Year 12 CIE Engineering Formula & Theorem Quick-Reference Handbook | CIE工程AS阶段公式定理速查手册

📚 Year 12 CIE Engineering Formula & Theorem Quick-Reference Handbook | CIE工程AS阶段公式定理速查手册

This handbook compiles the essential formulas, theorems and key relations encountered in the Year 12 CIE Engineering (AS Level) syllabus. It is designed as a rapid revision tool for end‑of‑topic tests and the final examination. Each section covers a core area, pairing concise English explanations with equivalent Chinese descriptions, and uses clearly formatted equations for quick scanning.

本手册汇集了 CIE 工程 AS 阶段（Year 12）所涉及的核心公式、定理和重要关系式，旨在为一课一练和最终考试提供快速复习工具。每个模块都以简洁的中英对照呈现，并通过加粗的公式排版让读者快速定位要点。

1. Linear Motion and Kinematics | 直线运动与运动学

The four equations of uniformly accelerated motion (SUVAT) describe the relationships between displacement, velocity, acceleration and time for constant acceleration.

匀加速运动的四个方程（SUVAT）描述了位移、速度、加速度和时间在恒定加速度下的关系。

v = u + a t

Equation 1: Final velocity equals initial velocity plus acceleration multiplied by time. v = u + a t

方程 1：末速度等于初速度加上加速度乘以时间。 v = u + a t

s = u t + ½ a t²

Equation 2: Displacement equals initial velocity times time plus half the acceleration times the square of time. s = u t + ½ a t²

方程 2：位移等于初速度乘以时间加上二分之一加速度乘以时间的平方。 s = u t + ½ a t²

s = ½ (u + v) t

Equation 3: Displacement equals the average velocity multiplied by time. s = ½ (u + v) t

方程 3：位移等于平均速度乘以时间。 s = ½ (u + v) t

v² = u² + 2 a s

Equation 4: The square of final velocity equals the square of initial velocity plus twice the product of acceleration and displacement. v² = u² + 2 a s

方程 4：末速度的平方等于初速度的平方加上两倍的加速度与位移的乘积。 v² = u² + 2 a s

Note: These equations apply only when acceleration a is constant. The sign convention must be consistent for all vectors.

注意：这些方程仅在加速度 a 恒定时适用，所有矢量的正负号必须前后一致。

Free fall and projectile motion: acceleration a = g (9.81 m s⁻² downward). The equations above are used separately in horizontal (a = 0) and vertical (a = g) directions.

自由落体与抛体运动：加速度 a = g（9.81 m s⁻² 向下）。将上述方程分别在水平方向 (a = 0) 和竖直方向 (a = g) 使用。

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2. Forces and Newton’s Laws | 力与牛顿定律

Newton’s First Law: An object remains at rest or in uniform motion in a straight line unless acted upon by a resultant external force.

牛顿第一定律：如果没有净外力作用，物体将保持静止或匀速直线运动状态。

F = m a

Newton’s Second Law: The net force acting on a body is equal to the product of its mass and acceleration. F = m a

牛顿第二定律：作用在物体上的净力等于其质量与加速度的乘积。 F = m a

Weight: The force due to gravity on a mass is W = m g, where g = 9.81 m s⁻² near the Earth’s surface.

重力：质量为 m 的物体所受重力 W = m g，地表附近 g = 9.81 m s⁻²。

Newton’s Third Law: For every action there is an equal and opposite reaction; forces occur in pairs acting on different bodies.

牛顿第三定律：每一个作用力都有一个大小相等、方向相反的反作用力；力成对出现，作用在不同的物体上。

Friction: The friction force f is given by f ≤ μ R, where μ is the coefficient of friction and R is the normal reaction. At the point of sliding, f = μ R.

摩擦力：摩擦力 f 满足 f ≤ μ R，μ 为摩擦系数，R 为法向反力。在即将滑动时，f = μ R。

Centripetal force: For an object moving in a circle of radius r with speed v, F = m v² / r or F = m r ω², directed towards the centre.

向心力：物体在半径为 r 的圆周上以速率 v 运动时，F = m v² / r 或 F = m r ω²，方向指向圆心。

Centripetal acceleration: a = v² / r and a = r ω².

向心加速度： a = v² / r 且 a = r ω²。

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3. Work, Energy and Power | 功、能与功率

Work done by a constant force F moving an object through displacement s at an angle θ is W = F s cos θ. When force and displacement are parallel, W = F s.

恒力 F 使物体发生位移 s，力与位移夹角 θ 时，做功 W = F s cos θ。当力与位移平行时，W = F s。

PE = m g h

Gravitational potential energy (GPE): near the Earth’s surface, PE = m g h, where h is the vertical height relative to a reference level.

重力势能：地表附近 PE = m g h，h 为相对于参考面的竖直高度。

KE = ½ m v²

Kinetic energy (KE): KE = ½ m v²

动能： KE = ½ m v²

Conservation of mechanical energy: In a closed system with no non‑conservative forces (e.g. friction), total mechanical energy (KE + PE) remains constant.

机械能守恒：在无摩擦等非保守力的封闭系统中，总机械能 (KE + PE) 保持不变。

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

功率：做功的速率。平均功率 P = W / t。若恒力与速度同向且匀速，P = F v。

Efficiency: η = (useful output energy / total input energy) × 100%, or in terms of power: η = useful output power / input power.

效率： η = (有用输出能量 / 总输入能量) × 100%，或用功率表示：η = 有用输出功率 / 输入功率。

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4. Moments and Equilibrium | 力矩与平衡

Moment of a force about a pivot = force × perpendicular distance from pivot to line of action: M = F d, unit N m.

力对支点的力矩 = 力 × 支点到力作用线的垂直距离：M = F d，单位 N m。

Principle of moments: For a body in rotational equilibrium, the sum of clockwise moments about any point equals the sum of anticlockwise moments about that point: Σ M_clockwise = Σ M_anticlockwise.

力矩原理：处于转动平衡的物体，对任一点，顺时针力矩之和等于逆时针力矩之和：Σ M_顺时针 = Σ M_逆时针。

Conditions for static equilibrium:
Conditions for static equilibrium:

• Resultant force in any direction is zero: Σ F_x = 0, Σ F_y = 0.

  任一方向的合力为零：Σ F_x = 0, Σ F_y = 0。

• Resultant moment about any point is zero: Σ M = 0.

  对任一点的合力矩为零：Σ M = 0。

Torque (T) is the rotational analogue of force: T = F r for a force applied tangentially at radius r. In rotational power transmission, P = T ω, where ω is angular velocity in rad s⁻¹.

扭矩 (T) 是力的转动等效量：力沿切线方向作用于半径 r 处时，T = F r。在旋转功率传递中，P = T ω，ω 为角速度 (rad s⁻¹)。

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

Direct stress (σ): σ = F / A, where F is the applied force and A is the original cross‑sectional area. Units: N m⁻² or Pa.

正应力 (σ)： σ = F / A，F 为外加载荷，A 为原始截面积。单位 N m⁻² 或 Pa。

Direct strain (ε): ε = ΔL / L₀, where ΔL is the change in length and L₀ is the original length. Strain is dimensionless often expressed as a percentage.

正应变 (ε)： ε = ΔL / L₀，ΔL 为长度变化量，L₀ 为原始长度。应变无量纲，常以百分数表示。

E = σ / ε

Young’s modulus (E): measures the stiffness of a material within the linear elastic region. E = σ / ε, units Pa. The relationship holds up to the limit of proportionality.

杨氏模量 (E)： 衡量材料在弹性线性阶段的刚度。E = σ / ε，单位 Pa。该关系在比例极限内成立。

Factor of safety (FoS): FoS = Ultimate tensile stress / Allowable working stress. It ensures the component operates well below the failure point.

安全系数 (FoS)： FoS = 极限抗拉应力 / 许用工作应力，确保构件在远低于破坏点的条件下工作。

Strain energy (elastic): When a material obeys Hooke’s law, the elastic strain energy stored is U = ½ F ΔL or U = ½ σ ε V, where V is the volume.

应变能（弹性）：材料遵从胡克定律时，储存的弹性应变能为 U = ½ F ΔL 或 U = ½ σ ε V，V 为体积。

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6. Beam Theory and Bending Moments | 梁理论与弯矩

A beam supports transverse loads, producing internal shear forces and bending moments. The bending moment M at a section is the algebraic sum of the moments of external forces on one side of the section.

梁承受横向载荷，产生内部剪力和弯矩。某截面的弯矩 M 是该截面任一侧外力矩的代数和。

Shear force (V): internal force perpendicular to the beam axis; Bending moment (M): internal moment causing bending. Sign convention: usually sagging moment (makes beam concave upward) is positive.

剪力 (V)：垂直于梁轴的内力；弯矩 (M)：使梁弯曲的内力矩。正负约定：一般使梁下凸（上凹）的弯矩为正。

Standard maximum bending moments for simply supported beams of length L:

• Point load P at midspan: M_max = P L / 4 at the centre.

  跨中集中荷载 P：M_max = P L / 4，出现在跨中。

• Uniformly distributed load w per unit length: M_max = w L² / 8 at the centre.

  均布载荷 w (单位长度)：M_max = w L² / 8，出现在跨中。

For a cantilever beam of length L with uniformly distributed load w, the maximum bending moment at the fixed support is M_max = w L² / 2.

悬臂梁长 L、均布载荷 w，固定端最大弯矩 M_max = w L² / 2。

The flexure formula relates bending stress σ at a distance y from the neutral axis to the bending moment M and the second moment of area I: σ = M y / I.

弯曲公式将距中性轴 y 处的弯曲应力 σ 与弯矩 M 和截面惯性矩 I 联系起来：σ = M y / I。

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7. Direct Current Circuits | 直流电路

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