📚 Pre-U OCR Physics: Formula and Theorem Quick Reference | Pre-U OCR 物理：公式定理速查手册
This article provides a concise yet comprehensive reference for the essential formulae and theorems required in the Pre-U OCR Physics syllabus. The content is organised by topic, from classical mechanics to modern quantum physics. Each section presents the key definitions, equations, and laws that form the foundation of physical reasoning, paired with practical applications frequently tested in examination contexts.
本文为 Pre-U OCR 物理课程提供了一份精炼而全面的公式定理速查手册。内容按主题编排，从经典力学一直延伸到现代量子物理。每一节都呈现了构成物理推理基础的关键定义、方程和定律，并穿插了考试中经常考查的实际应用。
1. Mechanics and Motion | 力学与运动
Kinematics describes the geometry of motion. For an object moving with uniform acceleration along a straight line, the SUVAT equations link displacement s, initial velocity u, final velocity v, acceleration a, and time t. These equations are derived from the definitions of velocity and acceleration, assuming the acceleration remains constant throughout the motion.
运动学描述的是运动的几何性质。对于一个沿直线做匀加速运动的物体，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。这些方程由速度和加速度的定义推导而来，前提是整个运动过程中加速度保持不变。
v = u + at
s = ut + ½at²
v² = u² + 2as
s = ½(u + v)t
Newton’s laws of motion establish the relationship between the forces acting on a body and its resulting motion. The first law defines inertia, the second law quantifies net force as the rate of change of momentum, and the third law states that forces occur in equal and opposite pairs acting on different bodies.
牛顿运动定律确立了作用于物体的力与其运动结果之间的关系。第一定律定义了惯性，第二定律将净力量化为动量的变化率，第三定律指出力以大小相等、方向相反的配对形式作用于不同物体上。
F = Δp/Δt = ma (for constant mass)
Momentum is a fundamental property of a moving object and is conserved in isolated systems. The principle of conservation of momentum is particularly powerful for analysing collisions and explosions, where internal forces are far greater than any external forces acting during the brief interaction time.
动量是运动物体的一个基本属性，在孤立系统中守恒。动量守恒定律在分析碰撞和爆炸时特别有效，因为在短暂的相互作用时间内，内力的作用远大于任何外力。
p = mv
Impulse delivered by a force equals the change in momentum it causes. This relationship is invaluable when the force varies over the collision time, as the area under a force-time graph gives the impulse directly.
力产生的冲量等于它引起的动量变化。当力在碰撞过程中随时间变化时，这一关系尤其重要，因为力-时间图下方的面积直接等于冲量。
Impulse = FΔt = Δp = mv – mu
2. Work, Energy, and Power | 功、能量与功率
Work is done when a force displaces its point of application in the direction of the force. Energy is the capacity to do work, and the principle of conservation of energy states that energy can be transformed from one form to another but cannot be created or destroyed in an isolated system.
当力使其作用点沿力的方向发生位移时，就做了功。能量是做功的能力，能量守恒定律指出，能量可以从一种形式转化为另一种形式，但在孤立系统中既不能被创造也不能被消灭。
W = Fd cosθ
Kinetic energy is the energy possessed by a body by virtue of its motion. Gravitational potential energy is the energy stored in a body due to its position in a gravitational field, with the zero level chosen arbitrarily for convenience.
动能是物体由于运动而具有的能量。重力势能是物体由于其在引力场中的位置而储存的能量，零点可以根据方便任意选取。
Eₖ = ½mv²
Eₚ = mgh
Power is defined as the rate at which work is done or energy is transferred. In the context of a constant force moving an object at a steady speed, power can also be expressed as the product of force and velocity in the direction of the force.
功率定义为做功或能量转移的快慢。在恒力以恒定速度移动物体的情况下，功率也可以表示为此力与该力方向上的速度的乘积。
P = ΔW/Δt = Fv
Efficiency measures the proportion of useful energy output to total energy input, always being less than 1 for real systems due to inevitable dissipative processes. This concept is tested frequently when analysing engines, motors, and energy conversion systems.
效率衡量有用能量输出占总能量输入的比例，由于不可避免的耗散过程，实际系统的效率总是小于 1。在分析发动机、电机和能量转换系统时，这一概念经常会考到。
η = Useful Power Output / Total Power Input
3. Circular Motion and Gravitation | 圆周运动与引力
An object moving in a circular path at constant speed experiences a centripetal acceleration directed towards the centre of the circle. This acceleration arises from a net centripetal force, which is not a distinct type of force but rather the resultant of existing forces providing the inward pull.
物体以恒定速率做圆周运动时，会经历指向圆心的向心加速度。这个加速度来源于净向心力，向心力不是一种独特的力类型，而是现有力中提供向内拉力的合力。
a = v²/r = ω²r
F = mv²/r = mω²r
The angular velocity ω relates the angle swept out per unit time to the linear speed. The period T of one complete revolution is inversely proportional to the angular velocity, forming a simple link between rotational frequency and the time for one cycle.
角速度 ω 将单位时间内扫过的角度与线速度联系起来。旋转一周的周期 T 与角速度成反比，在旋转频率与单圈时间之间建立了一个简单的联系。
ω = 2π/T = 2πf
v = ωr
Newton’s law of universal gravitation states that every point mass attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres. This law governs planetary motion and satellite orbits.
牛顿万有引力定律指出，每个质点都以一个力吸引另一个质点，这个力与两质点质量的乘积成正比，与它们中心之间距离的平方成反比。这一定律支配着行星运动和卫星轨道。
F = Gm₁m₂/r²
Gravitational field strength g at a point is defined as the gravitational force per unit mass experienced by a small test mass placed at that point. In a radial field, the field strength follows an inverse-square relation with distance from the centre of the mass.
引力场强度 g 定义为置于该点的一个小检验质量所受到的每单位质量的引力。在径向场中，场强与到质心距离的平方成反比关系。
g = F/m = GM/r²
Kepler’s third law for satellites in circular orbits links the orbital period to the radius of the orbit. Equating the centripetal force required for circular motion to the gravitational force provides the derivation of the relationship, extensively used for calculating satellite altitudes and planetary masses.
开普勒第三定律适用于圆形轨道上的卫星，将轨道周期与轨道半径联系起来。令圆周运动所需的向心力等于引力，即可推导出这一关系式，广泛用于计算卫星高度和行星质量。
T² = (4π²/GM) r³
4. Oscillations and Simple Harmonic Motion | 振动与简谐运动
Simple harmonic motion (SHM) is a special type of oscillation where the acceleration of the body is directly proportional to its displacement from the equilibrium position and is always directed towards that equilibrium. The defining equation can be written as a differential equation that yields sinusoidal solutions for displacement, velocity, and acceleration.
简谐运动是一种特殊的振动，其物体的加速度与它偏离平衡位置的位移成正比，并且总是指向平衡点。这一定义方程可以写成一个微分方程，从而得出位移、速度和加速度的正弦形式解。
a = –ω²x
The solutions to the SHM equation involve sinusoidal functions. The displacement varies as a cosine or sine function of time, with the amplitude A representing the maximum displacement and the phase constant determined by the initial conditions at time zero.
简谐运动方程的解包含正弦和余弦函数。位移随时间按余弦或正弦函数变化，振幅 A 代表最大位移，相位常数由零时刻的初始条件决定。
x = A cos(ωt) or x = A sin(ωt)
The velocity in SHM reaches a maximum as the oscillator passes through the equilibrium position and falls to zero at the extreme positions. These turning points of maximum displacement are where the restoring force and acceleration are greatest in magnitude.
简谐运动的速度在振子经过平衡位置时达到最大值，在极端位置处降为零。这些最大位移的转折点也正是回复力和加速度大小最大的地方。
v = ±ω√(A² – x²)
vₘₐₓ = ωA
The period of a simple pendulum undergoing small oscillations is independent of its amplitude, a property known as isochronism. For a mass-spring system, the period depends solely on the mass and the spring constant, with larger masses producing slower oscillations and stiffer springs producing faster ones.
做小幅振动的单摆周期与振幅无关，这一特性称为等时性。对于弹簧-振子系统，周期仅取决于质量和弹簧常数，质量越大振动越慢，弹簧越硬振速越快。
T = 2π√(l/g) (simple pendulum)
T = 2π√(m/k) (mass-spring system)
The total energy of an undamped SHM system remains constant, continuously transforming between kinetic and potential forms. At the equilibrium position, energy is entirely kinetic; at the extremes of displacement, it is entirely stored as potential energy within the spring or gravitational field.
无阻尼简谐振动系统的总能量保持不变，不断地在动能和势能之间转化。在平衡位置，能量全部为动能；在位移极值处，能量全部储存为弹簧或引力场中的势能。
Eₜₒₜ = ½mω²A²
5. Thermal Physics and Ideal Gases | 热物理与理想气体
The kinetic theory of gases models gas particles as point masses in random motion, undergoing perfectly elastic collisions with the container walls. The pressure exerted by a gas on its container arises from a vast number of microscopic molecular impacts, each imparting a tiny impulse.
气体动理论将气体粒子建模为做随机运动的质点，它们与容器壁发生完全弹性碰撞。气体施加在容器壁上的压强源于大量微观分子碰撞，每次碰撞都产生一个微小的冲量。
pV = ⅓ Nm⟨c²⟩
The ideal gas equation links the macroscopic state variables of pressure p, volume V, and absolute temperature T. The amount of gas is measured in moles n, and the universal gas constant R bridges the microscopic connection to the macroscopic behaviour through Boltzmann’s constant k.
理想气体状态方程将宏观状态变量压强 p、体积 V 和绝对温度 T 联系在一起。气体的量以摩尔数 n 度量，通用气体常数 R 通过玻尔兹曼常数 k 在微观与宏观行为之间架起了桥梁。
pV = nRT
pV = NkT
The mean translational kinetic energy of a gas molecule depends only on the absolute temperature of the gas. This fundamental result shows that temperature is a direct measure of the average random kinetic energy per particle, independent of the molar mass of the gas.
气体分子的平均平动动能仅取决于气体的绝对温度。这个基本结论表明，温度是每个粒子平均随机动能的直接量度，与气体的摩尔质量无关。
⟨Eₖ⟩ = ³⁄₂ kT
The first law of thermodynamics formalises the conservation of energy in a thermodynamic system. The change in internal energy of a system equals the net heat supplied to the system minus the net work done by the system on the surroundings.
热力学第一定律将热力学系统中的能量守恒形式化。系统内能的变化等于供给系统的净热量减去系统对外界做的净功。
ΔU = Q – W
Specific heat capacity characterises how much thermal energy is required to raise the temperature of a unit mass of a substance by one kelvin. The specific latent heat defines the energy required to change the phase of a unit mass of material without any temperature change.
比热容表征了使单位质量的物质温度升高一开尔文所需的热能。比潜热则定义了使单位质量的物质在温度不发生变化的情况下发生相变所需的能量。
Q = mcΔθ
Q = mL
6. Electric Fields and Circuits | 电场与电路
Coulomb’s law quantifies the electrostatic force between two point charges. The force is directly proportional to the product of the charges and inversely proportional to the square of their separation. The permittivity of free space ε₀ is a fundamental constant that determines the strength of the electrostatic interaction in a vacuum.
库仑定律量化了两个点电荷之间的静电力。此力与两电荷量的乘积成正比，与它们距离的平方成反比。真空介电常数 ε₀ 是一个基本常数，它决定了真空中静电相互作用的大小。
F = q₁q₂/(4πε₀r²)
The electric field strength E at a point is defined as the force per unit positive test charge placed at that point. In a uniform field between two parallel charged plates, the field strength simplifies to the potential difference divided by the plate separation distance.
电场强度 E 定义为置于该点的每单位正检验电荷所受的力。在两块平行带电板之间的均匀电场中，场强可以简化为电势差除以板间距。
E = F/q
E = V/d (uniform field)
Ohm’s law states that for a metallic conductor at constant temperature, the current flowing is directly proportional to the potential difference across it. Resistance quantifies the opposition to current flow, and resistivity links the resistance of a material to its geometry and intrinsic conductive properties.
欧姆定律指出，在温度不变的金属导体中，流过的电流与它两端的电势差成正比。电阻量化了对电流的阻碍，而电阻率则将材料的电阻与其几何形状和内在导电性质联系起来。
V = IR
R = ρL/A
Electrical power dissipated in a component is the product of the current through it and the potential difference across it. Using Ohm’s law, alternative forms express power in terms of current and resistance or voltage and resistance, each suited to different circuit analysis scenarios.
元件中耗散的电功率是流过它的电流与其两端电势差的乘积。利用欧姆定律，可以将功率表示成电流与电阻、或电压与电阻的替代形式，各自适用于不同的电路分析场景。
P = IV = I²R = V²/R
Kirchhoff’s circuit laws provide the fundamental rules for analysing complex circuits. The junction rule arises from the conservation of electric charge, and the loop rule is a consequence of the conservation of energy applied to a closed conducting path.
基尔霍夫电路定律为分析复杂电路提供了基本法则。节点定则由电荷守恒导出，回路定则是在闭合导电路径上应用能量守恒的直接结果。
ΣIᵢₙ = ΣIₒᵤₜ (Junction rule)
Σε = ΣIR (Loop rule)
7. Capacitance | 电容
A capacitor stores charge and electric potential energy. The capacitance C quantitatively measures how much charge a capacitor can store per unit potential difference applied across its plates. Capacitance depends on the geometry of the plates and the dielectric material separating them.
电容器储存电荷和电势能。电容 C 定量描述电容器每单位电势差下能够储存的电荷量。电容取决于板板的几何结构以及将其隔开的电介质材料。
C = Q/V
For a parallel-plate capacitor in vacuum, capacitance is proportional to the plate area and inversely proportional to the plate separation. Inserting a dielectric material with relative permittivity εᵣ increases the capacitance by that factor, as the dielectric reduces the effective electric field.
对于真空中的平行板电容器，电容与板面积成正比，与板间距成反比。插入相对介电常数为 εᵣ 的电介质可以将电容增大该倍数，因为电介质降低了有效电场。
C = ε₀A/d (vacuum)
C = εᵣε₀A/d (with dielectric)
The energy stored in a charged capacitor resides in the electric field between its plates. The energy can be expressed using combinations of capacitance, charge, and voltage, with each form being particularly convenient depending on which quantities are held constant in a given circuit.
充电电容器中储存的能量存在于其极板之间的电场中。该能量可以用电容、电荷量和电压的组合来表示，在特定电路中，根据被保持恒定的量选择便捷的表达形式。
E = ½QV = ½CV² = ½Q²/C
When capacitors discharge through a resistor, the charge, current, and voltage all decay exponentially. The time constant RC characterises the rate of decay: a larger resistance or capacitance produces a slower discharge as the time constant increases.
电容器通过电阻放电时，电荷、电流和电压均按指数规律衰减。时间常数 RC 决定了衰减的快慢：电阻或电容越大，时间常数越大，放电越慢。
Q = Q₀ e⁻ᵗ/ᴿᶜ
τ = RC
8. Magnetic Fields and Electromagnetism | 磁场与电磁学
A charged particle moving through a magnetic field experiences a force perpendicular to both its velocity and the field direction. This magnetic Lorentz force does no work since it is always perpendicular to the velocity, and it causes the particle to follow a circular or helical trajectory.
带电粒子在磁场中运动时，会受到一个同时垂直于其速度和磁场方向的力。这个磁洛伦兹力不做功，因为它始终垂直于速度，它会使粒子沿圆周或螺旋轨迹运动。
F = qvB sinθ
F = BIL sinθ (for a current-carrying wire)
Magnetic flux density B characterises the strength of a magnetic field. Magnetic flux Φ through a surface is the product of the component of B perpendicular to the surface and the area, and flux linkage multiplies this by the number of turns in a coil.
磁通量密度 B 描述磁场的强弱。通过一个面的磁通量 Φ 是 B 垂直于该面的分量与面积的乘积，而磁链则再乘以线圈的匝数。
Φ = BA cosθ
Flux linkage = NΦ
Faraday’s law of electromagnetic induction states that the induced emf in a circuit equals the negative rate of change of magnetic flux linkage. Lenz’s law, contained within the negative sign, states that the induced current flows in a direction that opposes the change causing it.
法拉第电磁感应定律指出，电路中产生的感应电动势等于磁链变化率的负值。包含在负号中的楞次定律指出，感应电流的方向总是使其对抗引发它的变化。
ε = –N dΦ/dt
A simple alternating current generator produces a sinusoidally varying emf when a coil rotates in a uniform magnetic field. The peak emf depends on the number of turns, the magnetic field strength, the coil area, and the angular speed of rotation.
当一个线圈在均匀磁场中旋转时，简单的交流发电机会产生按正弦变化的电动势。峰值电动势取决于匝数、磁场强度、线圈面积以及旋转的角速度。
ε = ε₀ sin(ωt)
ε₀ = NABω
Transformers operate on the principle of mutual induction, where a changing current in the primary coil generates a changing magnetic flux that links the secondary coil. For an ideal transformer with no flux leakage, the ratio of secondary to primary voltage equals the turns ratio.
变压器利用互感原理工作，初级线圈中变化的电流产生变化的磁通量，该磁通与次级线圈交链。对于无漏磁的理想变压器，次级电压与初级电压之比等于匝数比。
Vₛ/Vₚ = Nₛ/Nₚ
IₚVₚ = IₛVₛ (ideal transformer, 100% efficiency)
9. Waves and Optics | 波与光学
A progressive wave transfers energy without the net transfer of matter. The wave equation relates the speed of propagation v, the frequency f, and the wavelength λ, and applies universally to mechanical, electromagnetic, and other wavelike phenomena.
行波在传播能量时并没有物质的净输运。波速方程将传播速度 v、频率 f 和波长 λ 联系起来，它普遍适用于机械波、电磁波以及其他类似波的现象。
v = fλ
When two waves of the same type and frequency superpose, the principle of superposition applies. Constructive interference occurs when the path difference is an integer multiple of the wavelength, while destructive interference occurs when the path difference is an odd multiple of half-wavelengths.
当两个同类型、同频率的波叠加时，叠加原理适用。当路程差为波长的整数倍时，发生相长干涉；当路程差为半波长的奇数倍时，发生相消干涉。
Path difference = nλ (constructive)
Path difference = (n+½)λ (destructive)
Young’s double-slit experiment provides evidence for the wave nature of light. The fringe spacing Δx is directly proportional to the wavelength and the screen distance and inversely proportional to the slit separation, enabling precise measurement of optical wavelengths.
杨氏双缝实验为光的波动性提供了证据。条纹间距 Δx 与波长和屏幕距离成正比，与缝间距成反比，这使得精密测量光波波长成为可能。
λ = ax/D
A diffraction grating produces much sharper and brighter fringes than a double slit. The grating equation relates the angle at which a maximum of order n occurs to the wavelength and the grating spacing d, which is the reciprocal of the number of lines per unit length.
衍射光栅产生的条纹比双缝更锐利、更明亮。光栅方程将第 n 级极大出现的角度与波长和光栅间距 d 联系起来，其中 d 是单位长度刻线数的倒数。
d sinθ = nλ
Refraction occurs when a wave changes speed as it crosses a boundary between two media. Snell’s law relates the angles of incidence and refraction to the refractive indices of the media, which themselves are the ratios of the speed of light in vacuum to the speed in the medium.
当波跨越两种介质的界面且速度发生变化时，就发生折射。斯涅耳定律将入射角和折射角与介质的折射率联系起来，而折射率本身就是真空中光速与介质中光速的比值。
n₁ sinθ₁ = n₂ sinθ₂
n = c/v
Total internal reflection occurs when light travels from a medium of higher refractive index to one of lower refractive index at an angle of incidence greater than the critical angle. The critical angle depends only on the refractive indices of the two media forming the boundary.
当光从折射率较高的介质入射到折射率较低的介质，且入射角大于临界角时，发生全内反射。临界角仅取决于构成界面的两种介质的折射率。
sinθ𝒸 = n₂/n₁
10. Quantum Physics | 量子物理
The photoelectric effect provided the first compelling evidence for the quantisation of light. Einstein proposed that light consists of photons, each carrying energy hf, and that a photon can eject an electron from a metal surface only if its energy exceeds the work function φ of the metal.
光电效应为光的量子化提供了第一个令人信服的证据。爱因斯坦提出光由光子组成，每个光子携带能量 hf，只有当光子能量超过金属的逸出功 φ 时，它才能从金属表面击出一个电子。
E = hf = hc/λ
hf = φ + Kₘₐₓ
The stopping potential in a photoelectric experiment is directly related to the maximum kinetic energy of the emitted photoelectrons. Plotting stopping potential against frequency yields Planck’s constant and the work function from the gradient and intercept respectively.
光电实验中的遏止电势差与发射出的光电子最大动能直接相关。将遏止电势差对频率作图，可以由斜率和截距分别得出普朗克常数和逸出功。
eVₛ = Kₘₐₓ = hf – φ
Matter exhibits wave-like properties, as encapsulated by de Broglie’s hypothesis that any particle with momentum p has an associated wavelength. This wave-particle duality has been verified through electron diffraction experiments, where the spacing of atomic planes serves as a natural diffraction grating.
物质表现出波动性，德布罗意关于任何具有动量 p 的粒子都有一个相应波长的假设概括了这一点。这种波粒二象性已通过电子衍射实验得到验证，其中原子平面间距充当了天然的衍射光栅。
λ = h/p = h/mv
The Bohr model of the hydrogen atom postulated that electrons occupy stable, stationary orbits where the angular momentum is an integer multiple of h/2π. Transitions between these allowed energy levels result in the emission or absorption of photons with specific quantised energies.
氢原子的玻尔模型假设电子占据稳定的定态轨道，在这些轨道上角动量是 h/2π 的整数倍。在这些允许的能级之间的跃迁导致发射或吸收具有特定量子化能量的光子。
mvr = nh/2π (n = 1, 2, 3…)
Eₙ = –13.6 eV / n² (hydrogen energy levels)
Spectra provide fingerprints of atomic and molecular structure. The photon energy emitted in a transition between two energy levels equals the difference between those levels, with the wavelength determined by the Planck relation and the speed of light.
光谱为原子和分子结构提供了指纹特征。两个能级之间跃迁所发射的光子能量等于这两个能级之差，波长则由普朗克关系和光速决定。
hf = E₂ – E₁
11. Nuclear Physics and Radioactivity | 核物理与放射性Published by TutorHao | Pre-U Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply