📚 Essential Physics Formulas and Application Tips | 物理常见公式核心梳理与应用技巧
Physics is built on a small set of powerful equations. Mastering the core formulas, understanding their limits, and knowing when to apply them is the key to solving exam problems quickly and accurately.
物理学科建立在少数几个强大的方程之上。掌握核心公式、理解它们的适用条件，并知道何时使用它们，是快速准确解答考试题目的关键。
1. Kinematics | 运动学
The three constant-acceleration equations connect displacement s, initial velocity u, final velocity v, acceleration a, and time t.
三个匀加速运动方程联系了位移 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
These equations only work when acceleration is constant. In projectile motion, treat vertical and horizontal motion independently; horizontal velocity remains unchanged while vertical motion follows free fall.
这些公式仅在加速度恒定时成立。在抛体运动中，应分别处理竖直和水平运动：水平速度保持不变，竖直运动遵循自由落体规律。
- Tip: Decide your sign convention first — usually upward is positive.
- 技巧：先确定正方向——通常取向上为正。
2. Newton's Laws and Forces | 牛顿定律与力
Newton's second law relates net force, mass, and acceleration.
牛顿第二定律将合外力、质量和加速度联系起来。
Fnet = ma
Weight is a specific force: W = mg. Friction often appears as F_friction ≤ μN, where μ is the coefficient of friction and N is the normal reaction force.
重力是一种特殊力：W = mg。摩擦力通常表示为 F_摩擦 ≤ μN，其中 μ 为摩擦系数，N 为法向支持力。
When drawing free-body diagrams, list all forces acting on the object, then resolve them into components. For inclined planes, choose axes parallel and perpendicular to the surface.
画受力分析图时，先列出物体所受的所有力，再将其分解为分量。对于斜面，选择平行和垂直于斜面的方向作为坐标轴。
- mmon error: forgetting that F in F = ma is the net force, not an individual applied force.
- 常见错误：忘记 F = ma 中的 F 是合外力，而不是某个单独的外力。
3. Work, Energy and Power | 功、能量与功率
Work done by a constant force is given by:
恒力做功的表达式为：
W = F s cos θ
Kinetic energy and gravitational potential energy are:
动能和重力势能分别为：
KE = ½mv², PE = mgh
The work–energy theorem states that net work equals change in kinetic energy: Wnet = ΔKE. Power is the rate of doing work:
动能定理指出，合外力做功等于动能的变化：W_合 = ΔKE。功率是做功的快慢：
P = W / t = F v
For conservative forces, mechanical energy is conserved. Use energy conservation when forces are not constant or when the path is curved.
对于保守力，机械能守恒。当力不恒定或路径弯曲时，优先使用能量守恒。
4. Momentum and Collisions | 动量与碰撞
Momentum is the product of mass and velocity: p = mv. The impulse–momentum theorem states:
动量是质量与速度的乘积：p = mv。冲量–动量定理为：
Impulse = F Δt = Δp
In any collision or explosion, total momentum is conserved if no external force acts. For perfectly elastic collisions, kinetic energy is also conserved; for inelastic collisions it is not.
在任何碰撞或爆炸中，如果不受外力，总动量守恒。对于完全弹性碰撞，动能也守恒；对于非弹性碰撞，动能不守恒。
- In one-dimensional elastic collisions of equal masses, the two objects simply exchange velocities.
- 在一维弹性碰撞中，若两物体质量相等，则它们交换速度。
5. Circular Motion and Gravitation | 圆周运动与万有引力
For uniform circular motion, centripetal acceleration is directed toward the center:
对于匀速圆周运动，向心加速度指向圆心：
a = v² / r = ω² r
Thus the centripetal force is:
因此向心力为：
F = mv² / r = m ω² r
Newton's law of gravitation gives the attractive force between two masses:
牛顿万有引力定律给出两个质量之间的吸引力：
F = G m₁m₂ / r²
For satellites in orbit, the gravitational force provides the required centripetal force. This leads to orbital speed v = √GM/r and orbital period T² ∝ r³ (Kepler's third law).
对于在轨卫星，万有引力提供所需的向心力。由此可得轨道速度 v = √GM/r 以及轨道周期 T² ∝ r³（开普勒第三定律）。
6. Simple Harmonic Motion | 简谐运动
Simple harmonic motion (SHM) occurs when acceleration is proportional to displacement and directed toward equilibrium:
当加速度与位移成正比且指向平衡位置时，物体做简谐运动：
a = −ω² x
The displacement, velocity, and acceleration equations for SHM are:
简谐运动的位移、速度和加速度方程为：
x = A cos(ωt), v = −Aω sin(ωt), a = −ω² x
The period depends on the physical system. For a mass–spring system: T = 2π√m/k. For a simple pendulum: T = 2π√l/g.
周期取决于具体的物理系统。对于弹簧振子：T = 2π√m/k。对于单摆：T = 2π√l/g。
- Energy in SHM shifts between kinetic and potential forms; total energy is proportional to A².
- 简谐运动的能量在动能和势能之间转化；总能量与 A² 成正比。
7. Electric Fields and Coulomb's Law | 电场与库仑定律
Two point charges exert forces on each other according to Coulomb's law:
两个点电荷之间的作用力遵循库仑定律：
F = k q₁q₂ / r²
Here k = 1/(4πε₀) ≈ 8.99 × 10⁹ N·m²/C². The electric field due to a point charge is:
其中 k = 1/(4πε₀) ≈ 8.99 × 10⁹ N·m²/C²。点电荷产生的电场强度为：
E = F / q = k Q / r²
For uniform fields, such as between parallel plates, E = V/d, where V is the potential difference and d is the plate separation. Electric potential energy is U = qV.
对于平行板之间的匀强电场，E = V/d，其中 V 为电势差，d 为板间距。电势能为 U = qV。
8. Electric Circuits | 电路
Ohm's law relates voltage, current, and resistance:
欧姆定律给出电压、电流和电阻之间的关系：
V = IR
Electrical power dissipated in a resistor can be written in three equivalent forms:
电阻消耗的电功率有三种等价表达形式：
P = VI = I² R = V² / R
For resistors in series, resistances add: Rtotal = R₁ + R₂ + …. For resistors in parallel, conductances add:
对于串联电阻，总电阻相加：R_总 = R₁ + R₂ + …。对于并联电阻，电导相加：
1/Rtotal = 1/R₁ + 1/R₂ + …
When analyzing circuits, use Kirchhoff's laws: the sum of currents entering a junction equals the sum leaving; the sum of potential differences around any closed loop is zero.
分析电路时，使用基尔霍夫定律：流入节点的电流之和等于流出之和；沿任意闭合回路的电势差之和为零。
9. Magnetic Fields and Forces | 磁场与磁力
A current-carrying conductor in a magnetic field experiences a force:
载流导体在磁场中会受到力的作用：
F = B I L sin θ
Here B is magnetic flux density, I is current, L is the length of conductor, and θ is the angle between the wire and the field. For a moving charge, the force is:
其中 B 为磁感应强度，I 为电流，L 为导体长度，θ 为导线与磁场之间的夹角。对于运动电荷，洛伦兹力为：
F = q v B sin θ
The direction of force is perpendicular to both the velocity/current and the magnetic field, as determined by Fleming's left-hand rule.
力的方向同时垂直于速度/电流和磁场方向，可用左手定则判断。
- rpendicular to a uniform magnetic field, it follows a circular path of radius r = mv/(qB).
- 当带电粒子垂直于匀强磁场运动时，它做圆周运动，半径 r = mv/(qB)。
10. Thermal Physics and Ideal Gases | 热学与理想气体
The ideal gas law combines pressure, volume, temperature, and the number of moles:
理想气体状态方程将压强、体积、温度和物质的量联系起来：
P V = n R T
In terms of the number of molecules, it becomes PV = NkT, where k = R/NA is Boltzmann's constant. The average kinetic energy of gas molecules is directly proportional to absolute temperature:
用分子数表示时，方程为 PV = NkT，其中 k = R/NA 是玻尔兹曼常数。气体分子的平均平动动能与热力学温度成正比：
⟨KE⟩ = ½ m ⟨v²⟩ = (3/2) kT
The first law of thermodynamics states that the change in internal energy equals heat added minus work done by the gas:
热力学第一定律指出，内能的变化等于吸收的热量减去气体对外做的功：
ΔU = Q − W
Always check whether work is done by the system or on the system, as different textbooks may define sign conventions differently.
一定要注意功是系统对外做功还是外界对系统做功，因为不同教材采用的符号约定可能不同。
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