Classical Mechanics represents the foundational bedrock of natural philosophy and modern engineering science. From computing macroscopic celestial orbits to designing vibration isolation damping mounts, each mechanical discipline is governed by rigorous dynamical equations and universal conservation principles.
This cheat sheet systematically organizes 20 core physical laws and foundational equations in classical mechanics and kinematics, partitioned into Newton's Laws of Motion & Momentum, Conservation Principles & Energy, Gravitation & Circular Motion, and Oscillations & Waves, clearly presenting standard mathematical expressions, key variable definitions, physical principles, validity domains, and industrial engineering applications.
🔗 Physics Laws & Equations Matrix: Electromagnetism · Fluid Dynamics & Thermodynamics · Optics & Acoustics · Nuclear & Particle Physics · Relativity & Cosmology · Astrophysics & Geophysics · Quantum Mechanics
1. Newton's Laws of Motion & Momentum (5 Laws)
Newton's laws of motion established the axiomatic framework of classical particle dynamics, defining inertial reference frames, the nature of force, and momentum conservation.
01 · Newton's First Law
Law of Inertia
$$\sum \mathbf{F} = 0 \implies \mathbf{v} = \text{const}$$
- ①Physical Essence: In the absence of external forces or when net force vanishes, a body continues in its state of rest or uniform rectilinear motion. Force is the cause of changing a body's kinematic state, not maintaining it.
- ②Variable Definitions: $\sum\mathbf{F}$ is net external force vector, and $\mathbf{v}$ is particle velocity vector (constant velocity encompasses rest and steady rectilinear translation).
- ③Fundamental Significance: The logical starting point of classical mechanics, qualitatively defining inertia and establishing the existence of inertial reference frames.
- ④Validity Domain & Applications: Strictly valid only in inertial frames; non-inertial frames require fictitious inertial forces (Coriolis, centrifugal). Applied in unpowered deep-space orbital coasting, seatbelt passive safety, and impact crumple zone design.
02 · Newton's Second Law
Fundamental Equation of Dynamics
$$\mathbf{F} = m\mathbf{a} = \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t}$$
- ①Physical Essence: Applied net force produces acceleration; acceleration is directly proportional to and collinear with net force, and inversely proportional to inertial mass. The generalized calculus formulation states that net force equals time rate of change of momentum.
- ②Variable Definitions: $\mathbf{F}$ is net force ($\text{N}$), $m$ is inertial mass ($\text{kg}$), $\mathbf{a}$ is acceleration ($\text{m/s}^2$), and $\mathbf{p} = m\mathbf{v}$ is linear momentum.
- ③Fundamental Significance: The central equation of classical dynamics, serving as the computational starting point for virtually all macroscopic motion analysis.
- ④Validity Domain & Applications: Holds in inertial frames at non-relativistic velocities ($v \ll c$); micro-scale high-speed systems require relativistic and quantum corrections. Applied in ballistic trajectory simulation, vehicle braking budgets, seismic structural loads, and variable-mass rocket thrust dynamics.
03 · Newton's Third Law
Action and Reaction
$$\mathbf{F}_{AB} = -\mathbf{F}_{BA}$$
- ①Physical Essence: When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude, opposite in direction, collinear, and identical in physical nature on the first body.
- ②Variable Definitions: $\mathbf{F}_{AB}$ is force exerted by body A on body B, and $\mathbf{F}_{BA}$ is reaction force exerted by body B on body A.
- ③Fundamental Significance: Cornerstone for system force balancing and deriving momentum conservation. Action and reaction act on separate bodies and **never cancel each other out** (fundamentally distinct from balanced equilibrium forces on a single object).
- ④Validity Domain & Applications: Universally holds for macroscopic contact forces and instantaneous central fields. Applied in rocket jet propulsion, aircraft propeller thrust, and free-body diagram isolation in structural mechanics.
04 · Impulse-Momentum Theorem
Impulse & Momentum Change
$$\mathbf{I} = \int_{t_1}^{t_2} \mathbf{F}\,\mathrm{d}t = \Delta\mathbf{p}$$
- ①Physical Essence: Impulse represents the cumulative time effect of applied force; the total impulse exerted by net external force on a body equals the change in its linear momentum.
- ②Variable Definitions: $\mathbf{I}$ is impulse ($\text{N}\cdot\text{s}$), $\mathbf{F}$ is net force, $t$ is duration, and $\Delta\mathbf{p}$ is momentum change. Under constant force: $\mathbf{F}\Delta t = \Delta\mathbf{p}$.
- ③Fundamental Significance: Core analytical tool for resolving transient forces, collisions, and impacts without requiring detailed transient acceleration time histories.
- ④Validity Domain & Applications: For a fixed momentum change, prolonging impact time $\Delta t$ significantly attenuates peak impact forces. Applied in automotive airbag deceleration cushioning, ballistic armor trauma liners, safety catch netting, and planetary lander crushable honeycomb dampers.
05 · Conservation of Linear Momentum
System Momentum Conservation
$$\sum \mathbf{F}_{\text{ext}} = 0 \implies \sum_{i=1}^n m_i\mathbf{v}_i = \text{const}$$
- ①Physical Essence: If the net external force acting on a system is zero, the total linear momentum of the system remains strictly conserved. If external force along a given axis vanishes, component momentum along that axis is conserved.
- ②Variable Definitions: $\mathbf{F}_{\text{ext}}$ is net external force, while $m_i$ and $\mathbf{v}_i$ represent the mass and velocity of the $i$-th particle.
- ③Fundamental Significance: Stems from spatial translational symmetry (Noether's theorem). Internal system forces can alter kinetic energy distribution, but internal mutual action cancels exactly, leaving total momentum unchanged.
- ④Validity Domain & Applications: Holds more broadly than mechanical energy conservation, remaining exact in relativistic physics, nuclear reactions, and particle physics. Applied in elastic/inelastic collisions, rocket propellant staging, artillery recoil attenuation, and high-energy particle collider analysis.
2. Conservation Principles & Energy (4 Laws)
Energy and angular momentum conservation provide powerful scalar and vector analytical tools for unravelling complex, non-steady, curvilinear, and rotational dynamics without resolving instantaneous accelerations.
06 · Definition of Mechanical Work
Work-Energy Integration
$$W = \int \mathbf{F}\cdot\mathrm{d}\mathbf{r} = F s \cos\theta$$
- ①Physical Essence: The spatial path integral of force along displacement, quantifying mechanical energy transfer and conversion across boundaries.
- ②Variable Definitions: $W$ is mechanical work ($\text{J}$), $\mathbf{F}$ is applied force, $s$ is displacement distance, and $\theta$ is angle between force and displacement vectors.
- ③Fundamental Significance: Foundational scalar quantity of energy methods. Work has sign: $\theta < 90^\circ$ does positive work (energy input), $\theta = 90^\circ$ does zero work (e.g., centripetal force), and $\theta > 90^\circ$ does negative work (dissipative braking).
- ④Validity Domain & Applications: Valid for constant and spatially varying forces. Applied in hoisting powertrain motor sizing, braking distance dissipation budgets, and machine drivetrain efficiency calculations.
07 · Work-Energy Theorem
Net Work & Kinetic Energy Change
$$W_{\text{net}} = \Delta E_k = \frac{1}{2}m v_2^2 - \frac{1}{2}m v_1^2$$
- ①Physical Essence: The net algebraic work performed by all external and internal forces acting on an object equals the change in its translational kinetic energy.
- ②Variable Definitions: $W_{\text{net}}$ is total work performed by all active forces (gravity, normal, friction, thrust), and $E_k = \frac{1}{2}mv^2$ is kinetic energy ($\text{J}$).
- ③Fundamental Significance: Directly links path integrals of force to initial and terminal velocities, eliminating the need to evaluate time histories or instantaneous accelerations.
- ④Validity Domain & Applications: Universally valid in inertial frames for rectilinear and curvilinear trajectories under conservative and non-conservative forces. Applied in vehicle crash stopping distances, roller coaster peak velocities, and hydro-turbine runner kinetic extraction.
08 · Conservation of Mechanical Energy
Mechanical Energy Invariance
$$E_k + E_p = \text{const} \quad (\Delta E_k + \Delta E_p = 0)$$
- ①Physical Essence: In an isolated system where only conservative forces (gravity, ideal spring elasticity) do work, kinetic energy and potential energy continuously interchange while total mechanical energy remains strictly constant.
- ②Variable Definitions: $E_k$ is kinetic energy, and $E_p$ is potential energy (gravitational $mgh$, elastic spring $\frac{1}{2}kx^2$).
- ③Fundamental Significance: Stemming from temporal translation symmetry (Noether's theorem). Dissipative forces (friction, drag) convert mechanical energy into internal thermal energy, preserving total universe energy.
- ④Validity Domain & Applications: **Strictly valid only when non-conservative work is negligible**. Applied in pendulum dynamics, roller coaster vertical drop design, hydroelectric head conversion, and pole-vault kinetic-elastic-gravitational energy transfers.
09 · Conservation of Angular Momentum
Rotational Invariance
$$\sum \boldsymbol{\tau}_{\text{ext}} = 0 \implies \mathbf{L} = I\boldsymbol{\omega} = \text{const}$$
- ①Physical Essence: When net external torque acting on a system is zero, total angular momentum (moment of momentum) is conserved. Decreasing moment of inertia necessitates an inverse proportional increase in angular spin rate.
- ②Variable Definitions: $\boldsymbol{\tau}$ is external torque, $\mathbf{L}$ is angular momentum, $I$ is moment of inertia ($\text{kg}\cdot\text{m}^2$), and $\boldsymbol{\omega}$ is rotational angular velocity ($\text{rad/s}$).
- ③Fundamental Significance: Stemming from spatial rotational isotropy (Noether's theorem). Kepler's second law (areal velocity conservation) is fundamentally the conservation of angular momentum in a central gravitational field.
- ④Validity Domain & Applications: Valid for particles and rigid body rotations. Applied in figure skaters spinning faster with pulled-in arms, gyroscopic inertial guidance, feline aerial righting reflexes, helicopter anti-torque tail rotors, and millisecond pulsar spin-up during core collapse.
3. Gravitation & Circular Motion (5 Laws)
Universal gravitation and orbital mechanics construct the theoretical foundation of astrophysics, planetary orbits, and modern space flight trajectories.
10 · Newton's Law of Universal Gravitation
Inverse-Square Gravitational Law
$$F = G \frac{m_1 m_2}{r^2}$$
- ①Physical Essence: Every particle in the universe attracts every other particle with a mutual force directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers of mass.
- ②Variable Definitions: $G \approx 6.6743 \times 10^{-11}\;\text{N}\cdot\text{m}^2/\text{kg}^2$ is Newton's gravitational constant, $m_1, m_2$ are masses, and $r$ is separation distance.
- ③Fundamental Significance: The unifying cornerstone governing macroscopic cosmic structure and celestial dynamics, providing an extraordinarily precise approximation in weak gravitational regimes.
- ④Validity Domain & Applications: Strong gravitational fields or relativistic speeds require general relativity (e.g., perihelion precession of Mercury, gravitational lensing). Applied in planetary orbital tracking, oceanic tide calculations, predicting Neptune's existence, and binary star mass estimations.
11 · Kepler's Third Law
Harmonic Law of Planetary Motion
$$\frac{a^3}{T^2} = \frac{G(M+m)}{4\pi^2} \approx \text{const}$$
- ①Physical Essence: The cube of the semi-major axis of a planet's orbit divided by the square of its orbital period is a constant for all bodies orbiting the same central primary, dependent solely on primary mass.
- ②Variable Definitions: $T$ is orbital period ($\text{s}$), $a$ is orbital semi-major axis (radius $r$ for circular orbits), $M$ is central body mass, and $m$ is orbiting satellite mass ($m \ll M$).
- ③Fundamental Significance: Historically allowed Newton to derive the inverse-square dependency of gravitation by synthesizing Keplerian kinematics with centripetal acceleration.
- ④Validity Domain & Applications: Universally holds for circular and elliptical two-body Keplerian orbits. Applied in determining geostationary orbital altitude ($35,786\;\text{km}$), exoplanet radial velocity detection, and weighing supermassive black holes at galactic cores.
12 · Centripetal Acceleration
Centripetal Kinematics
$$a_n = \frac{v^2}{r} = \omega^2 r = v\omega$$
- ①Physical Essence: For a particle describing circular or curved motion, acceleration points radially inward along the radius of curvature toward the instantaneous center of curvature, **altering velocity direction without changing velocity magnitude**.
- ②Variable Definitions: $a_n$ is normal centripetal acceleration ($\text{m/s}^2$), $v$ is tangential orbital speed, $r$ is path radius of curvature, and $\omega$ is angular velocity ($\text{rad/s}$).
- ③Fundamental Significance: Because centripetal acceleration is strictly orthogonal to tangential velocity, **centripetal force does zero mechanical work**. In uniform circular motion, acceleration magnitude is invariant while direction continuously rotates.
- ④Validity Domain & Applications: Normal kinematic decomposition for circular and curvilinear trajectories. Applied in civil engineering banked roadway curves, high-speed rail cant compensation, centrifuge design, and cyclotron beam bending radii.
13 · First Cosmic Velocity
Orbital Velocity
$$v_1 = \sqrt{\frac{GM}{R}} = \sqrt{gR} \approx 7.91\text{ km/s}$$
- ①Physical Essence: The minimum circular orbital speed required for an unpowered spacecraft to skim just above the surface of a celestial body without falling back, where gravity exactly provides necessary centripetal force.
- ②Variable Definitions: $G$ is gravitational constant, $M$ is Earth's mass, $R$ is mean Earth radius ($\sim 6,371\;\text{km}$), and $g$ is surface gravitational acceleration.
- ③Fundamental Significance: The lowest velocity threshold for artificial orbital insertion; higher circular orbits require lower orbital velocities ($v \propto 1/\sqrt{r}$).
- ④Validity Domain & Applications: Near-Earth low Earth circular orbits (LEO). Applied in orbital booster payload delta-v budgets and the International Space Station baseline orbit ($400\;\text{km}$ altitude at $7.66\;\text{km/s}$).
14 · Second Cosmic Velocity / Escape Velocity
Gravitational Escape Velocity
$$v_2 = \sqrt{\frac{2GM}{R}} = \sqrt{2} v_1 \approx 11.2\text{ km/s}$$
- ①Physical Essence: The minimum launch speed required for a projectile to overcome the gravitational potential well of a celestial body and coast to infinity with zero remaining velocity, exactly $\sqrt{2}$ times the first cosmic velocity.
- ②Variable Definitions: $M$ is celestial body mass, $R$ is celestial body radius; escape velocity is independent of spacecraft launch mass.
- ③Fundamental Significance: Derived from mechanical energy conservation ($\frac{1}{2}mv^2 - GMm/R = 0$). Setting escape velocity equal to light speed $c$ directly yields the Schwarzschild black hole event horizon radius $R_s = 2GM/c^2$.
- ④Validity Domain & Applications: Gravitational potential wells of planets and stars. Applied in lunar and interplanetary probe escape burns, Voyager interstellar trajectories, and planetary atmospheric retention dynamics.
4. Oscillations & Waves (6 Laws)
Oscillation and wave mechanics govern restoring force responses, periodic resonances, and the propagation of energy across continuum media.
15 · Hooke's Law
Elastic Constitutive Law
$$\mathbf{F} = -k \mathbf{x} \quad (\sigma = E\varepsilon)$$
- ①Physical Essence: Within the linear elastic proportional limit, restoring force exerted by an elastic body is directly proportional and opposite in direction to displacement from equilibrium.
- ②Variable Definitions: $k$ is spring stiffness ($\text{N/m}$), $\mathbf{x}$ is displacement vector, and the negative sign indicates restoring action toward equilibrium; in 1D continuum mechanics, $\sigma$ is stress, $E$ is Young's modulus, and $\varepsilon$ is strain.
- ③Fundamental Significance: Quadratic approximation of potential energy minima near stable equilibrium, providing the starting point for continuum elasticity.
- ④Validity Domain & Applications: **Holds strictly within the linear elastic limit**; exceeding yield stress induces permanent plastic deformation. Applied in spring scales, vehicle suspension coil springs, elastomeric seismic bearings, and structural rigidity checks.
16 · Differential Equation of Simple Harmonic Motion
Simple Harmonic Motion (SHM)
$$m\ddot{x} + kx = 0 \iff \ddot{x} + \omega^2 x = 0$$
- ①Physical Essence: Restoring force is proportional and opposite to displacement; acceleration is proportional and opposite to position, producing sinusoidal harmonic solutions $x(t) = A\cos(\omega t + \varphi)$.
- ②Variable Definitions: $m$ is oscillator mass, $k$ is stiffness, $\ddot{x}$ is acceleration ($\mathrm{d}^2 x/\mathrm{d}t^2$), and $\omega = \sqrt{k/m}$ is natural angular frequency.
- ③Fundamental Significance: The universal first-order canonical model for small oscillations throughout physical science; incorporates damping and forcing terms to model damped oscillations and driven resonance.
- ④Validity Domain & Applications: Linear undamped free vibrations. Applied in acoustic tuning forks, small-amplitude pendulums, LC electrical resonant tanks, and MEMS vibratory gyroscopes.
17 · Period of a Simple Pendulum
Simple Pendulum Equation
$$T = 2\pi \sqrt{\frac{l}{g}}$$
- ①Physical Essence: Under small-angle approximations, tangential gravity components provide a linear restoring force; period depends exclusively on pendulum length and local gravitational acceleration, independent of swing amplitude and bob mass (isochronism).
- ②Variable Definitions: $T$ is full oscillation period ($\text{s}$), $l$ is effective string length ($\text{m}$), and $g$ is local gravitational acceleration ($\text{m/s}^2$).
- ③Fundamental Significance: The earliest high-precision timekeeping standard in human technological history and the classical experimental method for measuring gravitational acceleration.
- ④Validity Domain & Applications: **Strictly valid for small angular amplitudes ($\theta \le 5^\circ\text{--}10^\circ$)**; larger swings introduce non-linear elliptic integrals. Applied in mechanical pendulum clocks, gravimeter local $g$-field mapping, and Foucault pendulums proving Earth's rotation.
18 · Period of a Mass-Spring Oscillator
Mass-Spring Oscillator Period
$$T = 2\pi \sqrt{\frac{m}{k}}$$
- ①Physical Essence: Natural period of a mass-spring system is determined solely by system inertia (mass $m$) and restoring stiffness (spring constant $k$), remaining invariant with respect to initial release displacement and velocity (isochronous).
- ②Variable Definitions: $T$ is period ($\text{s}$), $m$ is oscillator mass ($\text{kg}$), and $k$ is spring constant ($\text{N/m}$).
- ③Fundamental Significance: Inherent natural characteristic of undamped lumped-parameter oscillator systems; significant spring mass requires Rayleigh equivalent correction ($m + \frac{1}{3}m_s$).
- ④Validity Domain & Applications: Valid within Hookean elastic deformation ranges. Applied in mechanical watch balance wheel hairspring regulation, optical isolation table resonant tuning, and seismometer inertial pickup calibration.
19 · Wave Speed Relation
Fundamental Wave Equation
$$v = f\lambda = \frac{\lambda}{T}$$
- ①Physical Essence: Phase velocity of wave propagation in a medium equals the product of source oscillation frequency and spatial wavelength, bridging temporal periodicity with spatial wavelength.
- ②Variable Definitions: $v$ is propagation wave speed ($\text{m/s}$), $f$ is source frequency ($\text{Hz}$), $\lambda$ is wavelength ($\text{m}$), and $T = 1/f$ is period.
- ③Fundamental Significance: Universal kinematic identity for all wave phenomena. **Wave speed is governed by medium physical properties (elastic modulus and density), while frequency is dictated by the wave source**; crossing media preserves frequency while wavelength scales with speed.
- ④Validity Domain & Applications: Universally holds for mechanical waves (acoustic, seismic, surface) and electromagnetic waves (light, RF). Applied in ultrasonic non-destructive testing, sonar range-finding, optical dispersion modeling ($n = c/v$), and RF antenna quarter-wave sizing.
20 · Doppler Effect Equation
Doppler Frequency Shift
$$f' = f \left(\frac{v \pm v_o}{v \mp v_s}\right)$$
- ①Physical Essence: Apparent frequency observed changes due to relative motion between wave source and observer: approaching motion compresses wave fronts (higher pitch/blue shift), while receding motion expands wave fronts (lower pitch/red shift).
- ②Variable Definitions: $f'$ is observed apparent frequency, $f$ is source natural frequency, $v$ is medium wave speed, $v_o$ is observer speed relative to medium (approaching is $+$, receding is $-$), and $v_s$ is source speed relative to medium (approaching is $-$, receding is $+$).
- ③Fundamental Significance: Acoustic Doppler depends on source/observer velocities relative to the acoustic medium; optical Doppler requires relativistic Lorentz corrections.
- ④Validity Domain & Applications: Relative motion between emitter and receiver. Applied in police Doppler traffic radars, medical color Doppler ultrasound hemodynamic velocimetry, astronomical galactic redshift measurements, and Hubble cosmic expansion verification.
5. Physical Quantities, Symbols, and SI Units
| Physical Quantity | Symbol | Scalar / Vector | SI Base or Derived Unit | SI Symbol | Dimensional Expression (M, L, T) |
|---|---|---|---|---|---|
| Displacement / Distance | $r, s, x$ | Vector (Displacement) / Scalar (Distance) | Meter | m | $\text{L}$ |
| Velocity / Speed | $v$ | Vector (Velocity) / Scalar (Speed) | Meters per second | m/s | $\text{L}\text{T}^{-1}$ |
| Acceleration | $a, g$ | Vector | Meters per second squared | $\text{m/s}^2$ | $\text{L}\text{T}^{-2}$ |
| Inertial Mass | $m, M$ | Scalar | Kilogram | kg | $\text{M}$ |
| Force / Weight | $F, G$ | Vector | Newton ($\text{kg}\cdot\text{m/s}^2$) | N | $\text{M}\text{L}\text{T}^{-2}$ |
| Linear Momentum | $p$ | Vector | Kilogram-meter per second ($\text{N}\cdot\text{s}$) | $\text{kg}\cdot\text{m/s}$ | $\text{M}\text{L}\text{T}^{-1}$ |
| Impulse | $I$ | Vector | Newton-second | $\text{N}\cdot\text{s}$ | $\text{M}\text{L}\text{T}^{-1}$ |
| Torque / Moment | $\tau, M$ | Vector | Newton-meter | $\text{N}\cdot\text{m}$ | $\text{M}\text{L}^2\text{T}^{-2}$ |
| Moment of Inertia | $I, J$ | Scalar (Fixed Axis) / Tensor (3D) | Kilogram square meter | $\text{kg}\cdot\text{m}^2$ | $\text{M}\text{L}^2$ |
| Angular Momentum | $L$ | Vector | Kilogram square meter per second | $\text{kg}\cdot\text{m}^2/\text{s}$ | $\text{M}\text{L}^2\text{T}^{-1}$ |
| Work / Energy | $W, E$ | Scalar | Joule ($\text{N}\cdot\text{m}$) | J | $\text{M}\text{L}^2\text{T}^{-2}$ |
| Power | $P$ | Scalar | Watt ($\text{J/s}$) | W | $\text{M}\text{L}^2\text{T}^{-3}$ |
| Spring Stiffness | $k$ | Scalar | Newtons per meter | N/m | $\text{M}\text{T}^{-2}$ |
| Frequency | $f, \nu$ | Scalar | Hertz ($1/\text{s}$) | Hz | $\text{T}^{-1}$ |
| Angular Frequency / Velocity | $\omega$ | Scalar (Frequency) / Vector (Velocity) | Radians per second | rad/s | $\text{T}^{-1}$ |