Year 13 Cambridge Physics: Formulas & Theorems Quick Reference | Year 13 Cambridge 物理：公式定理速查手册

📚 Year 13 Cambridge Physics: Formulas & Theorems Quick Reference | Year 13 Cambridge 物理：公式定理速查手册

This quick-reference guide brings together the essential formulas and theorems for the Cambridge International A Level Physics (9702) Year 13 syllabus. Use it to reinforce your understanding, speed up problem-solving, and have a reliable formula companion at your fingertips.

本快速参考手册汇集了剑桥国际A Level物理（9702）Year 13课程的核心公式与定理。用它来巩固理解、提高解题速度，并作为随手可查的公式伴侣。

1. Circular Motion | 圆周运动

In uniform circular motion, an object travels in a circle at constant speed. The velocity vector is always tangent to the circle, so there is a centripetal acceleration directed towards the centre.

在匀速圆周运动中，物体以恒定速率沿圆周运动。速度矢量始终与圆相切，故存在指向圆心的向心加速度。

Angular displacement (in radians) is defined as arc length divided by radius:

θ = s / r

角位移（弧度）定义为弧长除以半径：

θ = s / r

Angular velocity ω = Δθ/Δt = 2π/T = 2πf, where T is the period and f is the frequency. The linear speed v is related by v = ωr.

角速度 ω = Δθ/Δt = 2π/T = 2πf，其中 T 是周期，f 是频率。线速度 v 的关系为 v = ωr。

Centripetal acceleration:

a = v² / r = ω² r

向心加速度：

a = v² / r = ω² r

Centripetal force for an object of mass m:

F = m a = m v² / r = m ω² r

质量为 m 的物体所受向心力：

F = m a = m v² / r = m ω² r

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2. Gravitational Fields | 引力场

Newton’s law of universal gravitation states that the force between two point masses is directly proportional to the product of their masses and inversely proportional to the square of their separation.

牛顿万有引力定律指出，两质点之间的引力与它们质量的乘积成正比，与它们距离的平方成反比。

F = G m&sub1; m&sub2; / r²

Gravitational field strength g at a point is the force per unit mass: g = F/m. For a point mass M,

g = G M / r²

引力场强 g 定义为每单位质量的力：g = F/m。对于点质量 M，

g = G M / r²

Gravitational potential V at a distance r from a point mass M is the work done per unit mass to bring a test mass from infinity to that point:

V = – G M / r

Gravitational potential energy of a system of two masses: U = – G M m / r.

引力势 V (距离点质量 M 为 r 处)是将单位质量从无穷远移到该点所做的功：

V = – G M / r

两质点系统的引力势能：U = – G M m / r。

Kepler’s third law for circular orbits gives T² ∝ r³. For a satellite orbiting a central mass M:

T² = (4π² / G M) r³

开普勒第三定律对于圆轨道：T² ∝ r³。对于绕中心质量 M 运行的卫星：

T² = (4π² / G M) r³

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3. Simple Harmonic Motion | 简谐运动

Simple harmonic motion (SHM) occurs when the acceleration a is directly proportional to the displacement x from equilibrium and is directed towards the equilibrium position:

a = – ω² x

当加速度 a 与离开平衡位置的位移 x 成正比且方向指向平衡点时，发生简谐运动：

a = – ω² x

Displacement can be written as x = x₀ sin(ωt) or x = x₀ cos(ωt), where x₀ is the amplitude. The maximum speed is v_max = ω x₀. The speed at displacement x is v = ± ω √(x₀² – x²).

位移可写作 x = x₀ sin(ωt) 或 x = x₀ cos(ωt)，其中 x₀ 为振幅。最大速率 v_max = ω x₀。位移为 x 时速率 v = ± ω √(x₀² – x²)。

Period T = 1/f = 2π/ω. For a simple pendulum of length L:

T = 2π √(L / g)

For a mass m on a spring of force constant k:

T = 2π √(m / k)

周期 T = 1/f = 2π/ω。摆长为 L 的单摆：

T = 2π √(L / g)

劲度系数为 k 的弹簧振子：

T = 2π √(m / k)

The total energy of an undamped harmonic oscillator is constant: E = ½ m ω² x₀².

无阻尼谐振子的总能量为常数：E = ½ m ω² x₀²。

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4. Thermal Physics & Ideal Gases | 热物理与理想气体

The ideal gas equation relates pressure p, volume V, amount n (in moles) and thermodynamic temperature T:

p V = n R T

也可写作 p V = N k T，其中 N 是分子数，k 是玻尔兹曼常数。

Also written as p V = N k T, where N is the number of molecules and k is Boltzmann’s constant.

理想气体状态方程关联压强 p、体积 V、物质的量 n（摩尔）和热力学温度 T：

p V = n R T

In the kinetic theory model, p V = (1/3) N m ⟨c²⟩, and the average kinetic energy of a molecule is ½ m ⟨c²⟩ = (3/2) k T. For a monatomic ideal gas, internal energy U = (3/2) n R T.

在分子运动论中，p V = (1/3) N m ⟨c²⟩，分子平均动能 ½ m ⟨c²⟩ = (3/2) k T。对于单原子理想气体，内能 U = (3/2) n R T。

The first law of thermodynamics (using work done ON the system as W):

ΔU = Q + W

If the system does work on the surroundings, W is negative. For a gas expanding at constant pressure, W = -p ΔV.

热力学第一定律（以对系统做的功 W 为正）：

ΔU = Q + W

若系统对外做功，W 为负值。对等压膨胀，

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