📚 Year 10 CCEA Engineering Formula & Theorem Quick Reference | CCEA 工程学公式定理速查手册
This quick reference handbook brings together the core formulas and theorems required for Year 10 CCEA Engineering. It covers mechanical, electrical, material and energy systems that you will use in design, production and analysis. The explanations pair each equation with its practical meaning so you can apply them accurately in problem solving and coursework.
本速查手册汇集了 CCEA 十年级工程学所需的核心公式与定理，涵盖机械、电气、材料及能量系统。每个公式都配以实际含义解释，帮助你在解题和课业中准确应用。
1. Forces and Moments | 力与力矩
Newton’s second law relates force, mass and acceleration. When several forces act on a body, the resultant force determines its motion. A moment measures the turning effect of a force about a pivot, and equilibrium requires both net force and net moment to be zero.
牛顿第二定律描述力、质量与加速度的关系。当多个力作用在物体上时，合力决定它的运动。力矩衡量一个力绕支点的转动效果；平衡要求合外力为零且合力矩为零。
F = m × a
M = F × d
ΣF = 0 ; ΣM = 0
- F – force (N), m – mass (kg), a – acceleration (m/s²)
- M – moment (Nm), d – perpendicular distance from pivot (m)
- F – 力（牛顿），m – 质量（千克），a – 加速度（米/秒²）
- M – 力矩（牛·米），d – 力臂，即到支点的垂直距离（米）
2. Motion and Kinematics | 直线运动学
The SUVAT equations describe uniform acceleration in a straight line. They link initial velocity (u), final velocity (v), acceleration (a), time (t) and displacement (s). Engineers use these to design mechanisms, lifting equipment and transport systems.
SUVAT 方程描述匀加速直线运动，联系初速度(u)、末速度(v)、加速度(a)、时间(t)和位移(s)。工程师用它设计机构、起重设备和运输系统。
v = u + a t
s = u t + ½ a t²
v² = u² + 2 a s
Choose the equation that contains the three known quantities and the one unknown you need. Always work in consistent SI units: m, s, m/s, m/s².
选择包含三个已知量和你需要的未知量的方程。始终使用一致的国际单位：米、秒、米/秒、米/秒²。
3. Energy, Work and Power | 能量、功与功率
Work is done when a force moves its point of application. Kinetic and gravitational potential energy are the two main mechanical energy stores. Power measures the rate of energy transfer, and efficiency shows how much input energy becomes useful output.
力移动其作用点时做功。动能和重力势能是两种主要的机械能。功率衡量能量传递的速率，效率表示有多少输入能量转化为有用输出。
W = F × d
KE = ½ m v²
GPE = m g h
P = W / t
Efficiency = (useful output ÷ total input) × 100%
g is gravitational field strength, taken as 9.8 N/kg (or 10 N/kg for estimates). Always express efficiency as a percentage.
g 为重力场强度，通常取 9.8 N/kg（估算时可取 10）。效率始终以百分比表示。
4. Stress, Strain and Young’s Modulus | 应力、应变与杨氏模量
When a material is loaded, internal resistive forces develop. Stress describes the load per unit area and strain measures the deformation relative to the original dimension. Young’s modulus links them within the elastic limit and indicates stiffness.
材料受载时内部产生抵抗力。应力描述单位面积上的载荷，应变描述相对于原尺寸的变形量。杨氏模量在弹性范围内将二者联系起来，反映材料的刚度。
σ = F / A
ε = ΔL / L₀
E = σ / ε
- σ – stress (Pa or N/m²), F – force (N), A – cross‑sectional area (m²)
- ε – strain (no units), ΔL – change in length (m), L₀ – original length (m)
- E – Young’s modulus (Pa)
- σ – 应力（帕或牛/米²），A – 横截面积
- ε – 应变（无量纲），ΔL – 长度变化，L₀ – 原长
- E – 杨氏模量（帕）
5. Mechanical Advantage and Efficiency | 机械利益与效率
Simple machines such as levers, pulleys and gears provide mechanical advantage (MA) – the ratio of load to effort. Velocity ratio (VR) depends on the geometry of the system. Their ratio gives the efficiency of the machine, accounting for friction losses.
杠杆、滑轮和齿轮等简单机械提供机械利益(MA)——负载与施力的比值。速比(VR)取决于系统的几何尺寸。二者之比给出机器的效率，考虑了摩擦损失。
MA = Load / Effort
VR = Distance moved by effort / Distance moved by load
Efficiency = (MA / VR) × 100%
Efficiency can never exceed 100% due to friction. In examination questions, you may be asked to calculate any of these three variables given the other two.
由于摩擦，效率永远不会超过 100%。考试中可能会要求你在已知其中两个量的情况下计算第三个量。
6. Electricity: Ohm’s Law and Power | 欧姆定律与电功率
Ohm’s law defines the relationship between voltage, current and resistance in a metallic conductor at constant temperature. Electric power is the rate at which electrical energy is converted; it can be expressed in three useful forms.
欧姆定律定义了恒温下金属导体中电压、电流与电阻的关系。电功率是电能转换的速率，可以用三种有用的形式表示。
V = I × R
P = I × V
P = I² × R = V² / R
V – voltage (V), I – current (A), R – resistance (Ω), P – power (W). When two of V, I, R are known, the third can be found; similarly for power calculations.
V – 电压（伏），I – 电流（安），R – 电阻（欧），P – 功率（瓦）。已知 V、I、R 中的两个，可求第三个；功率计算同理。
7. Series and Parallel Circuits | 串联与并联电路
In series circuits, current is the same through all components and voltages add. In parallel circuits, voltage is the same across each branch and currents add. The rules for total resistance are different and are essential for circuit design.
串联电路中电流处处相等，电压相加。并联电路中各支路电压相等，电流相加。总电阻的规则不同，这对电路设计至关重要。
Series: Rtotal = R₁ + R₂ + R₃ + ...
Parallel: 1/Rtotal = 1/R₁ + 1/R₂ + 1/R₃ + ...
For two resistors in parallel, a quick formula is Rtotal ember that total parallel resistance is always less than the smallest individual resistor.
对于两个并联电阻，可用速算公式 Rtotal = (R₁ × R₂) / (R₁ + R₂)。记住并联总电阻始终小于最小的单个电阻。
8. Hydraulics: Pressure and Pascal’s Principle | 液压：压强与帕斯卡原理
Pressure in a fluid acts equally in all directions. Pascal’s principle states that a pressure change applied to an enclosed fluid is transmitted undiminished throughout the fluid. This is the basis of hydraulic jacks and brakes, where a small force on a small piston produces a large force on a large piston.
流体中的压强向各个方向均等作用。帕斯卡原理指出，施加在封闭流体上的压力变化会大小不变地向流体各处传递。这是液压千斤顶和制动器的基础——小活塞上的小力可在大活塞上产生大力。
P = F / A
F₁ / A₁ = F₂ / A₂
P – pressure (Pa), F – force (N), A – area (m²). In a hydraulic system, if the area ratio is 10:1, the output force is ten times the input force (ignoring friction). The volume of fluid displaced is the same at both pistons, so the smaller piston must move a greater distance.
P – 压强（帕），F – 力（牛），A – 面积（米²）。在液压系统中，面积比为 10:1 时，输出力是输入力的十倍（忽略摩擦）。两个活塞排出的流体体积相等，因此小活塞需移动更大的距离。
9. Thermodynamics: Heat Transfer and Specific Heat Capacity | 热传递与比热容
When a material gains or loses heat energy, its temperature changes unless a phase change occurs. The amount of energy stored or released depends on its mass, specific heat capacity and temperature change. This is crucial in selecting materials for engines and thermal protection.
材料获得或失去热量时，温度会发生变化（除非发生相变）。储存或释放的能量取决于质量、比热容和温度变化。这在选择发动机材料和热防护时至关重要。
Q = m × c × Δθ
Q – heat energy (J), m – mass (kg), c – specific heat capacity (J/kg°C), Δθ – temperature change (°C or K). For cooling systems and solar panels, use energy balances to model performance.
Q – 热量（焦），m – 质量（千克），c – 比热容（焦/千克·摄氏度），Δθ – 温度变化（°C 或 K）。在冷却系统和太阳能电池板中，利用能量平衡来模拟性能。
10. Materials: Density and Strength Properties | 材料：密度与强度特性
Density helps engineers choose lightweight or heavy materials for a task. Strength properties, such as tensile and compressive strength, describe the maximum stress a material can withstand before failure. These are not always given as simple formulas but are read from stress–strain graphs or data tables.
密度帮助工程师为特定任务选择轻质或重质材料。强度特性（如抗拉强度和抗压强度）描述材料失效前能承受的最大应力。这些不一定以简单公式给出，而是从应力–应变图或数据表中读取。
ρ = m / V
Tensile Strength = Maximum Load / Original Cross‑sectional Area
ρ – density (kg/m³), m – mass (kg), V – volume (m³). Safety factors are used to ensure a member’s working stress is well below its strength limit.
ρ – 密度（千克/米³），m – 质量（千克），V – 体积（米³）。采用安全系数可确保构件的工作应力远低于其强度极限。
11. Gears and Pulleys: Velocity Ratio | 齿轮与滑轮：速比
Gear trains and belt‑pulley systems change speed and torque. The velocity ratio (VR) is determined by the number of teeth on meshing gears or by the diameters of the pulleys. This ratio is the inverse of the speed ratio, allowing engineers to predict output speed from input speed.
齿轮系和带‑滑轮系统改变转速和扭矩。速比取决于啮合齿轮的齿数或滑轮的直径。这个比率是转速比的反比，使工程师能从输入转速推算出输出转速。
VR = Number of teeth on driven gear / Number of teeth on driver gear
VR = Diameter of driven pulley / Diameter of driver pulley
If VR > 1, the system reduces speed and increases torque (mechanical advantage). If VR < 1, speed is increased at the expense of torque.
若 VR > 1，系统减速增扭（机械利益提高）；若 VR < 1，系统增速但扭矩降低。
12. Trigonometric Relations for Engineering | 工程中的三角函数
Trigonometry is essential for resolving forces, calculating resultant vectors and determining unknown lengths and angles in structures. The sine, cosine and tangent ratios, together with Pythagoras’ theorem, form the foundation of many engineering calculations.
三角学对于分解力、计算合矢量以及确定结构中的未知长度和角度至关重要。正弦、余弦和正切比例，连同勾股定理，构成许多工程计算的基础。
sin θ = Opposite / Hypotenuse
cos θ = Adjacent / Hypotenuse
tan θ = Opposite / Adjacent
a² + b² = c²
θ and Fᵧ = F sin θ. Use inverse trig functions to find angles when side lengths are known.
当力 F 与水平方向成 θ 角时，其分量为 Fₓ = F cos θ，Fᵧ = F sin θ。已知边长求角时，使用反三角函数。
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