Phyiscs-Intro Heat, Waves, Fluids

Useful math functions/approximations:

cos( A ) + cos( B ) = 2 cos [ ( A + B ) / 2 ] cos [ ( A - B ) / 2 ]
sin( A ) + sin( B ) = 2 sin [ ( A + B ) / 2 ] cos [ ( A - B ) / 2 ]
1/L - 1 / ( L + ΔL ) ≈ ΔL / L ² 
for x << 1
(1 + x) ^1/2 ≈ 1 + x/2
ln (1 + x) ≈ x
due to taylor expansion

Constants:

ρ = 1 x 10 ³ kg/m ³ 
g = 9.8 m/s ² 
p ₀ = 1 x 10 ⁵

Speed of sound:

v _air = 340 m/s
v _water = 1440 m/s
1 atmosphere = 1 x 10 ⁵ = 76 cm Hg
R = N _A = 8.3 J/(mol K)
k _B = 1.4 x 10 ^-23 
N _A = 6.0 x 10 ²³ 
1 atomic mass unit = 1.7 x 10 ^-27 kg
1 cal = 4.2 J

Water

C = 4.2 kJ/kg-K
L _F = 330 kJ/kg
L _V = 2.2 x 10 ³ kJ/kg

Pressure:

ρAv = const
P + ρ v ² / 2 + ρgy = const

Simple Harmonic Oscillator:

d ² z / d t ² + cz = 0
z = z ₀ cos(ωt + φ), and ω = c ^1/2

For a spring:

E = m v ² / 2 + k x ² / 2

For damped spring:

d ² y / dt ² + b dy / m dt + k y / m = 0
y = y ₀ e ^{-α t} cos ( ω ' t )
α = b / 2 m
ω ' ≈ ω ₀ = ( k / m ) ^1/2

Forced oscillator

F = F _{0 e ^{i ω t}}
y = [(F ₀ / m) e ^{i ω t} ] / [ - ω ² + ω ² ₀ + 2 i α ω ]

Waves on a string

v = (T / μ)
ε ' = (1/2) μ ω ² D _m ² 
P = (1 / 2) μ ω ² D _m ² v _p

For sound waves

v _s = (B / ρ) ^1/2 
Δ p = - B (dD / dx)
I = (1 / 2) ρ ω ^{2 D _m 2 v _s}

Decibels

Β = 10 log ₁₀ (I / I₀)
I ₀ = 1 x 10 ^-12 W/m ²

Wave refraction, Snell's law

sin(θ ₁ ) / v ₁ = sin( θ ₂ ) / v ₂ 
sin( θ ₁ ) / sin( θ ₂ ) = v ₁ / v ₂ = n ₁ / n ₂

Beats in waves f ₁ - f ₂

D _m cos(ω ₁ t) + D _m cos(ω ₂ t) = 2 D _m cos[(ω ₁ t + ω ₂ t) / 2] cos[(ω ₁ t - ω ₂ t) / 2]
f = f ₀ (v _s +/- v _d ) / (v _s +/- v _s0 )
B = - V (dp / dV) _T

Thermodynamics

α = (1 / L) (dL / dT) _p 
Β = (1 / V) (dV / dT) _p 
P V = n R T
(m v ² _avg ) / 2 = (3 k _B T) / 2
l _mfp = 1 / [2 ^1/2 4 π r ² (N / V)]
dQ = dE + dW

c _v = (3 / 2) R , (5 / 2) R , 3 R
atoms, diatomic, polyatomic

c _p = c _v + R for ideal gas
P V ^γ = const. γ = c _p / c _v

W = [p ₁ V ₁ / (γ -1)] [1 - (V ₁ / V ₂ ) ^{γ - 1} ]
dQ / dt = e σ A T ⁴ 
σ = 5.7 x 10 ^-8 W/m ² K ⁴ 
dQ / dt = -k A dT / dx
ΔS ≥ 0
dQ = T dS
ΔS = ∫ dQ / T
S = k _B ln (W)
P _{b - > a} = exp(-ΔS _ab / k _B )
e = W / Q _H ≤ 1 - T _L / T _H 
COP = Q _L / W ≤ T _L / (T _L - T _H )