📚 A-Level Eduqas Engineering: Formulas and Theorems Quick Reference Handbook | A-Level Eduqas 工程：公式定理速查手册
This comprehensive quick-reference handbook distills the essential formulas and theorems required for the A-Level Eduqas Engineering syllabus. It is designed to provide clear, bilingual explanations to reinforce understanding and boost exam confidence.
这本全面的速查手册提炼了 A-Level Eduqas 工程课程所需的核心公式与定理，以清晰的双语解释强化理解、提升考试信心。
Each section pairs the governing equation with the physical meaning behind it, so that revision becomes a process of understanding rather than memorising symbols. Use the sections as a checklist, and return to any topic where the definition feels uncertain.
每一节都把主导方程与它背后的物理含义配对呈现，让复习成为理解的过程，而不仅仅是记忆符号。你可以把这些小节当作清单使用，凡是定义仍感模糊的主题就回头再读一遍。
1. Mechanics: Forces and Moments | 力学：力与力矩
For an object to be in static equilibrium, the vector sum of all forces acting upon it must be zero, and the sum of the moments about any point must also be zero. These two conditions underpin almost every structural question in the specification.
要使物体处于静力平衡状态，作用其上的所有力的矢量和必须为零，且对任意点的力矩之和也必须为零。这两个条件支撑着考纲中几乎所有结构类题目。
ΣF = 0 and ΣM = 0
The moment of a force about a pivot is calculated as the product of the force magnitude and the perpendicular distance from the pivot to the line of action of the force.
力对支点的力矩等于力的大小与支点到力作用线的垂直距离的乘积。
M = F × d
A couple is a pair of equal and opposite parallel forces that produce rotation but no net translational force. The torque of a couple is the product of one force and the perpendicular distance between them.
力偶是一对大小相等、方向相反的平行力，产生转动而无净平移力。力偶的扭矩等于其中一个力与两力之间垂直距离的乘积。
T = F × d
When a force acts at an angle θ to the horizontal, resolve it into perpendicular components before summing, because only components along the same line may be added algebraically.
当力与水平方向成 θ 角作用时，先把它分解为互相垂直的分量再求和，因为只有同一直线上的分量才能代数相加。
Fₓ = F cos θ and Fy = F sin θ
2. Stress, Strain and Young's Modulus | 应力、应变与杨氏模量
Tensile stress is the force applied per unit cross-sectional area, while tensile strain is the extension per unit original length.
拉伸应力是单位横截面积上所受的力，而拉伸应变是单位原长上的伸长量。
Stress: σ = F / A Strain: ε = x / L
Young's modulus is the ratio of tensile stress to tensile strain, and it is valid only within the linear elastic region where the stress-strain graph is a straight line through the origin.
杨氏模量是拉伸应力与拉伸应变之比，仅在应力—应变图为过原点直线的线弹性区域内成立。
E = σ / ε = F L / (A x)
10⁹ Pa). Strain has no unit and is often quoted as a percentage or in microstrain (×10⁻⁶).
常见工程单位有帕斯卡（Pa）、兆帕（MPa = 10⁶ Pa）和吉帕（GPa = 10⁹ Pa）。应变没有单位，常用百分数或微应变（×10⁻⁶）表示。
The standard tensile test curve identifies five key features in order: the limit of proportionality, the elastic limit, the yield point, the ultimate tensile strength and the fracture point.
标准拉伸试验曲线依次标出五个关键特征：比例极限、弹性极限、屈服点、极限抗拉强度和断裂点。
3. Beams, Reactions and Bending Moments | 梁、支座反力与弯矩
For a simply supported beam, the vertical reactions must balance the total downward load, and taking moments about one support reveals the reaction at the other.
对于简支梁，竖向支座反力必须与总向下荷载平衡；对其中一个支座取矩即可求出另一端支座的反力。
ΣR = ΣW and ΣM (about a support) = 0
The bending moment at any section is the algebraic sum of the moments of all forces to one side of that section, taken about the section itself.
任一截面处的弯矩等于该截面一侧所有力对该截面取矩的代数和。
M = R × x (maximum for a central point load: M = W L / 4)
For a cantilever carrying a point load at the free end, the maximum bending moment occurs at the fixed support, where the lever arm is greatest.
对于自由端承受集中荷载的悬臂梁，最大弯矩出现在固定支座处，因为那里的力臂最长。
Mmax = F × L
Shear force diagrams change value abruptly at every concentrated load, whereas bending moment diagrams are continuous and fall to zero at simple supports.
剪力图在每个集中荷载处发生突变，而弯矩图是连续的，并且在简支端处降为零。
4. Friction and Inclined Planes | 摩擦与斜面
Dry friction opposes relative motion, and in the limiting case just before sliding begins it equals the coefficient of friction multiplied by the normal reaction.
干摩擦阻碍相对运动；在即将开始滑动的临界状态下，摩擦力等于摩擦系数乘以法向反力。
F = μ R
On an inclined plane the weight resolves into a component parallel to the slope and a component perpendicular to the slope, both measured from the same angle θ.
在斜面上，重力分解为平行于斜面与垂直于斜面的两个分量，两者都由同一个角度 θ 确定。
F_parallel = W sin θ and F_perpendicular = W cos θ
At the angle of repose the block sits on the point of sliding, so the coefficient of friction equals the tangent of the slope angle.
在休止角处物块处于即将滑动的临界状态，因此摩擦系数等于斜面倾角的正切。
μ = tan θ
Static friction is generally larger than kinetic friction, which explains why the force needed to start a body moving exceeds the force needed to keep it moving.
静摩擦通常大于动摩擦，这正好解释了为什么启动一个物体所需的力大于维持其运动所需的力。
5. Work, Energy, Power and Efficiency | 功、能、功率与效率
Work is done when a force moves its point of application through a distance in the direction of the force.
当力的作用点沿力的方向移动一段距离时，就说该力做了功。
W = F d cos θ
Kinetic energy depends on mass and the square of speed, while gravitational potential energy depends on mass, gravitational field strength and change in height.
动能取决于质量与速度的平方，重力势能取决于质量、重力场强度和高度变化。
Ek = ½ m v² and Ep = m g h
Power is the rate of doing work or transferring energy, and for a body moving at constant velocity it equals force multiplied by speed.
功率是做功或传递能量的速率；对匀速运动的物体，功率等于力乘以速度。
P = W / t = F v
Efficiency compares useful output energy with total input energy, and in real machines it is always below 100 per cent because friction and other losses dissipate energy.
效率比较有用输出能量与总输入能量；在实际机器中由于摩擦及其他损耗会耗散能量，效率总是低于 100%。
η = (useful output / total input) × 100%
6. Rotational Motion and Power Transmission | 转动运动与动力传动
Angular velocity may be expressed in radians per second or in revolutions per minute, and the two are linked through a factor of 2π and 60.
角速度可用弧度每秒或每分钟转数表示，两者之间通过 2π 与 60 的因子换算。
ω = 2π f = 2π n / 60
The linear speed of a point on a rotating body equals the angular velocity multiplied by the radius measured from the axis of rotation.
旋转体上某点的线速度等于角速度乘以该点到转轴的半径。
v = ω r
The torque carried by a shaft is obtained from the transmitted power and the angular velocity, which is why a low-speed shaft must be larger than a high-speed one for the same power.
轴所承受的扭矩由传递功率与角速度求得，这正是相同功率下低速轴必须比高速轴做得更大的原因。
T = P / ω
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