📚 Year 10 OCR Engineering: Formulas & Theorems Quick Reference Handbook | Year 10 OCR 工程：公式定理速查手册
This handbook compiles the essential formulas, laws and theorems you will encounter throughout the Year 10 OCR Engineering curriculum. Use it as a rapid reference while completing design tasks, analysing mechanical systems or preparing for written assessments. Each entry pairs a clear statement of the principle with the corresponding mathematical expression, allowing you to check relationships quickly and deepen your understanding of how engineered products behave.
本手册汇编了你将在 Year 10 OCR 工程课程中遇到的所有核心公式、定律和定理。你可以将其作为快速参考，在完成设计任务、分析机械系统或准备书面评估时随手查阅。每个条目都将原理的清晰陈述与对应的数学表达式配对呈现，帮助你快速核对关系，加深对工程产品行为方式的理解。
1. Mechanics and Forces | 力学与力
Newton’s Second Law tells us that the resultant force on an object equals its mass multiplied by its acceleration. This relationship underpins all structural and dynamic calculations in engineering.
F = m × a
牛顿第二定律指出，作用在物体上的合力等于物体质量与其加速度的乘积。该关系是工程中所有结构和动力学计算的基础。
F = m × a
Weight is the force due to gravity acting on a mass. On Earth, the gravitational field strength g is approximately 9.8 N/kg.
W = m × g
重量是重力作用在质量上的力。在地球上，重力场强度 g 约为 9.8 N/kg。
W = m × g
Pressure measures how a force is distributed over an area. A small area subjected to a given force yields a high pressure, which is critical in cutting tools and hydraulic systems.
P = F / A
压强衡量力在面积上的分布情况。给定力作用在较小面积上会产生高压，这在切削刀具和液压系统中至关重要。
P = F / A
The moment of a force about a pivot is the product of the force and the perpendicular distance from the pivot to the line of action. Moments govern levers, spanners and the stability of structures.
M = F × d
力对支点的力矩等于力与支点到力作用线的垂直距离的乘积。力矩支配着杠杆、扳手和结构的稳定性。
M = F × d
2. Motion and Kinematics | 运动与运动学
Average speed is the total distance travelled divided by the time taken. It is a scalar quantity and describes how fast an object moves regardless of direction.
v = s / t
平均速率等于总路程除以所用时间。它是标量，描述物体运动的快慢而不计方向。
v = s / t
Acceleration is the rate of change of velocity. When an object changes its velocity from an initial value u to a final value v over time t, the uniform acceleration is given by:
a = (v – u) / t
加速度是速度的变化率。当物体在时间 t 内从初始速度 u 变化到最终速度 v 时，匀加速度由下式给出：
a = (v – u) / t
The SUVAT equations describe uniformly accelerated motion in a straight line. The three most commonly used forms are:
v = u + a t
s = u t + ½ a t²
v² = u² + 2 a s
SUVAT 方程描述匀加速直线运动。最常用的三种形式为：
v = u + a t
s = u t + ½ a t²
v² = u² + 2 a s
3. Energy, Work and Power | 能量、功与功率
Work done by a constant force acting in the direction of displacement is the product of the force and the distance moved. Work is a scalar quantity measured in joules (J).
W = F d
恒力在位移方向所做的功等于力与移动距离的乘积。功是标量，单位为焦耳 (J)。
W = F d
Kinetic energy is the energy possessed by an object due to its motion. It depends on the mass and the square of the velocity.
Eₖ = ½ m v²
动能是物体由于运动而具有的能量，其数值取决于质量和速度的平方。
Eₖ = ½ m v²
Gravitational potential energy is stored when an object is raised against gravity. It is proportional to mass, height and the gravitational field strength.
Eₚ = m g h
重力势能是物体克服重力升高时储存的能量，与质量、高度和重力场强度成正比。
Eₚ = m g h
Power is the rate at which work is done or energy is transferred. A high power output means a large amount of work is performed in a short time.
P = W / t
功率是做功或能量转移的速率。高功率输出意味着在短时间内完成大量功。
P = W / t
Efficiency compares the useful output to the total input energy (or power) of a system. It is often expressed as a percentage and can never exceed 100%.
Efficiency = (useful output energy / total input energy) × 100%
效率将系统有用输出与总输入能量（或功率）进行比较。它常以百分比表示，且永不超过 100%。
效率 = ( 有用输出能量 / 总输入能量 ) × 100%
4. Material Properties and Stress–Strain | 材料性质与应力–应变
Stress measures the internal force carried by a material per unit cross‑sectional area. It helps engineers predict whether a component will fail under tension or compression.
σ = F / A
应力衡量材料单位横截面积所承受的内力，帮助工程师预测部件在拉伸或压缩下是否会失效。
σ = F / A
Strain quantifies the deformation experienced by a material relative to its original length. Being a ratio, it has no units.
ε = ΔL / L₀
应变量化材料相对于原始长度所经历的变形。由于是比值，它没有单位。
ε = ΔL / L₀
Young’s modulus describes the stiffness of a material in the elastic region where stress and strain are directly proportional. A steep gradient indicates a stiff material.
E = σ / ε
杨氏模量描述材料在弹性区域内（应力与应变成正比）的刚度。梯度越大表示材料越刚硬。
E = σ / ε
The factor of safety is a design margin that ensures a component can withstand loads well beyond the expected maximum. It is the ratio of ultimate tensile stress to the allowable working stress.
Factor of Safety = Ultimate Stress / Working Stress
安全系数是为确保部件能承受远超预期最大载荷而设置的设计裕度，即极限拉伸应力与许用工作应力的比值。
安全系数 = 极限应力 / 工作应力
5. Electricity and Circuits | 电与电路
Ohm’s Law is the fundamental relationship linking voltage, current and resistance in a metallic conductor at constant temperature. It allows engineers to design safe and functional electrical systems.
V = I R
欧姆定律是在恒温条件下金属导体中联系电压、电流和电阻的基本关系，使工程师能够设计安全且功能正常的电气系统。
V = I R
Electrical power can be calculated using any combination of voltage, current and resistance. The three equivalent forms are used depending on the known quantities.
P = V I P = I² R P = V² / R
电功率可以利用电压、电流和电阻的任意组合进行计算。三种等价形式根据已知量选用。
P = V I P = I² R P = V² / R
In a series circuit, the total resistance is simply the sum of the individual resistances. In a parallel circuit, the reciprocal of the total resistance equals the sum of the reciprocals of each branch resistance.
Series: Rₜ = R₁ + R₂ + R₃ + …
Parallel: 1/Rₜ = 1/R₁ + 1/R₂ + 1/R₃ + …
在串联电路中，总电阻等于各电阻之和。在并联电路中，总电阻的倒数等于各支路电阻倒数之和。
串联：Rₜ = R₁ + R₂ + R₃ + …
并联：1/Rₜ = 1/R₁ + 1/R₂ + 1/R₃ + …
6. Electronics and Components | 电子学与元件
A potential divider uses two resistors to produce a fraction of the supply voltage. It is the basis of many sensor circuits in electronic products.
Vₒᵤₜ = Vᵢₙ × (R₂ / (R₁ + R₂))
分压器利用两个电阻产生电源电压的一部分，是许多电子产品中传感器电路的基础。
Vₒᵤₜ = Vᵢₙ × (R₂ / (R₁ + R₂))
Kirchhoff’s Current Law states that the total current entering a junction equals the total current leaving it. This conservation principle is essential when analysing parallel branches.
Σ Iᵢₙ = Σ Iₒᵤₜ
基尔霍夫电流定律指出，流入节点的总电流等于流出节点的总电流。这一守恒原理在分析并联支路时至关重要。
Σ Iᵢₙ = Σ Iₒᵤₜ
Kirchhoff’s Voltage Law asserts that the sum of the potential differences around any closed loop in a circuit is zero. It ensures energy is conserved around a loop.
Σ V = 0
基尔霍夫电压定律断言，电路中任一闭合回路的电位差之和为零。它确保能量在回路中守恒。
Σ V = 0
7. Thermal Physics for Engineers | 工程热物理
Linear thermal expansion occurs when a material’s length changes due to a temperature variation. The expansion depends on the original length, the coefficient of linear expansion and the temperature change.
ΔL = α L₀ ΔT
线性热膨胀是指材料长度因温度变化而改变的现象。膨胀量取决于原始长度、线膨胀系数和温度变化。
ΔL = α L₀ ΔT
Heat energy required to change an object’s temperature is calculated using specific heat capacity. Different materials need different amounts of energy to raise the temperature of 1 kg by 1 °C.
Q = m c Δθ
改变物体温度所需的热能用比热容计算。不同材料每千克每升高 1 °C 所需能量不同。
Q = m c Δθ
The rate of conductive heat transfer through a slab is proportional to the temperature difference, the area and the thermal conductivity, and inversely proportional to the thickness.
Q / t = k A ΔT / d
通过平板的导热速率与温差、面积和导热系数成正比，与厚度成反比。
Q / t = k A ΔT / d
8. Fluid Mechanics | 流体力学
The pressure at a depth in a static fluid is due to the weight of the fluid above. This expression is critical in designing dams, storage tanks and hydraulic systems.
P = ρ g h
静态流体中某深度的压强源于上方流体的重量。该表达式在设计水坝、储罐和液压系统时至关重要。
P = ρ g h
Pascal’s principle states that a pressure change applied to an enclosed incompressible fluid is transmitted undiminished to every part of the fluid. This explains the multiplication of force in hydraulic presses.
F₁ / A₁ = F₂ / A₂
帕斯卡原理表明，施加于封闭不可压缩流体上的压强变化会等值地传递至流体各处。这解释了液压机中力的放大作用。
F₁ / A₁ = F₂ / A₂
Archimedes’ principle: a body immersed in a fluid experiences an upthrust equal to the weight of the fluid it displaces. This determines whether an object floats or sinks.
Upthrust = ρ_fluid × V_displaced × g
阿基米德原理：浸入流体中的物体受到向上的浮力，其大小等于物体排开流体的重量。这决定物体是浮还是沉。
浮力 = ρ_流体 × V_排开 × g
9. Simple Machines and Mechanical Advantage | 简单机械与机械效益
The law of the lever states that a lever is balanced when the clockwise moments equal the anticlockwise moments about the pivot. This is the foundation of all lever-based tools.
F₁ d₁ = F₂ d₂
杠杆定律指出，当绕支点的顺时针力矩等于逆时针力矩时，杠杆平衡。这是所有杠杆类工具的基础。
F₁ d₁ = F₂ d₂
Mechanical advantage (MA) compares the load force overcome by a machine to the effort applied. A value greater than 1 indicates a force multiplier.
MA = Load / Effort
机械效益 (MA) 是将机器克服的负载力与所施加的动力进行对比。数值大于 1 表明是力放大器。
MA = 负载 / 动力
Velocity ratio (VR) relates the distance moved by the effort to the distance moved by the load in the same time. It is determined solely by the geometry of the machine.
VR = distance moved by effort / distance moved by load
速度比 (VR) 将动力的移动距离与同一时间内负载的移动距离关联起来，该值仅由机器的几何结构决定。
VR = 动力的移动距离 / 负载的移动距离
Gear ratio describes the relationship between two meshing gears. A ratio greater than 1 indicates a speed increase and torque reduction, while a ratio less than 1 indicates a torque multiplication.
Gear Ratio = Teeth on driven gear / Teeth on driver gear
齿轮比描述两个啮合齿轮之间的关系。比值大于 1 表示增速降扭，小于 1 表示增扭减速。
齿轮比 = 从动轮齿数 / 主动轮齿数
The efficiency of a simple machine can also be expressed in terms of mechanical advantage and velocity ratio.
Efficiency = (MA / VR) × 100%
简单机械的效率也可用机械效益与速度比表示。
效率 = (MA / VR) × 100%
10. Engineering Geometry and Trigonometry | 工程几何与三角
In any right‑angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. This theorem is used routinely in vector resolution and structural design.
c² = a² + b²
在任何直角三角形中，斜边的平方
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