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

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

Engineering at Year 13 CAIE level demands rapid and accurate recall of key formulas and theorems across mechanics, materials, thermodynamics, fluid dynamics, and electrical principles. This quick reference handbook distils the essential equations you will need for problem solving, clearly linking each concept to its physical meaning.

Year 13 CAIE 工程学要求快速准确地回忆力学、材料、热力学、流体动力学和电学等领域的核心公式与定理。这份速查手册浓缩了解题必备的关键方程，并将每个概念与其物理含义清晰对应。

1. Kinematics Equations | 运动学方程

For motion with constant acceleration, the four SUVAT equations link initial velocity u, final velocity v, acceleration a, displacement s, and time t. They are fundamental to solving linear motion problems when three of the five variables are known.

对于匀加速运动，四个 SUVAT 方程将初速度 u、末速度 v、加速度 a、位移 s 和时间 t 联系起来。当已知五个变量中的三个时，它们是求解直线运动问题的基础。

v = u + at

s = ut + ½at²

s = ½(u + v)t

v² = u² + 2as

In free fall under gravity, a is replaced by g = 9.81 m s⁻² and s becomes vertical displacement h. For projectile motion, horizontal velocity is constant (vₓ = u cosθ) while vertical motion obeys the SUVAT equations with a = -g.

在重力作用下的自由落体中，a 用 g = 9.81 m s⁻² 代替，s 变为竖直位移 h。对于抛体运动，水平速度保持不变 (vₓ = u cosθ)，而竖直运动遵循 a = -g 的 SUVAT 方程。

vᵧ = u sinθ – gt,   sᵧ = u t sinθ – ½gt²

sₓ = u t cosθ

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2. Newton’s Laws & Force Analysis | 牛顿定律与受力分析

Newton’s three laws govern the relationship between force and motion. The second law is the cornerstone of dynamics: net force equals mass times acceleration.

牛顿三大定律支配着力与运动的关系。第二定律是动力学的基石：合力等于质量乘以加速度。

F = m a

Weight is a gravitational force, and friction depends on the normal reaction R and the coefficient of friction μ. Static friction satisfies f ≤ μR; kinetic friction is fₖ = μₖR. The moment of a force about a point is force times perpendicular distance.

重力是一种引力，摩擦力取决于法向反作用力 R 和摩擦系数 μ。静摩擦力满足 f ≤ μR；动摩擦力为 fₖ = μₖR。力对点的力矩等于力乘以垂直距离。

W = m g

τ = F d

Conditions for equilibrium in a static system: resultant force in any direction is zero, and resultant moment about any point is zero.

静力系统的平衡条件：任何方向的合力为零，且对任意点的合力矩为零。

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3. Momentum, Impulse & Collisions | 动量、冲量与碰撞

Linear momentum p is the product of mass and velocity. Impulse is the change in momentum caused by a force acting over time.

线动量 p 是质量与速度的乘积。冲量是力在一段时间内作用引起的动量变化。

p = m v

Impulse = F Δt = Δp

The principle of conservation of momentum states that for a closed system with no external forces, total momentum before collision equals total momentum after collision. For two bodies:

动量守恒定律指出，在无外力的封闭系统中，碰撞前的总动量等于碰撞后的总动量。对于两个物体：

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

The coefficient of restitution e, ranging from 0 (perfectly inelastic) to 1 (perfectly elastic), describes the relative speed after and before impact.

恢复系数 e 的取值范围为 0（完全非弹性）到 1（完全弹性），描述碰撞前后相对速度的比值。

e = (v₂ – v₁) / (u₁ – u₂)

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4. Work, Energy & Power | 功、能量和功率

Work is done when a force causes displacement. Kinetic energy and gravitational potential energy are the two primary mechanical energy stores.

当力产生位移时做功。动能和重力势能是两种主要的机械能形式。

W = F s cosθ

KE = ½ m v²,   GPE = m g h

The work–energy principle states that the net work done on an object equals its change in kinetic energy. Power is the rate of doing work or transferring energy.

功-能定理指出，对物体所做的净功等于其动能的变化量。功率是做功或传递能量的速率。

W_net = ΔKE

P = W / t = F v

Efficiency η compares useful output to total input, always less than 1 in real systems.

效率 η 比较有用输出与总输入，在实际系统中始终小于 1。

η = Useful work output / Total energy input

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5. Circular Motion | 圆周运动

For uniform circular motion, angular velocity ω relates angle swept to time, and linear speed v is tied to radius r. Centripetal acceleration points towards the centre and keeps the object in its circular path.

对于匀速圆周运动，角速度 ω 与转过的角度和时间相关，线速度 v 与半径 r 有关。向心加速度指向圆心，使物体保持在圆周路径上。

ω = θ / t,   v = r ω

a = v² / r = r ω²

The centripetal force required is provided by tension, friction, gravity, or a normal reaction depending on the context. Period T is the time for one complete revolution.

所需的向心力根据情境由拉力、摩擦力、重力或法向反作用力提供。周期 T 是完成一整圈的时间。

F = m v² / r = m r ω²

T = 2π / ω

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6. Stress, Strain & Young’s Modulus | 应力、应变和杨氏模量

Stress σ measures the internal force per unit area, while strain ε is the fractional change in length. Hooke’s law applies within the elastic limit, linking stress and strain through Young’s modulus E.

应力 σ 衡量单位面积上的内力，而应变 ε 是长度的相对变化。在弹性限度内，胡克定律通过杨氏模量 E 将应力与应变联系起来。

σ = F / A,   ε = ΔL / L₀

E = σ / ε

Poisson’s ratio ν describes the negative ratio of transverse to axial strain. In beam bending, the simple bending equation relates bending moment M, second moment of area I, stress at a distance y from the neutral axis, and the radius of curvature R.

泊松比 ν 描述横向应变与轴向应变的负比值。在梁弯曲中，简单弯曲方程将弯矩 M、截面二次矩 I、距中性轴距离 y 处的应力以及曲率半径 R 关联起来。

ν = – εₗ / εₐ

M / I = σ / y = E / R

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7. Thermodynamics | 热力学

Heat energy Q required to change temperature or state is given by specific heat capacity c and latent heat L. The first law of thermodynamics relates internal energy change, heat added, and work done on the system.

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