📚 Year 9 CCEA Engineering: Quick Reference Formula & Theorem Handbook | CCEA 九年级工程：公式定理速查手册
This handbook provides a concise summary of essential formulas and principles for Year 9 CCEA Engineering. Understanding these relations will help you solve problems in mechanics, electronics, materials and energy systems. Keep this guide handy for revision and practical projects.
本手册为 CCEA 九年级工程课程汇编了核心公式与定理。掌握这些关系式有助于你解决机械、电子、材料与能源系统中的问题。复习和完成实践项目时请随身参考。
1. Forces and Motion | 力与运动
Newton's Second Law relates the force applied to an object, its mass and the resulting acceleration. The formula is central to all dynamic analysis.
牛顿第二定律将施加在物体上的力、物体的质量以及产生的加速度联系起来。该公式是所有动力学分析的核心。
F = m × a
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).
F 表示净力，单位为牛顿 (N)；m 为质量，单位为千克 (kg)；a 为加速度，单位为米每二次方秒 (m/s²)。
force of gravity acting on its mass. On Earth, gravitational field strength g ≈ 9.8 N/kg.
物体的重量是地球引力对其质量的作用力。在地球上，重力场强度 g ≈ 9.8 N/kg。
W = m × g
Average speed is total distance travelled divided by time taken. For constant acceleration starting from an initial velocity u to a final velocity v over time t, acceleration is the rate of change of velocity.
平均速度等于总路程除以所用时间。对于从初速度 u 匀加速到末速度 v、历时 t 的运动，加速度等于速度的变化率。
v = s / t and a = (v - u) / t
s stands for displacement or distance (m), t for time (s). These equations are the foundation of kinematics in engineering.
s 代表位移或路程 (m)，t 代表时间 (s)。这两个方程是工程运动学的基础。
2. Work and Energy | 功与能
Work is done when a force moves an object in the direction of the force. One joule (J) of work is done when a force of one newton moves an object one metre.
当力使物体沿力的方向移动时，力就对物体做了功。用 1 牛顿的力使物体移动 1 米，所做的功为 1 焦耳 (J)。
W = F × d
W is work (J), F is force (N), d is distance moved in the direction of the force (m). If force and displacement are at an angle θ, the effective component is F cos θ, but for most Year 9 cases they are parallel.
W 表示功 (J)，F 表示力 (N)，d 表示沿力方向移动的距离 (m)。若力与位移之间存在夹角 θ，则有效分量为 F cos θ，但在九年级多数情形下二者平行。
Gravitational potential energy is stored when an object is raised to a height against gravity. Kinetic energy is the energy of motion.
物体克服重力被举升到一定高度时储存了重力势能。动能则是物体由于运动而具有的能量。
Eₚ = m × g × h and Eₖ = ½ m v²
eight (m). Eₖ is kinetic energy (J), v is speed (m/s). Energy is always conserved, so useful work can be analysed by energy transfers.
Eₚ 为重力势能 (J)，m 为质量 (kg)，g ≈ 9.8 N/kg，h 为高度 (m)。Eₖ 为动能 (J)，v 为速率 (m/s)。能量总是守恒的，因此可通过能量转换分析有用功。
3. Power | 功率
Power is the rate at which work is done or energy is transferred. A more powerful machine does the same amount of work in less time.
功率是做功或能量传递的速率。一台功率更大的机器能在更短的时间内完成同等数量的功。
P = W / t or P = E / t
ower in watts (W), where 1 W = 1 J/s. W is work done or energy transferred, t is time (s).
P 为功率，单位瓦特 (W)，1 W = 1 J/s。W 为功或转化的能量，t 为时间 (s)。
When a constant force moves an object at a steady speed, power can also be calculated directly from force and velocity.
当恒力使物体以稳定速度移动时，功率还可直接由力和速度求出。
P = F × v
F is the force (N), v is velocity (m/s). This form is useful when analysing vehicles or moving conveyor systems.
F 为力 (N)，v 为速度 (m/s)。在分析交通工具或移动式传送系统时，这一形式非常实用。
4. Levers and Moments | 杠杆与力矩
The turning effect of a force is called a moment. It depends on the size of the force and the perpendicular distance from the pivot to the line of action of the force.
力的转动效应称为力矩。力矩的大小取决于力的量值和从支点到力作用线的垂直距离。
τ = F × d
τ is the moment (Nm), F is force (N), d is the perpendicular distance from pivot (m). For an object in rotational equilibrium, total clockwise moments equal total anticlockwise moments.
τ 为力矩 (牛·米，Nm)，F 为力 (N)，d 为支点到力作用线的垂直距离 (m)。当物体处于转动平衡时，顺时针方向的总力矩等于逆时针方向的总力矩。
Levers are simple machines that amplify force or distance. Mechanical advantage (MA) is the ratio of load to effort. Velocity ratio (VR) is the ratio of distance moved by effort to distance moved by load.
杠杆是能够放大力或距离的简单机械。机械效益 (MA) 是负载与施力的比值。速比 (VR) 是施力移动距离与负载移动距离的比值。
MA = Load / Effort and VR = Effort arm / Load arm
tice friction always reduces MA. Levers are classed into three types depending on the relative positions of fulcrum, load and effort.
对于理想机械，MA = VR，但在实际中摩擦总会降低机械效益。根据支点、负载和施力的相对位置，杠杆分为三类。
5. Gear and Pulley Systems | 齿轮与滑轮系统
Gears transmit rotary motion and torque between shafts. The gear ratio is determined by the number of teeth on the driven gear compared to the driver gear.
齿轮用于在两轴之间传递旋转运动和扭矩。传动比由从动齿轮齿数与主动齿轮齿数的比值决定。
Gear Ratio = Teeth on driven / Teeth on driver
If the driven gear has more teeth, the system reduces speed but increases torque. Speed ratio is the inverse of gear ratio:
若从动齿轮齿数较多，则系统降速增扭。速比是传动比的倒数：
Speed Ratio = Speed of driver / Speed of driven = Teeth on driven / Teeth on driver
Torque ratio is larger when output speed is smaller, following the relation: Torque out = Torque in × (Teeth out / Teeth in).
当输出转速较低时扭矩比较大，遵循公式：输出扭矩 = 输入扭矩 × (从动齿数 / 主动齿数)。
Pulley systems use ropes and wheels to change the direction of a force and multiply effort. The mechanical advantage of an ideal pulley equals the number of rope segments supporting the load.
滑轮系统利用绳索和轮子改变力的方向并放大施力。理想滑轮的机械效益等于支撑负载的绳索段数。
MA (pulley) = Number of supporting ropes
Velocity ratio for a pulley block is the distance the effort rope is pulled divided by the distance the load rises, which equals the number of rope segments.
滑轮组的速比等于施力端绳索被拉动的距离除以负载上升的距离，该值等于绳索段数。
6. Electrical Quantities and Ohm's Law | 电量与欧姆定律
Voltage (V) is the energy per unit charge, current (I) is the rate of flow of charge, and resistance (R) opposes current. Ohm's Law links these three fundamental quantities.
电压 (V) 是单位电荷所具有的能量，电流 (I) 是电荷流动的速率，电阻 (R) 则阻碍电流。欧姆定律将这三个基本量联系起来。
V = I × R
V is measured in volts (V), I in amperes (A) and R in ohms (Ω). This relationship holds for ohmic conductors at constant temperature.
V 的单位为伏特 (V)，I 为安培 (A)，R 为欧姆 (Ω)。该关系在温度恒定时适用于欧姆导体。
Electric charge transferred over time is simply current multiplied by time. Electric power is the product of voltage and current.
传输的电荷量就是电流与时间的乘积。电功率则是电压与电流的乘积。
Q = I × t and P = I × V
Q is charge in coulombs (C), t is time (s). Using Ohm's Law, power can also be expressed as P = I²R or P = V² / R.
Q 为电荷量，单位库仑 (C)，t 为时间 (s)。代入欧姆定律，电功率还可表示为 P = I²R 或 P = V² / R。
Electrical energy consumed is power multiplied by time, commonly measured in kilowatt-hours (kWh) for mains electricity.
消耗的电能为功率乘以时间，对于市电通常以千瓦时 (kWh) 计量。
E = P × t
7. Resistance in Series and Parallel | 串联与并联电阻
Components in a circuit can be connected in series (one after another) or in parallel (side by side). The total resistance depends on the arrangement.
电路中元件可以串联（一个接一个）或并联（并列连接）。总电阻取决于连接方式。
For resistors in series, the total resistance is the sum of individual resistances. The same current flows through each component, but the voltage is divided.
串联电阻的总电阻等于各个电阻值之和。流过每个元件的电流相同，但电压被分配到各个电阻上。
Rtotal = R₁ + R₂ + R₃ + ...
For resistors in parallel, the reciprocal of the total resistance equals the sum of the reciprocals of each resistance. The voltage across each branch is the same, while the current divides.
并联电阻的总电阻的倒数等于各支路电阻倒数之和。各支路两端的电压相同，但电流被分流。
1 / Rtotal = 1 / R₁ + 1 / R₂ + 1 / R₃ + ...
For two resistors in parallel, a simplified formula can be used: Rtotal = (R₁ × R₂) / (R₁ + R₂). This reduces calculation steps in practical engineering problems.
对于两个并联电阻，可采用简化公式：Rtotal = (R₁ × R₂) / (R₁ + R₂)。这能简化实际工程问题中的计算步骤。
8. Pressure and Hydraulics | 压强与液压
Pressure is defined as force acting per unit area. A large pressure can be generated by a small force concentrated on a small area.
压强定义为单位面积上所受的力。将小力集中在小面积上即可产生高压强。
p = F / A
p is pressure in pascals (Pa), where 1 Pa = 1 N/m², F is force (N), A is area (m²).
p 为压强，单位帕斯卡 (Pa)，1 Pa = 1 N/m²；F 为力 (N)，A 为面积 (m²)。
In a liquid, pressure increases with depth and depends on fluid density. This is important for designing dams, tanks and underwater equipment.
在液体中，压强随深度增加而增大，并与液体密度相关。这对于设计水坝、储罐和水下设备至关重要。
p = ρ × g × h
ρ is the density of the fluid (kg/m³), g is gravitational field strength (N/kg), h is depth (m).
ρ 为流体密度 (kg/m³)，g 为重力场强度 (N/kg)，h 为深度 (m)。
Hydraulic systems use Pascal's principle: pressure applied to an enclosed fluid is transmitted equally in all directions. This principle allows a small input force to be amplified.
液压系统运用帕斯卡原理：施加于密闭液体上的压强会大小不变地向各个方向传递。该原理使得较小的输入力得以放大。
F₁ / A₁ = F₂ / A₂
The output force F₂ can be much larger than F₁ if area A₂ is larger than A₁. The distance moved by the pistons follows an inverse relationship to the area ratio.
若面积 A₂ 大于 A₁，则输出力 F₂ 可远大于 F₁。活塞移动距离与面积比成反比关系。
9. Efficiency | 效率
Efficiency measures how well a system converts input energy or power into useful output. No real machine can be 100% efficient due to friction, heat losses and deformation.
效率衡量系统将输入能量或功率转化为有用输出的程度。由于摩擦、热损耗和变形，实际机器的效率无法达到 100%。
Efficiency (%) = (Useful output / Total input) × 100%
Output and input can be expressed as energy (J) or power (W). For mechanical systems, efficiency is often less than 50% in simple setups; precision-engineered systems can exceed 90%.
输出和输入可用能量 (J) 或功率 (W) 表示。在简单装置中，机械效率常低于 50%；经精密设计的系统可超过 90%。
In machines such as levers and pulleys, efficiency can also be linked to mechanical advantage (MA) and velocity ratio (VR).
在杠杆、滑轮等机械中，效率还可以与机械效益 (MA) 和速比 (VR) 联系起来。
Efficiency = MA / VR × 100%
If the measured MA is less than VR, energy is being lost. Engineers aim to maximise efficiency by reducing friction and using appropriate lubrication.
若测得的 MA 小于 VR，说明存在能量损耗。工程师们通过减小摩擦和合理润滑来最大限度地提高效率。
10. Material Stress and Strain | 材料的应力与应变
When a material is loaded, internal forces resist deformation. Stress describes the internal force per unit area, while strain measures how much the material deforms relative to its original dimension.
材料承受载荷时，内部会产生抵抗变形的力。应力描述单位面积上的内力，而应变衡量材料相对于原始尺寸的变形程度。
σ = F / A and ε = ΔL / L₀
σ is stress (Pa or N/m²), A is cross-sectional area (m²). ε is strain (no units), ΔL is change in length (m), L₀ is original length (m).
σ 表示应力 (Pa 或 N/m²)，A 为横截面积 (m²)。ε 为应变（无量纲），ΔL 为长度变化量 (m)，L₀ 为原始长度 (m)。
Young's modulus (E) is a measure of a material's stiffness. It is the ratio of stress to strain in the elastic region, where the material returns to its original shape after unloading.
杨氏模量 (E) 是衡量材料刚度的指标。它是弹性区域内应力与应变的比值，在该区域内卸载后材料可恢复原状。
E = σ / ε
A high Young' rmation, e.g. steel (≈ 200 GPa). A low modulus indicates a flexible material, e.g. rubber.
杨氏模量高意味着材料刚度大、抗变形能力强，例如钢的模量约 200 GPa。模量低则表明材料柔韧，如橡胶。
Understanding stress and strain helps engineers select suitable materials for structures, springs, beams and machine components, ensuring safety and performance.
理解应力与应变有助于工程师为结构件、弹簧、梁和机械零件选择合适的材料，确保安全与性能。
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