Year 8 Cambridge Engineering: Quick Reference Handbook of Formulas and Theorems | 八年级剑桥工程：公式定理速查手册

📚 Year 8 Cambridge Engineering: Quick Reference Handbook of Formulas and Theorems | 八年级剑桥工程：公式定理速查手册

This handbook brings together the essential formulas and theorems you will meet in Year 8 Cambridge Engineering. It is designed as a quick revision tool covering mechanics, electricity, materials and simple machines. Each concept is presented with its formula, units and a short explanation. Use it to check your understanding before tests, while working on design challenges, or when you need a fast reminder of how physical quantities are linked.

这本手册汇集了八年级剑桥工程课程中所有的核心公式与定理。它是一份快速复习工具，涵盖力学、电学、材料与简单机械。每个概念都给出了公式、单位与简要说明。你可以在考前检查理解、在设计挑战中随时查阅，或在需要快速回忆起物理量之间关系时使用。

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1. Forces and Newton's Laws of Motion | 力与牛顿运动定律

A force is a push or a pull that can change an object's motion, shape or direction. Force is measured in newtons (N). Newton's Second Law describes how force, mass and acceleration are related.

力是能改变物体运动状态、形状或方向的推或拉。力以牛顿（N）为单位。牛顿第二定律描述了力、质量和加速度之间的关系。

F = m a

In this equation, F is net force (N), m is mass (kg) and a is acceleration (m/s²). One newton is the force needed to give a 1 kg mass an acceleration of 1 m/s².

在此等式中，F 为净力（N），m 为质量（kg），a 为加速度（m/s²）。1 牛顿的定义是使 1 千克质量产生 1 米每二次方秒加速度所需的力。

Weight is the gravitational force pulling on a mass. Near Earth's surface the gravitational field strength g is approximately 9.8 N/kg. Weight can be calculated with:

重量是作用在质量上的重力。地球表面附近的重力场强度 g 约为 9.8 N/kg。重量可通过下式计算：

W = m g

newtons, m is mass in kilograms and g = 9.8 m/s² (or N/kg). Do not confuse mass and weight — mass stays the same everywhere, weight changes with gravity.

其中 W 为重量（N），m 为质量（kg），g = 9.8 m/s²（或 N/kg）。请勿混淆质量与重量：质量在任何地方都不变，重量会随重力改变。

Newton's Third Law tells us that forces always come in pairs: if object A pushes on object B, object B pushes back on A with an equal and opposite force.

牛顿第三定律告诉我们力总是成对出现：如果物体 A 对物体 B 施加一个力，那么物体 B 必定同时对物体 A 施加一个大小相等、方向相反的力。

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2. Work, Energy and Power | 功、能与功率

Work is done when a force moves an object in the direction of the force. The amount of work transfers energy from one store to another. Work and energy are both measured in joules (J).

当力使物体沿力的方向移动时，力做了功。做功的大小将能量从一个储存库转移到另一个。功和能量都以焦耳（J）为单位。

W = F d

W is work done (J), F is the constant force (N) and d is the distance moved in the direction of the force (m). If the force and motion are at an angle, only the component of force in the direction of motion does work.

W 为所做的功（J），F 为恒力（N），d 为物体沿力的方向移动的距离（m）。如果力与运动方向存在夹角，只有力沿运动方向的分量才会做功。

When a mass is lifted against gravity, gravitational potential energy (GPE) increases. For small height changes near Earth's surface:

当物体被克服重力举起时，重力势能增加。在地球表面附近，高度变化较小时：

Eₚ = m g h

where Eₚ is gravitational potential energy (J), m is mass (kg), g is 9.8 m/s² and h is the change in height (m).

其中 Eₚ 为重力势能（J），m 为质量（kg），g 为 9.8 m/s²，h 为高度变化（m）。

Power is the rate at which work is done or energy is transferred. A machine is powerful if it can do a lot of work in a short time.

功率是做功或能量传递的速率。如果一台机器能在短时间内完成大量功，它的功率就大。

P = W / t

P is power in watts (W), W is work done or energy transferred (J), and t is time (s). 1 watt equals 1 joule per second.

P 为功率（W），W 为所做的功或传递的能量（J），t 为时间（s）。1 瓦特等于 1 焦耳每秒。

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3. Simple Machines: Levers and Moments | 简单机械：杠杆与力矩

A lever is a rigid bar that turns about a pivot (fulcrum). The turning effect of a force is called a moment. For a lever to be balanced, the total clockwise moment must equal the total anticlockwise moment.

杠杆是绕着支点转动的刚性杆。力产生的转动效应称为力矩。要使杠杆平衡，顺时针力矩总和必须等于逆时针力矩总和。

M = F × d

M is moment (N m), F is the applied force (N) and d is the perpendicular distance from the pivot to the line of action of the force (m). Always measure d at right angles to the force.

M 为力矩（N m），F 为施加的力（N），d 为从支点到力作用线的垂直距离（m）。请始终测量与力方向垂直的垂直距离。

Levers allow a small effort to lift a large load by trading distance for force. Mechanical advantage (MA) compares the load to the effort, while velocity ratio (VR) compares the distances moved.

杠杆通过用距离换取力，使得较小的施力就能举起较大的负载。机械利益（MA）比较负载与施力，而速度比（VR）比较移动的距离。

MA = Load / Effort

VR = distance moved by effort / distance moved by load

In an ideal machine with no friction, MA would equal VR. In reality some energy is always wasted, so we define efficiency:

在无摩擦的理想机器中，MA 等于 VR。实际上总会有能量浪费，因此我们定义效率：

Efficiency = (MA / VR) × 100%

or equivalently Efficiency = (useful work output / total work input) × 100%.

或等效 效率 =（有用输出功 / 总输入功）× 100%。

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4. Inclined Planes | 斜面

An inclined plane is a flat surface set at an angle to the horizontal. It reduces the effort needed to raise a load by increasing the distance over which the force acts. The mechanical advantage of an ideal inclined plane is the ratio of the slope length to the vertical height.

斜面是一块与水平面呈一定角度的平面。它通过增大力的作用距离来减小举升负载所需的力。理想斜面的机械利益等于斜面长度与垂直高度之比。

MA (inclined plane) = length of slope / height

The velocity ratio for an inclined plane is the same ratio because the effort moves along the slope while the load rises vertically. Friction will reduce the real MA.

斜面的速度比与此相同，因为施力沿斜面移动，而负载垂直上升。摩擦会降低实际机械利益。

In practical design, engineers often use ramps with a shallow gradient to move heavy objects with a smaller force, following the same trade-off principle as levers.

在实际设计中，工程师经常利用缓坡度斜坡以小力移动重物，遵循着与杠杆相同的权衡原理。

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5. Pulley Systems | 滑轮系统

A pulley changes the direction of a force and can multiply the force when multiple rope segments support the load. The velocity ratio of a pulley system equals the number of rope sections that actually support the moving load.

滑轮能改变力的方向，并且当有多股绳索承载负载时能产生省力效果。滑轮系统的速度比等于实际支撑动负载的绳索段数。

VR (pulley) = number of supporting rope strands

For a single fixed pulley, VR = 1 and MA ≤ 1 — vable pulley, the load is supported by two rope strands, so VR = 2, and the ideal MA is 2.

对于单个定滑轮，VR = 1、MA ≤ 1 —— 它只改变力的方向。对于单个动滑轮，负载由两股绳索支撑，因此 VR = 2，理想 MA 为 2。

In block-and-tackle systems, several pulleys are combined to give a high VR and MA, though friction reduces efficiency. Engineers select pulleys to match the force available to the task.

在滑轮组（复滑车）系统中，多个滑轮组合可获得较大的 VR 和 MA，但摩擦会降低效率。工程师根据可用力来匹配任务的滑轮组合。

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6. Motion: Speed and Acceleration | 运动：速率与加速度

Speed is a scalar quantity that tells us how fast an object moves. Velocity is speed in a given direction. For motion in a straight line, the average speed can be found by dividing distance travelled by time taken.

速率是标量，描述物体运动的快慢。速度是带有方向的速率。对于直线运动，平均速率可通过距离除以时间得到。

v = s / t

v is speed or velocity (m/s), s is distance moved (m) and t is time (s). If direction changes, velocity changes even if speed stays constant.

v 为速率或速度（m/s），s 为移动距离（m），t 为时间（s）。如果方向改变，即使速率不变，速度也会变化。

Acceleration is the rate at which velocity changes. It can be an increase or a decrease in speed (deceleration). Acceleration is measured in metres per second squared (m/s²).

加速度是速度变化的快慢，可以是加速或减速，单位为米每二次方秒（m/s²）。

a = (v - u) / t

where a is acceleration, v is final velocity, u is initial velocity and t is the time taken for the change. This formula applies when acceleration is uniform.

其中 a 为加速度，v 为末速度，u 为初速度，t 为速度变化所用时间。此公式适用于匀加速度的情况。

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7. Ohm's Law and Electrical Circuits | 欧姆定律与电路

Ohm's law is one of the most important relationships in electrical engineering. It links voltage, current and resistance and helps predict how circuits will behave.

欧姆定律是电气工程中最重要的关系之一，它将电压、电流和电阻联系起来，并有助于预测电路的行为。

V = I R

V is voltage or potential difference in volts (V), I is current in amperes (A), and R is resistance in ohms (Ω). For a given resistance, increasing voltage drives more current.

V 为电压或电势差（V），I 为电流（A），R 为电阻（Ω）。在给定电阻下，电压升高会导致电流增大。

In a series circuit, the current is the same everywhere and the total resistance is the sum of individual resistances. The supply voltage is shared across components.

在串联电路中，电流处处相等，总电阻等于各电阻之和。电源电压被分配到各元件上。

R_total = R₁ + R₂ + R₃ ... (series)

In a parallel circuit, the voltage across each branch is the same, and the total resistance is less than any individual branch. The rule for two resistors in parallel is:

在并联电路中，各支路电压相同，总电阻小于任何单个支路的电阻。两个电阻并联时的规则为：

1/R_total = 1/R₁ + 1/R₂ (parallel)

These principles are essential for designing everything from lighting systems to control circuits.

这些原理对于设计从照明系统到控制电路的各种工程至关重要。

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8. Material Properties: Hooke's Law | 材料性质：胡克定律

Many solid materials obey Hooke's Law up to a certain point called the limit of proportionality. This law states that the extension of an elastic object is directly proportional to the applied force.

许多固体材料在称为比例极限的范围内遵循胡克定律。该定律指出，弹性物体的伸长量与施加的力成正比。

F = k x

F is the applied force (N), x is the extension (m) from the original length, and k is the spring constant (N/m), which measures stiffness. A stiffer spring has a larger k value.

F 为施加的力（N），x 为相较于原始长度的伸长量（m），k 为弹性常数（N/m），用于衡量劲度。劲度越大的弹簧 k 值越大。

Engineers use Hooke's Law when designing springs, suspension systems and any component that must return to its original shape after being loaded. Beyond the elastic limit, permanent deformation occurs.

工程师在设计弹簧、悬挂系统以及任何受载后必须恢复原形的部件时都会用到胡克定律。一旦超过弹性极限，就会发生永久变形。

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9. Stress and Pressure | 应力与压强

Stress describes how internal force is distributed within a material, while pressure describes how a force is spread over a surface. Both concepts use the idea of force per unit area, but in different contexts.

应力描述材料内部力的分布方式，压强则描述力如何在表面上分布。两者都使用“单位面积上的力”这一概念，但应用背景不同。

Tensile stress σ = F / A

N/m². Strain ε is the ratio of extension to original length, ε = ΔL / L, and has no units.

此处 F 为施加的力（N），A 为横截面积（m²）。应力的单位是帕斯卡（Pa）或 N/m²。应变 ε 是伸长量与原始长度之比，ε = ΔL / L，无单位。

Pressure is used for fluids and contact between surfaces:

压强则用于流体及表面接触：

p = F / A

p is pressure (Pa), F is force acting perpendicular to the surface (N), and A is the area of contact (m²). A sharp knife cuts better because the same force is concentrated on a smaller area, producing a larger pressure.

p 为压强（Pa），F 为垂直作用于表面的力（N），A 为接触面积（m²）。锋利的刀能更好地切割，是因为同样的力集中在更小的面积上，产生更大的压强。

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10. Gears and Transmission Ratios | 齿轮与传动比

Gears are toothed wheels that transmit rotary motion and torque between shafts. The gear ratio compares the number of teeth on two meshing gears and determines whether the output shaft turns faster or slower than the input.

齿轮是带齿的轮子，用于在轴之间传递旋转运动和扭矩。齿轮比比较两个啮合齿轮的齿数，决定输出轴比输入轴转得更快还是更慢。

Gear Ratio = N_driven / N_driver

N_driven is the number of teeth on the gear being driven (output), N_driver is the number of teeth on the driving gear (input). If the driven gear has more teeth, the output speed decreases but torque increases.

N_driven 是从动齿轮（输出）的齿数，N_driver 是主动齿轮（输入）的齿数。如果从动齿轮齿数更多，输出转速降低但扭矩增大。

The velocity ratio of a gear pair is the same as the gear ratio, and the speed of the output can be found from:

齿轮副的速度比等于齿轮比，输出转速可通过下式求出：

Speed_output = Speed_input × (N_driver / N_driven)

These relationships form the basis of gearboxes in bicycles, cars and factory machinery. Gears let engineers trade speed for force exactly like levers and pulleys.

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