📚 Year 11 OCR Engineering: Formula & Theorem Quick Reference Handbook | Year 11 OCR 工程：公式定理速查手册
This comprehensive quick reference handbook collates the essential formulas, theorems, and principles required for the Year 11 OCR Engineering curriculum. It has been designed as a one-stop revision tool, enabling you to swiftly locate and apply key relationships in mechanics, electronics, materials, and thermodynamics. Each entry is accompanied by clear definitions and practical contexts drawn from engineering science.
这本全面的速查手册汇编了 Year 11 OCR 工程课程所需的基本公式、定理和原理。它被设计为一站式复习工具，使你能够快速查找并应用力学、电子学、材料学与热力学中的关键关系。每个条目都配有清晰的定义和来自工程科学的实际应用情境。
1. Ohm's Law & Electrical Power | 欧姆定律与电功率
Ohm's Law is the cornerstone of circuit analysis, stating that the potential difference (V) across a conductor is directly proportional to the current (I) flowing through it, provided the temperature remains constant. The constant of proportionality is the resistance (R).
欧姆定律是电路分析的基石，它指出在温度保持恒定的条件下，导体两端的电势差 (V) 与流过导体的电流 (I) 成正比，其比例常数即为电阻 (R)。
V = I × R
| Symbol | Name (EN) | 名称 (中文) | Unit |
|---|---|---|---|
| V | Voltage | 电压 | Volt (V) |
| I | Current | 电流 | Ampere (A) |
| R | Resistance | 电阻 | Ohm (Ω) |
Electrical power (P) quantifies the rate at which electrical energy is transferred. In a resistive load, it can be expressed in several equivalent forms by combining Ohm's Law with the power definition.
电功率 (P) 量化了电能传递的速率。对于电阻性负载，可以通过结合欧姆定律与功率定义将其表达为几个等价的形式。
P = I × V
P = I²R
P = V² / R
- P: Power (Watt, W)
- I: Current (A)
- V: Voltage (V)
- R: Resistance (Ω)
- P：功率（瓦特，W）
- I：电流（A）
- V：电压（V）
- R：电阻（Ω）
2. Resistors in Series & Parallel | 电阻的串联与并联
When resistors are connected in series, the total resistance is the sum of the individual resistances. This arrangement increases the overall resistance and divides the voltage.
当电阻以串联方式连接时，总电阻为各个电阻值之和。这种布局会增大总电阻并分压。
Rₜₒₜₐₗ = R₁ + R₂ + R₃ + …
For resistors in parallel, the reciprocal of the total resistance equals the sum of the reciprocals of the individual resistances. The total resistance is always less than the smallest individual resistor, and the current divides among the branches.
对于并联电阻，总电阻的倒数等于各个电阻倒数之和。总电阻始终小于最小的单个电阻，并且电流在各支路中分配。
1/Rₜₒₜₐₗ = 1/R₁ + 1/R₂ + 1/R₃ + …
For two parallel resistors, a simplified product-over-sum formula is often used.
对于两个并联电阻，常使用简化的积除以和公式。
Rₜₒₜₐₗ = (R₁ × R₂) / (R₁ + R₂)
3. Stress, Strain & Young's Modulus | 应力、应变与杨氏模量
Engineering stress (σ) measures the internal force acting per unit area within a material. It is calculated by dividing the applied force by the cross-sectional area over which it acts.
工程应力 (σ) 衡量材料内部单位面积上作用的内力，由施加的力除以受力横截面积计算得出。
σ = F / A
Engineering strain (ε) is the measure of deformation representing the relative change in length. It is dimensionless and often expressed as a percentage.
工程应变 (ε) 是衡量变形的量，表示长度的相对变化。它是一个无量纲量，通常以百分数表示。
ε = ΔL / L₀
- ΔL: Change in length (m)
- L₀: Original length (m)
- ΔL：长度变化量（米）
- L₀：原始长度（米）
Young's Modulus (E) is a measure of a material's stiffness. It is the ratio of tensile stress to tensile strain within the linear elastic region.
杨氏模量 (E) 是衡量材料刚度的量，为线弹性范围内拉伸应力与拉伸应变之比。
E = σ / ε
The unit of Young's Modulus is the Pascal (Pa) or N/m².
杨氏模量的单位是帕斯卡 (Pa) 或 N/m²。
4. Moments & Equilibrium | 力矩与平衡
A moment is the turning effect of a force about a pivot. Its magnitude depends on the size of the force and the perpendicular distance from the pivot to the line of action of the force.
力矩是力绕支点产生的转动效应，其大小取决于力的大小和从支点到力作用线的垂直距离。
M = F × d
| Symbol | Name (EN) | 名称 (中文) | Unit |
|---|---|---|---|
| M | Moment | 力矩 | Newton metre (Nm) |
| F | Force | 力 | Newton (N) |
| d | Perpendicular distance | 垂直距离 | Metre (m) |
The principle of moments states that for a system to be in rotational equilibrium, the sum of the clockwise moments about a pivot must equal the sum of the anticlockwise moments.
力矩原理指出，要使系统处于转动平衡，绕支点的顺时针力矩之和必须等于逆时针力矩之和。
∑ M꜀ₗₒ꜀ₖwᵢₛₑ = ∑ Mₐₙₜᵢ꜀ₗₒ꜀ₖwᵢₛₑ
5. Mechanical Advantage & Velocity Ratio | 机械效益与速度比
Mechanical Advantage (MA) is the factor by which a mechanism multiplies the input force to overcome a larger load. It is the ratio of load to effort.
机械效益 (MA) 是机构将输入力放大以克服更大负载的倍数，即负载与施力之比。
MA = Load / Effort
Velocity Ratio (VR) indicates how the distance moved by the effort compares with the distance moved by the load in the same time, reflecting the ideal mechanical advantage without friction.
速度比 (VR) 表示在相同时间内施力端移动距离与负载端移动距离的比值，反映了无摩擦时的理想机械效益。
VR = Distance moved by effort / Distance moved by load
For simple machines like levers, pulleys, and gears, both MA and VR are essential for analysing performance.
对于杠杆、滑轮和齿轮等简单机械，MA 和 VR 对于分析性能都至关重要。
6. Efficiency | 效率
Efficiency (η) expresses the proportion of useful work output compared with the total energy input. It is commonly given as a percentage.
效率 (η) 表示有用功输出与总能量输入的比值，通常以百分数表示。
Efficiency = (Useful work output / Total energy input) × 100%
In mechanical systems, efficiency can also be calculated from the mechanical advantage and velocity ratio.
在机械系统中，效率也可通过机械效益和速度比计算。
Efficiency = (MA / VR) × 100%
An ideal, frictionless machine would have an efficiency of 100%, but real systems always have losses due to friction, heat, and sound.
理想的无摩擦机器效率为 100%，但实际系统总会因摩擦、发热和噪音而产生损耗。
7. Gear Ratios & Pulley Systems | 齿轮比与滑轮系统
The gear ratio determines the relationship between the rotational speeds of two meshing gears. It is inversely proportional to the number of teeth.
齿轮比决定了两个啮合齿轮转速之间的关系，与齿数成反比。
Gear Ratio = Number of teeth on driven gear / Number of teeth on driver gear
GR = N₂ / N₁
Where N₂ is the driven gear and N₁ is the driver. A gear ratio greater than 1 indicates an increase in torque and a reduction in speed.
其中 N₂ 是从动齿轮，N₁ 是主动齿轮。齿轮比大于 1 表示扭矩增大、转速降低。
In a pulley system, the velocity ratio is equal to the number of rope sections supporting the load (for a simple block and tackle).
在滑轮系统中，速度比等于支撑负载的绳索段数（对于简单的滑车组）。
VR (Pulley) = Number of supporting rope segments
8. Equations of Motion (SUVAT) | 运动学方程 (SUVAT)
For objects moving with uniform acceleration in a straight line, the SUVAT equations link displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t).
对于沿直线做匀加速运动的物体，SUVAT 方程将位移 (s)、初速度 (u)、末速度 (v)、加速度 (a) 和时间 (t) 联系起来。
v = u + a t
s = u t + ½ a t²
v² = u² + 2 a s
s = ½ (u + v) t
These equations are used extensively in dynamics, vehicle analysis, and projectile motion studies.
这些方程在动力学、车辆分析和抛体运动研究中被广泛使用。
9. Work, Energy & Power | 功、能量与功率
Work done (W) is the product of the force applied and the distance moved in the direction of the force. It represents a transfer of energy.
做功 (W) 是施加的力与沿力方向移动的距离的乘积，表示能量的转移。
W = F × d
Kinetic energy (KE) is the energy possessed by a moving body, and gravitational potential energy (GPE) is the energy stored due to an object's height in a gravitational field.
动能 (KE) 是运动物体所具有的能量，重力势能 (GPE) 是因物体在重力场中的高度而储存的能量。
KE = ½ m v²
GPE = m g h
- m: mass (kg)
- v: velocity (m/s)
- g: gravitational field strength (9.8 m/s² on Earth)
- h: height (m)
- m：质量（千克）
- v：速度（米/秒）
- g：重力场强度（地球约为 9.8 m/s²）
- h：高度（米）
Power (P) is the rate of doing work. For a constant force moving at a steady speed, power can also be expressed as the product of force and velocity.
功率 (P) 是做功的速率。对于恒力稳态运动，功率也可表示为力与速度的乘积。
P = W / t
P = F × v
10. Pressure & Density | 压强与密度
Pressure (p) is defined as the normal force applied per unit area. It explains how concentrated a force is on a surface and is critical in hydraulic systems and material testing.
压强 (p) 定义为单位面积上施加的法向力。它解释了力在表面上的集中程度，在液压系统和材料测试中至关重要。
p = F / A
The unit of pressure is the Pascal (Pa), equivalent to 1 N/m².
压强的单位是帕斯卡 (Pa)，等于 1 N/m²。
Density (ρ) is mass per unit volume and is a key property for material selection and buoyancy calculations.
密度 (ρ) 是单位体积的质量，是材料选择和浮力计算的关键属性。
ρ = m / V
| Symbol | Name (EN) | 名称 (中文) | Unit |
|---|---|---|---|
| ρ | Density | 密度 | kg/m³ |
| m | Mass | 质量 | kg |
| V | Volume | 体积 | m³ |
11. Specific Heat Capacity & Thermal Energy | 比热容与热能
The quantity of thermal energy (Q) required to change the temperature of a substance depends on its mass (m), specific heat capacity (c), and the temperature change (Δθ).
改变物质温度所需的热能 (Q) 取决于其质量 (m)、比热容 (c) 和温度变化量 (Δθ)。
Q = m c Δθ
Specific heat capacity is the energy needed to raise 1 kg of a material by 1 °C (or 1 K). It is an important parameter in heat sinks, engine cooling, and manufacturing processes.
比热容是使 1 千克材料升高 1 °C（或 1 K）所需的能量，是散热器、发动机冷却和制造过程中的重要参数。
