AS Cambridge Engineering: Formula & Theorem Quick Reference Handbook | AS剑桥工程：公式定理速查手册

📚 AS Cambridge Engineering: Formula & Theorem Quick Reference Handbook | AS剑桥工程：公式定理速查手册

This quick reference handbook gathers the essential formulas, laws and definitions required for the AS Cambridge Engineering syllabus. It covers mechanics, materials, thermodynamics and electrical principles, providing a concise tool for revision and problem-solving.

本速查手册汇集了AS剑桥工程课程所需的关键公式、定律和定义。内容涵盖力学、材料、热力学和电气原理，为复习与解题提供简明的工具。

1. Vectors and Equilibrium | 矢量与平衡

Scalar quantities possess magnitude only, whereas vector quantities have both magnitude and direction. Examples of vectors include displacement, velocity, acceleration and force.

标量仅有大小，矢量则既有大小又有方向。位移、速度、加速度和力均为矢量。

When two vectors act at a point, the resultant can be found using the parallelogram law. The magnitude of the resultant R of two vectors A and B with angle θ between them is given by:

当两个矢量作用于一点时，可用平行四边形法则求合力。两矢量 A 和 B 之间夹角为 θ，合力 R 的大小为：

R = √(A² + B² + 2AB cos θ)

For an object in static equilibrium, the net force in any direction must be zero and the net moment about any point must also be zero.

物体处于静力平衡时，任意方向上的合力必为零，且对任意点的合力矩也为零。

Σ F = 0   and   Σ M = 0

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2. Kinematics of Linear Motion | 直线运动学

For uniform acceleration a, the kinematic equations link initial velocity u, final velocity v, displacement s and time t.

在匀加速直线运动中，运动学方程联系了初速度 u、末速度 v、位移 s 和时间 t。

v = u + a t

s = u t + ½ a t²

v² = u² + 2 a s

s = ½ (u + v) t

The gradient of a displacement–time graph gives velocity; the area under a velocity–time graph gives displacement. The gradient of a velocity–time graph gives acceleration.

位移-时间图的斜率表示速度；速度-时间图下面积表示位移；速度-时间图的斜率表示加速度。

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3. Dynamics and Newton’s Laws | 动力学与牛顿定律

Newton’s second law states that the net force acting on an object equals the product of its mass and acceleration.

牛顿第二定律指出，物体所受合力等于其质量与加速度的乘积。

F = m a

The frictional force f between two solid surfaces is proportional to the normal reaction R, up to a limit: f ≤ μ R, where μ is the coefficient of friction.

两固体表面间的摩擦力 f 与法向反力 R 成正比，且不超过最大值：f ≤ μ R，μ 为摩擦系数。

Weight W is the gravitational force on a mass m and is given by W = m g, where g ≈ 9.81 m s⁻².

重量 W 是质量为 m 的物体所受重力，W = m g，g ≈ 9.81 m s⁻²。

Momentum p is the product of mass and velocity: p = m v. The impulse of a force equals the change in momentum: F Δt = Δp.

动量 p 为质量与速度的乘积：p = m v。力的冲量等于动量的变化：F Δt = Δp。

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4. Work, Energy and Power | 功、能和功率

Work done by a constant force F moving its point of application through a distance d in the direction of the force is:

恒力 F 使其作用点沿力的方向移动距离 d 所做的功为：

W = F d

When the force is at an angle θ to the displacement, W = F d cos θ.

当力与位移方向成 θ 角时，W = F d cos θ。

Gravitational potential energy near Earth’s surface: Eₚ = m g h, where h is the vertical height.

近地表面的重力势能：Eₚ = m g h，h 为竖直高度。

Kinetic energy: Eₖ = ½ m v².

动能：Eₖ = ½ m v²。

Power is the rate of doing work: P = W / t. For a constant force moving at constant speed v, P = F v.

功率是做功的速率：P = W / t。对于恒力作用下的匀速运动，P = F v。

Efficiency = (useful energy output / total energy input) × 100%.

效率 = (有用能量输出 / 总能量输入) × 100%。

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5. Moments and Torque | 力矩与转矩

The moment of a force about a point is the product of the force and the perpendicular distance from the point to the line of action of the force.

力对一点的力矩等于力的大小乘以该点到力作用线的垂直距离。

Moment = F × d

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

力矩原理指出，物体处于转动平衡时，对任意点顺时针力矩之和等于逆时针力矩之和。

A couple consists of two equal and opposite parallel forces; the torque (or moment of a couple) is:

力偶由一对大小相等、方向相反的平行力组成；力偶矩（转矩）为：

Torque = F × s

where s is the perpendicular distance between the forces.

其中 s 为两力之间的垂直距离。

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

Direct stress (σ) is the force per unit cross-sectional area: σ = F / A (unit: Pa or N m⁻²).

正应力 (σ) 是单位横截面积上的力：σ = F / A（单位：Pa 或 N m⁻²）。

Direct strain (ε) is the ratio of extension to original length: ε = ΔL / L₀ (dimensionless).

正应变 (ε) 是伸长量与原长之比：ε = ΔL / L₀（无量纲）。

Young modulus E measures the stiffness of a material in the linear elastic region:

杨氏模量 E 表征材料在线弹性区的刚度：

E = σ / ε = (F / A) / (ΔL / L₀)

Elastic strain energy stored in a stretched material can be found from the area under the force–extension graph. For Hookean deformation, U = ½ F ΔL = ½ k (ΔL)².

弹性应变能可由力-伸长图下的面积求得。对于胡克变形，U = ½ F ΔL = ½ k (ΔL)²。

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7. Pressure and Fluid Statics | 压强与流体静力学

Pressure is defined as force acting per unit area normal to the surface: p = F / A (unit: Pa).

压强定义为垂直于表面单位面积上的力：p = F / A（单位：Pa）。

The pressure at a depth h in a fluid of density ρ due to the fluid alone is:

密度为 ρ 的流体中，深度 h 处由流体自身产生的压强为：

p = ρ g h

Pascal’s principle states that a change in pressure applied to an enclosed fluid is transmitted undiminished to all parts of the fluid and to the walls of its container.

帕斯卡原理指出，施加在密闭流体上的压强变化会等值地传递到流体的各个部分及容器壁。

Upthrust (buoyancy force) on an immersed object equals the weight of fluid displaced (Archimedes’ principle): F_up = ρ_fluid × V_displaced × g.

浸入物体所受的浮力等于排开流体的重量（阿基米德原理）：F_up = ρ_fluid × V_displaced × g。

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8. Thermal Properties and Gas Laws | 热性质与气体定律

When the temperature of a solid or liquid changes by ΔT, its length changes by:

固体或液体的温度变化 ΔT 时，其长度变化为：

ΔL = α L₀ ΔT

where α is the coefficient of linear expansion.

其中 α 为线膨胀系数。

The heat energy required to change the temperature of a substance is: Q = m c ΔT, where c is the specific heat capacity.

改变物质温度所需的热能为：Q = m c ΔT，c 为比热容。

For an ideal gas, the relationship between pressure p, volume V and thermodynamic temperature T is given by the ideal gas law:

对于理想气体，压强 p、体积 V 和热力学温度 T 之间的关系由理想气体定律给出：

p V = n R T

where n is the number of moles and R = 8.31 J mol⁻¹ K⁻¹. For a fixed mass of gas, pV/T = constant; this leads to Boyle’s law (pV = constant at constant T), Charles’ law (V/T = constant at constant p) and the pressure law (p/T = constant at constant V).

其中 n 为物质的量，R = 8.31 J mol⁻¹ K⁻¹。对于一定质量的气体，pV/T = 常量；由此可导出波义耳定律（恒温下 pV = 常量）、查理定律（恒压下 V/T = 常量）和压强定律（恒容下 p/T = 常量）。

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9. DC Circuits and Ohm’s Law | 直流电路与欧姆定律

Ohm’s law states that the current I through a conductor is directly proportional to the potential difference V across it, provided temperature and other physical conditions remain constant.

欧姆定律指出，在温度和物理条件不变时，流过导体的电流 I 与其两端的电位差 V 成正比。

V = I R

Resistors in series: R_total = R₁ + R₂ + …

串联电阻：R_total = R₁ + R₂ + …

Resistors in parallel: 1 / R_total = 1 / R₁ + 1 / R₂ + …

并联电阻：1 / R_total = 1 / R₁ + 1 / R₂ + …

Electric power dissipated in a resistor:

电阻上消耗的电功率：

P = I V = I² R = V² / R

Energy transfer: W = I V t.

能量转换：W = I V t。

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10. Resistivity and Kirchhoff’s Laws | 电阻率与基尔霍夫定律

The resistance R of a uniform wire depends on its length l, cross-sectional area A and the material’s resistivity ρ:

均匀导线的电阻 R 取决于其长度 l、横截面积 A 与材料的电阻率 ρ：

R = ρ l / A

Resistivity is a material property measured in Ω m. Conductivity σ is the reciprocal of resistivity: σ = 1/ρ.

电阻率是材料特性，单位为 Ω m。电导率 σ 是电阻率的倒数：σ = 1/ρ。

Kirchhoff’s current law (KCL): the sum of currents entering a junction equals the sum of currents leaving it.

基尔霍夫电流定律：进入节点的电流之和等于离开节点的电流之和。

Σ I_in = Σ I_out

Kirchhoff’s voltage law (KVL): the sum of the electromotive forces in any closed loop equals the sum of the potential drops (products of I R) around that loop.

基尔霍夫电压定律：任一闭合回路中，电动势代数和等于该回路中各元件上电压降（I R）的代数和。

Σ ε = Σ I R

The potential divider (voltage divider) rule: for two resistors in series, the output voltage is V_out = V_in × R₂ / (R₁ + R₂).

分压器原理：对于两个串联电阻，输出电压为 V_out = V_in × R₂ / (R₁ + R₂)。

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