Misc Equations

 Stellar magnitude Fourier's Law Heat Flow Black Body Radiation Center of Mass Skin effect Distance to horizon Springs and weight Carnot cycle Youngs Modulus
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Stellar magnitude
M = m – 5((logD) – 1)
M is absolute magnitude
m is apparent magnitude
D is distance in Parsecs

Fourier's Law Heat Flow (1 dimensional differential form)
q = k(dT/dx)
q is heat flux in w/m²
k is material's conductivity in w/mK
dT/dx is the rate of temperature change along the axis

P = σAT⁴
P is power emitted
σ is the Stefan–Boltzmann constant 5.67×10^−8 W/m²·K⁴
T is temperature in Kelvins
A is surface area in m²

Center of mass
For a simple two body system, the CG is cal. from this:
Setting the origin at the center of the larger.
r = a / (1 + (m₁/m₂))
a = distance between centers
r = distance from center of larger to CG
m₁ and m₂ are the two masses, m₁ the larger

Center of mass for particles or regular shapes
Xcm = (1/M)(m₁x₁ + m₂x₂ + m₃x₃ + )
Ycm = (1/M)(m₁y₁ + m₂y₂ + m₃y₃ + )
M is total mass
mᵢxᵢ is the moment along the x axis
mᵢyᵢ is the moment along the y axis
Xcm, Ycm is the effective distance for the total mass
or the distance to the center of mass

Center of Mass in one dimension, for a object of
uniform thickness: sum of moments divided by sum of
areas, where moments are areas x distances.

Distance to horizon is d = 1.323√h
d in miles and h in feet
or d = 3.856√h
d in km and h in meters

Springs and weight
k = F/d
k is spring constant in N/m
F is force and d is deflection
ω = √(k/M)
f = (1/2π)√(k/M)
ω is angular frequency of bounce
k is spring constant in N/m
M is mass in kg
ω = 2πf
PE = ½kd²

Carnot cycle
Efficiency of a Carnot cycle heat engine is
Efficiency η = 1 – (Tc/Th)
Where Tc is absolute temperature of cold reservoir
Where Th is absolute temperature of hot reservoir

The Coefficient of Performance (COP) of the heat
engine is the reciprocal of its efficiency

Youngs Modulus
Amount a material stretches or compresses, in a
linear direction, force required
F = EA∆L/L
E is young's modulus in Pa
A is cross-sectional area
∆L/L is ratio stretched or compressed

nylon 2-3 GPa
Oak along grain 11 GPa
Mg 45 GPa
Al 69 GPa
Glass 50-90 GPa
Brass and bronze 100-125 GPa
Cu 117 GPa
Wrought iron 190-210 GPa
steel 200 GPa
tungsten 400 GPa
diamond 1220 GPa

Skin depth δ is approximated as follows:
It is the depth at which the current density has fallen to
1/e of that at the surface due to frequency.
δ = √(2ρ / ωµᵣµ₀)
where
δ = skin depth in meters
ρ = resistivity of the conductor in Ωm
ω = angular frequency of current = 2π × frequency in rad/s
µᵣ = relative magnetic permeability of the conductor in henries per meter
µ₀ = the permeability of free space 1.257×10−6 henries per meter

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