Fluids
pressure: $p=\frac{\Delta F}{\Delta A}$ (all direction)
pressure in liquid: $p=p_0+\rho gh$
Pascal's principle: $\Delta p_{\textrm{int}}=\Delta p_{\textrm{ext}}$
Archimede's principle: $F_{\textrm{buoyancy}}=G_{\textrm{displaced water}}$
equation of continuity:
volume flow rate $R=Av=\textrm{constant}$
mass flow rate $m=Av\rho =\textrm{constant}$
Bernoulli's equation: $p+\frac{1}{2}\rho v^2+\rho gy=\textrm{constant}$
Transverse waves
transerverse displacement: $y(x,t)=y_m\mathrm{sin}(kx-\omega t)$
angular wave number: $k=\frac{2\pi }{\lambda }$, waver number: $\kappa =\frac{1}{\lambda }$
angular frequancy: $\omega =\frac{2 \pi}{T}$, frequency: $f=\frac{1}{T}=\frac{\omega }{2 \pi}$
wave speed: $v=\frac{\omega }{k}=\lambda f$
―material expression: $v=\sqrt{\frac{\tau \textrm{(tension)}}{\mu (\textrm{density of media})}}$
transverse speed: $u=\frac{\partial y}{\partial t}$
average power: $\bar{P}=\frac{1}{2}\mu v\omega ^2y_m^2$
adding waves y=y1+y2=ymsin(kx-ωt+φ)+ymsin(kx-ωt)
result: $y=(2y_m\mathrm{cos}\frac{1}{2}\phi)\mathrm{sin}(kx-\omega t+\frac{1}{2}\phi)$
new amplitude: $2y_m\mathrm{cos}\frac{1}{2}\phi$
phase shift: $+\frac{1}{2}\phi$
standing waves y=y1+y2=ymsin(kx-ωt)+ymsin(kx+ωt)
result: $y=[(2y_m)\mathrm{sin}kx]\mathrm{cos}\omega t$
new amplitude: $2y_m\mathrm{sin}kx$
nodes: $x=n\frac{\lambda }{2}, n=0,1,2,...$
antinodes: $x=(n+\frac{1}{2})\frac{\lambda }{2}, n=0,1,2,...$
resonant frequency: $f_r=\frac{v}{\lambda }=\frac{v}{2l}n, n=1,2,3,...$
Longitudinal waves
speed of sound: $v=\sqrt{\frac{B}{\rho }}$, bulk modulus: $B=-\frac{\Delta p}{\Delta V/V}(=\rho v^2)$
longitudinal displacement: $s=s_m\mathrm{cos}(kx- \omega t)$
air pressure: $\Delta p=\Delta p_m\mathrm{sin}(kx-\omega t)$
―relation: $\Delta p_m=(v\rho \omega )s_m$
interference
phase shift: $\phi=\frac{\Delta d}{\lambda }2\pi$
fully constructive: $\phi=m2\pi, m=0,1,2,...$
fully destructive: $\phi=(m+\frac{1}{2})2\pi, m=0,1,2,...$
sound intensity: $I=\frac{1}{2}\rho v\omega ^2s_m^2$
sound level: $\beta =(10\textrm{ dB})\mathrm{log}(\frac{I}{I_0})$
―standard reference intensity: I0=10-12W/m2
resonant frequency
pipe, two opens: $f_r=\frac{v}{\lambda }=\frac{v}{2L}n,n=1,2,3$
pipe, one open: $f_r=\frac{v}{\lambda }=\frac{v}{4L}n,n=1,3,5,...$
beat frequency: $f_{beat}=f_1-f_2$
doppler effect: $f'=f\frac{v\pm v_D}{v\mp v_S}$
cone angle at supersonic speed: $\mathrm{sin}\theta =\frac{v}{v_s}$
Heat
coefficient of linear expansion: $\alpha =\frac{\Delta L/L}{\Delta T}$
―area: $\beta =2\alpha $
―volume: $\gamma =3\alpha $
heat capacity: $Q=cm(T_f-T_i)=C(T_f-T_i)$
heat of tranformation: $Q=Lm$
volume work: $W=\int_{V_i}^{V_f}pdV$
first law of thermodynamics: $\Delta E_\textrm{int}=E_{\textrm{int,f}}-E_{\textrm{int,i}}=Q-W$
rate of heat transfer: $H=\frac{Q}{t}=kA\frac{T_H-T_C}{L}$, k: media's thermal conductivity
―multiple slabs: $H=A\frac{T_H-T_C}{\sum (L/k)}$
Kinetic theory of gases
ideal gas law: $pV=nRT$, gas constant R=8.31J/mol·K
expansion at constant pressure: $W=\int \frac{nRT}{V}dV=nRT \, \mathrm{ln}(\frac{V_f}{V_i})$
gas pressure: $p=\frac{nMv_{\textrm{rms}}^2}{3V}$
translational kinetic energy: $\bar{K}=\frac{3}{2}kT$, Boltzman constant k=R/NA
mean free path: $\lambda =\frac{1}{\sqrt{2}\pi dN/V}$, d: diameter, N: number of molecules
Maxwell's speed distribution: $P(v)=4\pi (\frac{M}{2 \pi RT})^{3/2}v^2e^{-Mv^2/2RT}$
―most propable speed: $v_p=\sqrt{\frac{2RT}{M}}$
―average speed: $\bar{v}=\sqrt{\frac{8RT}{\pi M}}$
―rms speed: $v_{\mathrm{rms}}=\sqrt{\frac{3RT}{M}}$
internal energy of monoatomic gas: $E_{\mathrm{int}}=(nN_A)\bar{K}=\frac{3}{2}nRT$
―monoatom: 3/2 (f=1); diatom: 5/2 (f=2), 5-atom: 3 (f=5)
molar specific heat of monoatomic gas at constant volume: $C_v=\frac{3}{2}R=12.5\textrm{J/molK}$
constant volume, change in internal energy: $\Delta E_{\textrm{int}}=Q=nC_v(T_f-T_i)$
molar specific heat of monoatomic gas at constant pressure: $C_p-C_v=R$
heat: $Q=nC_p(T_f-T_i)$
work: $W=nR(T_f-T_i)$
law of adiabatic expansion: $pV^\gamma =\textrm{constant}$ or $TV^{\gamma -1}=\textrm{constant}$
$\gamma =C_p/C_v=1+2/f$
Second law of thermodynamics
thermal effiency of engine: $e=\frac{|W|}{|Q_H|}=\frac{|Q_H|-|Q_C|}{|Q_H|}$
―max: $e_{\mathrm{Car}}=\frac{T_H-T_C}{T_H}$
coefficient of performance of refrigerator: $e=\frac{|Q_C|}{|W|}=\frac{|Q_C|}{|Q_H|-|Q_C|}$
―max: $e_{\mathrm{Car}}=\frac{T_C}{T_H-T_C}$
closed system, first law of thermodynamics: $|W|=|Q_H|-|Q_C|$
entropy: $dS=\frac{dQ}{T}$ and $∮ dS\leq 0$
reversible process: $S_f-S_i=\int _i^f dS=\int _i^f \frac{dQ}{T}$
free expansion: $S_f-S_i=\frac{1}{T}\int _i^f dQ=nR \, \mathrm{ln}\frac{V_f}{V_i}$
irreversible heat transfer: $S_f-S_i=cm\, \mathrm{ln}\frac{T^2}{T^2-\Delta T^2}$