Year 10 AQA Engineering: Formula & Theorem Quick Reference Guide | Year 10 AQA 工程：公式定理速查手册

📚 Year 10 AQA Engineering: Formula & Theorem Quick Reference Guide | Year 10 AQA 工程：公式定理速查手册

This quick reference handbook compiles the essential formulas and theorems that Year 10 AQA Engineering students are expected to recall and apply. Covering forces and motion, moments, material properties, energy, electrical principles, fluids and mechanical systems, it provides a concise but complete review of the equations used in problem-solving, practical work and design tasks. Keep it beside you during revision, homework and coursework.

这本速查手册为学习 Year 10 AQA 工程的学生整理了必须能够回忆并运用的核心公式与定理。内容涵盖力与运动、力矩、材料性能、能量、电学原理、流体与机械系统，以简明而完整的方式呈现你在解题、实验和设计任务中用到的关键方程。复习、作业和课程作业时都可随时查阅。

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1. Forces and Linear Motion | 力与直线运动

F = m × a　　W = m × g　　v = s / t　　a = (v − u) / t

Newton' sultant force in newtons (N), m is mass in kilograms (kg) and a is acceleration in metres per second squared (m/s²). The object always accelerates in the direction of the resultant force, so the equation only works when F is the vector sum of every force acting on the body.

牛顿第二定律将力、质量和加速度联系起来：F = m × a，其中 F 是合力，单位为牛顿（N），m 是质量，单位为千克（kg），a 是加速度，单位为米/秒²（m/s²）。物体总是沿合力方向加速，因此只有当 F 是作用在物体上所有力的矢量和时，该式才成立。

to 10 m/s² for quick estimation. Weight, like any force, is measured in newtons, never in kilograms.

物体的重量是作用在其质量上的重力：W = m × g。在地球上 g ≈ 9.8 m/s²，快速估算时通常取 10 m/s²。重量和任何力一样，单位是牛顿，绝不是千克。

teadily, use acceleration a = (v − u) / t, where u is the initial velocity and v is the final velocity.

对于匀速运动，平均速度等于距离除以时间：v = s / t，其中 s 的单位是米（m），t 的单位是秒（s），v 的单位是米/秒（m/s）。若速度均匀变化，则使用加速度 a = (v − u) / t，其中 u 是初速度，v 是末速度。

ocity-time graph the gradient gives acceleration and the area under the line gives distance travelled.

匀加速运动的完整方程组在课程作业的测试任务中很有用：v = u + a × t，s = u × t + ½ × a × t²，以及 v² = u² + 2 × a × s。在速度—时间图像上，斜率代表加速度，图线下的面积代表行驶距离。

Always state the direction of a force or acceleration in your answer, because AQA mark schemes award a mark for recognising that force and acceleration are vector quantities.

作答时务必写明力或加速度的方向，因为 AQA 评分标准会为"识别出力与加速度是矢量"这一点给分。

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2. Moments and Equilibrium | 力矩与平衡

M = F × d　　Σ M (clockwise) = Σ M (anticlockwise)

nd d is the perpendicular distance from the pivot to the line of action of the force, in metres. Moment is measured in newton-metres (Nm), not in joules.

力对支点的力矩为 M = F × d，其中 F 是力（牛顿），d 是从支点到力作用线的垂直距离（米）。力矩的单位是牛顿·米（Nm），而不是焦耳。

evers, beams, cranes and wheelbarrows all follow this principle, and the distance used must always be perpendicular to the force.

对于处于转动平衡的系统，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。杠杆、横梁、起重机和小推车都遵循这一原理，所用的距离必须始终垂直于力的方向。

d, where d is the perpendicular distance between the two forces rather than from a pivot.

力偶是一对大小相等、方向相反且平行的力，只产生转动而不产生合力。其转动效果为 M = F × d，其中 d 是两个力之间的垂直距离，而不是到支点的距离。

ut any point is zero (Σ M = 0). If either condition fails, the object will accelerate or start to rotate.

完整的平衡需要同时满足两个条件：各方向上所有力的和为零（Σ F = 0），以及绕任意点的所有力矩之和为零（Σ M = 0）。若任一条件不满足，物体就会加速或开始转动。

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3. Stress, Strain and Material Properties | 应力、应变与材料性能

σ = F / A　　ε = ΔL / L₀　　E = σ / ε　　ρ = m / V

quare metres (m²). The result is in N/mm² or pascals (Pa), where 1 N/mm² = 1 MPa = 1 × 10⁶ Pa.

正应力定义为 σ = F / A，其中 F 是施加的力（牛顿），A 是原始横截面积，单位为平方毫米（mm²）或平方米（m²）。结果单位为 N/mm² 或帕斯卡（Pa），其中 1 N/mm² = 1 MPa = 1 × 10⁶ Pa。

ion and L₀ is the original length. Strain has no units because it is a ratio of two lengths, and it is often expressed as a percentage.

应变是长度的相对变化量：ε = ΔL / L₀，其中 ΔL 是伸长量，L₀ 是原始长度。应变没有单位，因为它是两个长度之比，通常也用百分比表示。

Young' , with units of N/mm² or GPa. A larger value of E means the material stretches less under the same stress, which is why steel (about 200 GPa) is far stiffer than aluminium (about 70 GPa).

杨氏模量衡量材料的刚度：E = σ / ε，单位为 N/mm² 或 GPa。E 值越大，表示在相同应力下材料伸长越少，这也是钢（约 200 GPa）比铝（约 70 GPa）刚度大得多的原因。

V, measured in kg/m³. Density allows you to compare materials for a design, for example choosing a low-density alloy to reduce the mass of a component without losing too much strength.

密度是单位体积的质量：ρ = m / V，单位为 kg/m³。密度可用于比较设计中的材料，例如选择低密度合金来减轻零件质量，同时不过多损失强度。

Hooke' fness in N/mm and x is the extension in mm. The factor of safety of a component is the failure stress divided by the working stress, and a value below 2 is rarely acceptable in a student design.

胡克定律在比例极限以内成立：F = k × x，其中 k 是刚度（N/mm），x 是伸长量（mm）。零件的安全系数等于破坏应力除以工作应力，在学生设计中低于 2 的取值通常不可接受。

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4. Energy, Work and Power | 能量、功与功率

W = F × s　　KE = ½ × m × v²　　GPE = m × g × h　　P = E / t

distance moved in metres. Work and energy are both measured in joules (J), and 1 J = 1 Nm.

当力使物体沿其作用线移动时，就说做了功：W = F × s，其中 s 是移动距离（米）。功和能量的单位都是焦耳（J），且 1 J = 1 Nm。

E = m × g × h. In an ideal system with no friction, the loss of GPE equals the gain in KE.

动能取决于质量和速度的平方：KE = ½ × m × v²。重力势能取决于高度：GPE = m × g × h。在没有摩擦的理想系统中，重力势能的减少量等于动能的增加量。

ed in watts (W), where 1 W = 1 J/s. For a moving object, an alternative form is P = F × v.

功率是做功或传递能量的速率：P = E / t 或 P = W / t，单位为瓦特（W），其中 1 W = 1 J/s。对于运动的物体，还有另一种形式：P = F × v。

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The law of conservation of energy states that energy cannot be created or destroyed, only transferred from one store to another. Energy can be wasted by friction, air resistance and sound, which is why the useful energy output of a real machine is always less than its total energy input.

能量守恒定律指出，能量既不能创生也不能消灭，只能从一种形式转移为另一种形式。摩擦、空气阻力和声音都会造成能量损耗，因此真实机械的有用能量输出总是小于总能量输入。

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5. Electrical Principles: Ohm's Law and Power | 电学原理：欧姆定律与功率

V = I × R　　P = V × I　　P = I² × R　　P = V² / R　　Q = I × t

Ohm' ifference in volts (V), I is current in amperes (A) and R is resistance in ohms (Ω). It holds for a component only if its resistance stays constant as the current changes.

欧姆定律把电压、电流和电阻联系起来：V = I × R，其中 V 是电势差（伏特），I 是电流（安培），R 是电阻（欧姆，Ω）。只有当元件的电阻随电流变化保持恒定时，该式才成立。

ches the quantities given in the question, which saves time and reduces arithmetic errors.

电功率有三种等价写法：P = V × I、P = I² × R 和 P = V² / R。选择与题目所给量相匹配的形式，可以节省时间并减少计算错误。

asured in joules, and this is why electricity bills are based on energy in kilowatt-hours.

电荷量等于电流与时间的乘积：Q = I × t，单位为库仑（C）。电学中传递的能量为 E = V × I × t 或 E = V × Q，单位为焦耳，这也是电费以千瓦时为单位计算能量的原因。

rrying 0.05 A dissipates P = I² × R = 0.05² × 240 = 0.6 W, so a 1 W resistor would be suitable.

电阻器把电能转化为热能，其额定功率必须大于实际使用中消耗的功率。例如，一个 240 Ω 的电阻通过 0.05 A 电流时消耗的功率为 P = I² × R = 0.05² × 240 = 0.6 W，因此选用 1 W 的电阻是合适的。

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6. Series and Parallel Circuits | 串联与并联电路

Series: R_T = R₁ + R₂ + R₃　　Parallel: 1 / R_T = 1 / R₁ + 1 / R₂ + 1 / R₃

tween the components, and the total resistance is the sum of the individual resistances: R_T = R₁ + R₂ + R₃.

在串联电路中，通过每个元件的电流相同，电源电压在各元件之间分配，总电阻等于各分电阻之和：R_T = R₁ + R₂ + R₃。

oss every branch is the same, the total current is the sum of the branch currents, and the reciprocal rule applies: 1 / R_T + 1 / R₃. The total resistance is always smaller than the smallest individual resistance.

在并联电路中，每条支路两端的电压相同，总电流等于各支路电流之和，并适用倒数法则：1 / R_T = 1 / R₁ + 1 / R₂ + 1 / R₃。总电阻总是小于其中最小的分电阻。

For two resistors in parallel only, a quick shortcut is the product-over-sum rule: R_T = (R₁ × R₂) / (R₁ + R₂). For example, a 120 Ω and a 60 Ω resistor in parallel give R_T = (120 × 60) / (120 + 60) = 40 Ω.

只对两个并联电阻，有一个快捷方法，即"积除以和"法则：R_T = (R₁ × R₂) / (R₁ + R₂)。例如 120 Ω 与 60 Ω 并联时，R_T = (120 × 60) / (120 + 60) = 40 Ω。

to a single resistance, add the series resistances, find the total current with I = V / R_T, then work back through the circuit to find individual voltages and currents.

混合电路题目最好分步求解：先把每组并联电阻化简为一个电阻，再与串联电阻相加，用 I = V / R_T 求出总电流，然后逐步回溯电路，求出各元件的电压和电流。

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7. Mechanical Systems: Gears, Pulleys and Ratios | 机械系统：齿轮、滑轮与传动比

Torque T = F × r　　Gear ratio = N_driven / N_driver　　VR = d_effort / d_load

is the radius in metres. A motor with a small torque can lift a heavy load if the drive system increases the torque through gearing.

力在某个半径处产生的转矩（转动力矩）为 T = F × r，其中 r 是半径（米）。如果传动系统通过齿轮机构放大了转矩，那么转矩较小的电动机也可以提起重物。

eeth on the driven gear divided by the number of teeth on the driving gear: Gear ratio = N_driven / N_driver ecreases and the output torque increases in the same proportion, ignoring friction losses.

传动比等于从动齿轮齿数除以主动齿轮齿数：传动比 = N_从动 / N_主动。若传动比大于 1，则输出转速降低，输出转矩按同一比例增大（忽略摩擦损失）。

hine is the distance moved by the effort divided by the distance moved by the load: VR = d_effort / d_load iprocal of the gear ratio when the system is driven from the small gear to the large gear.

任何机械的速比等于施力点移动距离除以负载移动距离：VR = d_施力 / d_负载。对于齿轮系统，当动力由小齿轮传向大齿轮时，速比等于传动比的倒数。

, and the block and tackle with a velocity ratio equal to the number of rope sections supporting the load.

常见的滑轮装置包括：定滑轮，速比 VR = 1，只改变施力方向；单个动滑轮，VR = 2；以及滑轮组，其速比等于支撑负载的绳子段数。

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8. Pressure, Density and Fluids | 压强、密度与流体

P = F / A　　P = ρ × g × h　　F₁ / A₁ = F₂ / A₂

large pressure, which is why a sharp blade cuts easily and a wide foundation spreads the load of a building.

压强是单位面积上的力：P = F / A，单位为帕斯卡（Pa），其中 1 Pa = 1 N/m²。接触面积越小，压强越大，这就是锋利的刀片容易切割、而宽大的地基能分散建筑物载荷的原因。

he fluid in kg/m³ and h is the depth in metres. The pressure depends only on depth and density, not on the shape or width of the container.

液柱产生的压强为 P = ρ × g × h，其中 ρ 是流体的密度（kg/m³），h 是深度（米）。压强只与深度和密度有关，与容器的形状或宽度无关。

all piston therefore produces a large force on a large piston, which is the principle behind the hydraulic jack and the car braking system.

液压系统通过不可压缩的流体传递力，由于各处压强相同，因此 F₁ / A₁ = F₂ / A₂。作用在小活塞上的小力可在大活塞上产生大力，这正是液压千斤顶和汽车制动系统的工作原理。

0 000 Pa, and the density of water is 1000 kg/m³, giving roughly 9.8 kPa of pressure for every metre of depth.

常用参考值：大气压约为 101 kPa，1 bar = 100 000 Pa，水的密度为 1000 kg/m³，因此每增加 1 米深度，压强约增加 9.8 kPa。

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9. Thermal Principles and Energy Transfer | 热学原理与能量传递

Q = m × c × ΔT　　Q = m × L　　ΔL = α × L₀ × ΔT

ere m is mass in kilograms, c is the specific heat capacity in J/(kg°C) and ΔT is the temperature change. Water has a high value of c, about 4200 J/(

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