Year 13 AQA Engineering: Quick Reference Formula & Theorem Handbook | AQA 工程：公式定理速查手册

📚 Year 13 AQA Engineering: Quick Reference Formula & Theorem Handbook | AQA 工程：公式定理速查手册

This comprehensive handbook gathers the essential formulas, theorems, and mechanical principles required for the Year 13 AQA Engineering curriculum. Use it as a quick reference to reinforce your understanding, tackle numerical problems, and prepare confidently for examinations. Each entry is paired with a concise explanation and a practical context where applicable.

这本综合手册汇集了 Year 13 AQA 工程课程所需的全部核心公式、定理和力学原理。将其用作快速参考，以巩固理解、解决计算题并自信地备考。每个条目都配有简洁的解释和实际应用场景。

1. Forces and Newton’s Laws | 力与牛顿定律

Newton’s Second Law states that the net force acting on a body is equal to the product of its mass and acceleration. This vector equation forms the foundation for analysing rectilinear motion.

牛顿第二定律指出，作用在物体上的合外力等于其质量与加速度的乘积。这个矢量方程是分析直线运动的基础。

Fₙₑₜ = m a

Fₙₑₜ (N): Net force acting on the body
m (kg): Mass of the body
a (m/s²): Acceleration produced

Fₙₑₜ (牛): 作用在物体上的合外力
m (千克): 物体的质量
a (米/秒²): 产生的加速度

For problems involving multiple forces, resolve them into perpendicular components using trigonometry. Equilibrium occurs when both the sum of horizontal forces and the sum of vertical forces equal zero.

对于涉及多个力的问题，使用三角法将力分解为互相垂直的分量。当水平方向合力与竖直方向合力均为零时，物体处于平衡状态。

2. Moments and Equilibrium | 力矩与平衡

The principle of moments states that for a body in rotational equilibrium, the sum of clockwise moments about any pivot equals the sum of anticlockwise moments about that same point.

力矩原理指出，对于处于旋转平衡的物体，绕任意支点的顺时针力矩之和等于绕同一点的逆时针力矩之和。

Σ M_cw = Σ M_acw

A moment is defined as the product of the force and the perpendicular distance from the line of action to the pivot.

力矩定义为力与力的作用线到支点的垂直距离的乘积。

M = F × d

M (Nm): Moment
F (N): Applied force
d (m): Perpendicular distance from pivot to line of action

M (牛·米): 力矩
F (牛): 施加的力
d (米): 支点到力作用线的垂直距离

A couple consists of two equal, parallel, but oppositely directed forces whose lines of action do not coincide. The moment of a couple is simply the product of one force and the perpendicular distance between the forces.

力偶由两个大小相等、方向相反且作用线不重合的平行力组成。力偶的力矩等于其中一个力与两力之间垂直距离的乘积。

3. Stress, Strain and Young’s Modulus | 应力、应变与杨氏模量

Direct stress quantifies the internal resistance per unit area to an externally applied load. It is measured in pascals (Pa) and indicates how intensely a material experiences force.

正应力量化了单位面积上对抗外部载荷的内阻力。其单位为帕斯卡 (Pa)，表明材料受力的强弱程度。

σ = F / A

σ (Pa): Direct stress
F (N): Applied tensile or compressive force
A (m²): Original cross-sectional area

σ (帕): 正应力
F (牛): 施加的拉伸或压缩力
A (米²): 原始横截面积

Tensile strain is the extension per unit original length, a dimensionless ratio describing deformation. Young’s modulus measures a material’s stiffness and is a constant within the elastic limit.

拉伸应变是单位原始长度的伸长量，是一个描述变形的无量纲比率。杨氏模量衡量材料的刚度，在弹性极限内为常数。

ε = ΔL / L₀

E = σ / ε

ε: Strain (no units)
ΔL (m): Change in length
L₀ (m): Original length
E (Pa): Young’s modulus

ε: 应变 (无单位)
ΔL (米): 长度变化量
L₀ (米): 原始长度
E (帕): 杨氏模量

4. Kinematic Equations of Linear Motion | 直线运动方程

For motion with constant acceleration along a straight line, four key equations link initial velocity, final velocity, acceleration, displacement, and time.

对于沿直线的匀加速运动，四个关键方程将初速度、末速度、加速度、位移和时间联系起来。

v = u + a t

s = u t + ½ a t²

v² = u² + 2 a s

s = ½ (u + v) t

v (m/s): Final velocity after time t
u (m/s): Initial velocity
a (m/s²): Constant acceleration
s (m): Displacement in the direction of motion
t (s): Time elapsed

v (米/秒): 时间 t 后的末速度
u (米/秒): 初速度
a (米/秒²): 恒定的加速度
s (米): 运动方向上的位移
t (秒): 经过的时间

These equations assume zero initial displacement and constant acceleration. Always inspect the sign conventions; typically upward or leftward vectors receive a negative sign.

这些方程假设初始位移为零且加速度恒定。务必检查符号约定；通常向上或向左的矢量取负号。

5. Work, Energy and Power | 功、能与功率

Work is done when a force moves its point of application in the direction of the force. The unit of work and energy is the joule (J). Power is the rate of doing work, measured in watts (W).

当力使其作用点沿力的方向移动时，就做了功。功和能量的单位是焦耳 (J)。功率是做功的速率，单位为瓦特 (W)。

W = F s cos θ

W (J): Work done
F (N): Constant force
s (m): Displacement of the point of application
θ: Angle between the force vector and the displacement direction

W (焦): 所做的功
F (牛): 恒定的力
s (米): 作用点的位移
θ: 力矢量与位移方向之间的夹角

Gravitational potential energy and kinetic energy are the two main mechanical energy forms. The work-energy principle equates net work to change in kinetic energy.

重力势能和动能是两种主要的机械能形式。功能原理将合外力做的功等同于动能的变化量。

Eₚ = m g h

Eₖ = ½ m v²

P = W / t = F v

Eₚ (J): Gravitational potential energy; g = 9.81 m/s²
Eₖ (J): Kinetic energy
P (W): Power; F v applies when force and velocity are parallel

Eₚ (焦): 重力势能；g = 9.81 米/秒²
Eₖ (焦): 动能
P (瓦): 功率；当力与速度同向时使用 P = F v

6. Momentum and Impulse | 动量与冲量

Linear momentum is a vector quantity defined as the product of a body’s mass and its velocity. The principle of conservation of momentum states that the total momentum of a closed system remains constant in the absence of external forces.

线动量是一个矢量，定义为一个物体的质量与其速度的乘积。动量守恒原理指出，在没有外力作用的情况下，封闭系统的总动量保持不变。

p = m v

Fₐᵥ = Δp / Δt

p (kg m/s): Momentum
Δp (kg m/s): Change in momentum
Fₐᵥ (N): Average force during impact
Δt (s): Duration of impact

p (千克·米/秒): 动量
Δp (千克·米/秒): 动量变化量
Fₐᵥ (牛): 冲击过程中的平均力
Δt (秒): 冲击持续时间

Impulse equals the change in momentum and also equals the area under a force-time graph. This relationship is crucial for analysing collisions and crash safety designs.

冲量等于动量的变化量，也等于力-时间图下的面积。这一关系对于分析碰撞和碰撞安全设计至关重要。

7. First Law of Thermodynamics | 热力学第一定律

The first law of thermodynamics is an expression of energy conservation applied to thermal systems. The internal energy of a closed system increases when heat is added or work is done on the system.

热力学第一定律是能量守恒定律在热力系统中的应用。当向系统加热或对系统做功时，封闭系统的内能增加。

ΔU = Q – W

ΔU (J): Change in internal energy
Q (J): Heat energy transferred into the system
W (J): Work done by the system on its surroundings

ΔU (焦): 内能的变化量
Q (焦): 传入系统的热量
W (焦): 系统对外界所做的功

For an ideal gas undergoing an isothermal process, ΔU = 0, so Q = W. In an adiabatic process, Q = 0, so ΔU = -W. Engineers apply this law when designing heat engines, compressors, and refrigeration cycles.

对于理想气体经历等温过程，ΔU = 0，因此 Q = W。在绝热过程中，Q = 0，因此 ΔU = -W。工程师在设计热机、压缩机和制冷循环时会应用该定律。

8. Fluid Statics and Hydrostatic Pressure | 流体静力学与静水压力

Hydrostatic pressure at a given depth in a static fluid depends only on the density of the fluid, the depth, and gravitational field strength. Pressure acts equally in all directions at a point within the fluid.

静止流体中给定深度处的静水压力仅取决于流体密度、深度和重力场强度。在流体内某一点，压力在各个方向上作用相等。

p = ρ g h

p (Pa): Hydrostatic pressure
ρ (kg/m³): Density of the fluid
g (m/s²): Acceleration due to gravity
h (m): Depth below the free surface

p (帕): 静水压力
ρ (千克/米³): 流体密度
g (米/秒²): 重力加速度
h (米): 自由表面以下的深度

The pressure in a connected body of fluid at the same horizontal level is constant. Upthrust on a submerged object equals the weight of fluid displaced, a principle described by Archimedes.

连通的流体中，同一水平高度上的压力是恒定的。作用在浸没物体上的浮力等于其排开流体的重量，这一原理由阿基米德描述。

F_upthrust = ρ_fluid × V_submerged × g

9. Ohm’s Law and DC Circuit Theorems | 欧姆定律与直流电路定理

Ohm’s law defines the linear relationship between potential difference, current, and resistance for metallic conductors at constant temperature. It serves as the cornerstone for analysing DC circuits.

欧姆定律定义了在恒定温度下金属导体的电势差、电流和电阻之间的线性关系。它是分析直流电路的基石。

V = I R

V (V): Potential difference across the conductor
I (A): Current flowing through the conductor
R (Ω): Resistance

V (伏): 导体两端的电势差
I (安): 流过导体的电流
R (欧): 电阻

Kirchhoff’s current law states that the sum of currents entering a junction equals the sum of currents leaving it. Kirchhoff’s voltage law states that the directed sum of potential differences around any closed loop is zero.

基尔霍夫电流定律指出，进入一个节点的电流之和等于离开该节点的电流之和。基尔霍夫电压定律指出，沿任意闭合回路的电势差代数和为零。

Σ I_in = Σ I_out

Σ V_loop = 0

Resistors in series simply add: R_total = R₁ + R₂ + … For parallel resistors, the reciprocal formula applies: 1/R_total = 1/R₁ + 1/R₂ + …

串联电阻直接相加：R_total = R₁ + R₂ + … 对于并联电阻，使用倒数公式：1/R_total = 1/R₁ + 1/R₂ + …

10. Factor of Safety and Material Failure | 安全系数与材料失效

The factor of safety is a design parameter that ensures a component can withstand loads greater than the expected maximum without failing. It is the ratio of the ultimate tensile stress (or yield stress) to the allowable working stress.

安全系数是一个设计参数，用于确保部件能够承受超过预期最大载荷的负荷而不失效。它是极限拉伸应力（或屈服应力）与许用工作应力之比。

FoS = σ_ultimate / σ_working

FoS: Factor of Safety (no units, typically between 1.5 and 10 depending on application)
σ_ultimate (Pa): Ultimate tensile stress the material can withstand
σ_working (Pa): Maximum stress expected in service

FoS: 安全系数 (无单位，通常根据应用在 1.5 到 10 之间)
σ_ultimate (帕): 材料能承受的极限拉伸应力
σ_working (帕): 使用中预期的最大应力

Materials fail through mechanisms such as ductile fracture, brittle fracture, fatigue, and creep. Fatigue failure is critically evaluated using the endurance limit derived from S-N curves, where the applied cyclic stress (S) is plotted against the number of cycles to failure (N).

材料通过韧性断裂、脆性断裂、疲劳和蠕变等机制失效。疲劳失效通过源自 S-N 曲线的耐久极限进行严格评估，其中施加的循环应力 (S) 与失效循环次数 (N) 对应绘制。

11. Angular Motion Parameters | 角运动参数

For a rigid body rotating about a fixed axis, angular displacement, angular velocity, and angular acceleration are the rotational analogues of linear kinematic quantities. The relationships mirror those for linear motion.

对于绕固定轴旋转的刚体，角位移、角速度和角加速度是直线运动学量的旋转对应量。它们的关系与直线运动相似。

ω = Δθ / Δt

α = Δω / Δt

θ (rad): Angular displacement
ω (rad/s): Angular velocity
α (rad/s²): Angular acceleration

θ (弧度): 角位移
ω (弧度/秒): 角速度
α (弧度/秒²): 角加速度

For uniform angular acceleration, the same kinematic structure applies: ω₂ = ω₁ + α t; θ = ω₁ t + ½ α t²; ω₂² = ω₁² + 2 α θ; θ = ½ (ω₁ + ω₂) t. Also, the torque required to produce an angular acceleration depends on the moment of inertia: T = I α.

对于匀角加速度，适用相同的运动学结构：ω₂ = ω₁ + α t；θ = ω₁ t + ½ α t²；ω₂² = ω₁² + 2 α θ；θ = ½ (ω₁ + ω₂) t。另外，产生角加速度所需的扭矩取决于转动惯量：T = I α。

12. Coefficient of Friction and Inclined Planes | 摩擦系数与斜面

Frictional force resists the relative motion of two surfaces in contact. It is proportional to the normal reaction force and can be static (no motion) or dynamic (during sliding).

摩擦力抵抗两个接触表面的相对运动。它与法向反作用力成正比，并且可以是静态的（无运动）或动态的（滑动中）。

F_friction ≤ μ_s R (static)

F_friction = μ_d R (dynamic)

F_friction (N): Frictional force
μ_s, μ_d: Coefficient of static/dynamic friction (no units)
R (N): Normal reaction force perpendicular to the surfaces

F_friction (牛): 摩擦力
μ_s, μ_d: 静/动摩擦系数 (无单位)
R (牛): 垂直于接触面的法向反作用力

When an object rests on an inclined plane, the condition for sliding down is tan θ > μ_s, where θ is the angle of inclination from the horizontal. Resolving weight components mg sin θ and mg cos θ is essential for any ramp analysis.

当物体静止在斜面上时，下滑的条件是 tan θ > μ_s，其中 θ 是相对于水平面的倾角。分解重力分量 mg sin θ 和 mg cos θ 对于任何斜面分析都至关重要。

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Year 13 AQA Engineering: Quick Reference Formula & Theorem Handbook | AQA 工程：公式定理速查手册