📚 Year 12 CCEA Engineering: Formula and Theorem Quick Reference Handbook | Year 12 CCEA 工程：公式定理速查手册
Welcome to the Year 12 CCEA Engineering Quick Reference Handbook. This resource compiles essential formulas, theorems and constants across the core engineering topics, including mechanics, materials, electronics and mathematics. Use it as a handy revision tool to reinforce your understanding and to solve problems efficiently, whether you are preparing for a class test, a practical assessment or the end-of-unit examination.
欢迎使用 Year 12 CCEA 工程公式定理速查手册。本资源汇编了核心工程主题（包括力学、材料、电子学和数学）中的基本公式、定理和常数。无论你是在准备随堂测验、实践评估还是单元末考试，都可以把它作为便捷的复习工具，以巩固理解并高效解决问题。
The handbook is organised in the same order that most CCEA teaching schemes follow, so you can read it straight through as a revision programme or jump directly to the section you need. Every formula is followed by a short explanation of when it applies and what each symbol means, because marks in CCEA papers are awarded for correct selection of a relationship as much as for correct substitution.
本手册的编排顺序与大多数 CCEA 教学方案一致，你既可以把它当作复习计划通读一遍，也可以直接跳到你需要的章节。每个公式后面都附有简短说明，解释它的适用条件以及每个符号的含义，因为在 CCEA 试卷中，正确选择关系式与正确代入数值同样得分。
1. Linear Motion and Kinematics | 直线运动与运动学
Kinematics describes the motion of objects without considering the forces causing it. For an object moving with constant acceleration along a straight line, the following SUVAT equations are fundamental. The five quantities involved are initial velocity u, final velocity v, acceleration a, time t and displacement s.
运动学描述物体的运动，而不考虑引起运动的力。对于沿直线以恒定加速度运动的物体，以下 SUVAT 方程是基础。涉及的五个量分别是初速度 u、末速度 v、加速度 a、时间 t 和位移 s。
v = u + a t
This equation relates final velocity (v), initial velocity (u), acceleration (a) and time (t). It is used when displacement is not required.
该方程关联末速度（v）、初速度（u）、加速度（a）和时间（t）。当不需要位移时使用。
s = (u + v) t ÷ 2
This relationship gives displacement from the average velocity and is the quickest route when acceleration is not mentioned in the question.
该关系式用平均速度求位移，当题目没有提到加速度时，这是最快的解法。
s = u t + ½ a t²
Use this form when time is known but final velocity is not. Note that the acceleration term carries the sign of a, so a deceleration must be entered as a negative value.
当已知时间而未知末速度时使用此式。注意加速度项带有 a 的符号，因此减速度必须以负值代入。
v² = u² + 2 a s
This is the only SUVAT equation that does not contain time, which makes it the natural choice for braking-distance and impact-speed problems.
这是唯一不含时间的 SUVAT 方程，因此在制动距离和撞击速度类问题中自然成为首选。
Motion graphs carry equal weight in the specification. On a displacement-time graph the gradient is velocity; on a velocity-time graph the gradient is acceleration and the area under the line is displacement. A curved line means the acceleration is changing, so instantaneous values must be found by drawing a tangent.
运动图像在考纲中同样重要。位移-时间图上的斜率是速度；速度-时间图上的斜率是加速度，线下的面积是位移。曲线意味着加速度在变化，因此瞬时值必须通过画切线求出。
1 m s⁻² and a body thrown upwards has u positive and a negative; forgetting this is the single most common source of lost marks in kinematics questions.
代入数值前一定要先声明正方向约定。取向上为正时，下落物体的 a = −9.81 m s⁻²，上抛物体的 u 为正而 a 为负；忘记这一点是运动学题目中最常见的失分原因。
2. Newton's Laws and Dynamics | 牛顿定律与动力学
Newton's first law states that a body remains at rest or moves with constant velocity unless acted upon by a resultant force. The second law quantifies that force, and the third law states that forces act in equal and opposite pairs on two different bodies.
牛顿第一定律指出，除非受到合力作用，物体保持静止或匀速直线运动。第二定律对该力进行定量描述，第三定律则指出力以大小相等、方向相反的形式作用在两个不同物体上。
F = m a
Resultant force (N) equals mass (kg) multiplied by acceleration (m s⁻²). The force and the acceleration are always in the same direction, and F must be the resultant, not any single applied force.
合力（N）等于质量（kg）乘以加速度（m s⁻²）。力与加速度方向始终相同，且 F 必须是合力，而不是某一个单独的施加力。
W = m g
Weight is the gravitational force on a mass, where g = 9.81 m s⁻² near the Earth' ms; weight is a vector and measured in newtons. Examiners regularly test this distinction.
重力是作用在质量上的引力，地球表面附近 g = 9.81 m s⁻²。质量是标量，单位为千克；重力是矢量，单位为牛顿。考官经常考查这一区别。
p = m v
Linear momentum is the product of mass and velocity and is conserved in all collisions and explosions provided no external resultant force acts on the system.
线动量是质量与速度的乘积，只要系统不受外力合力作用，它在所有碰撞和爆炸中都守恒。
F t = m v − m u
This is the impulse-momentum equation. The product Ft is the impulse, and the right-hand side is the change in momentum. It explains why crumple zones and airbags reduce injury: they increase t and therefore reduce F.
这是冲量-动量方程。乘积 Ft 是冲量，右侧是动量的变化量。它解释了溃缩区和安全气囊为何能减轻伤害：它们延长了 t，因此减小了 F。
F = μ R
incline of angle θ, R = m g cos θ and the component of weight down the slope is m g sin θ.
极限摩擦力等于摩擦系数乘以法向反力。在水平面上 R = m g；在倾角为 θ 的斜面上，R = m g cos θ，而重力沿斜面向下的分量为 m g sin θ。
For a system of connected bodies, such as two masses joined by a string over a pulley, treat the whole system as one to find the acceleration and then isolate one body to find the tension. This two-stage method avoids the algebra errors that come from writing simultaneous equations for every mass.
对于连接体系统，例如跨过滑轮的细绳连接的两个物体，先把整体视为一个物体求出加速度，再隔离其中一个物体求出绳中张力。这种两步法可以避免为每个物体分别写联立方程所带来的代数错误。
3. Work, Energy and Power | 功、能量与功率
Energy is the capacity to do work and is measured in joules. In mechanical problems energy is conserved when friction and air resistance are negligible, and the work-energy principle provides a powerful alternative to solving with forces and acceleration.
能量是做功的能力，单位为焦耳。在机械问题中，当摩擦和空气阻力可忽略时能量守恒，而功能原理提供了替代用力与加速度求解的有力方法。
W = F s cos θ
, cos 90° = 0 and no work is done, which is why a centripetal force never changes the kinetic energy of a body.
功等于力沿运动方向的分量乘以位移。若力垂直于运动方向，cos 90° = 0，不做功，这正是向心力从不改变物体动能的原因。
Eₖ = ½ m v²
Kinetic energy depends on the square of the speed, so doubling the speed quadruples the energy and roughly quadruples the braking distance. This relationship is examined almost every year in vehicle-safety contexts.
动能与速度的平方成正比，因此速度加倍会使动能变为四倍，制动距离也大致变为四倍。这一关系几乎每年都在车辆安全情境中考查。
Eₚ = m g h
Gravitational potential energy is measured relative to a chosen reference level. Whatever datum you choose, the change in potential energy between two heights is the same, so pick the convenient level, usually the lowest point in the problem.
重力势能是相对于所选参考平面测量的。无论选择哪个基准面，两高度之间的势能变化都相同，因此应选择方便的水平面，通常取问题中的最低点。
P = W ÷ t = E ÷ t
put power is always greater than useful output power because of friction and other losses.
功率是做功或传递能量的速率，单位为瓦特，1 W = 1 J s⁻¹。对于机械而言，由于摩擦和其他损耗，输入功率总是大于有用输出功率。
P = F v
This form is used whenever a constant driving force produces motion at constant speed, for example a vehicle on a level road or a winch lifting a load at steady velocity.
当恒定驱动力使物体匀速运动时使用此式，例如水平路面上的车辆或以稳定速度提升载荷的绞车。
η = (useful output ÷ total input) × 100%
Efficiency is a ratio with no unit and can never exceed 100 per cent. Always calculate it from energy or power values, not from force values, unless the distances are equal.
效率是无单位的比值，永远不可能超过 100%。计算时务必使用能量或功率值，除非位移相同，否则不要用力的值。
4. Stress, Strain and Material Properties | 应力、应变与材料特性
When a component is loaded it deforms, and the relationship between load and deformation determines whether the design is safe. These definitions and the Young's modulus relationship form the backbone of the materials section of the specification.
当构件受载时会发生变形，而载荷与变形之间的关系决定了设计是否安全。以下定义以及杨氏模量关系式构成考纲材料部分的主干。
σ = F ÷ A
Direct stress is the load per unit cross-sectional area, measured in N m⁻² or pascals. Working in N mm⁻² is often more convenient because it is numerically equal to MPa.
正应力是单位横截面积上的载荷，单位为 N m⁻² 或帕斯卡。使用 N mm⁻² 往往更方便，因为它在数值上等于 MPa。
ε = x ÷ L
length and has no units. It is usually quoted as a percentage or in microstrain (× 10⁻⁶).
应变是伸长量除以原始长度，没有单位。通常以百分数或微应变（× 10⁻⁶）表示。
E = σ ÷ ε
Young's modulus is the gradient of the straight-line portion of a stress-strain graph, so the stiffer the material the steeper the line. Its unit is the same as stress, since strain is dimensionless.
杨氏模量是应力-应变图上直线部分的斜率，因此材料越刚硬，直线越陡。由于应变无量纲，它的单位与应力相同。
For a wire or bar within the elastic limit, combining the previous three relationships gives a single design equation that you should be able to derive quickly in an examination:
对于处于弹性极限内的金属丝或杆件，把上述三个关系式组合起来可以得到一个单一的设计方程，你应该能在考试中迅速推导出来：
E = (F L) ÷ (A x)
Rearranging this allows you to find the extension of a member of given size, or to select the cross-sectional area needed to limit deflection to a specified value.
通过移项可以求出给定尺寸构件的伸长量，或者确定把挠度限制在某一规定值以内所需的横截面积。
Key points read from a stress-strain curve include the limit of proportionality, the elastic limit, the yield point, the ultimate tensile strength (the highest point) and the fracture point. Ductile materials such as mild steel show a long plastic region before breaking, whereas brittle materials such as cast iron or glass show almost none and fail suddenly.
从应力-应变曲线上读取的关键点包括比例极限、弹性极限、屈服点、抗拉强度极限（最高点）和断裂点。低碳钢等韧性材料在断裂前有很长的塑性区，而铸铁或玻璃等脆性材料几乎没有塑性区，并且会突然断裂。
| Quantity 物理量 | Symbol 符号 | Unit 单位 |
| Density 密度 | ρ = m ÷ V | kg m⁻³ |
| Pressure 压强 | P = F ÷ A | Pa |
| Shear stress 切应力 | τ = F ÷ A | N m⁻² |
| Modulus of rigidity 切变模量 | G = τ ÷ γ | N m⁻² |
5. Thermal Physics and Fluid Mechanics | 热学与流体力学
Thermal and fluid topics appear both in the examined theory and in practical coursework, where specific heat capacity and viscosity measurements are common tasks.
热学与流体主题既出现在笔试理论中，也出现在实践课程作业中，其中最常做的实验是比热容和黏度的测量。
Q = m c Δθ
Heat energy supplied or removed equals mass multiplied by specific heat capacity multiplied by the temperature change. Use Δθ in kelvin or degrees Celsius, since only the change matters.
供给或放出的热量等于质量乘以比热容再乘以温度变化。Δθ 用开尔文或摄氏度均可，因为只有变化量才有意义。
Q = m L
During a change of state the temperature stays constant while the latent heat L is absorbed or released. Use the specific latent heat of fusion for melting and solidification, and of vaporisation for boiling and condensing.
在状态变化过程中，温度保持不变，同时吸收或放出潜热 L。熔化与凝固用熔化比潜热，沸腾与冷凝用汽化比潜热。
bining the two relationships allows specific heat capacity to be measured. The main practical error is heat loss to the surroundings, which is reduced by lagging the block or vessel and by taking the temperature rise from the extrapolated cooling curve.
以恒定速率供能的电加热器给出 Q = V I t，因此把两个关系式结合即可测量比热容。主要的实验误差是向周围环境的热损失，可以通过给金属块或容器加保温层、并利用外推冷却曲线求得温升来减小。
P = ρ g h
Hydrostatic pressure at depth h depends only on the depth and the density of the fluid, not on the shape of the container. This is the basis of manometer and dam-design questions.
深度 h 处的流体静压强只取决于深度和流体密度，与容器形状无关。这是压强计和坝体设计题目的基础。
Upthrust = ρ g V
Archimedes' principle states that the upthrust on a submerged body equals the weight of fluid displaced. A floating body displaces its own weight of fluid; a submerged body displaces its own volume.
阿基米德原理指出，浸没物体所受浮力等于所排开流体的重力。漂浮物体排开与自身重力相等的流体；完全浸没的物体则排开与自身体积相等的流体。
A₁ v₁ = A₂ v₂
The continuity equation expresses conservation of mass for an incompressible fluid in steady flow: where the pipe narrows, the velocity increases. Bernoulli's principle extends this by linking pressure, height and velocity along a streamline.
连续性方程表达不可压缩流体稳定流动时的质量守恒：管道变窄处流速增大。伯努利原理通过沿流线联系压强、高度和速度，进一步扩展了这一关系。
6. Electrical Circuits and Ohm's Law | 电路与欧姆定律
Direct-current circuit theory is the most heavily examined area of the electrical unit. Nearly every numerical question can be solved with Ohm's law combined with the series and parallel rules for resistance.
直流电路理论是电学单元中考查最密集的部分。几乎所有计算题都可以用欧姆定律加上电阻串联、并联规则来解决。
V = I R
Potential difference in volts equals current in amperes multiplied by resistance in ohms. The relationship holds for ohmic conductors at constant temperature; filament lamps and diodes are non-ohmic and their resistance changes with operating conditions.
电压（伏特）等于电流（安培）乘以电阻（欧姆）。该关系适用于恒定温度下的欧姆导体；白炽灯丝和二极管是非欧姆元件，其电阻随工作条件变化。
P = V I = I² R = V² ÷ R
Choose the form that matches the quantities you already know. The I² R form is preferred when discussing power lost in transmission cables, because it shows the loss falls by a factor of four when the current is halved.
选择与你已知量相匹配的形式。在讨论输电电缆的功率损耗时首选 I² R 形式，因为它表明电流减半时损耗降为四分之一。
R = R₁ + R₂ + R₃ (series)
In a series circuit the current is the same through every component, the total pd is the sum of the individual pds, and resistances simply add.
在串联电路中，通过每个元件的电流相同，总电压等于各元件电压之和，电阻直接相加。
1 ÷ R = 1 ÷ R₁ + 1 ÷ R₂ + 1 ÷ R₃ (parallel)
In a parallel circuit the pd across every branch is the same and the currents add. The combined resistance is always smaller than the smallest individual resistance, which is a useful check on your answer.
在并联电路中，每个支路两端的电压相同，电流相加。总电阻总是小于其中最小的单个电阻，这是检验答案是否合理的好方法。
Q = I t
ies 1.60 × 10⁻¹⁹ C, so the number of electrons passing a point is Q divided by that value.
电荷量（库仑）等于电流乘以时间（秒）。一个电子带 1.60 × 10⁻¹⁹ C，因此通过某点的电子数等于 Q 除以该值。
For a source of emf ε and internal resistance r, the terminal pd is V = ε − I r. Kirchhoff' on equals the sum out, and the sum of emfs around a closed loop equals the sum of the pds.
对于电动势为 ε、内阻为 r 的电源，端电压为 V = ε − I r。基尔霍夫两条定律把你已经使用的规则形式化：流入节点的电流之和等于流出之和，闭合回路中各电动势之和等于各电压之和。
capacitors in series combine like resistors in parallel. The energy stored is E = ½ C V².
对于电容器，Q = C V；电容并联直接相加,而电容串联的组合方式与电阻并联相同。储存的能量为 E = ½ C V²。
7. Electromagnetism and Induction | 电磁学与电磁感应
Magnetic effects arise whenever charge moves, and the relationships below connect current, field strength and mechanical force. They underpin the operation of motors, generators, relays and transformers.
只要电荷运动就会产生磁效应，以下关系式把电流、磁场强度和机械力联系起来。它们是电动机、发电机、继电器和变压器工作原理的基础。
F = B I L
The force on a current-carrying conductor is greatest when the conductor is perpendicular to the magnetic field and zero when it is parallel. Use Fleming's left-hand rule for the direction of the force on the conductor.
载流导体所受的力在导体垂直于磁场时最大，平行时为零。判断导体受力方向使用弗莱明左手定则。
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