📚 SQA Engineering Formula & Theorem Quick Reference Guide | SQA 工程公式定理速查手册
This quick reference guide covers the essential formulas and theorems required for SQA National 5 Engineering Science. Whether you are designing a mechanical system, analysing an electronic circuit, or calculating structural loads, these equations form the backbone of your problem-solving toolkit. Keep this handbook handy as you practise past paper questions and build confidence for your assessments.
本速查手册涵盖 SQA National 5 工程科学的核心公式与定理。无论你是在设计机械系统、分析电子电路还是计算结构载荷，这些方程式都是你问题求解工具箱的基础。在练习历年真题时随身备查，为你的考核建立信心。
1. Newton's Laws of Motion | 牛顿运动定律
Newton's First Law (the law of inertia) states that an object will remain at rest or move with constant velocity unless acted upon by an unbalanced external force. This concept explains why seat belts are necessary in a car: your body wants to continue moving forward even when the vehicle stops.
牛顿第一定律（惯性定律）指出，除非受到非平衡外力的作用，否则物体将保持静止或匀速直线运动状态。这解释了为什么汽车中需要安全带：即使车辆停止，你的身体仍想继续向前运动。
Newton's Second Law quantifies the relationship between force, mass and acceleration. It is the most widely used equation in dynamics problems.
牛顿第二定律定量地描述了力、质量和加速度之间的关系。它是动力学问题中使用最广泛的方程。
F = m a
Newton's Third Law reminds us that forces always occur in pairs: if body A exerts a force on body B, then body B exerts an equal and opposite force on body A. Reaction forces in structures and thrust in rockets are direct applications of this law.
牛顿第三定律提醒我们，力总是成对出现的：如果物体A对物体B施加一个力，那么物体B也会对物体A施加一个大小相等、方向相反的力。结构中的支座反力和火箭推力都是该定律的直接应用。
2. Force, Mass and Acceleration (F = m a) | 力、质量与加速度
In the SQA exam, you will often resolve forces acting on a single object or system. The net force is the vector sum of all external forces, and it determines the acceleration of the centre of mass. Always identify the direction of motion and treat it as positive to simplify calculations.
在 SQA 考试中，你常常需要分析作用在单个物体或系统上的力。净力是所有外力的矢量和，它决定了质心的加速度。始终明确运动方向并将其视为正方向，以简化计算。
a = F / m, showing that acceleration is inversely proportional to mass for a given force.
当包含摩擦力或空气阻力时，净力等于施加的力减去阻力。重新整理 F = m a 可得到 a = F / m，表明在给定力的情况下，加速度与质量成反比。
A common pitfall is mixing up mass and weight. Weight is a force and is calculated as:
一个常见误区是混淆质量与重量。重量是一种力，其计算公式为：
W = m g
ways use units: newtons (N) for force, kilograms (kg) for mass, and m/s² for acceleration.
其中 g = 9.8 m/s²（地球上）。务必使用正确的单位：力的单位是牛顿（N），质量的单位是千克（kg），加速度的单位是 m/s²。
3. Moments and Equilibrium | 力矩与平衡
A moment is the turning effect of a force about a pivot. It depends on two factors: the magnitude of the force and the perpendicular distance from the pivot to the line of action of the force.
力矩是力对支点的转动效应。它取决于两个因素：力的大小以及从支点到力作用线的垂直距离。
M = F × d
For an object to be in static equilibrium, two conditions must be met: the sum of all forces in any direction must equal zero, and the sum of clockwise moments about any point must equal the sum of anticlockwise moments about the same point. This principle is used to calculate unknown forces in beams, levers and bridges.
物体要保持静力平衡，必须满足两个条件：任意方向上所有力的代数和必须为零，且对任意点而言，顺时针力矩之和必须等于逆时针力矩之和。该原理被用于计算梁、杠杆和桥梁中的未知力。
When tackling moment questions, always mark a clear pivot and indicate perpendicular distances. If a force is applied at an angle, resolve it into components to find the perpendicular component that creates the moment.
解答力矩问题时，要明确标出支点并指明垂直距离。如果力以某个角度施加，应先将其分解为分量，从而找到产生力矩的垂直分量。
4. Work, Energy and Power | 功、能量与功率
Work is done when a force moves an object through a distance in the direction of the force. The most fundamental relation is:
当一个力使物体沿力的方向移动一段距离时，就做了功。最基本的公式为：
W = F s
Energy is the capacity to do work. Kinetic energy and gravitational potential energy are two mechanical forms you must know:
能量是做功的能力。你必须掌握的两种机械能形式是动能和重力势能：
Eₖ = ½ m v²
Eₚ = m g h
Conservation of energy is a powerful concept: in the absence of friction, the total mechanical energy (Eₖ + Eₚ) remains constant. Real systems lose energy as heat, which leads us to efficiency calculations.
能量守恒是一个强大的概念：在没有摩擦的情况下，总机械能（Eₖ + Eₚ）保持不变。真实系统会以热能形式损失能量，这就引出了效率计算。
Power is the rate at which work is done or energy is transferred. In mechanical systems, average power is given by:
功率是做功或能量传递的速率。在机械系统中，平均功率由下式给出：
P = W / t
Understanding the distinction between energy (joules) and power (watts) is essential for interpreting motor ratings and energy consumption.
理解能量（焦耳）和功率（瓦特）之间的区别，对于解读电机额定值和能耗至关重要。
5. Ohm's Law and Electrical Power | 欧姆定律与电功率
Ohm's Law is the foundation of circuit analysis. It states that the current through a resistor is directly proportional to the voltage across it, provided the temperature remains constant.
欧姆定律是电路分析的基础。它表明，在温度保持不变的条件下，通过电阻器的电流与其两端的电压成正比。
V = I R
rms depending on which quantities are known. These are derived from the combination of Ohm's Law and the power equation P = I V.
电功率可以根据已知量的不同，用三种实用的形式表达。它们由欧姆定律和功率方程 P = I V 组合推导而来。
P = I V = I² R = V² / R
Pay close attention to unit conversions: resistors are rated in ohms (Ω), current in amperes (A), and power in watts (W). When rearranging, double-check that you are not accidentally dividing by zero or misplacing the squared terms.
特别注意单位换算：电阻以欧姆（Ω）为单位，电流以安培（A）为单位，功率以瓦特（W）为单位。在移项变换时，要再次检查是否有除以零的情况或平方项错位的问题。
6. Resistors in Series and Parallel | 串联与并联电阻
Combining resistors is a routine task in engineering circuits. The total resistance depends on whether the components share the same current (series) or the same voltage (parallel).
合并电阻是工程电路中的常规操作。总电阻取决于元件是共享相同的电流（串联）还是相同的电压（并联）。
| Series | Rtotal = R₁ + R₂ + … |
| Parallel | 1/Rtotal = 1/R₁ + 1/R₂ + … |
For two resistors in parallel, a convenient shortcut is Rtotal rmula only works for exactly two resistors; always use the reciprocal method for three or more.
对于两个并联电阻，便捷的公式是 Rtotal = (R₁ × R₂) / (R₁ + R₂)。但该公式仅适用于两个电阻；对于三个或更多电阻，始终使用倒数法。
Remember that adding resistors in series always increases total resistance, while adding resistors in parallel always decreases total resistance. This is because parallel connections provide additional paths for current to flow.
请牢记，串联增加电阻总会提高总电阻，而并联增加电阻总会降低总电阻。这是因为并联连接为电流提供了额外的流通路径。
7. Mechanical Advantage and Efficiency | 机械优势与效率
Simple machines such as levers, pulleys and gears are designed to multiply force or alter direction. Mechanical advantage (MA) quantifies the multiplication of the effort force.
诸如杠杆、滑轮和齿轮等简单机械旨在放大力量或改变方向。机械优势（MA）量化了动力被放大的倍数。
MA = Load / Effort
Velocity ratio (VR) is the ratio of the distance moved by the effort to the distance moved by the load in the same time. It describes the ideal mechanical advantage if no energy were lost.
速度比（VR）是同一时间内动力移动的距离与负载移动的距离之比。它描述了在无能量损失情况下的理想机械优势。
VR = Distance moved by effort / Distance moved by load
Efficiency compares the useful work output to the work input, or practically, MA to VR. Because friction always dissipates some energy, MA is always less than VR, and efficiency is always less than 100%.
效率将有用功输出与功输入，或实际中将 MA 与 VR 进行比较。由于摩擦总会耗散部分能量，MA 总是小于 VR，效率总是低于 100%。
Efficiency = (MA / VR) × 100% or η = (Useful output / Total input) × 100%
When tackling multi-stage systems, calculate the overall efficiency by multiplying the efficiencies of each stage rather than adding them.
在处理多级系统时，应通过将各级效率相乘（而非相加）来计算总效率。
8. Stress, Strain and Young's Modulus | 应力、应变与杨氏模量
Engineers must understand how materials deform under load. Stress measures the internal force per unit area and is analogous to pressure.
工程师必须了解材料在载荷下如何变形。应力衡量单位面积上的内力，类似于压力。
σ = F / A
Strain is a dimensionless ratio that describes the extension per unit original length.
应变是一个无量纲的比值，它描述的是单位原始长度的伸长量。
ε = ΔL / L₀
For many materials, the initial part of the stress-strain curve is linear, and the material obeys Hooke's Law. Young's modulus (E) is the gradient of this linear portion and is a measure of stiffness.
对于许多材料，应力-应变曲线的起始部分是线性的，且材料服从胡克定律。杨氏模量（E）是这一段直线的斜率，是刚度的度量。
E = σ / ε
A steel wire with a high Young's modulus will stretch very little under tension, whereas a rubber band has a much lower modulus. Always check units: stress is in pascals (Pa), and strain has no units, so Young's modulus is also in pascals.
杨氏模量高的钢丝在拉力作用下伸长很小，而橡皮筋的模量要低得多。务必检查单位：应力以帕斯卡（Pa）为单位，应变无单位，因此杨氏模量的单位也是帕斯卡。
9. Gear Ratio and Velocity Ratio | 齿轮比与速度比
Gear systems transmit rotational motion and torque. The gear ratio is determined by the number of teeth on each gear, and it directly affects the output speed and power transmission.
齿轮系统传递旋转运动和扭矩。齿轮比由每个齿轮的齿数决定，并直接影响输出转速和动力传递。
Gear Ratio = Tdriven / Tdriver
In an ideal gear train with no friction, the velocity ratio equals the gear ratio. The velocity ratio can also be expressed in terms of angular velocities or rotational speeds.
在无摩擦的理想齿轮系中，速度比等于齿轮比。速度比也可以用角速度或转速来表示。
VR = ωdriver / ωdriven = Ndriver / Ndriven
If the driven gear has more teeth than the driver, the output speed decreases but torque increases. This trade-off is critical when designing winches, bicycles, and industrial machinery.
如果从动轮的齿数多于主动轮，输出转速会降低，但扭矩会增大。在设计绞车、自行车和工业机械时，这种权衡关系至关重要。
Compound gear trains use multiple gears on the same shaft to achieve larger overall ratios. To find the total velocity ratio, multiply the individual ratios of each stage.
复合轮系在同一轴上使用多个齿轮以获得更大的总比值。要计算总速度比，应将每一级的个体比相乘。
10. Trigonometric Functions in Engineering | 工程中的三角函数
Trigonometry is used to resolve forces, calculate dimensions in structures, and analyse motion along inclined planes. The three primary ratios are defined for a right-angled triangle.
三角函数用于分解力、计算结构尺寸以及分析斜面运动。针对直角三角形，定义了三个基本比值。
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
When a force acts at an angle to a surface, the horizontal and vertical components can be found using these functions. For example, a force F at an angle θ can be resolved into F cos θ (along the slope) and F sin θ (perpendicular).
当力与表面成一定角度作用时，可以用这些函数求出水平和垂直分量。例如，一个大小为 F、与水平成 θ 角的力可分解为 F cos θ（沿斜面方向）和 F sin θ（垂直方向）。
Pythagoras' theorem also frequently appears alongside trigonometry to find unknown side lengths:
毕达哥拉斯定理也常与三角函数一同出现，用于求解未知边长：
a² + b² = c²
Mastering these relationships will help you solve resultant force problems, frame analysis, and vector addition with confidence.
掌握这些关系将帮助你自信地求解合力问题、框架分析以及矢量加法。
Published by TutorHao | Engineering Revision Series | aleveler.com
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