Quick Reference Handbook of Formulas and Theorems | 公式定理速查手册

📚 Quick Reference Handbook of Formulas and Theorems | 公式定理速查手册

This quick reference handbook provides Year 9 SQA engineering students with a concise summary of essential formulas and theorems. Use it to revise key concepts in mechanics, materials, and electrical circuits. Each section pairs theory with practical equations to support problem-solving.

本速查手册为 Year 9 SQA 工程学生提供重要公式和定理的简明总结。用它来复习力学、材料和电路等关键概念。每一节都将理论与实用方程式相结合，以帮助解决问题。

1. Units and Conversions | 单位与换算

All engineering calculations rely on standard SI units. Common base units include metre (m) for length, kilogram (kg) for mass, second (s) for time, and ampere (A) for electric current. Derived units such as newton (N) for force and pascal (Pa) for pressure are built from these.

所有工程计算都依赖于标准国际单位制。常见的基单位包括长度米 (m)、质量千克 (kg)、时间秒 (s) 和电流安培 (A)。导出单位如力的牛顿 (N) 和压力的帕斯卡 (Pa) 均由这些基单位构成。

Frequently used prefixes are essential: kilo- (k) means ×10³, mega- (M) means ×10⁶, and milli- (m) means ×10⁻³. For example, 1 kN = 1000 N and 1 MPa = 1×10⁶ Pa = 1 N/mm². Always convert to base units before substituting into formulas.

常用词头非常重要：千 (k) 表示 ×10³，兆 (M) 表示 ×10⁶，毫 (m) 表示 ×10⁻³。例如，1 kN = 1000 N，1 MPa = 1×10⁶ Pa = 1 N/mm²。代入公式前务必转换为基单位。

Quantity	Unit / Conversion
Length	1 m = 100 cm = 1000 mm
Mass	1 kg = 1000 g
Force	1 N = 1 kg m/s²; 1 kN = 1000 N
Pressure / Stress	1 Pa = 1 N/m²; 1 MPa = 1×10⁶ Pa = 1 N/mm²
Energy / Work	1 J = 1 N m
Power	1 W = 1 J/s

2. Kinematics Formulas | 运动学公式

Speed is the distance travelled per unit time. For constant speed, the formula is v = d / t, where v is speed, d is distance, and t is time. Typical units are m/s.

速度是单位时间内移动的距离。对于恒定速度，公式为 v = d / t，其中 v 是速度，d 是距离，t 是时间，常用单位 m/s。

v = d / t

When motion is uniformly accelerated, three equations link initial velocity u, final velocity v, acceleration a, time t, and displacement s. All use consistent SI units.

当物体做匀加速运动时，有三个方程将初速度 u、末速度 v、加速度 a、时间 t 和位移 s 联系起来，均采用一致的国际单位。

The first equation is v = u + a t. It is used when time is known and acceleration is constant.

第一个方程 v = u + a t，用于已知时间且加速度恒定的情景。

v = u + a t

The second equation is s = u t + ½ a t². It gives displacement when initial velocity and acceleration are known.

第二个方程 s = u t + ½ a t²，已知初速度和加速度时用来求位移。

s = u t + ½ a t²

The third equation is v² = u² + 2 a s. It avoids time and is useful when only distances are given.

第三个方程 v² = u² + 2 a s，不包含时间变量，在仅给出距离时非常有用。

v² = u² + 2 a s

Acceleration is the rate of change of velocity: a = (v – u) / t. It is measured in m/s².

加速度是速度的变化率：a = (v – u) / t，单位是 m/s²。

3. Newton’s Laws and Force | 牛顿定律与力

Newton’s first law states that an object remains at rest or in uniform motion unless acted upon by an external resultant force. The second law quantifies this: resultant force = mass × acceleration, F = m a. Force is measured in newtons (N).

牛顿第一定律指出，除非受到外力作用，物体将保持静止或匀速直线运动。第二定律将其量化：合力 = 质量 × 加速度，F = m a，力的单位是牛顿 (N)。

F = m a

Weight is a special case: the gravitational force on a mass is W = m g, where g is gravitational field strength (on Earth, g ≈ 9.8 N/kg or 10 N/kg for approximate calculations). Weight always acts vertically downwards.

重力是一个特例：作用在物体上的重力为 W = m g，其中 g 是引力场强度（地球上 g ≈ 9.8 N/kg，近似计算时可取 10 N/kg）。重力方向总是竖直向下。

W = m g

Newton’s third law states that for every action there is an equal and opposite reaction. In engineering, this principle is key to understanding forces in structures and machines.

牛顿第三定律指出，每一个作用力都有一个大小相等、方向相反的反作用力。在工程中，这一原理是理解结构体和机器中力的关键。

4. Work, Energy and Power | 功、能与功率

Work is done when a force moves an object. Work W = F × d, where F is the force in the direction of movement and d is the displacement. The unit of work and energy is the joule (J).

当力使物体移动时，力做功。功 W = F × d，其中 F 是运动方向上的力，d 是位移。功和能量的单位都是焦耳 (J)。

W = F d

Gravitational potential energy is the energy stored due to an object’s height: Eₚ = m g h, where h is the vertical height above a reference level.

重力势能是物体由于高度而储存的能量：Eₚ = m g h，其中 h 是高于参考面的垂直高度。

Eₚ = m g h

Kinetic energy is the energy of motion: Eₖ = ½ m v², where v is the instantaneous speed.

动能是物体运动具有的能量：Eₖ = ½ m v²，其中 v 是瞬时速度。

Eₖ = ½ m v²

Power is the rate of doing work or transferring energy. P = W / t, or in mechanical systems, P = F v when force and velocity are parallel. The unit of power is the watt (W).

功率是做功或转换能量的速率。P = W / t，在机械系统中，当力与速度同向时，P = F v。功率的单位是瓦特 (W)。

P = W / t

P = F v

5. Efficiency | 效率

Efficiency measures the effectiveness of energy conversion. It is the ratio of useful output energy (or power) to the total input energy (or power), expressed as a percentage. No real machine can be 100% efficient due to friction and other losses.

效率衡量能量转换的有效程度。它是有用输出能量（或功率）与总输入能量（或功率）的比值，用百分比表示。由于摩擦和其他损耗，任何实际机器的效率都不可能达到 100%。

Efficiency = (Useful output energy / Total input energy) × 100%

Equivalently, efficiency can be written as η = (P_out / P_in) × 100%. The symbol η is the Greek letter eta.

同样，效率也可写为 η = (P_out / P_in) × 100%。符号 η 是希腊字母 eta。

η = (P_out / P_in) × 100%

In machines, efficiency also relates mechanical advantage (MA) and velocity ratio (VR): Efficiency = (MA / VR) × 100%. This will be explored later.

在机械中，效率还与机械利益 (MA) 和速度比 (VR) 相关：效率 = (MA / VR) × 100%。后文将详细说明。

6. Moments and Levers | 力矩与杠杆

A moment is the rotational effect of a force about a pivot. It is calculated as the product of the force and the perpendicular distance from the line of action to the pivot: M = F × d. The unit of moment is the newton metre (N m).

力矩是力对支点产生的转动效应，计算公式为力乘以力作用线到支点的垂直距离：M = F × d。力矩的单位是牛顿·米 (N m)。

M = F × d

The principle of moments states that for an object in rotational equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments about any pivot. Mathematically, F₁ × d₁ = F₂ × d₂ for two opposing forces.

力矩原理指出，对于处于转动平衡的物体，对任意支点的顺时针力矩总和等于逆时针力矩总和。数学上，对于两个相反的力，有 F₁ × d₁ = F₂ × d₂。

F₁ × d₁ = F₂ × d₂

Levers are simple machines that use the principle of moments to amplify force. A first-class lever has the pivot between effort and load; a second-class lever has the load between pivot and effort; a third-class lever has the effort between pivot and load.

杠杆是利用力矩原理放大力的简单机械。第一类杠杆支点在施力点和负载之间；第二类杠杆负载在支点和施力点之间；第三类杠杆施力点在支点和负载之间。

7. Mechanical Advantage and Velocity Ratio | 机械利益与速度比

Mechanical advantage (MA) describes how much a machine multiplies the effort force. It is the ratio of load (output force) to effort (input force): MA = Load / Effort. It has no units.

机械利益 (MA) 描述机器将施力放大了多少倍，它是负载（输出力）与施力（输入力）的比值：MA = 负载 / 施力。它没有单位。

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: VR = d_effort / d_load. VR is determined by the geometry of the machine and does not change with friction.

速度比 (VR) 是在相同时间内施力点移动的距离与负载移动的距离之比：VR = d_effort / d_load。VR 由机器的几何结构决定，不随摩擦而改变。

VR = d_effort / d_load

For an ideal (frictionless) machine, MA = VR. In real machines, MA is always less than VR, and efficiency = (MA / VR) × 100%. These concepts are fundamental when analysing pulley systems, gears and inclined planes.

对于理想（无摩擦）机器，MA = VR。在实际机器中，MA 总是小于 VR，效率 = (MA / VR) × 100%。这些概念是分析滑轮组、齿轮和斜面的基础。

8. Stress and Strain | 应力与应变

When a material is subjected to a force, stress σ is defined as the internal force per unit area: σ = F / A, where F is the applied force and A is the cross-sectional area. The unit is the pascal (Pa) or N/m², but engineering stresses are often given in MPa.

当材料受力时，应力 σ 定义为单位面积上的内力：σ = F / A，其中 F 是施加的力，A 是横截面积。单位是帕斯卡 (Pa) 或 N/m²，但工程应力常用 MPa 表示。

σ = F / A

Strain ε is the extension per unit original length, a measure of deformation: ε = ΔL / L₀, where ΔL is the change in length and L₀ is the original length. Strain is a ratio and has no units; it is often given as a percentage.

应变 ε 是单位原始长度的延伸量，衡量变形程度：ε = ΔL / L₀，其中 ΔL 是长度变化量，L₀ 是原始长度。应变是一个比值，无单位，常以百分比表示。

ε = ΔL / L₀

Young’s modulus E is a measure of stiffness, defined by E = σ / ε within the elastic limit. The unit is Pa. A high Young’s modulus indicates a stiff material, e.g. steel has a much higher E than rubber.

杨氏模量 E 是衡量材料刚度的量，在弹性限度内定义为 E = σ / ε。单位是 Pa。高的杨氏模量表示材料刚性大，例如钢的 E 远大于橡胶。

E = σ / ε

9. Electrical Circuits: Ohm’s Law and Power | 电路：欧姆定律与电功率

Ohm’s law governs the relationship between voltage, current and resistance in a conductor at constant temperature. It states that the potential difference V across a resistor is directly proportional to the current I flowing through it: V = I R. Resistance R is measured in ohms (Ω).

欧姆定律描述了在恒温下导体中电压、电流和电阻的关系。它指出电阻两端的电势差 V 与流过的电流 I 成正比：V = I R。电阻 R 的单位是欧姆 (Ω)。

V = I R

Electrical power P is the rate at which electrical energy is converted. It can be calculated using P = V I. Combining with Ohm’s law gives two alternative forms: P = I² R and P = V² / R. Power is measured in watts (W).

电功率 P 是电能转化的速率。可用 P = V I 计算。结合欧姆定律得到另外两种形式：P = I² R

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Quick Reference Handbook of Formulas and Theorems | 公式定理速查手册