Year 12 OCR Engineering: Formula & Theorem Quick-Reference Handbook | Year 12 OCR 工程：公式定理速查手册

📚 Year 12 OCR Engineering: Formula & Theorem Quick-Reference Handbook | Year 12 OCR 工程：公式定理速查手册

This handbook provides a concise compilation of essential formulas and theorems for Year 12 OCR Engineering, covering mechanics, materials, and electrical principles. It is designed for quick revision and reference during problem-solving and exam preparation.

本手册为 Year 12 OCR 工程课程提供核心公式与定理的简明汇编，涵盖力学、材料与电学原理，用于在解题和备考过程中快速复习与查阅。

1. Forces and Free-Body Diagrams | 力与受力分析图

For a body in static equilibrium, the vector sum of all forces acting upon it must be zero. This can be expressed in two perpendicular components, typically horizontal and vertical.

处于静态平衡的物体，作用在其上的所有力矢量和必须为零。这个条件可以分解为两个相互垂直的分量，通常取水平和竖直方向。

ΣFₓ = 0    ΣFᵧ = 0

Resolving a force F at an angle θ to the horizontal gives the components: Fₓ = F cosθ, Fᵧ = F sinθ. Free-body diagrams isolate the body and show all external forces, including weight (W = mg), normal reaction, friction, and applied forces.

将一个与水平面成 θ 角的力 F 分解可得分量：Fₓ = F cosθ, Fᵧ = F sinθ。受力分析图将物体隔离，显示出所有外力，包括重力（W = mg）、法向反力、摩擦力和施加的力。

量 / Quantity	符号 / Symbol	单位 / Unit
合力 / Resultant force	F	N
质量 / Mass	m	kg
重力加速度 / Gravitational field strength	g	N kg⁻¹ or m s⁻²

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2. Moments and Equilibrium of Rigid Bodies | 力矩与刚体平衡

The moment of a force about a point is a measure of its turning effect. For equilibrium, the sum of clockwise moments must equal the sum of anticlockwise moments about any pivot.

力对某点的力矩是力产生转动效应的量度。当物体平衡时，绕任意支点，顺时针力矩之和必须等于逆时针力矩之和。

M = F × d    ΣM꜀ⱼ = ΣMₐ꜀ⱼ

Here d is the perpendicular distance from the line of action of the force to the pivot. A couple consists of two equal, parallel but opposite forces, producing a pure torque with no net force.

式中 d 为力的作用线到支点的垂直距离。力偶由两个大小相等、平行且方向相反的力组成，产生纯扭矩而合力为零。

量 / Quantity	符号 / Symbol	单位 / Unit
力矩 / Moment	M	N m
力 / Force	F	N
垂直距离 / Perpendicular distance	d	m

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3. Hooke’s Law and Elastic Potential Energy | 胡克定律与弹性势能

Hooke’s law states that the extension of a spring is directly proportional to the applied force, provided the elastic limit is not exceeded. The spring constant k characterises stiffness.

胡克定律指出，在不超过弹性极限的情况下，弹簧的伸长量与所施加的力成正比。劲度系数 k 表征弹簧的刚度。

F = k Δx

The elastic potential energy stored in a stretched spring is equal to the area under the force-extension graph. For a linear spring, this area is a triangle.

拉伸的弹簧中储存的弹性势能等于力-伸长图下的面积。对于线性弹簧，该面积为一个三角形。

Eₑ = ½ k (Δx)² = ½ F Δx

量 / Quantity	符号 / Symbol	单位 / Unit
力 / Force	F	N
劲度系数 / Spring constant	k	N m⁻¹
伸长量 / Extension	Δx	m
弹性势能 / Elastic potential energy	Eₑ	J

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4. Stress, Strain and Young’s Modulus | 应力、应变与杨氏模量

Stress is the internal resistive force per unit area, while strain is the fractional change in length. These quantities describe how materials deform under load without reference to sample dimensions.

应力是单位面积上的内部阻力，应变是长度的相对变化量。这些量描述材料在载荷下变形的行为，而不依赖于试样的尺寸。

σ = F / A    ε = ΔL / L₀

For many materials the initial region of the stress-strain curve is linear, obeying Hooke’s law in its material form. Young’s modulus E is the ratio of stress to strain in this linear region.

许多材料应力-应变曲线的初始段呈线性，遵从材料形式的胡克定律。杨氏模量 E 是该线性段应力与应变的比值。

E = σ / ε = (F L₀) / (A ΔL)

量 / Quantity	符号 / Symbol	单位 / Unit
应力 / Stress	σ	Pa (N m⁻²)
应变 / Strain	ε	无量纲 / dimensionless
横截面积 / Cross-sectional area	A	m²
原始长度 / Original length	L₀	m
伸长量 / Change in length	ΔL	m
杨氏模量 / Young’s modulus	E	Pa

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5. Equations of Motion (SUVAT) | 匀加速运动方程

For an object moving with constant acceleration in a straight line, five kinematic quantities are related. The SUVAT equations allow calculation of any unknown when three are known.

对于沿直线以恒定加速度运动的物体，五个运动学量之间存在关联。当已知其中三个量时，SUVAT 方程可用于求解任意未知量。

v = u + a t

s = u t + ½ a t²

v² = u² + 2 a s

s = ½ (u + v) t

These equations assume constant acceleration and no change in direction unless indicated. Vector sign conventions must be applied consistently for displacement, velocity and acceleration.

这些方程假设加速度恒定且除特别说明外方向不变。对于位移、速度和加速度需要一致地应用矢量正负号规则。

量 / Quantity	符号 / Symbol	单位 / Unit
位移 / Displacement	s	m
初速度 / Initial velocity	u	m s⁻¹
末速度 / Final velocity	v	m s⁻¹
加速度 / Acceleration	a	m s⁻²
时间 / Time	t	s

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6. Newton’s Laws and Momentum | 牛顿运动定律与动量

Newton’s second law states that the rate of change of momentum is directly proportional to the resultant force and occurs in the direction of that force. This underpins dynamic analysis.

牛顿第二定律指出，动量的变化率与合外力成正比，并发生在该力的方向上。这是动力学分析的基石。

F = m a    F = Δp / Δt

Momentum p = m v is a vector quantity. In the absence of external forces, momentum is conserved. Impulse is the product of force and the time for which it acts, equal to the change in momentum.

动量 p = m v 是一个矢量。在没有外力作用时，动量守恒。冲量是力与其作用时间的乘积，等于动量的变化量。

p = m v    Impulse = F Δt = Δp

量 / Quantity	符号 / Symbol	单位 / Unit
动量 / Momentum	p	kg m s⁻¹
冲量 / Impulse	I	N s
时间间隔 / Time interval	Δt	s

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7. Work, Energy and Power | 功、能与功率

Work is done when a force moves its point of application in the direction of the force. Energy is the capacity to do work, and power is the rate at which work is done.

当力使其作用点沿力的方向移动时，力做了功。能量是做功的能力，功率则是做功的速率。

W = F s cosθ    P = W / t = F v

Kinetic energy is associated with motion, while gravitational potential energy is stored by virtue of an object’s position in a gravitational field. Efficiency assesses how much input energy is converted to useful output.

动能与运动相关，重力势能是因物体在引力场中的位置而储存的能量。效率衡量有多少输入能量转化为有用输出。

Eₖ = ½ m v²    ΔEₚ = m g Δh

量 / Quantity	符号 / Symbol	单位 / Unit
功 / Work	W	J
动能 / Kinetic energy	Eₖ	J
势能变化 / Change in potential energy	ΔEₚ	J
功率 / Power	P	W

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8. Mechanical Advantage, Velocity Ratio and Efficiency | 机械效益、速比与效率

Mechanical systems such as levers, pulleys and gears multiply force or distance. The mechanical advantage (MA) compares output force to input force, while the velocity ratio (VR) compares distances moved.

杠杆、滑轮和齿轮等机械系统可以放大力或距离。机械效益（MA）将输出力与输入力进行比较，而速比（VR）则比较移动的距离。

MA = Fₒᵤₜ / Fᵢₙ    VR = dᵢₙ / dₒᵤₜ

Efficiency is the ratio of useful output work to input work, often expressed as a percentage. In an ideal, frictionless machine MA = VR, but in practice efficiency is always less than 100%.

效率是有用输出功与输入功的比值，通常以百分比表示。在理想的、无摩擦的机器中 MA = VR，但实际上效率始终低于 100%。

η = (Wₒᵤₜ / Wᵢₙ) × 100% = (MA / VR) × 100%

量 / Quantity	符号 / Symbol	单位 / Unit
机械效益 / Mechanical advantage	MA	无量纲 / dimensionless
速比 / Velocity ratio	VR	无量纲 / dimensionless
效率 / Efficiency	η	% 或无量纲

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9. Ohm’s Law and Electrical Power | 欧姆定律与电功率

Ohm’s law relates the potential difference across a conductor to the current flowing through it, provided temperature and other physical conditions remain constant. Resistance measures opposition to current.

欧姆定律给出了导体两端电势差与流过电流之间的关系，前提是温度和其他物理条件保持恒定。电阻衡量对电流的阻碍作用。

V = I R

Electrical power is the product of current and voltage. Using Ohm’s law, alternative expressions can be derived, which are useful when some quantities are unknown.

电功率是电流与电压的乘积。利用欧姆定律可以推导出其他形式，这在某些量未知时非常有用。

P = I V = I² R = V² / R

量 / Quantity	符号 / Symbol	单位 / Unit
电压 / Potential difference	V	V
电流 / Current	I	A
电阻 / Resistance	R	Ω
功率 / Power	P	W

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10. Kirchhoff’s Laws and Resistor Combinations | 基尔霍夫定律与电阻组合

Kirchhoff’s current law (KCL) states that the total current entering a junction equals the total current leaving it, a consequence of charge conservation. Kirchhoff’s voltage law (KVL) states that the algebraic sum of all voltages around any closed loop is zero.

基尔霍夫电流定律（KCL）指出，流入节点的总电流等于流出节点的总电流，这是电荷守恒的结果。基尔霍夫电压定律（KVL）指出，沿任意闭合回路的所有电压代数和为零。

ΣIᵢₙ = ΣIₒᵤₜ    ΣV = 0 (around loop)

Resistors in series carry the same current and their total resistance is the sum of individual resistances. Resistors in parallel share the same voltage; the reciprocal of the total resistance equals the sum of reciprocals.

串联电阻中流过相同的电流，总电阻等于各个电阻之和。并联电阻两端电压相同，总电阻的倒数等于各电阻倒数之和。

Rₜₒₜₐₗ = R₁ + R₂ + … (series)

1/Rₜₒₜₐₗ = 1/R₁ + 1/R₂ + … (parallel)

These laws are essential for simplifying circuits and analysing voltage dividers, which are widely used in sensor and control applications.

这些定律对于简化电路和分析分压器至关重要，分压器在传感器和控制应用中广泛使用。

量 / Quantity	符号 / Symbol	单位 / Unit
总电阻 (串联) / Total resistance (series)	Rₜₒₜₐₗ	Ω
支路电流 / Branch current	I₁, I₂ …	A

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