📚 Pre-U CAIE Engineering: Quick Reference Formula & Theorem Handbook | Pre-U CAIE 工程：公式定理速查手册
This concise handbook collates the core formulas and theorems required for the Pre-U CAIE Engineering syllabus. Each entry is presented with its symbolic notation, a brief contextual explanation, and the relevant SI units. Use this as a daily revision aid or a pre‑exam refresher to strengthen your problem‑solving confidence.
这份简明手册汇集了 Pre-U CAIE 工程课程的核心公式与定理。每一条目都包含符号表示、简要的上下文说明以及相关国际单位。可将它用作日常复习帮手或考前快速回顾，以增强解题信心。
1. Resolving Forces | 力的分解
Any force F acting at an angle θ to a reference axis can be resolved into perpendicular components. The horizontal component is F cosθ and the vertical component is F sinθ. This principle is essential for analysing structures, inclined planes, and equilibrium problems.
任一与参考轴成θ角的力 F 可分解为相互垂直的分力。水平分量为 F cosθ，垂直分量为 F sinθ。此原理对分析结构、斜面和平衡问题至关重要。
Fₓ = F cosθ
Fᵧ = F sinθ
- θ is measured from the positive x‑axis.
- θ 从正 x 轴量起。
2. Conditions for Static Equilibrium | 静力平衡条件
A body remains in static equilibrium when the resultant force in any direction is zero and the resultant moment about any point is zero. For a coplanar force system, these conditions yield three independent equations.
当任意方向上的合力为零且对任一点的合力矩为零时，物体保持静力平衡。对于共面力系，这些条件给出三个独立方程。
ΣFₓ = 0, ΣFᵧ = 0, ΣM = 0
Moments should be taken about a point that eliminates unknown forces where possible.
力矩应尽可能绕可消去未知力的点求取。
3. Stress and Strain | 应力与应变
Direct stress σ is the internal force per unit cross‑sectional area. Direct strain ε is the extension per unit original length. These definitions underpin all tensile, compressive, and shear load calculations.
正应力 σ 是单位横截面积上的内力。正应变 ε 是单位原始长度的伸长量。这些定义为所有拉伸、压缩和剪切载荷计算奠定基础。
σ = F / A (Pa)
ε = ΔL / L₀ (dimensionless)
| Tensile stress | σ = F/A |
| Shear stress | τ = F/A |
| Shear strain | γ = x / L |
4. Young’s Modulus and Hooke’s Law | 杨氏模量与胡克定律
Within the elastic limit, stress is directly proportional to strain. The constant of proportionality is Young’s modulus E. This linear relationship is Hooke’s Law and it governs the spring‑like behaviour of engineering materials.
在弹性极限内，应力与应变成正比。比例常数即杨氏模量 E。这一线性关系即胡克定律，支配着工程材料类弹簧的行为。
σ = E ε
F = k x (for springs)
Strain energy stored in an elastic body: U = ½ F x = ½ k x²
弹性体中储存的应变能：U = ½ F x = ½ k x²
5. Moments of Inertia | 惯性矩（截面二次矩）
The second moment of area I quantifies a cross‑section’s resistance to bending. For common shapes, standard formulas apply. The parallel axis theorem allows calculation about any axis parallel to the centroidal axis.
截面二次矩 I 量化横截面抵抗弯曲的能力。对于常见形状，有标准公式适用。平行轴定理允许对平行于形心轴的任意轴进行计算。
I = Σ (b h³ / 12) (rectangle about centroid)
I = I_c + A d² (parallel axis)
- Rectangle: I = b h³ / 12 about centroidal axis parallel to base.
- 矩形：绕平行于底边的形心轴 I = b h³ / 12。
- Circle: I = π d⁴ / 64.
- 圆形：I = π d⁴ / 64。
6. Bending Stress Formula | 弯曲应力公式
The bending stress at any point in a beam is given by the flexure formula. It relates bending moment M, Young’s modulus E, radius of curvature R, and distance from the neutral axis y.
梁中任一点的弯曲应力由弯曲公式给出。它将弯矩 M、杨氏模量 E、曲率半径 R 以及与中性轴的距离 y 联系起来。
σ / y = M / I = E / R
Maximum bending stress occurs at the extreme fibre, where y = y_max.
最大弯曲应力发生在最外层纤维处，即 y = y_max。
7. Torsion Formula | 扭转公式
For a circular shaft under pure torsion, shear stress varies linearly from zero at the centre to a maximum at the outer surface. The torsion equation links torque T, polar second moment of area J, shear stress τ, and angle of twist.
对于纯扭转下的圆轴，剪应力从中心为零线性变化至外表面处的最大值。扭转方程将扭矩 T、极惯性矩 J、剪应力 τ 和扭转角联系起来。
τ / r = T / J = G θ / L
Where G is the shear modulus, θ the angle of twist in radians, L the shaft length. Polar J for a solid circle: J = π d⁴ / 32.
其中 G 为剪切模量，θ 为扭转角（弧度），L 为轴长。实心圆的极惯性矩 J = π d⁴ / 32。
8. Linear Motion Equations | 直线运动方程
For uniform acceleration, the SUVAT equations relate displacement s, initial velocity u, final velocity v, acceleration a, and time t. They are fundamental to kinematics problems in engineering dynamics.
对于匀加速运动，SUVAT 方程关联位移 s、初速度 u、末速度 v、加速度 a 和时间 t。它们是工程动力学中运动学问题的基础。
v = u + a t
s = u t + ½ a t²
v² = u² + 2 a s
s = (u + v) t / 2
These apply where acceleration is constant. For free fall near Earth’s surface, a = g = 9.81 m s⁻².
这些公式适用于加速度恒定的情形。近地表面自由落体时，a = g = 9.81 m s⁻²。
9. Newton’s Laws and Impulse | 牛顿定律与冲量
Newton’s second law states that net force equals rate of change of momentum. Impulse is the product of force and the time for which it acts, equalling the change in momentum.
牛顿第二定律指出，合力等于动量变化率。冲量是力与其作用时间的乘积，等于动量的变化。
F = m a = Δp / Δt
Impulse = F Δt = Δp = m v − m u
For a system of particles with no external forces, total momentum is conserved: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂.
对于无外力作用的质点系，总动量守恒：m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂。
10. Work, Energy and Power | 功、能与功率
Work done by a constant force is the product of the force and the displacement in the direction of the force. Kinetic energy and gravitational potential energy are scalar quantities that simplify many analyses.
恒力所做的功是力与沿力方向位移的乘积。动能和重力势能是标量，可简化许多分析。
W = F s cosθ
KE = ½ m v²
GPE = m g h
Power is the rate of doing work: P = W / t = F v (for constant force and velocity in the same direction).
功率是做功的速率：P = W / t = F v（适用于恒力且速度与力同向）。
11. Electric Circuits and Ohm’s Law | 电路与欧姆定律
Ohm’s law relates potential difference V, current I, and resistance R for ohmic conductors. Power dissipated in a resistor can be expressed in three useful forms. Kirchhoff’s laws govern current and voltage in networks.
欧姆定律关联欧姆导体两端的电势差 V、电流 I 和电阻 R。电阻器消耗的功率可用三种实用形式表示。基尔霍夫定律支配网络中的电流和电压。
V = I R
P = I V = I² R = V² / R
Resistors in series: R_total = R₁ + R₂ + …
Resistors in parallel: 1/R_total = 1/R₁ + 1/R₂ + …
串联电阻：R_total = R₁ + R₂ + …
并联电阻：1/R_total = 1/R₁ + 1/R₂ + …
12. Thermodynamic Laws and Processes | 热力学定律与过程
The First Law of Thermodynamics is a statement of energy conservation. For a closed system, the change in internal energy equals heat added minus work done by the system. Ideal gas processes (isothermal, adiabatic, isobaric, isochoric) follow specific relationships.
热力学第一定律是能量守恒的表述。对于封闭系统，内能变化等于加入的热量减去系统对外做的功。理想气体过程（等温、绝热、等压、等容）遵循特定关系。
ΔU = Q − W
p V = n R T (ideal gas law)
Adiabatic process for ideal gas: p V^γ = constant, T V^(γ−1) = constant, where γ = c_p / c_v.
理想气体绝热过程：p V^γ = 常数，T V^(γ−1) = 常数，其中 γ = c_p / c_v。
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