Year 10 WJEC Engineering: Quick Reference Handbook of Formulas and Theorems | Year 10 WJEC 工程：公式定理速查手册

📚 Year 10 WJEC Engineering: Quick Reference Handbook of Formulas and Theorems | Year 10 WJEC 工程：公式定理速查手册

This handbook provides a concise summary of essential formulas, principles and theorems for Year 10 WJEC Engineering. Use it as a quick reference when solving problems, designing systems or revising for examinations. Each section presents key equations, their definitions and practical applications.

本手册简要总结了 Year 10 WJEC 工程课程的核心公式、原理和定理。当你在解题、设计系统或备考复习时，可将它作为快速查阅工具。每个单元都列出了关键公式、其定义和实际应用。

1. Mechanical Advantage, Velocity Ratio and Efficiency | 机械效益、速比与效率

Mechanical Advantage (MA) tells you how much a machine multiplies the input force. It is the ratio of the load (output force) to the effort (input force).

机械效益（MA）表示机器将输入力放大了多少倍。它是负载（输出力）与驱动力（输入力）的比值。

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 depends only on the design of the machine.

速比（VR）是同一时间内驱动力移动的距离与负载移动的距离之比。它只取决于机器的设计。

VR = Distance moved by effort / Distance moved by load

Efficiency measures how well a machine converts input work into useful output work. It is often expressed as a percentage.

效率衡量机器将输入功转化为有用输出功的能力。它通常用百分数表示。

Efficiency = (MA / VR) × 100%

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2. Moments and Equilibrium | 力矩与平衡

A moment is the turning effect of a force about a pivot. Its size depends on the force and the perpendicular distance from the pivot to the line of action of the force.

力矩是力绕支点的转动效应。其大小取决于力和支点到力作用线的垂直距离。

Moment = Force × Perpendicular distance

For a system to be in rotational equilibrium, the sum of clockwise moments must equal the sum of anticlockwise moments about any pivot point (principle of moments).

要使系统处于转动平衡，对任意支点，顺时针力矩之和必须等于逆时针力矩之和（力矩原理）。

Σ M_clockwise = Σ M_{anticlockwise}

es can be used to calculate the mechanical advantage directly: MA = effort arm / load arm.

在杠杆系统中，驱动力臂、负载臂和支点距离可用来直接计算机械效益：MA = 驱动力臂 / 负载力臂。

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3. Forces and Newton's Second Law | 力与牛顿第二定律

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

牛顿第二定律指出，物体的加速度与作用在其上的净力成正比，与物体的质量成反比。

F = m × a

Weight is the force due to gravity on a mass. Near Earth's surface, the gravitational field strength is approximately 9.8 N/kg.

重量是重力作用在质量上的力。在地球表面附近，重力场强度约为 9.8 N/kg。

W = m × g

When multiple forces act on one body, the resultant force can be found by adding vectors. If the resultant is zero, the body remains at rest or moves with constant velocity (Newton's First Law).

当多个力作用于同一物体时，可通过矢量相加求出合力。若合力为零，物体保持静止或匀速直线运动（牛顿第一定律）。

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4. Work, Energy and Power | 功、能与功率

Work is done when a force moves an object through a distance in the direction of the force. It is a scalar quantity measured in joules (J).

当一个力使物体沿力的方向移动一段距离时，就做了功。功是标量，单位为焦耳 (J)。

Work (W) = Force × Distance

Gravitational potential energy gained by an object lifted to a height is:

物体被举升到一定高度所获得的重力势能为：

E_p = m × g × h

Kinetic energy of a moving object is:

运动物体的动能为：

E_k = ½ × m × v²

f doing work or transferring energy. The standard unit is the watt (W), where 1 W = 1 J/s.

功率是做功或能量传递的速率。标准单位是瓦特 (W)，1 W = 1 J/s。

P = W / t = F × v

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5. Stress, Strain and Young Modulus | 应力、应变与杨氏模量

Stress is the internal force per unit area within a material when an external load is applied. It is measured in pascals (Pa) or N/m².

应力是材料受外力时单位面积上的内力。它以帕斯卡 (Pa) 或 N/m² 为单位。

σ = F / A

Strain is the ratio of extension to original length. It has no units because it is a ratio.

应变是伸长量与原始长度之比，没有单位，因为是一个比值。

ε = ΔL / L₀

Young Modulus (E) describes the stiffness of a material. It applies only within the elastic region where Hooke’s Law holds.

杨氏模量 (E) 描述材料的刚度。它仅适用于遵循胡克定律的弹性区域。

E = σ / ε

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6. Ohm's Law and Electrical Power | 欧姆定律与电功率

Ohm's Law states that the current through a conductor is directly proportional to the voltage across it, provided the temperature remains constant.

欧姆定律指出，在温度保持不变的条件下，通过导体的电流与其两端的电压成正比。

V = I × R

Electrical power is the rate at which electrical energy is converted. It can be calculated using any of the three equivalent formulas.

电功率是电能转换的速率。可以使用以下三个等价公式中的任意一个进行计算。

P = I × V = I² × R = V² / R

The energy consumed over time is power multiplied by time. Utility companies measure energy in kilowatt‑hours (kWh).

消耗的电能是功率乘以时间。公用事业公司以千瓦时 (kWh) 为单位计量电能。

E = P × t

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

In a series circuit, components are connected end to end. The current is the same at all points, and the total resistance is the sum of the individual resistances.

在串联电路中，元件逐个首尾连接。各点电流相同，总电阻等于各电阻之和。

R_total = R₁ + R₂ + R₃ + …

In a parallel circuit, components are connected across common points. The voltage across each branch is the same, and the total resistance is given by the reciprocal formula.

在并联电路中，元件连接在共同的节点之间。各支路电压相同，总电阻由倒数公式给出。

1 / R_total = 1 / R₁ + 1 / R₂ + 1 / R₃ + …

For two resistors in parallel, a simplified formula can be used:

两个电阻并联时，可使用简化公式：

R_total = (R₁ × R₂) / (R₁ + R₂)

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8. Gear Ratios and Drive Systems | 齿轮比与传动系统

Gears transmit rotary motion and torque. The gear ratio is the ratio of the number of teeth on the driven gear to the number of teeth on the driving gear.

齿轮传递旋转运动和扭矩。齿轮比是从动齿轮的齿数与主动齿轮的齿数之比。

Gear Ratio = Teeth on driven / Teeth on driver

A ratio greater than 1 gives a speed reduction but increases torque; a ratio less than 1 increases speed but reduces torque.

齿轮比大于 1 会减速但增扭；小于 1 会增速但降扭。

In belt and pulley systems, the velocity ratio can be found from the diameters of the pulleys:

在带传动和滑轮系统中，速比可通过带轮直径求得：

VR = Diameter of driven pulley / Diameter of driver pulley

ound gear trains multiply ratios: overall ratio = product of individual mating gear pairs.

复式齿轮系的总传动比为各对啮合齿轮比的乘积。

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9. Hydraulics and Pneumatics Fundamentals | 液压与气动基础

Pascal’s Principle states that pressure applied to an enclosed fluid is transmitted undiminished to every part of the fluid and to the walls of its container.

帕斯卡原理指出，施加在密闭流体上的压强会大小不变地向流体各部分及容器壁传递。

Pressure is defined as force per unit area. This relationship is the basis of hydraulic force multiplication.

压强定义为单位面积上的力。这一关系是液压增力的基础。

P = F / A

In a hydraulic press, a small effort on a small piston generates high pressure, which acts on a larger piston to produce a larger force.

在液压机中，小活塞上的小驱动力产生高压，作用在大活塞上以产生更大的力。

F₂ / F₁ = A₂ / A₁

Pneumatic systems use compressed air. They operate on similar principles but are softer and safer for light loads because air is compressible.

气动系统使用压缩空气。其原理相似，但因为空气可压缩，气动系统更柔和，适用于轻载且更安全。

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10. Material Properties and Destructive Testing | 材料性能与破坏性测试

Tensile tests measure how materials respond to a pulling force. Key values include ultimate tensile strength (UTS), yield strength and percentage elongation.

拉伸测试测量材料对拉力的响应。关键指标包括极限抗拉强度 (UTS)、屈服强度和延伸率。

Hardness is tested by indenting the surface (Brinell, Rockwell, Vickers). Toughness indicates energy absorbed before fracture, while ductility is the ability to stretch into a wire.

硬度通过压痕法测试（布氏、洛氏、维氏）。韧性表示断裂前吸收的能量，而延展性是指材料被拉拔成丝的能力。

Density is mass per unit volume and is important for selecting lightweight materials in design.

密度是单位体积的质量，在设计中选择轻质材料时非常重要。

ρ = m / V

Material selection also considers stiffness, corrosion resistance and thermal conductivity, which are evaluated using standard test pieces.

材料选择还需考虑刚度、耐腐蚀性和导热性，这些性能通过标准试样进行评估。

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11. Engineering Drawings and Standards | 工程图纸与标准

Engineering drawings use orthographic projection (front, top, side views) and isometric views to represent 3D objects. All dimensions are in millimetres unless stated otherwise.

工程图纸使用正投影（主视图、俯视图、侧视图）和等轴测图来表现三维物体。除特别说明外，所有尺寸以毫米为单位。

British Standards (BS 8888) define line types, symbols and dimensioning rules. Hidden edges are shown with dashed lines, and centre lines with long‑dash‑dot lines.

英国标准 (BS 8888) 规定了线型、符号和尺寸标注规则。隐藏边用虚线表示，中心线用长点划线表示。

Scale drawings allow large or small components to be represented proportionally. For example, a scale of 1:2 means the drawing is half the actual size.

比例图纸允许按比例表现大型或小型部件。例如，1:2 的比例表示图纸尺寸为实际尺寸的一半。

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12. Measurement Tools and Uncertainty | 测量工具与不确定度

Vernier calipers (resolution 0.02 mm) and micrometers (resolution 0.01 mm) are used for precise linear measurements. Digital instruments can improve readability and reduce human error.

游标卡尺（分辨率 0.02 mm）和千分尺（分辨率 0.01 mm）用于精密线性测量。数字仪表可提高可读性并减少人为误差。

cale division or half the resolution. Repeated measurements help identify random errors and allow an average to be calculated.

每次测量都有不确定度，通常取最小刻度值或半分辨率的 ±。重复测量有助于识别随机误差，并可计算平均值。

dimension written as 20 ± 0.1 mm means the part must be made between 19.9 mm and 20.1 mm.

工程中的公差是尺寸允许的变动范围，对于互换性零件至关重要。尺寸标注为 20 ± 0.1 mm 表示该零件的制作尺寸必须在 19.9 mm 到 20.1 mm 之间。

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