CAIE Year 11 Engineering: Formula & Theorem Quick Reference Handbook | CAIE 11年级工程学：公式定理速查手册

📚 CAIE Year 11 Engineering: Formula & Theorem Quick Reference Handbook | CAIE 11年级工程学：公式定理速查手册

This quick reference handbook compiles the essential formulas, principles, and theorems that underpin the CAIE Year 11 Engineering syllabus. From mechanics and electricity to materials and fluid systems, having these equations at your fingertips supports efficient problem-solving and deeper understanding. Bookmark, memorise, and apply them with confidence during your revision and examinations.

本速查手册汇集了支撑CAIE 11年级工程学教学大纲的核心公式、原理与定理。从力学、电学、材料到流体系统，掌握这些方程式能帮助您高效解题并深化理解。请在复习和考试期间标记、熟记并自信运用它们。

1. Kinematics and Dynamics | 运动学与动力学

v = u + a t

This linear motion equation links final velocity (v), initial velocity (u), constant acceleration (a), and time (t). It is used to find how fast an object is moving after a period of uniform acceleration.

该直线运动方程将末速度(v)、初速度(u)、恒定加速度(a)和时间(t)联系起来，用于求物体在一段匀加速运动后的速度。

s = u t + ½ a t²

Displacement (s) under uniform acceleration can be calculated when initial velocity, time, and acceleration are known. The term ½at² accounts for the additional distance covered due to acceleration.

当已知初速度、时间和加速度时，可用此式计算匀加速运动下的位移(s)。½at²项表示因加速度而产生的额外距离。

v² = u² + 2 a s

This equation is independent of time and relates final velocity squared, initial velocity squared, acceleration, and displacement. It is especially handy for braking distance or launch calculations.

该方程不含时间，将末速度平方、初速度平方、加速度和位移联系起来，在计算制动距离或发射问题时特别方便。

F = m a

Newton’s second law states that the net force (F) acting on a body equals its mass (m) multiplied by its acceleration (a). Force is measured in newtons (N).

牛顿第二定律指出，作用在物体上的净力(F)等于其质量(m)乘以其加速度(a)。力以牛顿(N)为单位。

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2. Moments and Equilibrium | 力矩与平衡

M = F d

A moment (M) is the turning effect of a force about a pivot. It is calculated as the product of the force and the perpendicular distance (d) from the pivot to the line of action of the force. Clockwise and anticlockwise moments must balance for a body in rotational equilibrium.

力矩(M)是力绕支点的转动效应，等于力与支点到力作用线垂直距离(d)的乘积。物体处于转动平衡时，顺时针力矩之和等于逆时针力矩之和。

Σ M꜀ₗₒ꜀ₖꜤᵢₛₑ = Σ Mₐₙₜᵢ꜀ₗₒ꜀ₖꜤᵢₛₑ

For a lever, beam, or any rigid body at equilibrium, the sum of all clockwise moments about any pivot equals the sum of all anticlockwise moments. This principle allows unknown forces or distances to be calculated.

对于杠杆、横梁或任何处于平衡状态的刚体，绕任一支点的顺时针力矩总和等于逆时针力矩总和。利用这一原理可求未知力或距离。

Σ F = 0, Σ M = 0

Translational equilibrium requires the net force in any direction to be zero, while rotational equilibrium requires the net moment about any point to be zero. Both conditions must hold for static structures.

平移平衡要求任意方向的合力为零，转动平衡要求绕任一点的合力矩为零。对于静态结构，这两个条件必须同时满足。

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3. Work, Energy and Power | 功、能与功率

W = F s

Work done (W) when a constant force (F) moves an object through a displacement (s) in the direction of the force. The SI unit is the joule (J).

当恒力(F)使物体沿力的方向发生位移(s)时所做的功(W)。国际单位是焦耳(J)。

KE = ½ m v²

Kinetic energy (KE) is the energy possessed by a body due to its motion. It depends on mass (m) and the square of its speed (v).

动能(KE)是物体因运动而具有的能量，取决于质量(m)和速度平方(v²)。

PE = m g h

Gravitational potential energy (PE) gained by an object of mass m lifted through a vertical height h in a gravitational field of strength g (9.8 m/s² on Earth).

重力势能(PE)是质量为m的物体在重力场强度g(地球上为9.8 m/s²)中被垂直举升高度h所获得的能量。

P = W / t and efficiency η = (useful output / total input) × 100%

Power (P) is the rate of doing work, measured in watts (W). Efficiency compares useful energy or power output to total input; no real system is 100% efficient due to losses like friction and heat.

功率(P)是做功的速率，单位为瓦特(W)。效率比较有用能量或功率输出与总输入的比值；由于摩擦和发热等损耗，真实系统的效率无法达到100%。

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4. Fundamentals of Electricity | 电学基础

V = I R

Ohm’s law states that the potential difference (V) across a conductor is directly proportional to the current (I) flowing through it, with resistance (R) as the constant. This holds for ohmic materials at constant temperature.

欧姆定律指出，导体两端的电势差(V)与流过它的电流(I)成正比，电阻(R)是比例常数。该定律在恒温条件下适用于欧姆材料。

P = I V = I² R = V² / R

Electrical power (P) can be expressed in three equivalent forms using voltage, current, and resistance. Use the form that best fits the known quantities in a circuit.

电功率(P)可用电压、电流和电阻的三种等价形式表示。请根据电路中已知量选择最合适的形式。

Series: Rₜₒₜₐₗ = R₁ + R₂ + … ; Parallel: 1/Rₜₒₜₐₗ = 1/R₁ + 1/R₂ + …

Resistors in series add directly to give a larger total resistance. For parallel resistors, the reciprocal of the total resistance equals the sum of the reciprocals; this always yields a smaller equivalent resistance.

串联电阻直接相加得到更大的总电阻。对于并联电阻，总电阻的倒数等于各电阻倒数之和，其等效电阻总是更小。

R = ρ L / A

The resistance (R) of a uniform wire depends on its resistivity (ρ), length (L), and cross‑sectional area (A). Resistivity is a material property measured in ohm‑metres (Ω·m).

均匀导线的电阻(R)取决于其电阻率(ρ)、长度(L)和横截面积(A)。电阻率是材料的固有属性，单位为欧姆·米(Ω·m)。

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5. Mechanics of Materials | 材料力学

Stress σ = F / A

Tensile or compressive stress (σ) is the force applied per unit cross‑sectional area. It is measured in pascals (Pa) or N/m². Understanding stress helps engineers ensure components do not deform or fail.

拉伸或压缩应力(σ)是施加在单位横截面积上的力，以帕斯卡(Pa)或牛/平方米为单位。理解应力有助于工程师确保构件不会变形或失效。

Strain ε = ΔL / L₀

Strain (ε) is the ratio of extension (or contraction) to the original length. It has no units. Strain quantifies how much a material deforms relative to its original size.

应变(ε)是伸长量(或缩短量)与原长之比，无量纲。应变量化了材料相对于原始尺寸的变形程度。

Young’s modulus E = σ / ε

The modulus of elasticity (E) measures stiffness; it is the ratio of stress to strain in the linear elastic region. A high Young’s modulus indicates a stiff material that resists deformation.

弹性模量(E)衡量刚度，是线弹性区域内应力与应变之比。高杨氏模量表明材料刚度大，不易变形。

Factor of Safety = Ultimate stress / Working stress

Engineers apply a factor of safety to account for uncertainties in load, material defects, and environmental conditions. It is the ratio of the material’s ultimate (breaking) stress to the maximum allowable working stress.

工程师采用安全系数来应对载荷不确定性、材料缺陷及环境条件，它是材料极限(断裂)应力与最大许用工作应力之比。

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6. Fluid Systems | 流体系统

Continuity equation: Q = A v = constant

For an incompressible fluid in a closed pipe, the volume flow rate (Q) remains constant. The product of cross‑sectional area (A) and flow velocity (v) is the same at all points along the pipe.

对于封闭管道中的不可压缩流体，体积流量(Q)恒定不变。横截面积(A)与流速(v)的乘积在管道各点均相同。

Bernoulli’s principle: P + ½ ρ v² + ρ g h = constant

Along a streamline, the sum of static pressure (P), dynamic pressure (½ρv²), and potential energy per unit volume (ρgh) remains constant. This explains lift on an aerofoil and venturi effects.

沿同一流线，静压(P)、动压(½ρv²)与单位体积势能(ρgh)之和保持恒定。这解释了机翼升力和文丘里效应。

Pressure in a liquid: p = ρ g h

The pressure at a depth h in a fluid of density ρ is given by ρgh, where g is the gravitational field strength. This does not include atmospheric pressure acting on the surface.

密度为ρ的流体中深度为h处的压强为ρgh，其中g为重力场强度。该值未包含作用于液面的大气压强。

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7. Mechanical Drives and Levers | 机械传动与杠杆

Gear ratio = Number of teeth on driven gear / Number of teeth on driver gear

The gear ratio determines the relationship between input and output speeds. A ratio greater than 1 reduces speed and increases torque; a ratio less than 1 increases speed and reduces torque.

齿轮比决定输入与输出转速的关系。比值大于1时减速增扭，比值小于1时增速减扭。

Velocity ratio = Distance moved by effort / Distance moved by load

For simple machines (levers, pulleys, gear systems), the velocity ratio (VR) compares the movement of the effort to that of the load. It is a geometric property unaffected by friction.

对于简单机械(杠杆、滑轮、齿轮系统)，速度比(VR)用于比较施力运动与负载运动的关系，是一个不受摩擦影响的几何特性。

Mechanical advantage MA = Load / Effort; Efficiency = MA / VR × 100%

Mechanical advantage tells you how much a machine multiplies the input force. Efficiency relates mechanical advantage to velocity ratio, reflecting energy losses in the system.

机械效益表示机器将输入力放大的倍数。效率将机械效益与速度比关联起来，反映系统中的能量损失。

Lever principle: Effort × Effort arm = Load × Load arm

For a lever in equilibrium, the product of the effort force and its perpendicular distance from the fulcrum equals the product of the load force and its distance from the fulcrum.

杠杆平衡时，动力与动力臂的乘积等于阻力与阻力臂的乘积。

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8. Thermodynamics and Energy Transfer | 热力学与能量传递

Thermal efficiency ηₜₕ = (Useful work output / Heat energy input) × 100%

In heat engines, thermal efficiency quantifies how well input heat is converted into useful mechanical work. The Second Law of Thermodynamics limits this efficiency; some energy is always rejected to a cold sink.

在热机中，热效率量化了输入热量转化为有用机械功的程度。热力学第二定律限制了这一效率，总有部分能量被排放至冷源。

Heat transfer rate: Q = m c Δθ

The amount of heat energy (Q) required to change the temperature of a mass m of a substance by Δθ depends on its specific heat capacity (c). This is fundamental for cooling system design.

使质量为m的物质温度变化Δθ所需的热量(Q)取决于其比热容(c)。这是冷却系统设计的基础。

Conduction: P = k A (ΔT / d)

The rate of heat transfer by conduction through a material is proportional to its thermal conductivity (k), the cross‑sectional area (A), and the temperature gradient (ΔT/d).

通过材料传导的热传递速率与其热导率(k)、横截面积(A)及温度梯度(ΔT/d)成正比。

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9. Systems, Measurement and Tolerances | 系统、测量与公差

Gain = Output / Input

In control systems, gain describes how much a signal is amplified. For a mechanical amplifier or electronic circuit, it is the ratio of output magnitude to input magnitude.

在控制系统中，增益描述信号的放大程度。对于机械放大器或电子电路，它是输出幅值与输入幅值之比。

Uncertainty = ± half the smallest division (for analogue instruments)

When using a ruler or analogue meter, the reading uncertainty is typically taken as ± half the scale division. This must be recorded alongside measurements to indicate precision.

使用直尺或模拟仪表时，读数不确定度通常取最小刻度的一半。这必须与测量值一同记录以表明精度。

Tolerance = Upper limit – Lower limit

In manufacturing, tolerance specifies the allowable variation in a dimension. It is the difference between the maximum and minimum permissible sizes, ensuring parts fit and function correctly.

在制造中，公差规定了尺寸允许的变动范围，即最大与最小许可尺寸之差，以确保零件正确配合与运行。

Output = G × (Set point – Feedback)

In a closed‑loop control system, the error signal (difference between desired set point and actual feedback) is multiplied by the system gain (G) to drive the actuator. This negative feedback loop stabilises the output.

在闭环控制系统中，误差信号(期望设定点与实际反馈之差)乘以系统增益(G)以驱动执行机构。该负反馈回路使输出趋于稳定。

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