📚 Year 11 WJEC Engineering: Formula & Theorem Quick Reference Handbook | Year 11 WJEC 工程：公式定理速查手册
This quick reference handbook summarises the essential formulas, theorems, and key relationships required for the Year 11 WJEC Engineering course. It covers mechanics, materials, electricity, energy, and machines, providing a compact revision aid. Students should familiarise themselves with each equation, its units, and its typical engineering application. All symbols and quantities follow standard convention; make sure you can manipulate each equation to find any unknown.
本速查手册总结了 Year 11 WJEC 工程课程所需的基本公式、定理和关键关系。内容涵盖力学、材料、电学、能量和机械，是一份紧凑的复习工具。学生应熟悉每一个方程、单位及其典型工程应用。所有符号和物理量遵循标准惯例；务必掌握对每个方程进行变形求解任意未知量的能力。
1. Fundamental Quantities and Units | 基本量与单位
Engineers work with a set of base and derived SI units. The fundamental base units relevant to this course are length (metre, m), mass (kilogram, kg), time (second, s), and electric current (ampere, A). All other mechanical and electrical units can be built from these.
工程师使用一组基本单位和导出国际单位。与本课程相关的基本单位是长度（米，m）、质量（千克，kg）、时间（秒，s）和电流（安培，A）。所有其他力学和电学单位均可由这些单位组合而成。
Derived quantities include force (newton, N, equivalent to kg·m/s²), stress (pascal, Pa, equivalent to N/m²), energy and work (joule, J, equivalent to N·m), power (watt, W, equivalent to J/s), voltage (volt, V), and resistance (ohm, Ω). Always check that the units you use are consistent before substituting numbers into a formula.
导出量包括力（牛顿，N，等于 kg·m/s²）、应力（帕斯卡，Pa，等于 N/m²）、能量和功（焦耳，J，等于 N·m）、功率（瓦特，W，等于 J/s）、电压（伏特，V）和电阻（欧姆，Ω）。在代入公式之前，请务必检查所用单位是否一致。
2. Stress and Strain | 应力与应变
Stress, σ = F / A
easured in pascals (Pa), where 1 Pa = 1 N/m². Stress describes the internal resistance of a material to deformation. In tensile or compressive loading, it is often called direct stress.
应力 (σ) 定义为垂直施加在物体上的力 (F) 除以受力作用的截面积 (A)，单位为帕斯卡 (Pa)，1 Pa = 1 N/m²。应力描述材料抵抗变形的内部抵抗力。在拉伸或压缩载荷下，常称为正应力。
Strain, ε = ΔL / L₀
Strain (ε) is the ratio of the change in length (ΔL) to the original length (L₀) of the material. It is a dimensionless quantity because it is a ratio of two lengths; therefore it has no units. Strain can be tensile (positive) or compressive (negative). The formula assumes uniform deformation along the sample’s gauge length.
应变 (ε) 是长度变化量 (ΔL) 与材料原始长度 (L₀) 之比。它无量纲，因为它是两个长度的比值；因此没有单位。应变可以是拉伸应变（正）或压缩应变（负）。该公式假设沿试样标距的变形是均匀的。
3. Young's Modulus | 杨氏模量
Young's modulus, E = σ / ε = (F L₀) / (A ΔL)
Young's modulus E is a measure of the stiffness of a solid material. It is defined as the ratio of stress to strain in the elastic (linear) region of a material’s stress–strain curve. Because strain is dimensionless, E has the same units as stress, typically pascals (Pa) or gigapascals (GPa) for metals. A high value of E indicates that the material is very stiff and undergoes small deformations under load.
杨氏模量 E 是衡量固体材料刚度的量度。它被定义为材料应力-应变曲线弹性（线性）区域内的应力与应变之比。由于应变无量纲，E 的单位与应力相同，金属材料通常用帕斯卡（Pa）或吉帕（GPa）。高 E 值表示材料非常刚硬，在载荷下变形很小。
This relationship is only valid within the limit of proportionality. Once the limit is exceeded, Hooke’s law no longer applies, and permanent deformation may occur. Engineers use Young’s modulus to select materials for beams, shafts and springs where deflection must be controlled.
该关系仅在比例极限内有效。一旦超出比例极限，胡克定律不再适用，可能发生永久变形。工程师利用杨氏模量来为需要控制挠度的梁、轴和弹簧选择材料。
4. Equations of Motion (SUVAT) | 运动学公式 (SUVAT)
The SUVAT equations describe uniformly accelerated linear motion. The five quantities are: s – displacement (m), u – initial velocity (m/s), v – final velocity (m/s), a – constant acceleration (m/s²), and t – time (s). They are only valid when acceleration is constant.
SUVAT 方程描述匀加速直线运动。五个物理量分别为：s — 位移（m），u — 初速度（m/s），v — 末速度（m/s），a — 恒定加速度（m/s²），t — 时间（s）。它们仅在加速度恒定时有效。
v = u + a t
This equation links final velocity, initial velocity, acceleration and time without involving displacement. It is the direct expression of the definition of constant acceleration.
该方程将末速度、初速度、加速度和时间联系起来，不涉及位移。它是恒定加速度定义的直接表达。
s = u t + ½ a t²
Use this when you know initial velocity, acceleration and time, and you need to find displacement. It is derived from integrating the velocity function.
当已知初速度、加速度和时间，需要求位移时使用该公式。它由速度函数积分得到。
s = ½ (u + v) t
This formula is useful when acceleration is unknown but the time and the initial and final velocities are known. It represents displacement as average velocity multiplied by time.
该公式在加速度未知但已知时间和初末速度时非常实用。它将位移表示为平均速度乘以时间。
v² = u² + 2 a s
This eliminates time and relates velocities to displacement and acceleration. It is particularly handy for braking distance calculations and vertical motion under gravity.
该公式消去了时间，将速度与位移和加速度关联起来。特别适用于制动距离计算和重力作用下的竖直运动问题。
5. Newton's Laws and Force | 牛顿定律与力
Newton's Second Law: F = m a
The resultant force F (in newtons) acting on an object is equal to the mass m (in kg) multiplied by the acceleration a (in m/s²). This law governs all mechanical systems, from vehicles to structural components. If multiple forces act, use the net force in the direction of acceleration.
作用在物体上的合力 F（单位为牛顿）等于质量 m（kg）乘以加速度 a（m/s²）。该定律支配着从车辆到结构部件的所有机械系统。若存在多个力，应沿加速度方向取净力。
Weight: W = m g
Weight is the gravitational force on a mass. On Earth, g is approximately 9.81 m/s². Weight is a force, so it is measured in newtons, not kilograms. In engineering calculations, always distinguish between mass (amount of matter) and weight (force due to gravity).
重量是作用在质量上的重力。在地球表面，g 约为 9.81 m/s²。重量是一种力，因此单位是牛顿，而非千克。在工程计算中，务必区分质量（物质的量）和重量（重力引起的力）。
Moment of a force: Moment = F × d
A moment (torque) is the turning effect of a force about a pivot point. It is the product of the force F and the perpendicular distance d from the pivot to the line of action of the force. The SI unit is newton-metre (N·m). For equilibrium, the sum of clockwise moments must equal the sum of anticlockwise moments.
力矩是力绕支点的转动效果。它是力 F 与从支点到力作用线的垂直距离 d 的乘积。国际单位为牛顿·米 (N·m)。要达到平衡，顺时针力矩之和必须等于逆时针力矩之和。
6. Work, Energy, and Power | 功、能与功率
Work done: W = F d cos θ
d the distance moved. If the force is parallel to displacement (θ = 0°), it simplifies to W = F d. Work is measured in joules (J).
当一个恒定力 F 使物体移动位移 d 时，所做的功 W 等于力沿位移方向的平行分量与移动距离的乘积。若力与位移平行（θ = 0°），则简化为 W = F d。功的单位是焦耳 (J)。
Kinetic energy: KE = ½ m v²
Kinetic energy is the energy an object possesses due to its motion. It depends on mass m and velocity v. It is a scalar quantity measured in joules. In collisions or braking, work done against friction converts KE into heat.
动能是物体由于运动而具有的能量。它取决于质量 m 和速度 v。动能是标量，单位为焦耳。在碰撞或制动中，克服摩擦力做的功将动能转化为热能。
Gravitational potential energy: PE = m g h
This is the energy stored in an object due to its height h above a reference level. Lifting an object increases its PE; lowering it releases PE. In many mechanical systems, such as hoists and lifts, the work done against gravity is given by m g h.
这是物体因高出参考水平面高度 h 而储存的能量。提升物体会增加其势能；下降则释放势能。在许多机械系统中，如卷扬机和电梯，克服重力做功为 m g h。
Power: P = W / t and P = F v
eady speed v, power can also be calculated as P = F v. This is particularly useful for vehicles and machinery.
功率是做功或能量传递的速率。单位是瓦特 (W)，相当于 J/s。如果一个恒定力 F 使物体以恒定速度 v 运动，功率也可按 P = F v 计算。这对车辆和机械特别有用。
7. Mechanical Advantage and Velocity Ratio | 机械利益与速度比
Mechanical Advantage (MA) = Load / Effort
Mechanical advantage tells you how much a machine multiplies the effort force. It is the ratio of the load (output force) to the effort (input force). An MA greater than 1 means a smaller effort can lift a larger load. For an ideal machine with no friction, MA would equal velocity ratio.
机械利益表示机器将输入力放大了多少。它是负载（输出力）与作用力（输入力）之比。MA 大于 1 意味着较小的作用力可以提升较大的负载。对于无摩擦的理想机器，MA 等于速度比。
Velocity Ratio (VR) = Distance moved by effort / Distance moved by load
Velocity ratio is a purely geometric property of a machine. For levers, it depends on the ratio of lever arms; for pulleys, on the number of rope segments supporting the load. VR is always constant for a given machine setup.
速度比是机器的纯几何属性。对于杠杆，它取决于力臂之比；对于滑轮组，则取决于支撑负载的绳索段数。在给定的机器配置下，VR 始终为常数。
Efficiency = (MA / VR) × 100%
The efficiency of a simple machine compares the actual mechanical advantage achieved to the ideal velocity ratio. Friction and other losses reduce MA, so efficiency is less than 100%. Engineers aim to minimise energy losses in linkages, bearings and gear trains.
简单机械的效率将实际达到的机械利益与理想速度比进行比较。摩擦和其他损失会降低 MA，因此效率始终低于 100%。工程师力求将连杆、轴承和齿轮系中的能量损失降至最低。
8. Electrical Principles (Ohm’s Law, Power) | 电学原理（欧姆定律、功率）
Ohm’s Law: V = I R
The voltage V (in volts) across a resistor is directly proportional to the current I (in amperes) flowing through it, provided temperature remains constant. The constant of proportionality is resistance R, measured in ohms (Ω). Ohm’s law is fundamental for analysing circuits, choosing resistors, and ensuring components operate within safe current limits.
电阻器两端的电压 V（伏特）与流过它的电流 I（安培）成正比，前提是温度保持恒定。比例常数是电阻 R，单位为欧姆 (Ω)。欧姆定律是分析电路、选择电阻器以及确保元件在安全电流范围内工作的基础。
Electrical Power: P = I V = I² R = V² / R
Electrical power P (watt, W) is the rate at which electrical energy is converted into other forms. The three equivalent forms allow you to calculate power when two of the three quantities V, I, R are known. When selecting components, always check the power rating to avoid overheating or failure.
电功率 P（瓦特，W）是电能转换为其他形式能量的速率。三种等效形式使你在已知 V、I、R 中任意两个量时均可计算功率。选择元件时，务必核对额定功率，以避免过热或故障。
9. Resistors in Series and Parallel | 串联与并联电阻
Series: Rtotal = R₁ + R₂ + R₃ + …
When resistors are connected end to end, the total resistance is simply the sum of the individual resistances. The same current passes through each resistor, and the supply voltage is divided among them. This arrangement is used to limit current or create a desired voltage drop.
当电阻器首尾相连时，总电阻等于各电阻之和。流过每个电阻的电流相同，电源电压在各电阻之间分配。这种布局常用于限制电流或产生所需的电压降。
Parallel: 1/Rtotal = 1/R₁ + 1/R₂ + 1/R₃ + …
For resistors in parallel, the total resistance is always less than the smallest individual resistance. The voltage across each branch is the same, while the total current is the sum of the branch currents. This configuration provides multiple independent paths and is typical in household wiring.
对于并联电阻，总电阻始终小于其中最小的单个电阻。各支路两端的电压相同，而总电流为各支路电流之和。这种配置提供了多个独立路径，常见于家庭布线。
Potential divider: Vout = Vin × (R₂ / (R₁ + R₂))
A potential (voltage) divider is a series circuit with two resistors. The output voltage Vout is taken across R₂ and depends on the ratio of R₂ to the total resistance. When one of the resistors is replaced by a sensor (e.g. thermistor or LDR), the output voltage varies with physical conditions, forming the basis of many control systems.
分压器是由两个电阻串联而成的电路。输出电压 Vout 取自 R₂ 两端，其大小取决于 R₂ 与总电阻的比值。当其中一个电阻替换为传感器（如热敏电阻或光敏电阻）时，输出电压随物理条件变化，构成许多控制系统的基础。
10. Efficiency and Energy Transfer | 效率与能量转换
Efficiency = (Useful energy output / Total energy input) × 100%
Efficiency measures how well a system converts input energy into useful output. No real system is 100% efficient because some energy is always dissipated as waste heat, sound, or friction. In engineering, improving efficiency reduces running costs and environmental impact. For machines, efficiency can also be expressed in terms of
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