📚 Year 10 OCR Engineering: Formula & Theorem Quick Reference Handbook | 公式定理速查手册
This quick reference handbook brings together the essential formulae and theorems covered in the Year 10 OCR Engineering course. Use it to revise key principles from mechanics, electronics, materials, thermodynamics and mechanical systems, all presented in clear, direct pairs of English and Chinese explanations.
本速查手册汇集了Year 10 OCR工程课程中必修的核心公式与定理。你可以用它来复习力学、电子学、材料学、热力学和机械系统等关键原理，所有内容均采用中英对照的解释方式，一目了然。
1. Basic Mathematical Foundations | 数学基础
A solid grasp of algebra, geometry and units is vital in engineering. The following are the most frequently used relationships.
牢固掌握代数、几何和单位对于工程至关重要。以下是最常用的关系式。
Pythagoras' theorem: in any right‑angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
a² + b² = c²
勾股定理：在任何直角三角形中，斜边的平方等于两直角边的平方和。
Area of a circle is pi times the radius squared. This appears in cylinder volume, pipe cross‑section and material stress calculations.
A = πr²
圆的面积等于圆周率乘以半径的平方。这常用于圆柱体积、管道截面积和材料应力计算。
Volume of a rectangular prism and a cylinder are both area of base multiplied by height.
V = l × w × h and V = πr²h
长方体和圆柱体的体积都等于底面积乘以高。
Percentage change and efficiency are often expressed as ratios. Know how to convert between fractions, decimals and percentages.
百分比变化和效率常以比率表示。要熟悉分数、小数和百分数之间的转换。
2. Forces and Motion | 力与运动
Newton's laws are the backbone of mechanical engineering. These equations link force, mass and acceleration and help analyse linear motion.
牛顿定律是机械工程的基石。以下方程将力、质量和加速度联系起来，用于分析直线运动。
Resultant force equals mass times acceleration (Newton's second law).
F = m × a
合力等于质量乘以加速度（牛顿第二定律）。
Weight is the gravitational force on a mass, where g ≈ 9.8 m/s² on Earth.
W = m × g
重量是作用在质量上的重力，地球上 g ≈ 9.8 m/s²。
displacement are related. The equation v² = u² + 2as is often used when time is not given.
v² = u² + 2as
对于匀加速运动：末速度、初速度、加速度和位移之间存在关系。当时间未知时，常用方程 v² = u² + 2as。
Average speed is total distance divided by total time. In uniform motion, speed is constant.
v = s / t
平均速率等于总距离除以总时间。在匀速运动中，速率保持不变。
3. Moments and Equilibrium | 力矩与平衡
Understanding moments is essential for analysing levers, beams and rotating components. A body is in equilibrium when there is no resultant force and no resultant moment.
理解力矩对于分析杠杆、横梁和旋转部件至关重要。物体在合力与合力矩均为零时处于平衡状态。
A moment is the turning effect of a force. It is calculated as force multiplied by the perpendicular distance from the pivot.
M = F × d
力矩是力的转动效果，等于力乘以到支点的垂直距离。
Principle of moments: for a lever in equilibrium, total clockwise moment equals total anticlockwise moment about the same pivot.
Σ Mclockwise = Σ Manticlockwise
力矩原理：杠杆平衡时，绕同一点的总顺时针力矩等于总逆时针力矩。
When two forces act in opposite directions and are not in line, they form a couple. The moment of a couple is force times the perpendicular distance between the forces.
Moment of couple = F × d
当两个大小相等、方向相反但不共线的力作用时，形成力偶。力偶矩等于力乘以两力之间的垂直距离。
4. Work, Energy and Power | 功、能量与功率
Energy can be stored, transferred and dissipated. Engineers calculate work done by forces, kinetic and potential energy, and power output of machines.
能量可以被储存、转移和耗散。工程师需要计算力所做的功、动能和势能，以及机器的输出功率。
Work done when a force moves an object through a distance in the direction of the force.
W = F × d
力使物体沿力的方向移动一段距离，所做的功等于力与位移的乘积。
Gravitational potential energy depends on mass, gravitational field strength and height.
Ep = m × g × h
重力势能取决于质量、重力场强度和高度。
Kinetic energy of a moving body is one half times mass times velocity squared.
Ek = ½ m v²
运动物体的动能等于二分之一乘以质量乘以速度的平方。
Power is the rate of doing work or transferring energy. In mechanics it can also be force multiplied by velocity.
P = W / t and P = F × v
功率是做功或转移能量的速率。在力学中，它也可以表示为力乘以速度。
Efficiency compares useful output work or power to total input, often expressed as a percentage.
Efficiency = (useful output / total input) × 100%
效率比较有用输出功或功率与总输入之比，通常以百分数表示。
5. Electrical Fundamentals: Ohm's Law and Power | 电学基础：欧姆定律与功率
Every electronic and electrical system follows a few fundamental laws. Ohm's law and the power rule are the most important for circuit analysis.
所有电子和电气系统都遵循几条基本定律。欧姆定律和功率公式是电路分析中最重要的。
Ohm's law states that the current through a conductor is directly proportional to the voltage across it, provided temperature remains constant.
V = I × R
欧姆定律指出，在温度不变的情况下，通过导体的电流与导体两端的电压成正比。
Electrical power dissipated in a resistor can be calculated using voltage and current, or current and resistance.
P = V × I and P = I² × R
电阻消耗的电功率可以用电压与电流，或电流与电阻来计算。
The conventional current flows from positive to negative, while electron flow is in the opposite direction. Circuit diagrams must use correct symbols.
传统电流方向是从正极到负极，而电子流动方向与之相反。电路图必须使用正确的符号。
Ammeters are placed in series to measure current, voltmeters in parallel to measure potential difference.
安培计串联在电路中测量电流，伏特计并联测量电压。
6. Resistance and Resistor Networks | 电阻与电阻网络
Combining resistors correctly is fundamental to designing voltage dividers, sensors and control circuits. The total resistance depends on whether they are connected in series or parallel.
正确组合电阻是设计分压器、传感器和控制电路的基础。总电阻取决于它们是串联还是并联。
Resistors in series: the total resistance is the sum of individual resistances.
Rtotal = R₁ + R₂ + R₃ + ...
电阻串联时，总电阻等于各个电阻之和。
Resistors in parallel: the reciprocal of total resistance equals the sum of the reciprocals of each resistance.
1 / Rtotal = 1 / R₁ + 1 / R₂ + 1 / R₃ + ...
电阻并联时，总电阻的倒数等于各电阻倒数之和。
For two parallel resistors, a shortcut formula is product over sum: Rtotal = (R₁ × R₂) / (R₁ + R₂).
Rtotal = R₁R₂ / (R₁ + R₂)
对于两个并联电阻，可以用积除以和的简捷公式：Rtotal = (R₁ × R₂) / (R₁ + R₂)。
Resistivity links a material's resistance to its length and cross‑sectional area. This helps select materials for wires and heating elements.
R = ρL / A
电阻率将材料的电阻与其长度和横截面积联系起来。这有助于选择导线和加热元件的材料。
7. Material Properties: Stress, Strain and Young's Modulus | 材料性能：应力、应变与杨氏模量
When a material is loaded, it deforms. Engineers use stress and strain to describe the intensity of force and the amount of deformation, then connect them via Young's modulus.
当材料受载时会发生变形。工程师用应力和应变描述力的强度和变形量，再通过杨氏模量将它们联系起来。
Tensile stress is the applied force divided by the original cross‑sectional area.
σ = F / A
拉伸应力等于施加的力除以原始横截面积。
Tensile strain is the change in length divided by the original length. It has no units.
ε = ΔL / L₀
拉伸应变等于长度变化量除以原始长度，无单位。
Young's modulus measures stiffness: it is the ratio of stress to strain within the elastic limit.
E = σ / ε
杨氏模量衡量刚度：它是在弹性限度内应力与应变的比值。
Elastic deformation is reversible; plastic deformation is permanent. The yield point marks the transition on a stress‑strain graph.
弹性变形是可逆的，塑性变形是永久的。屈服点在应力-应变图上标志着这一过渡。
Factor of safety is the ultimate tensile strength divided by the maximum expected working stress. It ensures a design does not fail unexpectedly.
Factor of safety = UTS / Working stress
安全系数是极限抗拉强度除以最大预期工作应力，确保设计不会意外失效。
8. Thermodynamics: Heat and Temperature | 热力学：热量与温度
Thermal energy transfer is central to engines, heating systems and material selection. The key concepts are specific heat capacity, thermal expansion and heat transfer modes.
热能在发动机、供暖系统和材料选择中至关重要。核心概念包括比热容、热膨胀和传热方式。
The energy required to change an object's temperature depends on mass, specific heat capacity and temperature change.
Q = m × c × Δθ
改变物体温度所需的能量取决于质量、比热容和温度变化。
Specific heat capacity c is the energy needed to raise 1 kg of a substance by 1 °C (or 1 K). Water has a high value: 4200 J/kg°C.
比热容 c 是使 1 kg 物质温度升高 1 °C（或 1 K）所需的能量。水的比热容很高，为 4200 J/kg°C。
Linear expansion: a solid bar increases in length when heated. The change is proportional to original length and temperature rise.
ΔL = α × L₀ × Δθ
线性膨胀：固体棒受热时长度增加。变化量与原始长度和温度升高成正比。
Conduction, convection and radiation are the three modes of heat transfer. Conduction follows Fourier's law in a simplified form for steady‑state plane walls.
Q / t = k × A × Δθ / d
传导、对流和辐射是三种传热方式。对于稳态平壁，传导遵循简化版的傅里叶定律。
9. Fluid Pressure and Hydraulic Systems | 流体压力与液压系统
Fluids transmit force in enclosed spaces, enabling hydraulic jacks, brakes and presses to multiply force. Pressure in a fluid acts equally in all directions (Pascal's principle).
流体在封闭空间中传递力，使液压千斤顶、制动器和压力机能够放大作用力。流体中的压强在所有方向上作用相等（帕斯卡原理）。
re is defined as force acting per unit area. The unit is Pascal (Pa), where 1 Pa = 1 N/m².
p = F / A
压强定义为单位面积上的作用力。单位为帕斯卡 (Pa)，1 Pa = 1 N/m²。
l force on a small piston creates a larger force on a larger piston because F₁/A₁ = F₂/A₂.
F₂ = F₁ × (A₂ / A₁)
在液压系统中，液体各处压强相等。小活塞上的小力会产生大活塞上的大力，因为 F₁/A₁ = F₂/A₂。
Liquid pressure at a depth depends on fluid density, gravity and depth; it is independent of the shape of the container.
p = ρ × g × h
液体在某一深度的压强取决于流体密度、重力加速度和深度，与容器形状无关。
Archimedes' principle: upthrust on a submerged object equals the weight of fluid displaced. This is the foundation of buoyancy calculations.
Fupthrust = ρfluid × g × Vdisplaced
阿基米德原理：浸入物体所受的浮力等于排开流体的重量。这是浮力计算的基础。
10. Mechanical Drives and Gear Ratios | 机械传动与齿轮比
Gears, pulleys and sprockets transmit rotary motion and torque. Understanding speed and torque ratios helps design efficient drive trains.
齿轮、带轮和链轮传递旋转运动和扭矩。理解转速比和扭矩比有助于设计高效的传动系统。
For a pair of gears, the gear ratio is the ratio of the number of teeth on the driven gear to the number of teeth on the driver gear.
Gear ratio = Tdriven / Tdriver
对于一对齿轮，齿轮比等于从动轮齿数除以主动轮齿数。
Speed output is inversely proportional to gear ratio. A large ratio reduces speed but increases torque.
Output speed = Input speed / Gear ratio
输出转速与齿轮比成反比。大齿轮比会降低转速，但增加扭矩。
For a belt and pulley system, the speed ratio is the diameter of the driven pulley divided by the diameter of the driver pulley.
Speed ratio = Ddriven / Ddriver
对于带传动系统，转速比等于从动带轮直径除以主动带轮直径。
Efficiency of a drive system is always less than 100% due to friction, heat and noise. Input power = output power + losses.
由于摩擦、发热和噪声，传动系统的效率始终低于 100%。输入功率等于输出功率加上损耗。
11. Trigonometry and Vector Resolution | 三角学与矢量分解
Forces, velocities and displacements are often vectors. Trigonometry allows engineers to resolve them into perpendicular components and find resultants.
力、速度和位移通常是矢量。三角学能帮助工程师将它们分解为相互垂直的分量并求出合力。
In a right‑angled triangle: sine, cosine and tangent ratios relate sides and angles.
osite / hypotenuse, cosθ = adjacent / hypotenuse, tanθ = opposite / adjacent
在直角三角形中：正弦、余弦和正切将边与角关联起来。
To resolve a force F at an angle θ: horizontal component is F cosθ, vertical component is F sinθ.
Fx = F cosθ, Fy = F sinθ
将一个与水平方向夹角为 θ 的力 F 分解：水平分量为 F cosθ，竖直分量为 F sinθ。
The resultant of two perpendicular vectors can be found using Pythagoras and the direction with tan inverse.
R = √Fx² + Fy², θ = tan⁻¹(Fy / Fx)
两个相互垂直矢量的合力可以用勾股定理求出大小，用反正切求出方向。
The sine rule and cosine rule are essential for non‑right‑angled triangles, often used in structural analysis and force polygons.
a/sinA = b/sinB = c/sinC and a² = b² + c² − 2bc cosA
正弦定理和余弦定理对于非直角三角形至关重要，常用于结构分析和力多边形。
12. Measurement, Unit Conversions and Scientific Notation | 测量、单位换算与科学计数法
Accurate measurement and correct units are the starting point of every engineering calculation. Standard form and SI prefixes keep numbers manageable.
精确测量和正确的单位是每一项工程计算的起点。标准形式和SI前缀能让数字更易于处理。
All calculations must use SI base units: metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol) and candela (cd). Derived units such as newton (N) and pascal (Pa) are built from them.
所有计算必须使用SI基本单位：米、千克、秒、安培、开尔文、摩尔和坎德拉。牛顿、帕斯卡等导出单位由它们组合而成。
quantities, such as the speed of light (3.0 × 10⁸ m/s) or electron charge (1.6 × 10⁻¹⁹ C).
标准形式将数字写为 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。对于极大或极小的量（如光速 3.0 × 10⁸ m/s、电子电荷 1.6 × 10⁻¹⁹ C）至关重要。
Common prefixes: kilo (k, 10³), mega (M, 10⁶), giga (G, 10⁹), milli (m, 10⁻³), micro (μ, 10⁻⁶), nano (n, 10⁻⁹). Always check that units are consistent before substituting values into a formula.
常见词头：千 (k, 10³)、兆 (M, 10⁶)、吉 (G, 10⁹)、毫 (m, 10⁻³)、微 (μ, 10⁻⁶)、纳 (n, 10⁻⁹)。在代入公式前务必检查单位是否一致。
e, to convert km/h to m/s multiply by 1000 and divide by 3600, giving 1 km/h = 0.2778 m/s.
1 km/h = (1000 m) / (3600 s) = 0.2778 m/s
复合单位转换：例如，将 km/h 转换为 m/s 需乘以 1000 再除以 3600，即 1 km/h = 0.2778 m/s。
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