📚 Year 8 SQA Engineering: Quick Reference Handbook of Formulas and Theorems | Year 8 SQA 工程：公式定理速查手册
This handbook offers a concise collection of essential formulas and theorems for Year 8 Engineering following the Scottish SQA framework. Use it to refresh your memory before tests, when tackling design challenges, or while exploring how things work. Every formula is paired with a clear explanation and practical examples to help you understand not just what the symbols mean, but why they matter in real engineering contexts.
本手册为遵循苏格兰 SQA 课程体系的 Year 8 工程学精选了核心公式与定理。你可以在测验前、处理设计难题时，或单纯在探索事物工作原理时快速查阅。每个公式都配有明确的解释和实例，帮助你不仅理解符号含义，更明白它们在真实工程情境中为何如此重要。
1. Ohm’s Law | 欧姆定律
Ohm’s Law describes the relationship between voltage, current and resistance in an electrical circuit. It states that the current flowing through a conductor is directly proportional to the voltage across it, provided the temperature remains constant. This is the fundamental rule for designing and analysing any circuit that uses resistors, lamps or heating elements.
欧姆定律描述了电路中电压、电流和电阻之间的关系：在温度不变的条件下，流过导体的电流与导体两端的电压成正比。这是设计和分析所有包含电阻器、灯泡或发热元件的电路时最基本的原则。
Voltage = Current × Resistance V = I × R
ected to a bulb with a resistance of 6 Ω, the current flowing is I = V ÷ R = 12 ÷ 6 = 2 A.
电压 (V) 的单位是伏特 (V)，电流 (I) 的单位是安培 (A)，电阻 (R) 的单位是欧姆 (Ω)。例如，一个 12 V 的电池连接一个 6 Ω 的灯泡时，流过的电流为 I = V ÷ R = 12 ÷ 6 = 2 A。
ng how to rearrange this formula is essential for troubleshooting circuits and selecting the right components in engineering projects.
这一规律同样允许你在已知电压和电流时计算元件的电阻：R = V ÷ I。学会变换该公式的形态，对于排查电路故障和工程项目中的元器件选型至关重要。
2. Electrical Power | 电功率
Electrical power measures the rate at which electrical energy is transferred in a circuit. It tells you how much work a component can do in a given time, whether it is lighting a lamp or turning a motor. The power formula helps engineers choose suitable wires, fuses and power supplies.
电功率衡量的是电路中电能传输的速率，它能表明一个元件在给定时间内能做多少功——无论那是点亮灯泡还是驱动电动机。功率公式帮助工程师选择合适的导线、熔断器和电源。
Power = Current × Voltage P = I × V
ful forms appear: P = I²R and P = V² ÷ R, which are handy when you know resistance instead of voltage or current.
功率 (P) 的单位是瓦特 (W)。一个工作在 230 V 下并吸取 0.5 A 电流的设备，其功率为 P = 0.5 × 230 = 115 W。结合欧姆定律，还会派生出另外两种常用的形式：P = I²R 和 P = V² ÷ R。当你知道的是电阻而非电压或电流时，这些公式会非常方便。
In engineering, power rating is critical for safety and efficiency. Overloading a resistor with too much power can cause overheating and failure, so always check the power rating marked on components before building a circuit.
在工程学中，功率额定值对安全性和效率至关重要。让电阻器承受过高功率会导致过热和损坏，因此在搭建电路前，务必核对元件上标注的功率额定值。
3. Work Done | 功
In engineering and physics, work is done when a force moves an object in the direction of the force. This concept underpins the design of every machine that lifts, pushes or pulls. Understanding work allows you to calculate the energy a device must supply to get a job done.
在工程与物理中，当一个力使物体沿力的方向移动时，我们就说它做了功。这个概念是设计一切能提升、推动或拖拽物体的机械的基础。理解了功，你就能计算出设备完成一项任务所需提供的能量。
Work = Force × Distance W = F × d
0 J of work. If the force is not applied entirely in the direction of motion, only the component of force along the direction counts.
功 (W) 的单位是焦耳 (J)，力 (F) 的单位是牛顿 (N)，距离 (d) 的单位是米 (m)。将一个 20 N 的工具箱竖直提起 1.5 m，做的功为 W = 20 × 1.5 = 30 J。如果力的方向与运动方向不完全一致，则只有沿运动方向的分力才会做功。
The idea of work links closely with energy transfer: whenever work is done, energy is transferred from one store to another. This makes the work equation essential when analysing mechanical systems, from simple levers to complex gear trains.
功的概念与能量转移密切相关：做功的过程始终伴随着能量从一个储存库转移到另一个。这使得功的方程在分析从简单杠杆到复杂齿轮系的各类机械系统时，都不可或缺。
4. Gravitational Potential Energy | 重力势能
Gravitational potential energy (GPE) is the energy stored in an object because of its height above the ground. In engineering, this is the energy that cranes must overcome, that roller coasters convert into thrill, and that dams use to generate electricity. The higher an object is, the more GPE it stores.
重力势能 (GPE) 是物体因其相对于地面的高度而储存的能量。在工程中，起重机需要克服的就是这种能量，过山车把它转化成刺激感，而水坝则利用它来发电。物体所处位置越高，储存的重力势能就越多。
GPE = Mass × Gravity × Height Eₚ = m × g × h
es (m). A 5 kg concrete block lifted 4 m high gains Eₚ = 5 × 10 × 4 = 200 J (using g = 10 m/s²).
质量 (m) 以千克 (kg) 为单位，地表重力加速度 (g) 取 9.8 m/s²（估算时常近似为 10 m/s²），高度 (h) 的单位是米 (m)。将一块 5 kg 的混凝土块提升 4 m，增加的重力势能为 Eₚ = 5 × 10 × 4 = 200 J（取 g = 10 m/s²）。
When designing lifting equipment, engineers calculate the GPE to determine the minimum work the hoist must do, and therefore the motor power required. This principle is also used to estimate the energy stored in water behind a hydroelectric dam.
设计起重设备时，工程师通过计算重力势能来判断起重机至少需要做多少功，从而确定所需的电动机功率。这一原理同样用于估算水电站大坝后方储存的水体能量。
5. Kinetic Energy | 动能
Kinetic energy is the energy an object possesses because of its motion. Every moving part in a machine, from gears to flywheels, carries kinetic energy. Knowing how to calculate it is essential for designing safe braking systems and for understanding vehicle collisions.
动能是物体由于运动而具备的能量。机器中每一个运动着的部分，无论是齿轮还是飞轮，都携带着动能。学会计算动能，是设计安全的制动系统和理解车辆碰撞问题的基础。
Kinetic Energy = ½ × Mass × Velocity² Eₖ = ½ × m × v²
tres per second (m/s). Doubling the speed quadruples the kinetic energy, which is why high‑ are so destructive. A 1000 kg car travelling at 10 m/s has Eₖ = ½ × 1000 × 10² = 50 000 J.
速度 (v) 必须以米每秒 (m/s) 为单位。速度加倍会使动能变成原来的四倍，这正是高速碰撞破坏力极强的原因。一辆 1000 kg 的轿车以 10 m/s 的速度行驶时，动能 Eₖ = ½ × 1000 × 10² = 50 000 J。
Engineers use the kinetic energy formula when designing crash barriers, airbags and vehicle crumple zones. The aim is to dissipate as much kinetic energy as possible in a controlled way during a collision, reducing the force transferred to passengers.
工程师在设计防撞护栏、安全气囊和汽车吸能区时都要用到动能公式，目标是在碰撞过程中以受控方式尽可能多地耗散动能，从而减小传递至乘员身上的作用力。
6. Principle of Moments (Levers) | 力矩原理（杠杆）
A moment is the turning effect of a force around a pivot. The principle of moments explains the balance of seesaws, the action of spanners, and the working of countless levers in engineering. When a system is stationary and balanced, the sum of clockwise moments equals the sum of anticlockwise moments.
力矩是力绕着一个支点产生的转动效果。力矩原理解释了跷跷板的平衡、扳手的施力效果以及工程中无数杠杆的工作原理。当一个系统静止且平衡时，所有顺时针力矩之和等于所有逆时针力矩之和。
Moment = Force × Perpendicular distance from pivot M = F × d
For a lever, this is often written as: Effort × Effort arm = Load × Load arm m. To balance the lever, an effort on the other side must provide an anticlockwise moment of exactly 12 N m.
对于杠杆，这常写作：动力 × 动力臂 = 阻力 × 阻力臂。如果一个 40 N 的载荷放在距支点 0.3 m 处，它产生的顺时针力矩为 40 × 0.3 = 12 N m。为了平衡杠杆，另一侧的动力必须恰好产生 12 N m 的逆时针力矩。
This theorem is used to design everything from scissors to excavator arms. By varying the distances, engineers can create a mechanical advantage that allows a small effort to lift a much heavier load, which is the essence of many hand tools and lifting machines.
从剪刀到挖掘机臂，这一原理广泛应用于各种设计中。通过改变力臂的距离，工程师可以创造出“机械效益”，让小力气举起大载荷——这正是许多手动工具和起重机械的核心所在。
7. Speed, Distance and Time | 速度、距离与时间
The relationship between speed, distance and time is one of the most frequently used equations in engineering design and testing. Whether you are calculating the cutting speed of a drilling machine, the conveyor belt rate, or the travel time of a robot arm, this formula is the starting point for motion analysis.
速度、距离与时间的关系是工程设计与测试中最常用的等式之一。无论你要计算钻床的切削速度、传送带的输送速率，还是机器人手臂的移动时间，这个公式都是运动分析的起点。
Speed = Distance ÷ Time s = d / t
s a speed of s = 15 ÷ 3 = 5 m/s. To convert m/s to km/h, multiply by 3.6: 5 m/s = 18 km/h.
务必保持单位一致。工程学中的速度常用米每秒 (m/s) 表示。一辆模型车在 3 秒内行驶 15 米，其速度为 s = 15 ÷ 3 = 5 m/s。要将 m/s 转换为 km/h，只需乘以 3.6：5 m/s = 18 km/h。
automated systems, timing calculations ensure that components meet at the right moment, preventing collisions and improving efficiency.
这个公式可以变形为求距离 (d = s × t) 或求时间 (t = d ÷ s)。在自动化系统中，时间计算能确保零部件在正确的时刻相遇，避免碰撞并提高效率。
8. Pressure | 压强
Pressure measures how much force acts on a unit area. A sharp knife cuts easily because it concentrates the applied force onto a very small area, creating high pressure. In civil and mechanical engineering, pressure determines foundation design, hydraulic system performance and even the grip of tyres on a road.
压强衡量的是单位面积上承受的力有多大。锋利的刀能轻松切开物体，是因为它将施加的力集中在极小的面积上，产生了很高的压强。在土木和机械工程中，压强决定了地基设计、液压系统性能，甚至轮胎在路面上的抓地力。
Pressure = Force ÷ Area P = F ÷ A
oot with a sole area of 0.03 m² exerts a pressure of P = 600 ÷ 0.03 = 20 000 Pa, which is enough to sink into soft ground.
力的单位是牛顿 (N)，面积的单位是平方米 (m²)。压强的单位是帕斯卡 (Pa)，1 Pa = 1 N/m²。一个体重 600 N 的人站在一只鞋底面积为 0.03 m² 的靴子上，施加的压强 P = 600 ÷ 0.03 = 20 000 Pa，足以让他陷入松软的地面。
Hydraulic presses and brake systems use Pascal’s principle: when a liquid is confined, a pressure change at one point is transmitted equally everywhere. By applying a small force to a small piston, a larger force can be created on a bigger piston, which is why hydraulic lifts can raise heavy vehicles.
液压机和制动系统利用了帕斯卡原理：液体在密闭容器内，一点处的压强变化会等大地传递到各处。通过对小活塞施加较小的力，可以在大活塞上产生更大的力，这正是液压起重机能举起重型车辆的原因。
9. Density | 密度
Density tells you how much mass is packed into a given volume. Engineers use density to choose materials: aluminium is preferred for aeroplanes because of its low density, while steel is used for building frames due to its high density and strength. It also explains whether objects float or sink.
密度表示单位体积内含有多少质量。工程师依据密度来选择材料：铝因其较低的密度而成为飞机的首选，钢材则因其高密度和高强度用于建筑物框架。密度还决定了物体的沉浮。
Density = Mass ÷ Volume ρ = m ÷ V
ensity of 1000 kg/m³. A block of wood with mass 0.2 kg and volume 500 cm³ (0.0005 m³) has a density of ρ = 0.2 ÷ 0.0005 = 400 kg/m³, so it floats.
希腊字母 ρ 代表密度，单位是千克每立方米 (kg/m³) 或克每立方厘米 (g/cm³)。淡水的密度是 1000 kg/m³。一块质量为 0.2 kg、体积为 500 cm³ (0.0005 m³) 的木块，密度为 ρ = 0.2 ÷ 0.0005 = 400 kg/m³，因此它会漂浮。
Mass, volume and density are linked, so if you know any two, you can calculate the third. In materials testing, density can help identify unknown substances or detect flaws such as air pockets inside a casting.
质量、体积和密度三者相互关联，只要知道其中任意两个，就能求出第三个。在材料测试中，密度有助于识别未知物质，或检测铸件内部是否存在气孔等缺陷。
10. Mechanical Advantage and Velocity Ratio | 机械效益与速度比
Mechanical advantage (MA) tells you how many times a machine multiplies the effort force you put into it. It is a key concept for evaluating the effectiveness of levers, pulleys, gears and other mechanisms. The velocity ratio (VR) compares the distance moved by the effort to the distance moved by the load in the same time.
机械效益 (MA) 表示一台机器将你输入的动力放大了多少倍。它是评估杠杆、滑轮、齿轮等机构效果的关键概念。速度比 (VR) 则比较了同一时间内动力移动的距离与载荷移动的距离。
Mechanical Advantage = Load ÷ Effort MA = L ÷ E
locity Ratio = Distance moved by effort ÷ Distance moved by load VR = dₑ / dₗ
sses. If a pulley system has a VR of 4 but an MA of 3.2, its efficiency is (MA ÷ VR) × 100% = (3.2 ÷ 4) × 100% = 80%.
在一个没有摩擦的理想世界中，机械效益等于速度比。现实中由于能量损失，MA 总是略小一些。假如一套滑轮组的 VR 是 4，但 MA 仅为 3.2，那么它的效率为 (MA ÷ VR) × 100% = (3.2 ÷ 4) × 100% = 80%。
Gear systems also follow this logic: the gear ratio is essentially the velocity ratio. A small gear driving a larger gear reduces speed but increases torque (turning force), which is exactly how bicycle gears help you cycle up a steep hill.
齿轮系统同样遵循这一逻辑：齿轮比本质上就是速度比。一个小齿轮带动一个大齿轮会降低转速，但增大扭矩（旋转力），这正是自行车变速器能帮助你在陡坡上骑行的工作原理。
11. Stress and Strain Basics | 应力与应变基础
Stress and strain describe how forces deform solid materials. Stress is the internal force per unit area that develops within a material when it is loaded, while strain is the measure of deformation relative to the original dimensions. These concepts allow engineers to predict whether a bridge beam or an engine component will break or bend permanently.
应力和应变描述了力如何使固体材料发生形变。应力是材料在受载时内部产生的每单位面积上的力，应变则是形变相对于原始尺寸的量度。这些概念让工程师能够预测桥梁梁体或发动机零件是会断裂还是发生永久弯曲。
Stress = Force ÷ Area σ = F ÷ A
Strain = Change in length ÷ Original length ε = ΔL ÷ L₀
stretches by 1.0 mm under load experiences a strain of ε = 0.001 ÷ 2.00 = 0.0005 (or 0.05%).
应力的单位是牛顿每平方米 (N/m²) 或帕斯卡，与压强相同。应变量则没有单位。一根长 2.00 m 的低碳钢杆在载荷作用下拉伸了 1.0 mm，其应变为 ε = 0.001 ÷ 2.00 = 0.0005（即 0.05%）。
lps engineers select materials that will deflect acceptably under load without permanent damage.
对许多材料而言，在弹性范围内应力与应变成正比，这一关系由杨氏模量描述：E = σ ÷ ε。这个刚度系数帮助工程师挑选在负载下能产生可接受变形而又不会永久损坏的材料。
12. Gear Ratio and Torque | 齿轮比与扭矩
Gear systems transmit rotational motion and force between shafts. The gear ratio determines how the speed and torque change from the input (driver) gear to the output (driven) gear. In vehicles, robotics and clocks, managing gear ratios is essential for matching motor speed to the desired mechanical action.
齿轮系统在轴与轴之间传递旋转运动和力。齿轮比决定了从主动轮（输入）到从动轮（输出）的速度和扭矩如何变化。在车辆、机器人和钟表中，调控齿轮比对于让电动机转速适配所需的机械动作至关重要。
Gear Ratio = Number of teeth on driven gear ÷ Number of teeth on driver gear
tio greater than 1 means the output turns more slowly but produces greater torque. If a 20‑tooth driver meshes with a 60‑tooth driven gear, the ratio is 60 ÷ 20 = 3:1. The output speed is one‑third of the input speed, while the output torque is roughly tripled (ignoring friction).
比值大于 1 表示输出转速较慢但能产生更大的扭矩。如果一个 20 齿的主动轮与一个 60 齿的从动轮啮合，齿轮比为 60 ÷ 20 = 3:1。输出转速是输入转速的三分之一，而输出扭矩大约增加为原来的三倍（忽略摩擦）。
Torque itself can be thought of as the turning force: Torque = Force × Lever arm. In gear design, engineers calculate torque to ensure shafts are strong enough and to design appropriate bearings. Simple compound gear trains can be analysed by multiplying individual ratios together.
扭矩本身可以理解为转动力：扭矩 = 力 × 力臂。在齿轮设计中，工程师通过计算扭矩来确保轴有足够的强度，并设计相配的轴承。简单的复式轮系可以通过将各级齿轮比相乘来进行分析。
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