📚 GCSE OCR Engineering: Essential Formula & Theorem Quick Reference Handbook | GCSE OCR 工程：公式定理速查手册
This quick reference handbook brings together the most important formulas, definitions, and theorems you will encounter in the GCSE OCR Engineering course. Each section presents key equations with clear notation, practical applications, and bilingual explanations to support your revision and exam preparation.
本速查手册汇集了 GCSE OCR 工程课程中最重要的公式、定义和定理。每个章节都以清晰的符号、实际应用和双语解释呈现核心公式，助力复习与备考。
1. Hooke’s Law | 胡克定律
Hooke’s Law describes the linear relationship between the force applied to a spring and its extension, provided the elastic limit is not exceeded. The spring constant k indicates the stiffness of the spring.
胡克定律描述了在不超过弹性极限时，施加在弹簧上的力与其伸长量之间的线性关系。弹簧常数 k 表示弹簧的刚度。
| Formula | 中文名称 | Equation | SI Units |
|---|---|---|---|
| Hooke’s Law | 胡克定律 | F = k x | N, N/m, m |
The extension x is measured from the original length; the force F is the applied load. The spring constant k has units of newtons per metre (N/m).
伸长量 x 从原长开始测量；力 F 为施加的载荷。弹簧常数 k 的单位是牛顿每米 (N/m)。
2. Stress and Strain | 应力与应变
Stress measures the internal force per unit area within a material, while strain is the ratio of extension to original length. These dimensionless quantities help engineers predict material behaviour under loading.
应力衡量材料内部单位面积上的内力，而应变是伸长量与原长之比。这些量纲一的量帮助工程师预测材料在载荷下的行为。
| Formula | 中文名称 | Equation | SI Units |
|---|---|---|---|
| Tensile Stress | 拉应力 | σ = F / A | Pa or N/m² |
| Tensile Strain | 拉应变 | ε = ΔL / L₀ | no unit |
Here σ is the stress, F is the applied force, A is the cross‑sectional area. ΔL is the change in length and L₀ is the original length. Strain is often expressed as a percentage.
式中 σ 为应力，F 为施加的力，A 为横截面积。ΔL 为长度变化量，L₀ 为原始长度。应变通常以百分比表示。
3. Young’s Modulus | 杨氏模量
Young’s modulus E describes the stiffness of a solid material. It is the gradient of the stress–strain graph in the linear region and is a material property that does not depend on shape or size.
杨氏模量 E 描述固体材料的刚度。它是应力-应变图在线性区域的斜率，是一种与形状尺寸无关的材料属性。
| Formula | 中文名称 | Equation | SI Units |
|---|---|---|---|
| Young’s Modulus | 杨氏模量 | E = σ / ε | Pa or N/m² |
A high modulus means the material is stiff and resists deformation. Engineering metals typically have E values of 70–200 GPa, while polymers are much lower.
高模量意味着材料刚硬，不易变形。工程金属的 E 值通常在 70–200 GPa，而聚合物则低得多。
4. Mechanical Advantage and Velocity Ratio | 机械利益与速度比
Mechanical advantage (MA) tells us how much a mechanism multiplies the effort force, while velocity ratio (VR) compares the distance moved by the effort to the distance moved by the load. They are crucial for analysing levers, pulleys, gears and other simple machines.
机械利益 (MA) 反映机械增力的倍数，而速度比 (VR) 比较动力移动的距离与负载移动的距离。它们是分析杠杆、滑轮、齿轮及其它简单机械的关键参数。
| Formula | 中文名称 | Equation | Units |
|---|---|---|---|
| Mechanical Advantage | 机械利益 | MA = Load / Effort | no unit |
| Velocity Ratio | 速度比 | VR = Distance moved by effort / Distance moved by load | no unit |
| Gear Ratio (simple) | 齿轮比 | Gear ratio = Teeth on driven gear / Teeth on driver gear | no unit |
For an ideal frictionless system MA equals VR, but real mechanisms always have MA < VR due to energy losses.
在理想无摩擦系统中 MA 等于 VR，但实际机械因能量损失总有 MA ＜ VR。
5. Efficiency | 效率
Efficiency measures how well a machine converts input energy into useful output. It is always less than 100% because of friction, heat and sound losses.
效率衡量机器将输入能量转化为有用输出的程度。由于摩擦、发热和噪声损失，效率永远低于 100%。
| Formula | 中文名称 | Equation | Units |
|---|---|---|---|
| Efficiency from energy | 能量效率 | η = (Useful energy output / Total energy input) × 100% | % |
| Efficiency from MA and VR | 机械效率 | η = (MA / VR) × 100% | % |
Improving lubrication, reducing friction and using lightweight components can increase the efficiency of a mechanical system.
改善润滑、减小摩擦以及使用轻量化部件可以提高机械系统的效率。
6. Work and Energy | 功与能
Work is done when a force moves an object in the direction of the force. Energy is the capacity to do work and exists in many forms, including kinetic energy and gravitational potential energy.
当力使物体沿力的方向移动时就做了功。能量是做功的能力，以多种形式存在，包括动能和重力势能。
| Formula | 中文名称 | Equation | SI Units |
|---|---|---|---|
| Work Done | 功 | W = F d (force parallel to displacement) | J (N m) |
| Kinetic Energy | 动能 | KE = ½ m v² | J |
| Gravitational Potential Energy | 重力势能 | GPE = m g h | J |
The constant g is the gravitational field strength, taken as 9.8 N/kg on Earth (often 10 N/kg in GCSE problems). Energy is always conserved; it is transferred or transformed, never destroyed.
常数 g 为重力场强度，在地球上取 9.8 N/kg（GCSE 题目中常用 10 N/kg）。能量总是守恒的，只能转移或转化，不会消失。
7. Power | 功率
Power is the rate at which energy is transferred or work is done. A machine with a high power rating can do the same amount of work in a shorter time.
功率是能量传递或做功的速率。额定功率高的机器能在更短的时间内完成同样的功。
| Formula | 中文名称 | Equation | SI Units |
|---|---|---|---|
| Mechanical Power | 机械功率 | P = W / t or P = E / t | W (J/s) |
Power can also be calculated for moving objects using P = F v, where v is the constant velocity. This is useful for vehicles and conveying systems.
对于运动物体还可用 P = F v 计算功率，其中 v 为恒定速度。这在交通工具和传送系统中很有用。
8. Levers and Moments | 杠杆与力矩
A moment is the turning effect of a force about a pivot. The principle of moments states that for a system in equilibrium, the total clockwise moments equal the total anticlockwise moments.
力矩是力绕支点产生的转动效应。力矩原理指出，对于平衡系统，顺时针力矩之和等于逆时针力矩之和。
| Formula | 中文名称 | Equation | SI Units |
|---|---|---|---|
| Moment of a force | 力矩 | M = F × d | N m |
| Equilibrium condition | 平衡条件 | Σ clockwise moments = Σ anticlockwise moments | – |
The perpendicular distance d must be measured from the line of action of the force to the pivot. If the force is not perpendicular, use the perpendicular component.
垂直距离 d 必须从力的作用线到支点测量。若力不垂直，则取其垂直分量。
9. Electrical Circuits and Ohm’s Law | 电路与欧姆定律
Ohm’s Law links voltage, current and resistance in a metallic conductor at constant temperature. Electrical power calculations are essential for selecting components and ensuring safe operation.
欧姆定律将金属导体在恒温下的电压、电流与电阻联系起来。电功率计算对于选择元件和确保安全运行至关重要。
| Formula | 中文
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