Misc Equations
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 Black Body Radiation 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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