📚 KS3 CCEA Engineering: Quick Reference Formula & Theorem Handbook | KS3 CCEA 工程：公式定理速查手册
Welcome to your KS3 CCEA Engineering quick reference handbook. This guide gathers the essential formulas, theorems and unit relationships you will meet across the CCEA Key Stage 3 Engineering and Technology course, from forces and motion to electrical circuits, materials and mechanisms. Use it for revision, homework and practical workshop projects, and always check the symbol list before you substitute numbers into an equation.
欢迎使用 KS3 CCEA 工程速查手册。本指南汇集了 CCEA 关键阶段 3 工程与技术课程中的核心公式、定理与单位关系，涵盖力与运动、电路、材料与机械结构。可用于复习、作业与车间实践项目；代入数值前，请务必先核对每节的符号说明。
1. Force and Newton's Second Law | 力与牛顿第二定律
F = m × a a = F ÷ m m = F ÷ a
Newton's Second Law links the resultant force acting on an object to its mass and the acceleration it produces. The greater the resultant force, the greater the acceleration; the greater the mass, the smaller the acceleration for the same force.
牛顿第二定律把作用在物体上的合力与其质量和所产生的加速度联系起来。合力越大，加速度越大；在相同合力下，质量越大，加速度越小。
- F = resultant force, measured in newtons (N)
- m = mass, measured in kilograms (kg)
- a = acceleration, measured in metres per second squared (m/s²)
- 1 N is the force that gives a 1 kg mass an acceleration of 1 m/s²
- F = 合力，单位为牛顿 (N)
- m = 质量，单位为千克 (kg)
- a = 加速度，单位为米每二次方秒 (m/s²)
- 1 N 即使 1 kg 的物体产生 1 m/s² 加速度的力
e resultant force, which is the overall force remaining after every push and pull has been taken into account.
示例：质量为 2.5 kg 的小车在桌面上受到 10 N 的合力推动，则加速度 a = F ÷ m = 10 ÷ 2.5 = 4 m/s²。计算时必须使用合力，即把所有推力和拉力都考虑之后剩下的总力。
2. Weight, Mass and Gravity | 重量、质量与重力
W = m × g
Mass is the amount of matter in an object and is measured in kilograms; it does not change from place to place. Weight is the force of gravity acting on that mass and is measured in newtons, so it does change with the strength of the gravitational field.
质量是物体所含物质的多少，单位为千克，在任何地方都不改变；重量是重力作用在该质量上的力，单位为牛顿，因此会随重力场强度而变化。
- g on Earth = 9.81 N/kg, often rounded to 10 N/kg for quick estimates
- g on the Moon = 1.62 N/kg, so objects weigh roughly one sixth as much
- = mass in kilograms (kg), g = gravitational field strength in newtons per kilogram (N/kg)
- 地球上的 g = 9.81 N/kg，快速估算时常取 10 N/kg
- 月球上的 g = 1.62 N/kg，因此物体的重量约为地球上的六分之一
- W = 重量 (N)，m = 质量 (kg)，g = 重力场强度 (N/kg)
same student weighs 60 × 1.62 = 97.2 N on the Moon. The mass stays at 60 kg in both cases.
示例：质量为 60 kg 的学生在地球上的重量为 W = 60 × 9.81 = 588.6 N，而在月球上只有 60 × 1.62 = 97.2 N。两种情况下质量都是 60 kg。
3. Density, Mass and Volume | 密度、质量与体积
ρ = m ÷ V m = ρ × V V = m ÷ ρ
Density tells you how much mass is packed into a given volume. It is a property of the material, not of the particular object, so a small steel bolt and a large steel beam have exactly the same density.
密度表示单位体积内包含多少质量。它是材料的属性，而不是某个具体物体的属性，因此一颗小钢螺栓和一根大钢梁的密度完全相同。
- rho) = density, in kilograms per cubic metre (kg/m³) or grams per cubic centimetre (g/cm³)
- m = mass in kilograms or grams; V = volume in cubic metres or cubic centimetres
- 1 g/cm³ = 1000 kg/m³
The Greek letter ρ (rho) is used for density. If you measure mass in grams and volume in cubic centimetres, your answer comes out in g/cm³; if you use kilograms and cubic metres, it comes out in kg/m³. Never mix the two systems in one calculation.
密度用希腊字母 ρ（rho）表示。若质量用克、体积用立方厘米，结果单位就是 g/cm³；若用千克和立方米，结果就是 kg/m³。同一次计算中切勿混用两套单位。
es the metal as aluminium. Useful reference values: water 1000 kg/m³, aluminium 2700 kg/m³, steel about 7850 kg/m³.
示例：体积为 0.002 m³ 的金属块质量为 5.4 kg，则 ρ = 5.4 ÷ 0.002 = 2700 kg/m³，可判断该金属为铝。常用参考值：水 1000 kg/m³，铝 2700 kg/m³，钢约 7850 kg/m³。
4. Pressure in Solids, Liquids and Gases | 固体、液体与气体压强
P = F ÷ A P = ρ × g × h F₁ ÷ A₁ = F₂ ÷ A₂
Pressure is the force spread over an area. A sharp point concentrates force onto a tiny area and therefore produces a very high pressure, which is why a drawing pin pierces wood while a thumb does not.
压强是分布在某个面积上的力。尖锐的尖端把力集中在极小的面积上，因此产生很高的压强——这就是图钉能刺入木头而拇指不能的原因。
- P = pressure in pascals (Pa), where 1 Pa = 1 N/m²
- F = force in newtons; A = area in square metres
- P = ρ × g × h gives the pressure due to a column of liquid at depth h (in metres)
- P = 压强，单位为帕斯卡 (Pa)，1 Pa = 1 N/m²
- F = 力 (N)；A = 面积 (m²)
- P = ρ × g × h 用来计算深度 h（米）处液体产生的压强
000 × 9.81 × 2 = 19 620 Pa ≈ 19.6 kPa. Atmospheric pressure at sea level is about 101 kPa.
示例：500 N 的力作用在 0.25 m² 的面积上，P = 500 ÷ 0.25 = 2000 Pa = 2 kPa。在水中 2 m 深处，P = 1000 × 9.81 × 2 = 19 620 Pa ≈ 19.6 kPa。海平面大气压约为 101 kPa。
In a hydraulic system, pressure is transmitted equally throughout the fluid, so a small force on a small piston can produce a large force on a large piston. This is the principle behind car brakes, jacks and digger arms.
在液压系统中，压强在液体内各处等值传递，因此作用在小活塞上的小力可以在大活塞上产生很大的力。这就是汽车刹车、千斤顶和挖掘机臂的工作原理。
5. Work, Energy and Power | 功、能量与功率
W = F × d
Eₚ = m × g × h Eₖ = ½ × m × v²
P = E ÷ t = W ÷ t efficiency = (useful output ÷ total input) × 100%
Work is done whenever a force moves an object through a distance. Energy is the capacity to do work and is measured in joules; power is the rate at which energy is transferred, measured in watts.
只要力使物体移动了一段距离，就做了功。能量是做功的能力，单位为焦耳；功率是能量传递的速率，单位为瓦特。
- F = force in newtons (N); d = distance moved in the direction of the force, in metres (m)
- Eₚ = gravitational potential energy; Eₖ = kinetic energy, both in joules
- P = power in watts (W), where 1 W = 1 J/s
Energy cannot be created or destroyed, only transferred or stored, so the total energy before and after any process is the same. In a real machine, some energy always escapes as heat or sound, which is why efficiency is never 100%.
能量既不能凭空产生，也不能凭空消失，只会转移或储存，因此任何过程中能量总量保持不变。在真实机械中，总有一部分能量以热或声的形式散失，所以效率永远达不到 100%。
. A car of mass 1200 kg travelling at 15 m/s has Eₖ = ½ × 1200 × 15² = 135 000 J = 135 kJ.
示例：把 20 kg 的箱子提升 1.5 m，储存的重力势能 Eₚ = 20 × 9.81 × 1.5 = 294.3 J。若用时 3 s，平均功率 P = 294.3 ÷ 3 = 98.1 W。质量 1200 kg、速度 15 m/s 的汽车动能为 Eₖ = ½ × 1200 × 15² = 135 000 J = 135 kJ。
6. Moments, Levers and Equilibrium | 力矩、杠杆与平衡
M = F × d
sum of clockwise moments = sum of anticlockwise moments
A moment is the turning effect of a force about a pivot. The further the force is applied from the pivot, the greater the turning effect, which is why a long spanner loosens a tight nut more easily than a short one.
力矩是力绕支点产生的转动效果。施力点离支点越远，转动效果越大——这就是长扳手比短扳手更容易拧松螺母的原因。
- M = moment in newton metres (N m)
- = perpendicular distance from the pivot to the line of action of the force, in metres (m)
- For a balanced beam, total clockwise moment = total anticlockwise moment
- M = 力矩，单位为牛顿米 (N m)
- F = 力 (N)；d = 支点到力的作用线的垂直距离（米）
- 杠杆平衡时，顺时针力矩总和 = 逆时针力矩总和
lance this, an adult of weight 600 N must sit at d = 450 ÷ 600 = 0.75 m on the other side.
示例：体重 300 N 的儿童坐在跷跷板支点 1.5 m 处，力矩 M = 300 × 1.5 = 450 N m。为使其平衡，体重 600 N 的成人应坐在另一侧 d = 450 ÷ 600 = 0.75 m 处。
Levers come in three classes: in a first-class lever the pivot lies between the effort and the load (scissors, crowbar); in a second-class lever the load lies between the pivot and the effort (wheelbarrow); in a third-class lever the effort lies between the pivot and the load (tweezers, fishing rod).
杠杆分为三类：第一类杠杆支点在施力点和负载之间（剪刀、撬棍）；第二类杠杆负载在支点和施力点之间（手推车）；第三类杠杆施力点在支点和负载之间（镊子、钓鱼竿）。
7. Speed, Velocity and Acceleration | 速率、速度与加速度
= s ÷ t a = (v − u) ÷ t v² = u² + 2 × a × s
Speed is how fast something moves; velocity is speed in a stated direction, which makes it a vector quantity. Acceleration is the rate of change of velocity and can mean speeding up, slowing down or changing direction.
速率表示物体运动的快慢；速度是带有方向的速率，因此是矢量。加速度是速度的变化率，可以是加速、减速或改变方向。
- v = final velocity in metres per second (m/s); u = initial velocity (m/s)
- s = distance or displacement in metres (m); t = time in seconds (s)
- a = acceleration in metres per second squared (m/s²); a negative value means deceleration
- v = 末速度 (m/s)；u = 初速度 (m/s)
- s = 路程或位移 (m)；t = 时间 (s)
- a = 加速度 (m/s²)；负值表示减速
On a distance-time graph, the gradient gives speed; on a velocity-time graph, the gradient gives acceleration and the area under the line gives the distance travelled. A straight horizontal line on a velocity-time graph means constant velocity and zero acceleration.
在路程—时间图像上，斜率表示速率；在速度—时间图像上，斜率表示加速度，图线下的面积表示路程。速度—时间图像上的水平直线表示匀速运动，加速度为零。
mly to rest over 450 m; using v² = u² + 2 × a × s gives 0 = 900 + 900 × a, so a = −1 m/s².
示例：自行车手在 8 s 内从 4 m/s 加速到 12 m/s，a = (12 − 4) ÷ 8 = 1 m/s²。列车以 30 m/s 行驶、匀减速 450 m 后停下，由 v² = u² + 2 × a × s 得 0 = 900 + 900 × a，故 a = −1 m/s²。
8. Stress, Strain and the Young Modulus | 应力、应变与杨氏模量
σ = F ÷ A ε = ΔL ÷ L₀ E = σ ÷ ε
When a material is stretched or compressed, engineers describe the load using stress (force per unit area) and the deformation using strain (fractional change in length). The ratio of the two is the Young modulus, a measure of stiffness.
当材料被拉伸或压缩时，工程师用应力（单位面积上的力）描述载荷，用应变（长度的相对变化）描述形变。两者之比即杨氏模量，用来衡量材料的刚度。
- σ (sigma) = stress in pascals (Pa) or newtons per square millimetre (N/mm²)
- ε (epsilon) = strain, which has no units because it is a ratio
- E = Young modulus in pascals; ΔL = extension in metres; L₀ = original length in metres
- σ（sigma）= 应力，单位
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