📚 Formula & Theorem Quick Reference Handbook | 公式定理速查手册
This quick reference handbook brings together the essential formulas, principles and theorems needed for Year 10 Cambridge Engineering. It covers core topics from mechanics, materials, electricity, thermal physics and machine analysis, serving as a compact revision aid for coursework and examinations. Each entry pairs a concise explanation in English with its Chinese equivalent, so you can master both technical vocabulary and the underlying concepts.
这本速查手册汇集了 Year 10 剑桥工程课程所必需的核心公式、原理和定理。内容涵盖力学、材料、电学、热物理和机械分析等基础领域，可作为课程作业和备考的紧凑型复习工具。每个条目均以英文简明解释与对应的中文配对呈现，帮助你同时掌握专业术语与根本概念。
1. Mechanics Essentials | 力学基础
Newton's Second Law is the cornerstone of dynamic analysis: F = m × a, where F is the net force in newtons (N), m is mass in kilograms (kg), and a is acceleration in metres per second squared (m/s²). Engineers apply this to predict how structures and machines respond to loads.
牛顿第二定律是动力学分析的基石：F = m × a，其中 F 为合力（牛顿），m 为质量（千克），a 为加速度（米/秒²）。工程师运用该定律预测结构和机械在荷载作用下的响应。
Weight is the gravitational pull on an object: W = m × g n Earth, g ≈ 9.8 m/s². Weight is treated as a concentrated force acting through the object's centre of gravity, and it appears in beam reactions and lifting calculations.
重量是物体受到的重力：W = m × g。在地球表面，g 约 9.8 米/秒²。重量被视为通过物体重心作用的集中力，常用于梁的反力计算和吊装分析。
The maximum static friction before slipping is Ff = μs × R, where μs is the coefficient of static friction and R is the normal reaction force. Understanding this limit helps engineers design safe braking systems and stable foundations.
滑动前的最大静摩擦力为 Ff = μs × R，其中 μs 是静摩擦系数，R 是法向反力。掌握这一极限有助于工程师设计安全的制动系统和稳固的基础。
2. Stress and Strain | 应力与应变
Direct stress measures internal force per unit area: σ = F / A, where σ (sigma) is stress in pascals (Pa) or N/m², F is applied force, and A is cross‑sectional area. When a component is in tension or compression, stress is used to check whether the material will yield.
正应力衡量单位面积上的内力：σ = F / A，其中 σ 是应力（帕斯卡或牛/米²），F 是作用力，A 是横截面积。当构件受拉或受压时，通过应力可判断材料是否会屈服。
Tensile strain describes the extension relative to the original length: ε = ΔL / L₀, where ε (epsilon) is strain (dimensionless), ΔL is the change in length, and L₀ is the original gauge length. Small strains often accompany elastic behaviour.
拉伸应变描述伸长量相对于原长的比值：ε = ΔL / L₀，其中 ε 为应变（无量纲），ΔL 为长度变化量，L₀ 为原始标距。小应变通常伴随着弹性行为。
Young's modulus links stress and strain in the linear elastic region: E = σ / ε. A high Young' GPa), means the material is stiff and resists deformation. This property is vital when selecting materials for beams, columns and shafts.
杨氏模量将线弹性范围内的应力与应变联系起来：E = σ / ε。高杨氏模量（如钢材约 210 GPa）意味着材料刚硬、不易变形。在梁、柱和轴的选材中，这一性质至关重要。
3. Work, Energy and Power | 功、能量与功率
Mechanical work is done when a force moves its point of application: W = F × d, where W is work in joules (J), F is the constant force in the direction of motion, and d is the displacement. Work is a scalar quantity that transfers energy into or out of a system.
当力使作用点移动时即做机械功：W = F × d，其中 W 为功（焦耳），F 为沿运动方向的恒力，d 为位移。功是一个标量，表示能量传入或传出系统。
Kinetic energy of a moving object is KE = ½ × m × v². Here m is mass and v is speed. In vehicle safety engineering, kinetic energy determines the severity of impacts and the energy that crumple zones must absorb.
运动物体的动能为 KE = ½ × m × v²。其中 m 为质量，v 为速率。在车辆安全工程中，动能决定了碰撞的严重程度以及吸能区需要吸收的能量。
Gravitational potential energy stored by raising an object is GPE = m × g × h, where h is the height change. This formula is used in designing lifts, cranes and pumped‑storage systems.
抬高物体所储存的重力势能为 GPE = m × g × h，其中 h 为高度变化。该公式用于设计电梯、起重机和抽水蓄能系统。
Power is the rate of doing work or transferring energy: P = W / t, measured in watts (W). Machines are rated by their power output; the same work done more quickly requires a higher power rating.
功率是做功或传递能量的速率：P = W / t，单位为瓦特。机器按其输出功率分级；相同功量在更短时间内完成则需要更高的额定功率。
4. Moments and Equilibrium | 力矩与平衡
The moment of a force about a pivot is its turning effect: M = F × d, where M is the moment in newton‑metres (Nm), F is the force, and d is the perpendicular distance from the pivot to the line of action. Moments cause rotation; engineers balance them to ensure structural stability.
力对支点的力矩是其转动效应：M = F × d，其中 M 为力矩（牛·米），F 为力，d 为支点到力作用线的垂直距离。力矩引起转动；工程师通过平衡力矩来保证结构稳定。
lockwise moments = Σ anticlockwise moments. This is applied to levers, seesaws, cantilever beams and bridge supports.
力矩原理指出，处于转动平衡的物体，顺时针力矩之和等于逆时针力矩之和。这一原理被应用于杠杆、跷跷板、悬臂梁和桥梁支座的分析中。
In static equilibrium, both the resultant force and the resultant moment must be zero. These conditions allow engineers to solve for unknown reaction forces at supports, ensuring that a structure will not move or tip under load.
在静力平衡中，合力与合力矩都必须为零。利用这些条件，工程师可以求解支座处的未知反力，确保结构在荷载下不发生移动或倾覆。
5. Linear Motion Equations | 直线运动方程
For uniform acceleration, the final velocity v is related to initial velocity u, acceleration a and time t by v = u + a × t. This equation is indispensable when analysing the start‑up or braking of vehicles and machinery.
对于匀加速运动，末速度 v 与初速度 u、加速度 a 和时间 t 的关系为 v = u + a × t。该方程在分析车辆和机械的启动或制动时必不可少。
Displacement under uniform acceleration is given by s = u × t + ½ × a × t². Here s is the distance travelled. This formula lets engineers calculate stopping distances or the stroke of a pneumatic cylinder.
匀加速运动的位移公式为 s = u × t + ½ × a × t²，其中 s 为行驶距离。通过该公式可计算制动距离或气动缸的行程。
When time is not known, the relationship v² = u² + 2 × a × s is used. It is particularly useful in crash‑test analysis to determine vehicle speed from skid marks and deceleration distance.
在时间未知时，使用关系式 v² = u² + 2 × a × s。该公式在碰撞试验分析中尤其有用，可由刹车痕迹和减速距离推算车速。
6. Electricity Fundamentals | 电学基础
Ohm's Law governs the relationship between voltage, current and resistance: V = I × R, where V is voltage in volts, I is current in amperes, and R is resistance in ohms. This law underpins circuit design, from simple wiring to electronic control systems.
欧姆定律描述了电压、电流和电阻的关系：V = I × R，其中 V 为电压（伏特），I 为电流（安培），R 为电阻（欧姆）。该定律是电路设计的基础，从简单布线到电子控制系统均适用。
Electrical power determines how quickly a device converts electrical energy: P = I × V and also P = I² × R. High‑power components, such as motors and heaters, must be selected with the correct voltage and current ratings to avoid overheating.
电功率决定了设备转换电能的快慢：P = I × V 以及 P = I² × R。像电动机和加热器之类的大功率部件必须根据正确的电压和电流额定值选用，以免过热。
Charge (Q) transferred by a steady current is Q = I × t, measured in coulombs (C). This concept links directly to battery capacity and the duration a circuit can operate before recharging.
稳恒电流所输送的电荷量为 Q = I × t，单位为库仑。这一概念直接关联到电池容量以及电路在充电前可持续运行的时间。
7. Material Properties | 材料性质
Density (ρ) is mass per unit volume: ρ = m / V, with units of kg/m³. In lightweight design, engineers choose low‑density materials such as aluminium alloys to reduce structural dead loads without sacrificing strength.
密度 ρ 为单位体积的质量：ρ = m / V，单位为千克/米³。在轻量化设计中，工程师选用铝合金等低密度材料来降低结构自重，同时不牺牲强度。
Elastic strain energy stored in a stretched spring or component obeying Hooke's Law is Eel = ½ × k × x², where k is the spring constant and x is the extension from equilibrium. This applies to suspension springs and shock‑absorbing mechanisms.
遵循胡克定律的弹簧或构件所储存的弹性势能为 Eel = ½ × k × x²，其中 k 为弹簧常数，x 为相对于平衡位置的伸长量。该公式适用于悬架弹簧和减震机构。
Hardness, toughness and ductility are also characterised through standard tests, but the numerical design often relies on the modulus of elasticity and density. Selecting a material always involves balancing performance, cost and sustainability.
硬度、韧性和延展性也通过标准试验来表征，但在数值设计中常依赖弹性模量和密度。材料的选择始终需要在性能、成本与可持续性之间取得平衡。
8. Thermal Physics in Engineering | 工程热物理
Linear thermal expansion is calculated by ΔL = α × L₀ × ΔT, where α is the coefficient of linear expansion, L₀ is the original length, and ΔT is the temperature change. Expansion joints in bridges and pipelines accommodate this effect to prevent stress build‑up.
线热膨胀的计算公式为 ΔL = α × L₀ × ΔT，其中 α 为线膨胀系数，L₀ 为原始长度，ΔT 为温度变化。桥梁和管道的伸缩缝即用于容纳这种效应，防止应力积聚。
Heat energy required to raise the temperature of a substance is Q = m × c × ΔT, where c is the
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