1206 FINAL EXAM Formula sheet - December 2009 Constants and conversion factors units given in [ ] mass of earth: ME = 5.98 x 1024 m radius of earth: rE = 6378 km 1 mi = 1609 m atmospheric pressure: 1 atm = 1.013 x 105 Pa 1 ft = 0.305 m 0 °C = 273.15 K T(K) = T(C) + 273 NA = 6.022 x 1023 mol-1 T(C) = (5/9)(F -32) R = 8.31 J/(K mol) 3 1 atm = 101.3x10 Pa = 14.7 psi k = 1.38 x 10-23 m2 kg s-2 K-1 σ = 5.67 x 10-8 J/(sm2K4) universal gravitation: G = 6.67 x 10-11 N m2/s2 MECHANICS: KINEMATICS, FORCES, WORK, ENERGY, POWER, MOMENTUM displacement x [m], velocity v [m/s], acceleration a [m/s2] and time t [s] subscript i means initial, subscript f means final Set of equations for constant acceleration: vf = vi + at xf = xi + vit + 1/2at2 xf = xi + 1/2(vi + vf)t vf2 = vi2 + 2a(xf-xi) constant circular acceleration, same set of equations with angular displacement Θ, angular velocity ω, angular acceleration α. tangential velocity vT = s/t (s arc length, t time) tangential acceleration aT = rα (α in rad/s2) centripetal acceleration aC = v2/r = rω2 ω = 2π/T (T period is time for one revolution) vT = rω (r radius) Newton’s Second Law unit for force is N [N = kgm/s2] ΣF = ma (m mass, F force) Στ = iα ( i moment of inertia, i = mr2 for solid sphere) Work [Joules J = Nm] W = F d cos(Θ) (d displacement, F force, Θ angle between the two) Work = final total mechanical energy minus initial total mechanical energy Power = Work per time (P = W/t) unit for Power is Watt W = J/s or Power = Force times Velocity (P = Fv) Energy unit for Energy is Joule J potential gravitational energy PE = mgh (m mass, h height, g accel. due to gravity) elastic potential energy PE = 1/2kx2 (k spring constant, x compression) 2 kinematic energy KE = 1/2mv (m mass, v velocity) 1206 FINAL EXAM Formula sheet - December 2009 Momentum p = mv Impuls J = Favg * Δt (m mass, v velocity) Impuls = change of momentum Friction force: Ff = μ FN (μ friction coefficient and FN normal force) Spring force: Fs = k x (k spring constant, x compression) Centripetal force: FC = G mME/r2 = mv2/r (ME mass of the earth) Torque = magnitude of force times lever arm (τ = F*l) Angular momentum L = i ω [Nm] (moment of inertia i, angular velocity ω) FLUIDS: Density ρ = m/V (m mass,V volume) Pressure P = F/A (F applied force, A area) pressure in dependence of depth: ΔP = ρgh (Pascal’s principle) Buoyant force FB = ρgV Archimedes principle: magnitude of buoyant force = weight of displaced fluid Equation of continuity: ρ1A1v1 = ρ2A2v2 (A area, v velocity at position 1 and 2) Bernoulli’s equation: P1 + 1/2ρv12 + ρgh1 = P2 + 1/2ρv22 + ρgh2 SOLIDS: linear thermal expansion of a solid: ΔL = α L0 ΔT (L length, T temperature) Elastic deformation F/A = Y (ΔL/L0) (A cross-sectional area,Y Young’s modulus) Shear deformation F/A = S (ΔX/L0) Volume deformation ΔP = -B (ΔV/V0) Stress proportional to Strain GAS: mparticle = Mass per mole/NA ideal gas law PV = nRT Stefan-Boltzmann Law: Q/t = eσT4A KE = 3/2 kT vrms = sqrt(2 KE/mparticle) (Q/t power, e emissivity, T temperature) First law of thermodynamics ΔU = Uf - Ui = Q - W Second law of thermodynamics ΔS = ΔQ/T (U internal energy) (Q heat) 1206 FINAL EXAM Formula sheet - December 2009 Type of thermal process Work done First law of Thermodynamics (ΔU = Q - W) Isobaric (constant pressure) W = P(Vf-Vi) and ΔU = Q - P(Vf - Vi) Isochoric (constant volume) W = 0 J and ΔU = Q Isothermal (constant temperature) W = nRT ln(Vf/Vi) and 0J = Q - nRT ln(Vf/Vi) Adiabatic (no heat flow) W = 3/2nR(Tf - Ti) and ΔU = -3/2nR(Tf - Ti) CARNOT ENGINE: efficiency e = 1 - QC/QH (all processes) also e = 1 - TC/TH (specific to Carnot process) Work performed W = QH e SIMPLE HARMONIC MOTION: Formulas for simple harmonic motion (see drawing): PENDULUM ω = 2πf = sqrt(mgL/I) 1206 FINAL EXAM Formula sheet - December 2009 WAVES: velocity v = λ/T = fλ Wave motion toward +x: y = A sin (2πft - 2πx/λ) Wave motion toward - x: y = A sin (2πft + 2πx/λ) Sound intensity I = P/A (Power over Area) decibel: β = (10 dB) log (I/I0) I0 = 1.00 x 10-12 W/m2 DOPPLER EFFECT Observer moving toward stationary source fo = fs (1 + vo/v) Observer moving away from stationary source fo = fs (1 - vo/v) observer and sound source are moving fo = fs { [1 ± (vo/v)]/[1 -/+ (vs/v)]} In the numerator, the plus sign applies when the observer moves toward the source, and the minus sign applies when the observer moves away from the source. In the denominator, the minus sign is used when the source moves toward the observer, and the plus sign is used when the source moves away form the observer. Diffraction (slit): sinΘ = λ/D (circular opening sinΘ = 1.22 λ/D) Sting fixed at both ends: fn = n(v/2L) f2/f1 = (v2/2L)/(v1/2L) = v2/v1