📚 Key Physics Models for Exam Preparation: Summary and Applications | 物理备考：重点物理模型归纳与应用
Physics problems in high-stakes exams often reduce to a small set of conceptual models. Mastering these models helps you identify the core mechanics, choose the right equations, and avoid common traps. This article summarizes the most frequently tested models across mechanics, electromagnetism, and thermal physics, with practical applications for your revision.
物理考试中的难题往往可以归结为少数几个核心模型。掌握这些模型，能够帮助你迅速抓住问题本质、选择正确的公式，并避开常见陷阱。本文汇总了力学、电磁学和热学中最常考查的模型，并给出备考应用建议。
1. Particle (Mass Point) Model | 质点模型
When the size and shape of an object do not affect the motion studied, treat the object as a particle with mass concentrated at one point. This approximation applies to translation, projectile motion, and orbital motion, provided that rotation or deformation is negligible.
当物体的形状和大小对所研究的运动没有影响时，可将其视为质量集中于一点的质点。这一近似适用于平动、抛体运动和轨道运动，前提是转动或形变可忽略。
- Applied in Newton's second law: F = ma, where force, mass, and acceleration all refer to the particle.
- 应用牛顿第二定律：F = ma，其中力、质量和加速度均针对质点。
- In projectile motion, decompose motion into horizontal uniform motion and vertical accelerated motion.
- 在抛体运动中，将运动分解为水平匀速运动和竖直加速运动。
x = v₀t, y = ½gt²
For a ball thrown horizontally, the flight time depends only on height, not on horizontal velocity.
对于水平抛出的球，飞行时间只取决于高度，与水平速度无关。
2. Spring Oscillator Model | 弹簧振子模型
le harmonic motion (SHM). The restoring force is F = −kx, and the angular frequency is ω = √k/m. Energy shifts between elastic potential energy and kinetic energy.
弹簧-质量系统是简谐运动的原型。回复力为 F = −kx，角频率为 ω = √k/m。能量在弹性势能与动能之间相互转化。
- Amplitude determines total energy: E = ½kA².
- 振幅决定总能量：E = ½kA²。
- In vertical oscillation, gravity shifts the equilibrium position but does not change the period.
- 在竖直振动中，重力会改变平衡位置，但不改变周期。
T = 2π√m/k
If the spring is cut in half, the spring constant doubles, so the period decreases by a factor of √2.
若将弹簧剪成两半，劲度系数变为原来的两倍，因此周期变为原来的 1/√2。
3. Simple Pendulum Model | 单摆模型
A simple pendulum is a mass on an inextensible, massless string. For small angles, the restoring torque is approximately linear, giving SHM. The period depends only on length and gravitational acceleration.
单摆由不可伸长、质量忽略的细绳和质量块构成。在小角度近似下，回复力矩近似线性，因此做简谐运动。周期仅取决于摆长和重力加速度。
T = 2π√L/g
In an accelerating lift, use an effective gravitational acceleration: g' = g ± a. A free-falling lift gives g' = 0, losing periodicity.
在加速升降机中，应使用等效重力加速度：g' = g ± a。自由下落的升降机中 g' = 0，单摆不再周期运动。
- Do not use this model if the angle exceeds about 5° where SHM no longer holds.
- 当摆角超过约 5° 时，简谐近似不再成立，此时不能使用该模型。
4. Connected Bodies (Pulley) Model | 连接体（滑轮）模型
Multiple objects connected by strings or rods share the same magnitude of acceleration if the string remains taut. The key is to treat the whole system as one object to find acceleration, then isolate a single object to find internal forces.
通过绳或杆连接的多个物体，当绳张紧时具有大小相等的加速度。关键步骤是：先整体求加速度，再隔离单个物体求内力。
a = (m₁ − m₂)g / (m₁ + m₂) (Atwood machine)
For a block on a frictionless table connected to a hanging mass, the hanging weight accelerates the whole system.
对于光滑水平桌面上的物块连接一个悬挂重物的系统，悬挂重物的重力使整个系统加速。
- Check whether the string is ideal (massless, inextensible) for the same tension.
- 检查绳是否为理想绳（质量零、不可伸长），以保证张力处处相同。
- When the floor is inclined, include components of gravity along the slope.
- 当接触面为斜面时，必须考虑重力沿斜面的分量。
5. Conveyor Belt Model | 传送带模型
Conveyor belt problems combine friction, kinematics, and relative motion. The friction direction is determined by the relative slide between the object and the belt. Once the object reaches belt speed, friction may vanish or change from kinetic to static.
传送带问题综合了摩擦、运动学和相对运动。摩擦方向取决于物体与传送带之间的相对滑动。当物体速度与传送带速度相同时，摩擦力可能消失或由滑动摩擦变为静摩擦。
- If the belt is horizontal, the object accelerates under friction until v = vbelt.
- 水平传送带：物体在摩擦力作用下加速，直到 v = v_带。
- On an inclined belt, compare the component of gravity along the slope with the maximum static friction.
- 倾斜传送带：需比较重力沿斜面分量与最大静摩擦力。
f = μmg (sliding) → 0 or ≤ μₛmg (static)
Calculate the relative displacement to find heat loss: Q = f · srel.
求相对位移可得摩擦生热：Q = f · s_相对。
6. Block on Block (Plate) Model | 滑块-木板模型
This model involves two contacting objects with possible relative sliding or sticking together. The critical condition is whether the friction between the blocks is enough to ensure common motion.
该模型涉及两个相互接触的物体，可能相对滑动或相对静止。临界条件是两物体间的摩擦力是否足以维持共同运动。
fmax = μₛN, acommon = F / (M + m)
If the lower block accelerates too fast, the upper block will slip. Determine the maximum applied force F that keeps them together.
如果下方木板加速度过大，上方物块将发生滑动。应求出能使两者保持相对静止的最大拉力 F。
- Draw free-body diagrams for each object separately.
- 分别对两个物体做受力分析。
- Use momentum and energy conservation only if no external impulse or work beyond gravity is present.
- 只有在无外力冲量或除重力外无其他做功时，才能使用动量守恒和能量守恒。
7. Collision Model | 碰撞模型
Collisions are separated into elastic, inelastic, and perfectly inelastic types. Momentum is always conserved in an isolated system; kinetic energy is conserved only in elastic collisions.
碰撞分为弹性碰撞、非弹性碰撞和完全非弹性碰撞。孤立系统中动量总守恒，但动能仅在弹性碰撞中守恒。
m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'
For a perfectly inelastic collision, the two objects move together with the same velocity.
完全非弹性碰撞中，两物体粘在一起以相同速度运动。
- Elastic collision formula: v₁' = (m₁−m₂)v₁/(m₁+m₂), v₂' = 2m₁v₁/(m₁+m₂) for m₂ initially at rest.
- 弹性碰撞公式（m₂初始静止）：v₁' = (m₁−m₂)v₁/(m₁+m₂)，v₂' = 2m₁v₁/(m₁+m₂)。
- In a one-dimensional collision, the relative speed of approach equals the relative speed of separation for elastic collisions.
- 在一维弹性碰撞中，接近的相对速度等于分离的相对速度。
8. Charged Particle in a Uniform Electric Field | 带电粒子在匀强电场中的运动
This model resembles projectile motion, with a constant electric force providing acceleration perpendicular or parallel to the initial velocity. It is central to cathode-ray tubes and deflection plates.
该模型类似于抛体运动，恒定电场力提供加速度，方向可与初速度垂直或平行。这是阴极射线管和偏转板的核心原理。
a = qE/m, y = ½at² = qUL²/(2mdv₀²)
A particle entering perpendicular to a uniform field follows a parabolic trajectory. The deflection depends on charge-to-mass ratio and plate geometry.
带电粒子垂直进入匀强电场时沿抛物线轨迹运动。偏转量取决于荷质比和极板几何参数。
- Always split the analysis into horizontal uniform motion and vertical accelerated motion.
- 始终将运动分解为水平匀速运动和竖直匀加速运动。
- If the particle exits the field, ignore the field after exit and continue with straight-line motion.
- 若粒子飞出电场，离开后不再受电场力，按匀速直线运动处理。
9. Charged Particle in a Uniform Magnetic Field | 带电粒子在匀强磁场中的运动
A charged particle moving perpendicular to a uniform magnetic field experiences a Lorentz force that provides centripetal acceleration. The speed remains constant, and the path is circular.
带电粒子垂直进入匀强磁场时，洛伦兹力提供向心力。速率保持不变，运动轨迹是圆。
qvB = mv²/r → r = mv/(qB), T = 2πm/(qB)
The radius is proportional to momentum, and the period is independent of speed. This is the basis of mass spectrometers and cyclotrons.
回旋半径与动量成正比，周期与速率无关。这是质谱仪和回旋加速器的基本原理。
- If the velocity has a component parallel to B, the motion is a helix.
- 若速度存在平行于 B 的分量，运动轨迹是螺旋线。
- In circular motion, the magnetic force does no work, so kinetic energy is conserved.
- 在圆周运动中，洛伦兹力不做功，因此动能守恒。
10. Ideal Transformer Model | 理想变压器模型
The ideal transformer assumes no energy loss, no leakage flux, and zero winding resistance. It changes AC voltage and current according to the turn ratio while conserving power.
理想变压器假设无能量损失、无漏磁、绕组电阻为零。它根据匝数比改变交流电压和电流，但功率守恒。
U₁/U₂ = n₁/n₂, I₁/I₂ = n₂/n₁, P₁ = P₂
For a step-up transformer, voltage increases but current decreases proportionally to keep power constant.
升压变压器中，电压升高，但电流成比例减小以保持功率恒定。
- This model only works for alternating current, not direct current.
- 此模型仅适用于交流电，不适用于直流电。
- Real transformers have core losses and copper losses; efficiency is less than 100%.
- 实际变压器存在铁损和铜损，效率小于 100%。
11. Ideal Gas Model | 理想气体模型
lecular forces, except during elastic collisions. It obeys the equation of state PV = nRT.
理想气体模型假设气体分子为质点，除弹性碰撞外无分子间作用力，分子体积可忽略。它遵守状态方程 PV = nRT。
PV = nRT, ΔU = Q + W (first law)
In an isothermal process, ΔU = 0, so Q = −W. In an adiabatic process, Q = 0, so ΔU = W.
等温过程中 ΔU = 0，因此 Q = −W。绝热过程中 Q = 0，所以 ΔU = W。
- Use the kinetic theory relation: average kinetic energy ∝ T.
- 利用分子动理论关系：平均动能 ∝ T。
- For a monatomic ideal gas, U = (3/2)nRT.
- 对单原子理想气体，U = (3/2)nRT。
12. Circuit Model with Internal Resistance | 含内阻的电路模型
A real battery is modeled as an ideal EMF source in series with an internal resistance r. The terminal voltage equals the EMF minus the voltage drop across r.
实际电池可建模为理想电动势源与内阻 r 串联。路端电压等于电动势减去内阻上的电压降。
U = E − Ir, Pmax = E²/(4r)
delivered to the load when the external resistance equals the internal resistance (R = r).
当外电阻等于内阻（R = r）时，负载获得最大功率。
- In a closed circuit, the current is I = E/(R + r).
- 闭合电路中电流为 I = E/(R + r)。
- Short circuit occurs when R = 0, giving a dangerously large current.
- 当 R = 0 时发生短路，电流极大，十分危险。
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