§F
Physics Foundations
§F · Cheat sheet

Cheat sheet

Every key formula, in canonical order, at a glance. Each formula links to the lesson that derives it.

1 — Mechanics

Newton’s second law

F=dpdt, p=mv\mathbf{F} = \frac{d\mathbf{p}}{dt},\ \mathbf{p} = m\mathbf{v}

Centre-of-mass momentum theorem

dPtotdt=Fext\frac{d\mathbf{P}_\text{tot}}{dt} = \mathbf{F}_\text{ext}

Impulse–momentum theorem

∫t1t2F dt=Δp\int_{t_1}^{t_2} \mathbf{F}\, dt = \Delta\mathbf{p}

Momentum conservation (collision)

m1v1i+m2v2i=m1v1f+m2v2fm_1 v_1^i + m_2 v_2^i = m_1 v_1^f + m_2 v_2^f

Coefficient of restitution

v2f−v1f=−e (v2i−v1i)v_2^f - v_1^f = -e\,(v_2^i - v_1^i)

Work–energy theorem

∫F⋅dr=12m∣v2∣2−12m∣v1∣2\int \mathbf{F}\cdot d\mathbf{r} = \tfrac12 m\lvert\mathbf{v}_2\rvert^2 - \tfrac12 m\lvert\mathbf{v}_1\rvert^2

Conservative force and potential

F=−∇U\mathbf{F} = -\nabla U

Conservation of mechanical energy

T+U=constantT + U = \text{constant}

Power

P=F⋅v=dW/dtP = \mathbf{F}\cdot\mathbf{v} = dW/dt

Rotational second law

dLdt=τ, L=r×p, τ=r×F\frac{d\mathbf{L}}{dt} = \boldsymbol\tau,\ \mathbf{L} = \mathbf{r}\times\mathbf{p},\ \boldsymbol\tau = \mathbf{r}\times\mathbf{F}

Rigid-body rotation

L=Iω, I=∑imiri2L = I\omega,\ I = \sum_i m_i r_i^2

Static equilibrium

∑F=0, ∑τ=0\sum \mathbf{F} = 0,\ \sum \boldsymbol\tau = 0

Lever balance

FL LL=FR LRF_L\,L_L = F_R\,L_R

Wave equation (continuum limit)

∂2u∂t2=c2 ∂2u∂x2, c=aκ/m\frac{\partial^2 u}{\partial t^2} = c^2\,\frac{\partial^2 u}{\partial x^2},\ c = a\sqrt{\kappa/m}

Continuum master equation

ρ DuDt=∇⋅σ\rho\,\frac{D\mathbf{u}}{Dt} = \nabla\cdot\boldsymbol\sigma

Euler–Lagrange equation

ddt∂L∂q˙−∂L∂q=0, L=T−U\frac{d}{dt}\frac{\partial L}{\partial \dot q} - \frac{\partial L}{\partial q} = 0,\ L = T - U

2 — Kinetic theory & equipartition

Pressure from collisions

p=13 n m ⟨v2⟩p = \tfrac13\, n\, m\, \langle v^2\rangle

Mean molecular kinetic energy

⟨ε⟩=32 kBT\langle\varepsilon\rangle = \tfrac32\, k_B T

Equipartition per quadratic DOF

⟨12αq2⟩=12kBT\big\langle \tfrac12\alpha q^2 \big\rangle = \tfrac12 k_B T

Internal energy and specific heats

U=d2NkBT, cv=d2R, γ=(d+2)/dU = \tfrac{d}{2} N k_B T,\ c_v = \tfrac{d}{2} R,\ \gamma = (d+2)/d

Maxwell–Boltzmann speed distribution

f(v)=4π(m2πkBT)3/2v2 e−mv2/(2kBT)f(v) = 4\pi \left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2\, e^{-m v^2/(2k_B T)}

Most-probable speed

vp=2kBT/mv_p = \sqrt{2 k_B T/m}

Mean speed

⟨v⟩=8kBT/(πm)\langle v\rangle = \sqrt{8 k_B T/(\pi m)}

Root-mean-square speed

vrms=3kBT/mv_\text{rms} = \sqrt{3 k_B T/m}

Boltzmann factor

P(E)∝e−E/kBTP(E) \propto e^{-E/k_B T}

Barometric formula

n(z)=n0 e−mgz/kBTn(z) = n_0\, e^{-mgz/k_B T}

Arrhenius law

k=A e−Ea/kBTk = A\, e^{-E_a/k_B T}

Mean free path

ℓ=12 n πd2\ell = \frac{1}{\sqrt2\, n\, \pi d^2}

Viscosity

η≈13 ρ ⟨v⟩ ℓ\eta \approx \tfrac13\,\rho\,\langle v\rangle\,\ell

Diffusion coefficient

D≈13⟨v⟩ ℓD \approx \tfrac13\langle v\rangle\,\ell

Thermal conductivity

κ≈13 n c ⟨v⟩ ℓ\kappa \approx \tfrac13\, n\, c\,\langle v\rangle\,\ell

Einstein relation

D=kBT/γD = k_B T/\gamma

3 — Thermodynamics

Ideal gas law

p=nkBT=ρRTMp = n k_B T = \dfrac{\rho R T}{M}

First law

dU=δQ−δW=T dS−p dVdU = \delta Q - \delta W = T\,dS - p\,dV

Heat capacity at constant volume

CV=(∂U∂T)VC_V = \left(\frac{\partial U}{\partial T}\right)_V

Heat capacity at constant pressure

Cp=(∂H∂T)pC_p = \left(\frac{\partial H}{\partial T}\right)_p

Mayer’s relation

cp−cv=Rc_p - c_v = R

Ratio of specific heats

γ≡cpcv=1+2d=d+2d\gamma \equiv \dfrac{c_p}{c_v} = 1 + \dfrac{2}{d} = \dfrac{d+2}{d}

Adiabatic equation of state

pVγ=constp V^\gamma = \text{const}

Speed of sound

c2=γ pρ=γRTMc^2 = \gamma\,\dfrac{p}{\rho} = \dfrac{\gamma R T}{M}

Carnot efficiency

η=1−QcQh=1−TcTh\eta = 1 - \dfrac{Q_c}{Q_h} = 1 - \dfrac{T_c}{T_h}

Entropy differential

dS=δQrevTdS = \dfrac{\delta Q_\text{rev}}{T}

Second law

ΔSisolated≥0\Delta S_\text{isolated} \ge 0

Boltzmann entropy

S=kBln⁡WS = k_B \ln W

Enthalpy

H≡U+pV, dH=T dS+V dpH \equiv U + pV,\ dH = T\,dS + V\,dp

Helmholtz free energy

F=U−TS, dF=−S dT−p dVF = U - TS,\ dF = -S\,dT - p\,dV

Gibbs free energy

G=H−TS, dG=−S dT+V dpG = H - TS,\ dG = -S\,dT + V\,dp

Maxwell relation (from FF)

(∂S∂V)T=(∂p∂T)V\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial p}{\partial T}\right)_V

4 — Free energy and phase transitions

Helmholtz free energy

F≡U−TSF \equiv U - TS

Gibbs free energy

G≡U+pV−TS=F+pVG \equiv U + pV - TS = F + pV

Free-energy criterion (fixed T,VT,V / T,pT,p)

ΔF≤0, ΔG≤0\Delta F \le 0,\ \Delta G \le 0

Free-energy differentials

dF=−S dT−p dV, dG=−S dT+V dpdF = -S\,dT - p\,dV,\ dG = -S\,dT + V\,dp

Stability condition

∂2G/∂x2>0⇒stable\partial^2 G/\partial x^2 > 0 \Rightarrow \text{stable}

Chemical potential

μ≡(∂G/∂N)T,p\mu \equiv \left(\partial G/\partial N\right)_{T,p}

Phase coexistence

μ1=μ2\mu_1 = \mu_2

Latent heat

L=T Δs=T (s2−s1)L = T\,\Delta s = T\,(s_2 - s_1)

Clausius–Clapeyron relation

dpdT=ΔsΔv=LT Δv\dfrac{dp}{dT} = \dfrac{\Delta s}{\Delta v} = \dfrac{L}{T\,\Delta v}

Vapour-pressure law

ln⁡p=−LMRT+const\ln p = -\dfrac{LM}{RT} + \text{const}

Nucleation free energy

ΔG(R)=−43πR3 Δp+4πR2 σ\Delta G(R) = -\tfrac43\pi R^3\,\Delta p + 4\pi R^2\,\sigma

Critical radius and barrier

R∗=2σΔp, ΔG∗=16πσ33(Δp)2R^* = \dfrac{2\sigma}{\Delta p},\ \Delta G^* = \dfrac{16\pi\sigma^3}{3(\Delta p)^2}

Nucleation rate factor

e−ΔG∗/kBTe^{-\Delta G^*/k_B T}

Two-state occupancy (Fermi function)

P(F)=11+e(ΔG0−αF)/kBTP(F) = \dfrac{1}{1 + e^{(\Delta G_0 - \alpha F)/k_B T}}

5 — Fluid mechanics

Material derivative

DϕDt≡∂ϕ∂t+u⋅∇ϕ\frac{D\phi}{Dt} \equiv \frac{\partial \phi}{\partial t} + \mathbf{u}\cdot\nabla\phi

Convective acceleration

DuDt=∂u∂t+(u⋅∇)u\frac{D\mathbf{u}}{Dt} = \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{u}

Continuity equation

∂ρ∂t+∇⋅(ρu)=0\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{u}) = 0

Incompressible condition

∇⋅u=0\nabla\cdot\mathbf{u} = 0

Euler’s equation

ρ DuDt=−∇p+ρg\rho\,\frac{D\mathbf{u}}{Dt} = -\nabla p + \rho\mathbf{g}

Bernoulli’s equation

12ρ∣u∣2+p+ρgz=const\tfrac12\rho\lvert\mathbf{u}\rvert^2 + p + \rho g z = \text{const} ρ DuDt=−∇p+μ ∇2u+ρg\rho\,\frac{D\mathbf{u}}{Dt} = -\nabla p + \mu\,\nabla^2\mathbf{u} + \rho\mathbf{g}

Kinematic viscosity

ν=μ/ρ\nu = \mu/\rho

Reynolds number

Re≡ρULμ=ULν\mathrm{Re} \equiv \frac{\rho U L}{\mu} = \frac{U L}{\nu}

Stokes equation

∇p=μ∇2u\nabla p = \mu\nabla^2\mathbf{u}

Stokes drag

F=6πμaUF = 6\pi\mu a U

Boundary-layer thickness

δ(x)∼νxU=xRex\delta(x) \sim \sqrt{\frac{\nu x}{U}} = \frac{x}{\sqrt{\mathrm{Re}_x}}

6 — Viscosity, diffusion, and transport

Newtonian shear stress

τ=μ dudy\tau = \mu\,\frac{du}{dy}

Kinematic viscosity

ν=μρ\nu = \frac{\mu}{\rho}

Fourier’s law of heat conduction

q=−k ∇T\mathbf{q} = -k\,\nabla T

Heat equation

∂T∂t=α ∇2T, α=kρcp\frac{\partial T}{\partial t} = \alpha\,\nabla^2 T,\ \alpha = \frac{k}{\rho c_p}

Fick’s law of diffusion

J=−D ∇c\mathbf{J} = -D\,\nabla c

Diffusion equation

∂c∂t=D ∇2c\frac{\partial c}{\partial t} = D\,\nabla^2 c

Diffusion coefficient from step variance

D=σ2/2D = \sigma^2/2

Mean-squared displacement (1-D)

⟨x2⟩=2Dt\langle x^2 \rangle = 2Dt

Stokes drag on a sphere

Fdrag=−6πμa U\mathbf{F}_\text{drag} = -6\pi\mu a\,\mathbf{U}

Stokes friction coefficient

γ=6πμa\gamma = 6\pi\mu a

Einstein relation

D=kBTγD = \frac{k_B T}{\gamma}

Stokes–Einstein relation

D=kBT6πμaD = \frac{k_B T}{6\pi\mu a}

Diffusion time over length LL

τ∼L2D\tau \sim \frac{L^2}{D}

Kirchhoff–Stokes absorption coefficient

αabs(f)=(2πf)22ρc3[43μ+(γ−1)kcp]\alpha_\text{abs}(f) = \frac{(2\pi f)^2}{2\rho c^3}\left[\tfrac{4}{3}\mu + (\gamma - 1)\frac{k}{c_p}\right]

7 — Elasticity and continuum mechanics

Fluid sound speed (bulk)

c=K/ρc = \sqrt{K/\rho}

String-wave equation

utt=c2uxxu_{tt} = c^2 u_{xx}

Chain wave speed

c=aκ/mc = a\sqrt{\kappa/m}

String wave speed

c=T/μlinc = \sqrt{T/\mu_\text{lin}}

Local resonant frequency

ω0(x)=k(x)/m\omega_0(x) = \sqrt{k(x)/m}

BM mechanical impedance

ZBM(x,ω)=b(x)+i ⁣(ωm−k(x)ω)Z_\text{BM}(x, \omega) = b(x) + i\!\left(\omega m - \frac{k(x)}{\omega}\right)

Cochlear long-wave dispersion

κ2(x,ω)=2iωρ/(AZBM)\kappa^2(x, \omega) = 2i\omega\rho/(A Z_\text{BM})

8 — Intermolecular forces and the liquid state

Bulk modulus from molecular scale

K∼ε/σ3K \sim \varepsilon/\sigma^3

Surface tension from cohesion

σ∼ε/σ2\sigma \sim \varepsilon/\sigma^2

van der Waals instability region

(∂p/∂v)>0(\partial p/\partial v) > 0

9 — Surface tension and capillarity

Young–Laplace pressure jump

Δp=2σ/R\Delta p = 2\sigma/R

Bubble-wall boundary condition

pB=p∞+ρRR¨+32ρR˙2+2σR+4μR˙Rp_B = p_\infty + \rho R \ddot R + \tfrac32 \rho \dot R^2 + \frac{2\sigma}{R} + \frac{4\mu \dot R}{R}

Crevice gas-trapping condition

θc+β>90∘\theta_c + \beta > 90^\circ

Weber number

We=ρU2L/σ\mathrm{We} = \rho U^2 L/\sigma

10 — Waves as physical objects

Reflection coefficient

R=(Z2−Z1)/(Z2+Z1)R = (Z_2 - Z_1)/(Z_2 + Z_1)

Power reflection coefficient

RP=R2R_P = R^2

WKB amplitude rule

A(x)∝1/κ(x)A(x) \propto 1/\sqrt{\kappa(x)}

Closed-pipe fundamental

f=c/(4L)f = c/(4L)

Intensity

I=⟨p′v′⟩I = \langle p'v' \rangle

Acoustic radiation pressure

Prad=I/cP_\text{rad} = I/c

11 — Electromechanics and electrochemistry

Membrane specific capacitance

C/A=ε0εrdC/A = \frac{\varepsilon_0 \varepsilon_r}{d}

Surface charge density

σ=CV/A\sigma = CV/A

Gating-spring force

F=KTL xF = K_\text{TL}\,x

12 — Scaling and dimensionless numbers

Local nonlinear sound speed

clocal≈c0+βv′c_\text{local} \approx c_0 + \beta v'

Shock-formation distance

Lshock∼c0/(βωv′)=1/(βωMaa)L_\text{shock} \sim c_0/(\beta \omega v') = 1/(\beta \omega \mathrm{Ma}_a)

Helmholtz number

ka=ωa/cka = \omega a/c

Bond number

Bo=ρgL2/σ\mathrm{Bo} = \rho g L^2/\sigma