📚 Essential Formulae and Theorems for OCR Pre-U Engineering | OCR Pre-U 工程：公式定理速查手册
This quick-reference handbook brings together the core mathematical relationships and scientific principles required for OCR Pre-U Engineering. It is designed for rapid revision before examinations, covering mechanics, materials, thermodynamics, and electrical systems. Each formula is paired with its physical meaning and standard units to help you apply it confidently under timed conditions.
这本速查手册汇集了 OCR Pre-U 工程课程所必需的核心数学关系与科学原理，专为考前快速复习设计，涵盖力学、材料、热力学与电气系统。每条公式均附物理意义与标准单位，帮助你在限时考试中有把握地正确运用。
1. Linear Motion and Kinematics | 直线运动与运动学
For constant acceleration along a straight line, where u is initial velocity (m/s), v is final velocity (m/s), a is acceleration (m/s²), t is time (s), and s is displacement (m):
对于直线上的匀加速度运动，其中 u 为初速度 (m/s)，v 为末速度 (m/s)，a 为加速度 (m/s²)，t 为时间 (s)，s 为位移 (m)：
v = u + a t
s = u t + ½ a t²
v² = u² + 2 a s
s = ½ (u + v) t
These equations are only valid when acceleration is uniform. Always check that the direction of motion is consistent when assigning positive and negative signs.
这些方程仅在加速度恒定时有效。解题时务必确认正方向设定一致，以正确处理正负号。
2. Forces and Newton's Laws | 力与牛顿定律
The resultant force F (N) acting on a body of mass m (kg) produces an acceleration a (m/s²) in the same direction:
作用在质量为 m (kg) 的物体上的合力 F (N) 使其沿同一方向产生加速度 a (m/s²)：
F = m a
an inclined plane at angle θ to the horizontal, the component of weight parallel to the plane is mg sin θ.
当所有力的矢量和为零（即 ΣF = 0）时，物体处于平衡状态。在倾角为 θ 的斜面上，重力沿斜面的分量为 mg sin θ。
Friction force Ff R is the normal reaction (N). For static friction, use μ_s ; for kinetic sliding, use μ_k.
摩擦力 Ff ≤ μ R，其中 μ 为摩擦系数，R 为法向反力 (N)。静摩擦对应静摩擦系数 μ_s，滑动摩擦对应动摩擦系数 μ_k。
3. Moments and Couples | 力矩与力偶
The moment of a force about a point is the product of the force and the perpendicular distance from the point to the line of action of the force. The unit is newton metre (N m).
力对某点的力矩等于力的大小乘以该点到力作用线的垂直距离。单位为牛顿米 (N m)。
M = F d
clockwise. A couple is a pair of equal and opposite parallel forces, producing a torque equal to one force × perpendicular distance between them.
物体处于转动平衡时，对任意支点的顺时针力矩之和等于逆时针力矩之和：ΣM_顺时针 = ΣM_逆时针。力偶是一对大小相等、方向相反的平行力，产生的转矩等于其中一力乘以两力作用线之间的垂直距离。
4. Work, Energy and Power | 功、能与功率
Work done W (J) when a force F (N) moves its point of application by a distance s (m) in the direction of the force:
当力 F (N) 的作用点沿力的方向移动距离 s (m) 时，所做的功 W (J)：
W = F s cos θ
where θ is the angle between the force and the displacement. Kinetic energy Ek = ½ m v². Gravitational potential energy ΔEp = m g Δh (g = 9.81 m/s²).
其中 θ 为力与位移之间的夹角。动能 Ek = ½ m v²。重力势能变化 ΔEp = m g Δh（g = 9.81 m/s²）。
onstant force and velocity, P = F v. Efficiency = useful work output / total energy input.
功率 P (W) 是做功的快慢：P = W / t，或者在恒力与速度情况下，P = F v。效率 = 有用输出功 / 总输入能。
5. Stress, Strain and Young's Modulus | 应力、应变与杨氏模量
e (N) and A is cross-sectional area (m²). Normal strain ε = ΔL / L₀ (dimensionless). Young's modulus E (Pa) describes material stiffness in the elastic region:
正应力 σ (Pa) = F / A，其中 F 为力 (N)，A 为横截面积 (m²)。正应变 ε = ΔL / L₀（无量纲）。杨氏模量 E (Pa) 描述材料在弹性区域的刚度：
E = σ / ε
= x / y (lateral displacement / perpendicular distance). Shear modulus G = τ / γ. Poisson's ratio ν = - lateral strain / longitudinal strain.
剪应力 τ = F / A（力平行于截面积）。剪应变 γ = x / y（横向位移 / 垂直距离）。剪切模量 G = τ / γ。泊松比 ν = - 横向应变 / 纵向应变。
6. Beams and Bending | 梁与弯曲
For a beam in pure bending, the simple bending equation relates bending moment M (N m), second moment of area I (m⁴), stress σ (Pa), Young's modulus E (Pa) and radius of curvature R (m):
对于纯弯曲的梁，简支弯曲方程将弯矩 M (N m)、截面二次矩 I (m⁴)、应力 σ (Pa)、杨氏模量 E (Pa) 与曲率半径 R (m) 联系起来：
M / I = σ / y = E / R
Here y is the distance from the neutral axis. Maximum stress occurs at the outermost fibre. The neutral axis passes through the centroid of the cross-section when the material is homogeneous.
其中 y 为到中性轴的距离。最大应力出现在最外层纤维。均质材料的中性轴通过截面形心。
a: rectangle about base: I = b d³ / 12; solid circle about centroidal axis: I = π d⁴ / 64.
常用截面二次矩：矩形绕基边：I = b d³ / 12；圆形绕形心轴：I = π d⁴ / 64。
7. Torsion of Circular Shafts | 圆轴扭转
For a solid or hollow circular shaft under torque T (N m), the torsion equation links torque, polar second moment of area J (m⁴), shear stress τ (Pa), shear modulus G (Pa), and angle of twist θ (rad) over length L (m):
对于承受扭矩 T (N m) 的实心或空心圆轴，扭转方程将扭矩、截面极二次矩 J (m⁴)、剪应力 τ (Pa)、剪切模量 G (Pa) 及长度 L (m) 上的扭转角 θ (rad) 联系起来：
T / J = τ / r = G θ / L
. Maximum shear stress occurs at the outer surface. Hollow shafts provide a higher strength-to-weight ratio when weight saving is critical.
对于实心圆轴，J = π d⁴ / 32。最大剪应力出现在外表面。在需要减轻重量的场合，空心轴提供更高的比强度。
8. Fluid Statics and Pressure | 流体静力学与压强
ρ g h, where ρ is fluid density (kg/m³) and g gravitational acceleration. Pressure at a point in a fluid acts equally in all directions. Buoyancy force (Archimedes' principle) equals weight of displaced fluid.
静止流体中的压强 p (Pa) 随深度 h (m) 增加，p = ρ g h，其中 ρ 为流体密度 (kg/m³)，g 为重力加速度。流体中某点的压强在各个方向上相等。浮力（阿基米德原理）等于排开流体的重量。
For a manometer, pressure difference Δp = ρ g Δh. Pascal' uid is transmitted undiminished to every part of the fluid and the walls of the container.
对于压力计，压强差 Δp = ρ g Δh。帕斯卡定律：施加于密闭流体的压强将等值地传递到流体各部分和容器壁。
9. Thermodynamics – First Law and Processes | 热力学 – 第一定律与过程
one BY the system (J). For gases, a reversible process pVⁿ = constant describes many processes.
对于封闭系统的热力学第一定律：ΔU = Q - W，其中 ΔU 为内能变化 (J)，Q 为加入系统的热量 (J)，W 为系统对外做的功 (J)。对于气体，可逆过程 pVⁿ = 常数 描述了多种过程：
- Isothermal: n=1, T constant, ΔU=0 so Q=W. pV = constant.
- Adiabatic: n=γ, Q=0, pV^γ = constant, where γ = cp / cv.
- Constant pressure: n=0, W = p ΔV.
- Constant volume: n=∞, W=0, ΔU = Q.
- 等温过程：n=1，T 恒定，ΔU=0 故 Q=W，pV = 常数。
- 绝热过程：n=γ，Q=0，pV^γ = 常数，其中 γ = cp / cv。
- 定压过程：n=0，W = p ΔV。
- 定容过程：n=∞，W=0，ΔU = Q。
Ideal gas equation: p V = n R T = m R_specific T, where R = 8.31 J/(mol K).
理想气体状态方程：p V = n R T = m R_比 T，其中 R = 8.31 J/(mol K)。
10. Electrical Circuits and Ohm's Law | 电路与欧姆定律
Ohm' tor: R = ρ L / A, where ρ is resistivity (Ω m), L length (m), A cross-sectional area (m²).
欧姆定律：V = I R，其中 V 为电压 (V)，I 为电流 (A)，R 为电阻 (Ω)。均匀导体的电阻：R = ρ L / A，其中 ρ 为电阻率 (Ω m)，L 为长度 (m)，A 为截面积 (m²)。
Resistors in series: Rtotal = R₁ + R₂ + ...
Resistors in parallel: 1/Rtotal = 1/R₁ + 1/R₂ + ...
Voltage divider: Vout = Vin × R₂/(R₁ + R₂).
串联电阻：R_总 = R₁ + R₂ + ...
并联电阻：1/R_总 = 1/R₁ + 1/R₂ + ...
分压电路：V_出 = V_入 × R₂/(R₁ + R₂)。
Kirchhoff's current law: ΣIin = ΣIout at a node. Kirchhoff' law: ΣV around any closed loop = 0. Capacitance C = Q / V; capacitor charge Q = C V (1 - e-t/RC).
基尔霍夫电流定律：任一节点上 ΣI_入 = ΣI_出。基尔霍夫电压定律：任一闭合回路中 ΣV = 0。电容 C = Q / V；电容器充电方程 Q = C V (1 - e-t/RC)。
11. AC Circuits and Impedance | 交流电路与阻抗
In an AC circuit with angular frequency ω = 2πf, the reactance of an inductor L is XL = ω L, and of a capacitor C is XC = 1/(ω C). Impedance Z for an RLC series circuit:
在角频率 ω = 2πf 的交流电路中，电感 L 的电抗为 XL = ω L，电容 C 的电抗为 XC = 1/(ω C)。RLC 串联电路的阻抗 Z：
Z = √(R² + (XL - XC)²)
Phase angle φ between voltage and current: tan φ = (XL - XC)/R. The rms value of a sinusoidal voltage V0 is Vrms = V0/√2. Average power P = Vrms Irms cos φ.
电压与电流之间的相角 φ：tan φ = (XL - XC)/R。正弦电压 V0 的有效值为 Vrms = V0/√2。平均功率 P = Vrms Irms cos φ。
12. Data Analysis and Error Propagation | 数据分析与误差传播
or Q = a - b, then absolute uncertainty ΔQ = Δa + Δb. If Q = k a (k constant), ΔQ = k Δa.
当物理量组合时，不确定度会传播。对于独立变量，若 Q = a + b 或 Q = a - b，则绝对不确定度 ΔQ = Δa + Δb。若 Q = k a（k 为常数），ΔQ = k Δa。
/ c, use relative uncertainties: ΔQ/Q = Δa/a + Δb/b + Δc/c. For a power law Q = aⁿ, ΔQ/Q = |n| /a. Always quote final values with absolute uncertainty to one or two significant figures.
对于乘除运算 Q = a × b / c，使用相对不确定度：ΔQ/Q = Δa/a + Δb/b + Δc/c。对于幂函数 Q = aⁿ，ΔQ/Q = |n| Δa/a。最终结果需用一至两位有效数字的绝对不确定度报告。
gradient m and intercept c are estimated with uncertainty; use min-max lines to determine error ranges.
绘制线性图：y = m x + c。斜率 m 和截距 c 需估计不确定度；使用最大/最小斜率线确定误差范围。
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