弹性变形构件的应变能：U = ½ F ΔL = ½ σ ε V。
5. Linear and Rotational Motion | 直线与旋转运动
Equations of uniform acceleration (suvat equations). (u initial velocity, v final velocity, a acceleration, t time, s displacement)
v = u + a t
s = u t + ½ a t²
v² = u² + 2 a s
s = ½ (u + v) t
匀加速运动方程 (suvat)。u 初速度，v 末速度，a 加速度，t 时间，s 位移。
For rotational motion about a fixed axis, analogous relationships hold using angular displacement θ, angular velocity ω and angular acceleration α.
ω = ω₀ + α t
θ = ω₀ t + ½ α t²
ω² = ω₀² + 2 α θ
绕固定轴的旋转运动，使用角位移 θ、角速度 ω 和角加速度 α，存在类似的运动学关系。
Linear velocity and angular velocity: v = r ω (where r is the radius).
v = r ω
线速度与角速度：v = r ω（r 为半径）。
6. Work, Energy and Power | 功、能量和功率
Work done by a constant force along a straight line: product of force and displacement in the direction of the force.
W = F s cos θ
恒力沿直线做功：力与沿力方向的位移的乘积。
Kinetic energy of a translating body.
Eₖ = ½ m v²
平移物体的动能。
Gravitational potential energy near the Earth’s surface.
Eₚ = m g h
近地表面的重力势能。
Mechanical power is the rate of doing work or transferring energy.
P = W / t = F v
机械功率是做功或传递能量的速率。
Efficiency (η) of a machine: ratio of useful output work/energy to input work/energy.
η = (useful output) / (input) × 100%
机器效率 (η)：有用输出功/能量与输入功/能量之比。
7. Thermal Expansion and Heat Transfer | 热膨胀与传热
Linear thermal expansion: change in length ΔL is proportional to the original length L₀ and the temperature change ΔT.
ΔL = α L₀ ΔT
线热膨胀：长度变化 ΔL 正比于原始长度 L₀ 和温度变化 ΔT，α 为线膨胀系数。
Volume thermal expansion for solids: ΔV = β V₀ ΔT, and for isotropic materials β ≈ 3α.
ΔV = β V₀ ΔT, β ≈ 3α
固体体热膨胀：ΔV = β V₀ ΔT，对各向同性材料 β ≈ 3α。
Heat energy required to change the temperature of a substance (specific heat capacity).
Q = m c ΔT
物质温度变化所需的热量（比热容）。
Thermal conductivity (k) governs conductive heat transfer: Q / t = k A ΔT / d, where d is the thickness.
Q / t = k A (ΔT / d)
导热系数 (k) 描述热传导：Q / t = k A (ΔT / d)，d 为厚度。
8. Fluid Pressure and Bernoulli’s Equation | 流体压力与伯努利方程
Pressure is force acting normally per unit area.
p = F / A
压强是单位面积上垂直作用的力。
Hydrostatic pressure at a depth h in a fluid of density ρ.
p = ρ g h
密度为 ρ 的流体在深度 h 处的静水压强。
Pascal’s principle: pressure applied to an enclosed fluid is transmitted undiminished in all directions.
帕斯卡原理：施加于密闭流体的压强会大小不变地向各个方向传递。
Flow rate for an incompressible fluid is constant along a stream tube (continuity equation).
A₁ v₁ = A₂ v₂
不可压缩流体沿流管的流量恒定（连续性方程）。
Bernoulli’s equation for steady, incompressible, non‑viscous flow along a streamline.
p + ½ ρ v² + ρ g h = constant
伯努利方程（适用于定常、不可压缩、无粘流沿流线）。
9. Electrical Circuits and Components | 电路与元器件
Ohm’s law relates potential difference V, current I and resistance R.
V = I R
欧姆定律表达电压 V、电流 I 与电阻 R 的关系。
Resistors in series: total resistance is the sum of individual resistances.
Rₜₒₜ = R₁ + R₂ + …
电阻串联：总电阻等于各电阻之和。
Resistors in parallel: the reciprocal of total resistance equals the sum of the reciprocals.
1/Rₜₒₜ = 1/R₁ + 1/R₂ + …
电阻并联：总电阻的倒数等于各电阻倒数之和。
Electrical power dissipated in a resistor can be expressed in three forms.
P = I V = I² R = V² / R
电阻消耗的电功率可用三种形式表达。
Kirchhoff’s current law (KCL): the sum of currents entering a junction equals the sum leaving. Kirchhoff’s voltage law (KVL): the directed sum of potential differences around any closed loop is zero.
Σ Iᵢₙ = Σ Iₒᵤₜ ; Σ V = 0
基尔霍夫电流定律 (KCL)：流入节点的电流之和等于流出之和。基尔霍夫电压定律 (KVL)：沿任一闭合回路，电压的代数和为零。
Capacitance relates stored charge to voltage: C = Q / V. Energy stored: E = ½ C V².
C = Q / V ; E = ½ C V²
电容：C = Q / V。储存能量：E = ½ C V²。
Inductance for a coil: V = L (ΔI / Δt). Energy in an inductor: E = ½ L I².
V = L (ΔI / Δt) ; E = ½ L I²
电感（线圈）：V = L (ΔI / Δt)。电感储存能量：E = ½ L I²。
10. Digital Logic and Boolean Algebra | 数字逻辑与布尔代数
Basic logic gates: AND (output is 1 only if all inputs are 1), OR (output is 1 if at least one input is 1), NOT (output is the inverse of the input).
基本逻辑门：与门 (AND，仅当所有输入均为 1 时输出 1)，或门 (OR，至少一个输入为 1 时输出 1)，非门 (NOT，输出为输入的取反)。
Boolean identities for simplification: A·0 = 0, A+1 = 1, A·A = A, A+A = A, A·A’ = 0, A+A’ = 1.
A·0 = 0, A+1 = 1, A·A = A, A+A = A, A·A’ = 0, A+A’ = 1
布尔化简恒等式：A·0 = 0，A+1 = 1，A·A = A，A+A = A，A·A’ = 0，A+A’ = 1。
De Morgan’s theorems allow conversion between AND and OR forms with inverted variables.
(A·B)’ = A’ + B’ ; (A+B)’ = A’·B’
德摩根定理实现与门和或门形式之间的带非变量转换。
An XOR (exclusive‑OR) gate gives output 1 when the inputs are different. Boolean expression: A ⊕ B = A·B’ + A’·B.
A ⊕ B = A·B’ + A’·B
异或门 (XOR) 当输入不同时输出 1。布尔表达式：A ⊕ B = A·B’ + A’·B。
A NAND gate is an AND gate followed by a NOT; a NOR gate is an OR followed by a NOT. Both are universal gates.
与非门 (NAND) 是与门后接非门；或非门 (NOR) 是或门后接非门。两者均为通用门。
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