📚 Year 10 Cambridge Engineering: Formulae & Theorems Quick Reference Guide | Year 10 剑桥工程：公式定理速查手册
This quick reference handbook compiles the essential formulas, principles, and theorems needed for the Year 10 Cambridge Engineering course. Use it to reinforce your understanding, check calculations, and prepare for assessments with confidence.
本速查手册汇编了 Year 10 剑桥工程课程所必需的核心公式、原理与定理。利用本手册巩固概念、核查计算，自信备战各类评估。
1. Fundamental Quantities & Units | 基础量与单位
All engineering measurements are built on SI base units: length in metres (m), mass in kilograms (kg), time in seconds (s), and electric current in amperes (A). Derived units such as the newton (N) for force and the pascal (Pa) for pressure are formed from combinations of these bases.
所有的工程测量都建立在国际单位制（SI）基本单位之上：长度用米 (m)，质量用千克 (kg)，时间用秒 (s)，电流用安培 (A)。诸如力（牛顿 N）和压强（帕斯卡 Pa）等导出单位则由这些基本单位组合而成。
Pressure is defined as the force acting perpendicularly per unit area. The formula is straightforward and essential for hydraulics and structural analysis:
压强定义为垂直作用在单位面积上的力。该公式是液压和结构分析的基础：
P = F / A
Here, P is pressure (Pa), F is the applied force (N), and A is the area over which the force is distributed (m²). For example, a force of 500 N acting on an area of 2 m² produces a pressure of 250 Pa.
其中 P 为压强 (Pa)，F 为作用力 (N)，A 为力分布的面积 (m²)。例如，500 N 的力作用在 2 m² 面积上，产生的压强为 250 Pa。
2. Kinematics Equations | 运动学公式
For an object moving with uniform acceleration in a straight line, four equations – often called SUVAT equations – link displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t).
对于沿直线做匀加速运动的物体，以下四个方程（常称为 SUVAT 方程）将位移 (s)、初速度 (u)、末速度 (v)、加速度 (a) 和时间 (t) 联系起来。
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 neglect air resistance. Choose the equation that includes the unknown you need and that does not require a quantity you have not been given.
这些方程假设加速度恒定且忽略空气阻力。选择包含未知量、且无需未给出物理量的那个方程即可求解。
3. Newton’s Laws of Motion | 牛顿运动定律
Newton’s First Law states that an object remains at rest or continues moving at constant velocity unless acted upon by a resultant external force. This property of an object is called inertia.
牛顿第一定律指出，物体在不受合外力作用时，将保持静止或匀速直线运动状态。物体的这种性质称为惯性。
Newton’s Second Law provides the quantitative link between force, mass and acceleration. It is the single most important relationship in dynamics:
牛顿第二定律给出了力、质量与加速度之间的定量关系，是动力学中最重要的关系式：
F = m a
Resultant force F (N) equals mass m (kg) times acceleration a (m/s²). A net force of 10 N acting on a 2 kg mass produces an acceleration of 5 m/s².
合外力 F (N) 等于质量 m (kg) 乘以加速度 a (m/s²)。10 N 的合力作用在 2 kg 的质量上将产生 5 m/s² 的加速度。
Newton’s Third Law reminds us that forces always occur in pairs: if body A exerts a force on body B, then body B exerts an equal and opposite force on body A.
牛顿第三定律指出力总是成对出现：若物体 A 对物体 B 施加一个力，则物体 B 必同时对物体 A 施加一个大小相等、方向相反的力。
4. Moments and Levers | 力矩与杠杆
The moment of a force about a pivot is the turning effect. It depends on both the magnitude of the force and the perpendicular distance from the pivot to the line of action of the force.
力对支点的力矩是其转动效应，取决于力的大小和支点到力作用线的垂直距离。
M = F × d
M is the moment (Nm), F is the force (N), and d is the perpendicular distance (m). For a balanced lever, the sum of clockwise moments equals the sum of anticlockwise moments about the pivot:
M 为力矩 (Nm)，F 为力 (N)，d 为垂直距离 (m)。对于平衡杠杆，绕支点的顺时针力矩总和等于逆时针力矩总和：
F₁ × d₁ = F₂ × d₂
This principle is used in beams, see-saws, and many simple machines. When designing structures, engineers must ensure that moments are balanced to prevent unwanted rotation.
此原理适用于横梁、跷跷板以及众多简单机械。工程师设计结构时，必须确保力矩平衡，以防止不希望发生的转动。
5. 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. Work is a scalar quantity measured in joules (J).
当力使其作用点沿力的方向发生移动时，即做了功。功是标量，单位为焦耳 (J)。
W = F × d
Where W is work done (J), F is the constant force in the direction of motion (N), and d is the distance moved (m).
其中 W 为做功 (J)，F 为沿运动方向的恒力 (N)，d 为移动距离 (m)。
The energy of a moving object – kinetic energy – and the energy stored due to an object’s height – gravitational potential energy – are given by:
运动物体具有的动能和因高度而储存的重力势能分别由以下公式给出：
Eₖ = ½ m v²
Eₚ = m g h
Here m is mass (kg), v is speed (m/s), g is the gravitational field strength (≈ 9.8 N/kg on Earth), and h is height (m). Power is the rate of doing work or transferring energy:
其中 m 为质量 (kg)，v 为速度 (m/s)，g 为重力场强度（地球约 9.8 N/kg），h 为高度 (m)。功率是做功或传递能量的速率：
P = W / t
For an object moving at constant velocity v under a constant force F, power can also be expressed as P = F v.
对于在恒力作用下做匀速运动的物体，功率也可表示为 P = F v。
6. Simple Machines: Mechanical Advantage, Velocity Ratio & Efficiency | 简单机械：机械效益、速度比与效率
A simple machine makes work easier by changing the magnitude or direction of the applied force. Its key parameters are mechanical advantage (MA), velocity ratio (VR), and efficiency (η).
简单机械通过改变作用力的大小或方向使工作变得省力。其关键参数为机械效益 (MA)、速度比 (VR) 和效率 (η)。
MA = Load / Effort
Load is the force overcome by the machine, and effort is the force applied. MA has no units and is typically greater than 1 for force-multiplying devices.
负载为机械克服的力，动力为输入的力。MA 无单位，在增力装置中通常大于 1。
VR = dₑ / dₗ
VR is the ratio of the distance moved by the effort (dₑ) to the distance moved by the load (dₗ) in the same time. For an ideal frictionless machine, MA = VR.
VR 是动力移动的距离 (dₑ) 与负载移动的距离 (dₗ) 在同一时间内的比值。在理想无摩擦情况下，MA = VR。
The efficiency of a machine tells us how well it converts input work into useful output work, and is usually expressed as a percentage:
效率反映机械将输入功转化为有用输出功的能力，通常以百分比表示：
η = (MA / VR) × 100%
Real machines always have friction, so their efficiency is less than 100%.
实际机械总有摩擦，因此效率始终低于 100%。
7. Mechanics of Materials: Stress and Strain | 材料力学：应力和应变
When an external force acts on a solid, internal forces resist deformation. Stress quantifies the internal resistance, while strain measures the resulting deformation.
当外力作用于固体时，内部分子力抵抗变形。应力量化了这种内部抗力，而应变则衡量产生的变形程度。
σ = F / A
Tensile or compressive stress σ (Pa) is the force F (N) applied perpendicularly to the cross-sectional area A (m²).
拉应力或压应力 σ (Pa) 是垂直作用于横截面积 A (m²) 上的力 F (N)。
ε = ΔL / L₀
Strain ε (no units) is the extension ΔL divided by the original length L₀. Strain is often given as a percentage. Under small loads, many materials behave elastically, meaning they return to their original shape when the load is removed.
应变 ε（无单位）是延伸量 ΔL 除以原始长度 L₀。应变常以百分比表示。在小载荷下，许多材料呈弹性行为，即卸除载荷后能恢复原状。
8. Hooke’s Law & Young’s Modulus | 胡克定律与杨氏模量
Hooke’s Law describes the linear relationship between the force applied to a spring or elastic object and its extension, up to the limit of proportionality.
胡克定律描述了施加在弹簧或弹性物体上的力与其伸长量之间的线性关系，该关系在比例极限内成立。
F = k x
F is the applied force (N), x is the extension (m), and k is the spring constant (N/m) – a measure of stiffness. A steep k value means a stiff spring.
F 为作用力 (N)，x 为伸长量 (m)，k 为弹簧常数 (N/m)——它是刚度的量度。k 值越大，弹簧越硬。
Young’s Modulus E links stress and strain for a given material within the elastic region, providing a measure of the material’s inherent stiffness:
杨氏模量 E 在弹性区域内将应力与应变联系起来，衡量材料的固有刚度：
E = σ / ε
E is measured in pascals (Pa). For instance, steel has a Young’s Modulus around 200 GPa, whereas polymers are much lower. Once the elastic limit is exceeded, permanent (plastic) deformation occurs, and Hooke’s Law no longer applies.
E 的单位为帕斯卡 (Pa)。例如，钢的杨氏模量约为 200 GPa，而聚合物则低得多。一旦超过弹性极限，将发生永久（塑性）变形，胡克定律不再适用。
9. Fluid Mechanics Principles | 流体力学原理
Pascal’s Principle states that an externally applied pressure on an enclosed incompressible fluid is transmitted equally and undiminished to all parts of the fluid. This is the foundation of hydraulic systems such as car brakes and jacks.
帕斯卡原理指出，施加在密闭不可压缩流体上的外部压强会等值且不减地传递到流体各处。这是汽车制动器和千斤顶等液压系统的基础。
P₁ = P₂ => F₁ / A₁ = F₂ / A₂
A small force F₁ on a small area A₁ can support a large force F₂ on a large area A₂, because the pressure is equal.
由于压强相等，小面积 A₁ 上的小力 F₁ 可支撑
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