📚 Year 9 WJEC Engineering: Formula & Theorem Quick Reference Handbook | 九年级 WJEC 工程：公式定理速查手册
Engineering in Year 9 under the WJEC specification blends physics, mathematics and design principles to help you understand how things work and how to build them. This quick reference handbook gathers the essential formulas and theorems you will encounter, organising them into clear topics from forces and motion to electricity and materials. Each section pairs a concise English explanation with a Chinese translation so that you can study bilingually and master both the language of engineering and the concepts themselves.
WJEC 九年级工程课程融合了物理、数学和设计原理，帮助你理解事物的工作原理以及如何构建它们。这本速查手册汇集了你会遇到的核心公式和定理，从力与运动到电学与材料，按主题清晰梳理。每个部分都将简明的英文解释与中文翻译配对，让你可以双语学习，同时掌握工程语言和概念本身。
1. Force, Mass and Acceleration | 力、质量与加速度
Newton's second law states that the force acting on an object is equal to its mass multiplied by its acceleration. This is the foundation for analysing how structures and moving parts behave.
牛顿第二定律指出，作用在物体上的力等于其质量乘以加速度。这是分析结构和运动部件行为的基础。
F = m × a
F is force in newtons (N), m is mass in kilograms (kg), and a is acceleration in metres per second squared (m/s²).
F 表示力，单位为牛顿 (N)；m 表示质量，单位为千克 (kg)；a 表示加速度，单位为米每平方秒 (m/s²)。
If the net force on a body is zero, the body remains at rest or moves at constant velocity (Newton's first law). This is crucial when designing stable bridges or stationary machines.
如果物体所受的净力为零，物体将保持静止或匀速直线运动（牛顿第一定律）。这在设计稳定的桥梁或固定式机械时至关重要。
2. Weight and Gravitational Field Strength | 重量与重力场强度
Weight is the force due to gravity acting on an object. It is directly proportional to the mass of the object and the gravitational field strength.
重量是由于重力作用在物体上的力。它与物体的质量和重力场强度成正比。
W = m × g
W is weight in newtons (N), m is mass in kilograms (kg), and g is gravitational field strength, which on Earth is approximately 9.8 N/kg (often rounded to 10 N/kg in classroom calculations).
W 表示重量，单位为牛顿 (N)；m 表示质量，单位为千克 (kg)；g 表示重力场强度，在地球上约为 9.8 N/kg（课堂计算中常取 10 N/kg）。
Understanding the difference between mass and weight prevents design mistakes. A component's mass stays the same on the moon, but its weight changes.
理解质量与重量的区别可以避免设计错误。一个零件的质量在月球上不变，但其重量会改变。
3. Moments and Turning Effect | 力矩与转动效应
The principle of moments states that for an object in equilibrium, the sum of clockwise moments about any pivot equals the sum of anticlockwise moments.
力矩原理指出，对于处于平衡状态的物体，绕任何支点的顺时针力矩之和等于逆时针力矩之和。
M = F × d
M is the moment in newton metres (Nm), F is the force in newtons (N), and d is the perpendicular distance from the pivot to the line of action of the force in metres (m).
M 表示力矩，单位为牛顿米 (Nm)；F 表示力，单位为牛顿 (N)；d 表示从支点到力的作用线的垂直距离，单位为米 (m)。
If a lever or beam is balanced, we can write:
F₁ × d₁ = F₂ × d₂
This formula helps calculate the force needed to lift a load with a crowbar or to keep a seesaw level.
该公式有助于计算用撬棍提起重物所需的力，或保持跷跷板水平。
4. Hooke's Law and Elasticity | 胡克定律与弹性
Hooke’s law describes the behaviour of springs and many elastic materials under tension. The extension of a spring is directly proportional to the applied force, provided the elastic limit is not exceeded.
胡克定律描述了弹簧和许多弹性材料在拉伸下的行为。只要不超过弹性极限，弹簧的伸长量与施加的力成正比。
F = k × e
F is force in newtons (N), k is the spring constant in newtons per metre (N/m), and e is extension in metres (m). k measures the stiffness of the spring.
F 表示力，单位为牛顿 (N)；k 表示弹簧常数，单位为牛顿每米 (N/m)；e 表示伸长量，单位为米 (m)。k 衡量弹簧的刚度。
Engineers use this law when designing vehicle suspension systems and measuring instruments like force meters.
工程师在设计车辆悬挂系统和测力计等测量仪器时会运用这一定律。
5. Pressure, Force and Area | 压强、力与面积
Pressure is defined as the force acting perpendicularly per unit area. It explains why sharp tools cut more easily and why foundations spread loads over large footprints.
压强定义为垂直作用在单位面积上的力。它解释了为什么锋利工具更容易切割，以及为什么地基要把载荷分散到较大的占地面积上。
P = F / A
P is pressure in pascals (Pa) or N/m², F is force in newtons (N), and A is area in square metres (m²).
P 表示压强，单位为帕斯卡 (Pa) 或 N/m²；F 表示力，单位为牛顿 (N)；A 表示面积，单位为平方米 (m²)。
In hydraulic systems, a small force applied over a small area can generate a large force over a larger area, because pressure is transmitted equally throughout an enclosed fluid.
在液压系统中，在小面积上施加较小的力可以在大面积上产生较大的力，因为压强在密闭液体中处处相等地传递。
6. Speed, Distance and Time | 速度、路程与时间
The relationship between distance travelled, speed and time is fundamental in analysing moving parts, conveyor belts or vehicles.
路程、速度与时间之间的关系是分析运动部件、传送带或车辆的基础。
s = v × t
s is distance in metres (m), v is average speed in metres per second (m/s), and t is time in seconds (s).
s 表示路程，单位为米 (m)；v 表示平均速度，单位为米每秒 (m/s)；t 表示时间，单位为秒 (s)。
If an object moves with constant acceleration from rest, distance also relates to acceleration:
s = ½ a t²
This is useful in predicting stopping distances or the travel of automated guided vehicles.
这在预测制动距离或自动导引车的行进时很有用。
7. Ohm's Law and Simple Circuits | 欧姆定律与简单电路
Ohm’s law links voltage, current and resistance in a circuit. It is the cornerstone of electrical and electronic engineering design.
欧姆定律将电路中的电压、电流与电阻联系起来，是电气与电子工程设计的基石。
V = I × R
V is voltage in volts (V), I is current in amperes (A), and R is resistance in ohms (Ω).
V 表示电压，单位为伏特 (V)；I 表示电流，单位为安培 (A)；R 表示电阻，单位为欧姆 (Ω)。
For an ohmic conductor at constant temperature, the resistance remains constant and the V–I graph is a straight line through the origin.
对于恒温下的欧姆导体，电阻保持不变，V–I 图线是一条过原点的直线。
Series circuits divide voltage, while parallel circuits divide current. Understanding these rules prevents overloads and ensures safe design.
串联电路分压，并联电路分流。理解这些规则可以防止过载并确保设计安全。
8. Power in Electrical Systems | 电学系统中的功率
Electrical power is the rate at which electrical energy is transferred. It helps an engineer select the right motor or heating element for a task.
电功率是电能转换的速率。它帮助工程师为某项任务选择合适的电机或加热元件。
P = V × I
P is power in watts (W), V is voltage in volts (V), and I is current in amperes (A).
P 表示功率，单位为瓦特 (W)；V 表示电压，单位为伏特 (V)；I 表示电流，单位为安培 (A)。
Combining Ohm’s law yields two alternative forms:
P = I² × R and P = V² / R
These equations allow you to calculate power losses in cables or the rating required for resistors.
这些公式可用于计算电缆中的功率损耗或电阻器所需的额定值。
9. Energy, Work and Efficiency | 能量、功与效率
Work is done whenever a force moves an object through a distance. The amount of work done measures the energy transferred.
只要力使物体移动一段距离，就做了功。做功的多少衡量了能量转移的量。
W = F × d
W is work in joules (J), F is force in newtons (N), and d is distance moved in the direction of the force in metres (m).
W 表示功，单位为焦耳 (J)；F 表示力，单位为牛顿 (N)；d 表示沿力方向移动的距离，单位为米 (m)。
Gravitational potential energy gained by raising an object:
Eₚ = m × g × h
Kinetic energy of a moving object:
Eₖ = ½ m v²
Efficiency of any system is given by useful output energy divided by total input energy, often expressed as a percentage.
Efficiency = (useful energy output / total energy input) × 100%
Engineers strive to maximise efficiency to reduce waste and cost.
工程师力求最大化效率以减少浪费和成本。
10. Density, Mass and Volume | 密度、质量与体积
Density links the mass of a material to the space it occupies. Selecting materials with appropriate density is a key part of engineering design.
密度将材料的质量与其所占空间联系起来。选择密度合适的材料是工程设计的关键部分。
ρ = m / V
ρ (rho) is density in kilograms per metre cubed (kg/m³), m is mass in kilograms (kg), and V is volume in cubic metres (m³).
ρ (rho) 表示密度，单位为千克每立方米 (kg/m³)；m 表示质量，单位为千克 (kg)；V 表示体积，单位为立方米 (m³)。
For regular shapes, volume can be calculated geometrically; for irregular objects, the displacement method is used.
对于规则形状，体积可通过几何方法计算；对于不规则物体，则使用排水法。
11. The Pythagorean Theorem in Engineering | 工程中的勾股定理
Right‑angled triangles appear frequently in structural design, from diagonal bracing to ramps and trusses. The Pythagorean theorem gives the relationship between the sides.
直角三角形在结构设计中经常出现，从对角支撑到坡道和桁架。勾股定理给出了边长之间的关系。
a² + b² = c²
c is the length of the hypotenuse (the longest side), while a and b are the other two sides.
c 是斜边（最长边）的长度，a 和 b 是另外两条边。
This theorem is used to calculate the true length of a diagonal member or the resultant of two perpendicular forces.
该定理用于计算对角构件的实际长度或两个垂直力的合力。
12. Trigonometry Ratios for Forces and Structures | 力和结构中的三角比
When forces act at angles, engineers resolve them into horizontal and vertical components using sine, cosine and tangent.
当力以一定角度作用时，工程师使用正弦、余弦和正切将其分解为水平和垂直分量。
For a right‑angled triangle with angle θ:
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
This is essential in analysing roof loads, cable tensions and framework stresses.
这在分析屋顶载荷、缆索张力和框架应力时至关重要。
Equilibrium of concurrent forces often requires that the sum of horizontal components equals zero and the sum of vertical components equals zero.
共点力的平衡通常要求水平分力之和为零，垂直分力之和为零。
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