Year 12 OCR Engineering: Quick Formula and Theorem Reference Guide | 12年级OCR工程：公式定理速查手册

📚 Year 12 OCR Engineering: Quick Formula and Theorem Reference Guide | 12年级OCR工程：公式定理速查手册

This handbook presents the essential formulas, theorems and constants required for the Year 12 OCR Engineering course. Whether you are tackling Engineering Principles exam papers, completing design reports or solving analytical problems, these equations underpin mechanics, electronics, materials and thermal systems. Use this reference to check your understanding and speed up numerical work.

本手册汇总了12年级OCR工程课程中必备的核心公式、定理和常数。不论你是在准备工程原理考试、完成设计报告还是解决分析性问题，这些方程都是力学、电子学、材料与热学系统的基础。利用这份速查手册检验你的理解并提升解题效率。

1. Mechanics and Statics | 力学与静力学

Newton’s Second Law – the net force on a body equals its mass times its acceleration:

ΣF = m a

Apply this vector equation independently in perpendicular directions. For constant mass, ΣF = m a; units are newtons (N).

牛顿第二定律——物体所受合外力等于其质量与加速度的乘积：

ΣF = m a

在相互垂直的方向上独立应用该矢量方程。质量不变时，ΣF = m a，单位为牛顿 (N)。

Torque (moment of a force) about a pivot:

τ = F d

where d is the perpendicular distance from the pivot to the line of action. For static equilibrium the total clockwise moment equals the total anticlockwise moment, giving ΣM = 0.

关于支点的扭矩（力矩）：

τ = F d

其中 d 为支点到力作用线的垂直距离。静力平衡时，顺时针总力矩等于逆时针总力矩，即 ΣM = 0。

Weight is the gravitational force acting on a mass:

W = m g

where g = 9.81 m/s² (standard value on Earth’s surface).

重量是作用在质量上的引力：

W = m g

g 取 9.81 米/秒²（地球表面的标准值）。

Friction force Ff opposes relative motion and is limited by the coefficient of static friction μ:

Ff ≤ μ R

where R is the normal reaction force. The limiting friction occurs just before sliding.

摩擦力 Ff 阻碍相对运动，受静摩擦系数 μ 的限制：

Ff ≤ μ R

R 为法向反作用力。极限摩擦力出现在即将滑动时。

Centre of gravity: for a uniform body the weight acts through its geometric centre. For composite shapes, take moments of individual weights to find the overall centre of gravity.

重心：均匀物体的重力作用线通过其几何中心。对于复合形状，对各部分重量取矩以确定整体重心。

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2. Kinematics of Linear Motion | 直线运动学

For constant acceleration in a straight line, use the SUVAT equations:

v = u + a t

s = u t + ½ a t²

v² = u² + 2 a s

s = ½ (u + v) t

where u = initial velocity (m/s), v = final velocity, a = constant acceleration, t = time, s = displacement.

对于匀加速直线运动，可应用匀加速运动方程（SUVAT）：

v = u + a t

s = u t + ½ a t²

v² = u² + 2 a s

s = ½ (u + v) t

其中 u = 初速度（米/秒），v = 末速度，a = 恒定加速度，t = 时间，s = 位移。

These equations are only valid when acceleration is constant. Choose the equation that includes the given quantities and the unknown you need to find.

以上方程仅在加速度恒定时有效。应选择包含已知量和待求量的方程进行求解。

Average speed:

v_avg = total distance / total time

匀速率：

v_avg = 总路程 / 总时间

For free fall under gravity near Earth’s surface, a = 9.81 m/s² (downwards); use the same equations with appropriate sign conventions.

近地表面自由落体中，a = 9.81 米/秒²（向下）；采用相同方程，注意正负号规定。

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3. Work, Energy and Power | 功、能量与功率

Work done by a constant force:

W = F d cos θ

where θ is the angle between the force and displacement. Work is measured in joules (J).

恒力所做的功：

W = F d cos θ

θ 为力与位移的夹角，单位为焦耳 (J)。

Kinetic energy:

Eₖ = ½ m v²

Gravitational potential energy change near Earth’s surface:

ΔEₚ = m g Δh

动能：

Eₖ = ½ m v²

近地表面重力势能的变化：

ΔEₚ = m g Δh

Power is the rate of doing work:

P = W / t = F v

where v is the velocity in the direction of the force. The SI unit is the watt (W).

功率为做功的速率：

P = W / t = F v

v 为沿力方向的速度。国际单位制为瓦特 (W)。

Efficiency of any energy conversion:

η = (useful output energy / total input energy) × 100%

For mechanical systems, efficiency can also be written as η = (MA / VR) × 100% where MA is mechanical advantage and VR is velocity ratio.

任何能量转换的效率：

η = (有用输出能量 / 总输入能量) × 100%

对机械系统，效率也可表示为 η = (MA / VR) × 100%，其中 MA 为机械效益，VR 为速比。

p>Conservation of energy: in a closed system energy cannot be created or destroyed, only transferred. For a falling object with no air resistance, m g Δh = ½ m v².

能量守恒：在封闭系统中能量不能被创造或消灭，只能被转移。对于无空气阻力的落体，m g Δh = ½ m v²。

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4. Material Properties and Mechanics of Solids | 材料性能与固体力学

Direct stress (tensile or compressive):

σ = F / A

where F is the applied force and A is the cross-sectional area. The unit is the pascal (Pa) or N/m².

正应力（拉伸或压缩）：

σ = F / A

F 为施加的力，A 为横截面积。单位为帕斯卡 (Pa) 或 牛/米²。

Direct strain:

ε = ΔL / L₀

Strain is dimensionless; ΔL is the change in length, L₀ the original length.

正应变：

ε = ΔL / L₀

应变为无量纲量；ΔL 为长度变化，L₀ 为原长。

Young’s modulus (stiffness of a material in the linear elastic region):

E = σ / ε

measured in pascals. For a straight, uniform wire, it also follows E = (F L₀) / (A ΔL).

杨氏模量（材料在线弹性区域的刚度）：

E = σ / ε

单位为帕斯卡。对于均匀直导线，也满足 E = (F L₀) / (A ΔL)。

Shear stress and shear strain are used for twisting and cutting; shear stress τ_s = F / A (parallel to area), shear strain γ = Δx / y. The shear modulus G = τ_s / γ.

剪切应力与剪切应变用于扭转和切割；剪切应力 τ_s = F / A（平行于面积），剪切应变 γ = Δx / y。剪切模量 G = τ_s / γ。

Factor of safety (FoS) for a component:

FoS = ultimate stress / allowable working stress

A higher factor of safety adds reliability against unexpected loads.

零部件的安全系数：

安全系数 = 极限应力 / 许用工作应力

较高的安全系数可增加克服意外载荷的可靠性。

Poisson’s ratio links lateral strain to axial strain:

ν = – (lateral strain / axial strain)

For most engineering metals ν ≈ 0.3.

泊松比将横向应变与轴向应变关联：

ν = – (横向应变 / 轴向应变)

大多数工程金属的泊松比约为0.3。

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5. Fluid Systems | 流体系统

Pressure exerted by a force on a surface:

p = F / A

The SI unit is the pascal (Pa), where 1 Pa = 1 N/m². Pressure in a static fluid increases with depth:

p = ρ g h

where ρ is the fluid density (kg/m³), g is 9.81 m/s², and h is the vertical depth.

力对表面产生的压强：

p = F / A

国际单位是帕斯卡 (Pa)，1 Pa = 1 N/m²。静止流体内压强随深度增加：

p = ρ g h

ρ 为流体密度（千克/米³），g 为9.81 m/s²，h 为竖直深度。

Volumetric flow rate Q (discharge):

Q = A v

where A is the cross-sectional area of the pipe/channel and v is the mean flow velocity. Continuity equation for an incompressible fluid: A₁ v₁ = A₂ v₂.

体积流量 Q：

Q = A v

A 为管道截面积，v 为平均流速。不可压缩流体的连续性方程：A₁ v₁ = A₂ v₂。

Bernoulli’s principle for steady, inviscid, incompressible flow along a streamline (with no height change):

p₁ + ½ ρ v₁² = p₂ + ½ ρ v₂²

When height differences are significant, include gravitational terms: p₁ + ½ ρ v₁² + ρ g h₁ = p₂ + ½ ρ v₂² + ρ g h₂.

伯努利原理（沿流线的定常、无粘、不可压缩流动，无高度变化时）：

p₁ + ½ ρ v₁² = p₂ + ½ ρ v₂²

当高度差显著时，加入重力项：p₁ + ½ ρ v₁² + ρ g h₁ = p₂ + ½ ρ v₂² + ρ g h₂。

Hydraulic systems rely on pressure transmission: for two connected cylinders, p₁ = p₂, so F₁ / A₁ = F₂ / A₂; this gives a mechanical advantage.

液压系统

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