📚 Pre-U OCR Physics: Core Concepts Overview | Pre-U OCR 物理：核心知识点梳理
The Pre-U OCR Physics course develops a deep understanding of fundamental physical principles, combining rigorous theoretical treatment with practical applications. This revision guide distils the core concepts across the syllabus, from classical mechanics to modern physics, to support exam preparation and conceptual clarity.
Pre-U OCR 物理课程旨在培养对基本物理原理的深刻理解，将严谨的理论处理与实际应用相结合。这份复习指南提炼了整个教学大纲中的核心概念，从经典力学到现代物理，以支持备考并澄清概念。
1. Mechanics: Kinematics and Dynamics | 力学：运动学与动力学
Kinematics describes motion without reference to its causes. Displacement, velocity and acceleration are vector quantities defined relative to a frame of reference. For constant acceleration in one dimension, the SUVAT equations link displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t).
运动学描述运动而不涉及其原因。位移、速度和加速度是相对于参考系定义的矢量。对于一维匀加速运动，SUVAT 方程联系了位移 (s)、初速度 (u)、末速度 (v)、加速度 (a) 和时间 (t)。
v = u + at s = ut + ½at² v² = u² + 2as s = ½(u+v)t
Dynamics stems from Newton’s laws of motion. The first law defines inertia, the second law states ΣF = ma, and the third law describes action–reaction pairs. Free-body diagrams are essential for resolving forces on objects on inclined planes, connected particles and systems in equilibrium or accelerating.
动力学源于牛顿运动定律。第一定律定义了惯性，第二定律指出 ΣF = ma，第三定律描述了作用力与反作用力对。受力图对于解决斜面物体、连接体以及平衡或加速系统中力的分解至关重要。
The coefficient of friction μ relates the maximum static friction and kinetic friction to the normal reaction force: F_fr = μR. When an object is on the point of sliding, friction reaches its limiting value.
摩擦系数 μ 将最大静摩擦力和动摩擦力与法向反作用力联系起来：F_fr = μR。当物体即将滑动时，摩擦力达到其极限值。
2. Momentum and Energy | 动量与能量
Linear momentum p = mv is conserved in a closed system when no external resultant force acts. The impulse-momentum theorem states that the change in momentum equals the impulse FΔt. In collisions, momentum is conserved for both elastic and inelastic interactions, but kinetic energy is only conserved in perfectly elastic collisions.
线动量 p = mv 在没有外部合力作用的封闭系统中守恒。冲量-动量定理指出动量的变化等于冲量 FΔt。在碰撞中，弹性和非弹性相互作用下动量都守恒，但动能仅在完全弹性碰撞中守恒。
Work done by a force is W = Fs cos θ, where θ is the angle between force and displacement. The principle of conservation of energy states that energy cannot be created or destroyed, only transferred between forms. Gravitational potential energy near Earth’s surface is mgh, and kinetic energy is ½mv².
力做的功为 W = Fs cos θ，其中 θ 是力与位移之间的夹角。能量守恒定律指出能量不能被创造或消灭，只能在形式之间转移。地球表面附近的重力势能为 mgh，动能为 ½mv²。
Power is the rate of doing work, P = W/t or P = Fv for a force moving at constant velocity. Understanding energy transfers helps analyse systems involving springs (elastic potential energy ½kx²) and dissipative forces such as air resistance.
功率是做功的速率，P = W/t，或对于匀速运动的力，P = Fv。理解能量转换有助于分析包含弹簧（弹性势能 ½kx²）和耗散力（如空气阻力）的系统。
3. Circular Motion and Simple Harmonic Motion | 圆周运动与简谐运动
Uniform circular motion requires a centripetal force directed towards the centre. The angular speed ω = Δθ/Δt = 2πf, linear speed v = ωr, and centripetal acceleration a = v²/r = ω²r. For a mass moving in a vertical circle, kinetic energy and gravitational potential energy exchange while the net radial force varies.
匀速圆周运动需要指向圆心的向心力。角速度 ω = Δθ/Δt = 2πf，线速度 v = ωr，向心加速度 a = v²/r = ω²r。对于在竖直面内做圆周运动的物体，动能和重力势能相互转换，径向合力则变化。
Simple harmonic motion (SHM) occurs when the restoring force is proportional to displacement and acts towards equilibrium: F = −kx, leading to a = −ω²x. The defining equation gives solutions x = A sin(ωt) or x = A cos(ωt). Velocity is v = ±ω√(A² − x²), with maximum speed v_max = ωA.
简谐运动 (SHM) 发生在恢复力与位移成正比且指向平衡位置时：F = −kx，导致 a = −ω²x。该定义方程的解为 x = A sin(ωt) 或 x = A cos(ωt)。速度 v = ±ω√(A² − x²)，最大速度 v_max = ωA。
The period of a mass-spring system is T = 2π√(m/k), and for a simple pendulum T = 2π√(l/g) for small amplitudes. Energy in SHM continuously interchanges between kinetic and potential, with total energy E = ½mω²A².
弹簧-质量系统的周期为 T = 2π√(m/k)，单摆在小振幅下的周期为 T = 2π√(l/g)。简谐运动中的能量在动能和势能之间连续交换，总能量 E = ½mω²A²。
Damping reduces amplitude over time; light damping produces a gradual loss, while critical damping returns to equilibrium without oscillation. Forced oscillations and resonance occur when the driving frequency matches the natural frequency, leading to sharp amplitude peaks.
阻尼使振幅随时间减小；轻阻尼产生逐渐衰减，临界阻尼则无振荡地回到平衡。受迫振动与共振发生在驱动频率等于固有频率时，导致尖锐的振幅峰值。
4. Gravitational Fields | 引力场
Newton’s law of universal gravitation states that two point masses attract each other with a force F = Gm₁m₂/r². The gravitational field strength g = F/m is a vector directed towards the source mass. For a spherical mass, the field outside is identical to that of a point mass at its centre.
牛顿万有引力定律指出两个质点以力 F = Gm₁m₂/r² 相互吸引。引力场强度 g = F/m 是指向场源质量的矢量。对于球体质量，外部的场等同于所有质量集中于中心的质点的场。
Gravitational potential V at a point is the work done per unit mass to bring a test mass from infinity to that point: V = −Gm/r. Field strength is the negative gradient of potential: g = −dV/dr. Equipotential surfaces are perpendicular to field lines, and no work is done when moving along them.
一点的引力势 V 是将单位质量从无穷远移至该点所做的功：V = −Gm/r。场强是势的负梯度：g = −dV/dr。等势面与场线垂直，沿等势面移动不做功。
Kepler’s laws describe planetary motion: orbits are ellipses with the Sun at one focus; a line joining a planet to the Sun sweeps out equal areas in equal times; T² ∝ r³ for elliptical orbits. These can be derived from Newtonian gravitation and conservation of angular momentum.
开普勒定律描述行星运动：轨道为椭圆，太阳位于一个焦点；行星与太阳的连线在相等时间内扫过相等面积；对于椭圆轨道，T² ∝ r³。这些可以从牛顿引力理论和角动量守恒推导得出。
5. Electric Fields and Capacitance | 电场与电容
Electric fields arise from charge distributions. Coulomb’s law gives the force between two point charges: F = kQ₁Q₂/r², where k = 1/(4πε₀). Electric field strength E = F/q, and for a uniform field between parallel plates, E = V/d, directed from positive to negative plate.
电场由电荷分布产生。库仑定律给出两点电荷之间的力：F = kQ₁Q₂/r²，其中 k = 1/(4πε₀)。电场强度 E = F/q，对于平行板之间的匀强电场，E = V/d，方向由正极板指向负极板。
Electric potential V = kQ/r for a point charge, and the relationship E = −dV/dr holds. The energy stored in a capacitor is W = ½QV = ½CV² = ½Q²/C. Capacitance C = Q/V, with a dielectric increasing capacitance by a factor ε_r.
点电荷的电势 V = kQ/r，且满足 E = −dV/dr。电容器储存的能量为 W = ½QV = ½CV² = ½Q²/C。电容 C = Q/V，电介质使电容增大 ε_r 倍。
Charging and discharging a capacitor through a resistor follow exponential laws: for discharge Q = Q₀ e^{−t/RC}, V = V₀ e^{−t/RC}, I = I₀ e^{−t/RC}. The time constant τ = RC determines how quickly the charge decays.
电容器通过电阻器的充电和放电遵循指数规律：对于放电，Q = Q₀ e^{−t/RC}，V = V₀ e^{−t/RC}，I = I₀ e^{−t/RC}。时间常数 τ = RC 决定了电荷衰减的快慢。
6. Magnetic Fields and Electromagnetic Induction | 磁场与电磁感应
A magnetic field exerts a force on a moving charge: F = Bqv sin θ for a charge q moving at velocity v, and on a current-carrying conductor F = BIL sin θ. Fleming’s left-hand rule gives the direction of force, magnetic field and conventional current.
磁场对运动电荷施加力：对于以速度 v 运动的电荷 q，F = Bqv sin θ；对于载流导体，F = BIL sin θ。弗莱明左手定则给出了力、磁场和常规电流的方向。
The Hall effect produces a transverse voltage across a conductor in a magnetic field, allowing determination of charge carrier density. Charged particles move in circular paths in uniform magnetic fields with radius r = mv/(Bq).
霍尔效应在磁场中的导体两端产生横向电压，可用于测定载流子浓度。带电粒子在匀强磁场中做圆周运动，半径 r = mv/(Bq)。
Faraday’s law states that the induced emf in a circuit is equal to the rate of change of magnetic flux linkage: ε = −d(NΦ)/dt. Lenz’s law gives the direction of the induced emf as opposing the change causing it. Applications include generators, transformers and eddy current braking.
法拉第定律指出回路中的感生电动势等于磁通链变化率的负值：ε = −d(NΦ)/dt。楞次定律给出感生电动势的方向，总是阻碍引起它的变化。应用包括发电机、变压器和涡流制动。
Self-inductance L is defined by ε = −L dI/dt, and the energy stored in an inductor is ½LI². The transformer equation V_s/V_p = N_s/N_p assumes ideal conditions with no flux leakage.
自感 L 的定义为 ε = −L dI/dt，电感器中储存的能量为 ½LI²。变压器方程 V_s/V_p = N_s/N_p 在无漏磁的理想条件下成立。
7. Wave Phenomena and Optics | 波动现象与光学
Progressive waves transfer energy without transferring matter. Key parameters are wavelength λ, frequency f, amplitude A and speed v = fλ. Transverse waves oscillate perpendicular to propagation direction, while longitudinal waves oscillate parallel. Electromagnetic waves are transverse and travel at speed c in vacuum.
行波传递能量而不传递物质。关键参数包括波长 λ、频率 f、振幅 A 和波速 v = fλ。横波的振动垂直于传播方向，纵波的振动平行于传播方向。电磁波是横波，在真空中以光速 c 传播。
Superposition leads to interference, diffraction and standing waves. Two coherent sources produce an interference pattern; constructive interference occurs when path difference is nλ, destructive when path difference is (n+½)λ. Young’s double-slit fringe spacing Δy = λD/d.
叠加导致干涉、衍射和驻波。两个相干波源产生干涉图样；当波程差为 nλ 时发生相长干涉，为 (n+½)λ 时发生相消干涉。杨氏双缝的条纹间距 Δy = λD/d。
Diffraction gratings produce sharper maxima at angles given by d sin θ = nλ. Single-slit diffraction yields a central maximum of width proportional to λ/a, where a is the slit width. The Rayleigh criterion for resolution states that two sources are just resolved when the central maximum of one coincides with the first minimum of the other.
衍射光栅在满足 d sin θ = nλ 的角度产生更锐利的极大。单缝衍射产生宽度与 λ/a 成正比的中央明纹，其中 a 为缝宽。瑞利判据指出，当一个光源的中央极大与另一个光源的第一极小重合时，两者刚好能被分辨。
Refraction obeys Snell’s law: n₁ sin θ₁ = n₂ sin θ₂. Total internal reflection occurs when the angle of incidence exceeds the critical angle sin θ_c = n₂/n₁ (for n₁ > n₂). Fibre optics exploit this principle for communication.
折射遵循斯涅尔定律：n₁ sin θ₁ = n₂ sin θ₂。当入射角超过临界角 sin θ_c = n₂/n₁（对于 n₁ > n₂）时发生全内反射。光纤利用这一原理进行通信。
8. Quantum Physics and Atomic Structure | 量子物理与原子结构
Photons have energy E = hf and momentum p = h/λ. The photoelectric effect provides evidence for the particle-like nature of light: electrons are emitted only when photon energy exceeds the work function φ, with maximum kinetic energy K_max = hf − φ.
光子具有能量 E = hf 和动量 p = h/λ。光电效应为光的粒子性提供了证据：仅当光子能量大于功函数 φ 时电子才会逸出，最大动能 K_max = hf − φ。
Matter waves are described by the de Broglie wavelength λ = h/p. Electron diffraction confirms wave-like behaviour of particles. The Bohr model of the hydrogen atom gives quantised energy levels E_n = −13.6 eV/n², with transitions producing photons of energy ΔE = hf.
物质波由德布罗意波长 λ = h/p 描述。电子衍射证实了粒子的波动性。氢原子的玻尔模型给出了量子化能级 E_n = −13.6 eV/n²，能级间的跃迁产生能量为 ΔE = hf 的光子。
Heisenberg’s uncertainty principle sets a fundamental limit on simultaneous measurements: ΔxΔp_x ≥ h/(4π). Energy-time uncertainty ΔEΔt ≥ h/(4π) allows the existence of virtual particles and explains the natural width of spectral lines.
海森堡不确定性原理对同时测量设定了基本限制：ΔxΔp_x ≥ h/(4π)。能量-时间不确定性 ΔEΔt ≥ h/(4π) 允许虚粒子的存在，并解释光谱线的自然宽度。
9. Nuclear and Particle Physics | 核物理与粒子物理
The nucleus consists of protons and neutrons, held together by the strong nuclear force. The mass defect Δm accounts for binding energy via E = Δmc². Binding energy per nucleon indicates nuclear stability, peaking around iron-56.
原子核由质子和中子组成，靠强核力结合在一起。质量亏损 Δm 通过 E = Δmc² 对应于结合能。比结合能反映核的稳定性，在铁-56 附近达到峰值。
Radioactive decay types include alpha (emission of helium nucleus), beta-minus (n → p + e⁻ + ν_e_bar), and gamma (photon from nuclear de-excitation). Activity A = λN follows the exponential decay law N = N₀ e^{−λt}, with half-life t_½ = ln2/λ.
放射性衰变类型包括 α 衰变（发射氦核）、β⁻ 衰变（n → p + e⁻ + ν̅ₑ）和 γ 衰变（核退激发射光子）。活度 A = λN 服从指数衰变规律 N = N₀ e^{−λt}，半衰期 t_½ = ln2/λ。
The Standard Model classifies particles into quarks and leptons. Protons and neutrons are baryons made of three quarks; mesons are quark-antiquark pairs. Conservation laws for baryon number, lepton number, charge, and strangeness govern particle interactions.
标准模型将粒子分为夸克和轻子。质子和中子是由三个夸克组成的重子；介子是夸克-反夸克对。重子数、轻子数、电荷和奇异数守恒定律支配着粒子相互作用。
Feynman diagrams represent exchange particles mediating forces: photons for electromagnetic, W/Z bosons for weak, and gluons for strong interactions. Pair production and annihilation illustrate energy–mass equivalence.
费曼图表示传递力的交换粒子：光子传递电磁力，W/Z 玻色子传递弱力，胶子传递强力。电子偶的产生和湮灭展示了质能等价。
10. Thermodynamics and Kinetic Theory | 热力学与分子运动论
Internal energy U is the sum of random kinetic and potential energies of molecules. The first law of thermodynamics states ΔU = Q − W, where Q is heat added to the system and W is work done by the system. Thermodynamic processes include isothermal (ΔU = 0), adiabatic (Q = 0), constant‑volume and constant‑pressure changes.
内能 U 是分子无规则动能与势能的总和。热力学第一定律为 ΔU = Q − W，其中 Q 为系统吸收的热量，W 为系统对外做的功。热力学过程包括等温过程 (ΔU = 0)、绝热过程 (Q = 0)、等容和等压变化。
The ideal gas law pV = nRT combines Boyle’s, Charles’s and the pressure law. For an ideal gas, the average translational kinetic energy of a molecule is ³/₂ kT, leading to pV = ⅓ Nm⟨c²⟩. The root-mean-square speed c_rms = √(3RT/M).
理想气体定律 pV = nRT 综合了玻意耳定律、查理定律和压力定律。对于理想气体，分子的平均平动动能为 ³/₂ kT，从而得出 pV = ⅓ Nm⟨c²⟩。方均根速率 c_rms = √(3RT/M)。
Specific heat capacities at constant volume C_V and constant pressure C_P relate to molar heat capacities, with C_P − C_V = R. Adiabatic changes obey pV^γ = constant, where γ = C_P/C_V. Kinetic theory explains gas pressure and temperature in terms of molecular motion.
定容热容 C_V 和定压热容 C_P 与摩尔热容相关，且 C_P − C_V = R。绝热变化遵循 pV^γ = 常数，其中 γ = C_P/C_V。分子运动论从分子运动的角度解释了气体压强和温度。
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