To save paper rubbish do not write into the formulas document. After the exam, please return the document. BFH/MNG, Dr. F. Löwenthal Version 4.2 ππ ππ πΌπΌ Formulas in physics ππππ ππππ ππππ πΌπΌ = ππππ Oscillations and waves ………………………………………….4 Geometrical optics ………………………………………………..5 Thermodynamics …………………………………………………..7 Atomphysics ………………………………………………………….9 Constants ……………………………………………………………..10 π₯π₯π₯π₯ π₯π₯π₯π₯ π₯π₯ = π£π£ ⋅ π‘π‘ displacement [m] Mean velocity [m/s] Movement on a circuit with constant acceleration π£π£ 2 ππ 2ππππ 1 2ππ = = π£π£ ππ ππ Angular acceleration Centripetal acceleration Vb: speed on the cicuit r: Radius Uniform movement on a circuit centripetal acceleration a Periode T Frequency f 1 πΈπΈππππππ = πΌπΌππ2 2 Energy of rotation πΌπΌπ΄π΄ = πΌπΌππ + ππππ 2 Theorem of Steiner πΏπΏοΏ½β = ππβ × ππβ angular momentum οΏ½οΏ½β = ππβ × πΉπΉβ ππ Moment of force Orbital angular momentum distance [m] velocity [m/s] time [s] πΏπΏ = πΌπΌ ⋅ ππ distance x [m] acceleration a [m/s^2] ππ = ππππ = πΌπΌ ⋅ πΌπΌ ππππ angular momentum of a solid Equation of circular motion Force and Movement Mean acceleration [m/s^2] πΉπΉβ = ππ ⋅ ππβ Moment in two and three dimensions ππβ = π₯π₯πποΏ½οΏ½οΏ½β1 + π¦π¦πποΏ½οΏ½οΏ½οΏ½β οΏ½οΏ½οΏ½βπ§π§ π¦π¦ + π§π§ππ 1 ππ = ππ0 + ππ ⋅ π‘π‘ + πΌπΌπ‘π‘ 2 2 π£π£ 2 ππ = ππ ππ = Moment with constant acceleration 1 π₯π₯ = π₯π₯0 + π£π£0 ⋅ π‘π‘ + πππ‘π‘ 2 2 π£π£(π‘π‘) = π£π£0 + ππ ⋅ π‘π‘ π£π£ 2 = π£π£02 + 2a(π₯π₯ − π₯π₯0 ) 1 π₯π₯ − π₯π₯0 = (π£π£0 + π£π£)π‘π‘ 2 π₯π₯π₯π₯ ππΜ = π₯π₯π₯π₯ Angular speed ππ = Movement in one dimension Angular speed and tangential speed ππ = π£π£ππ = ππ ⋅ ππ Mechanics π£π£Μ = radian π£π£ = ππ ⋅ ππ Electronics …………………………………………………………….3 π₯π₯π₯π₯ = π₯π₯2 − π₯π₯1 π π ππ ππ = Mechanis ………………………………………………………………1 angle in [rad] angular speed [rad/s] angular acceleration Reibung vector πππ π ,πππππ₯π₯ = πππ π ππ displacement π₯π₯ππβ = οΏ½οΏ½οΏ½β ππ2 − οΏ½οΏ½οΏ½β ππ1 throw π₯π₯ − π₯π₯0 = (π£π£0 πππππππ©π©0 )π‘π‘ 1 π¦π¦ − π¦π¦0 = (π£π£0 π π π π π π π©π©0 )π‘π‘ − πππ‘π‘ 2 2 ππππ 2 π¦π¦(π₯π₯) = (π‘π‘π‘π‘π‘π‘π©π©0 )π₯π₯ − 2(π£π£0 cosπ©π©0 )2 2 π£π£0 π π = sin2π©π©0 Horizontal reach ππ ππππ = ππππ ππ Resistance of fluid 1 πΉπΉ = πΆπΆπ€π€ πππππ£π£ 2 2 Movement on a circle 1 π£π£π‘π‘ = οΏ½ 2πΉπΉππ πΆπΆπ€π€ ππππ 2. axiom of Newton F [N] Maximal friction N: Normal force Dynamic friction πΆπΆπ€π€ : constant ππ: density of the fluide A: front surface v: speed π£π£π‘π‘ Maximal speed To save paper rubbish do not write into the formulas document. After the exam, please return the document. BFH/MNG, Dr. F. Löwenthal Version 4.2 πΉπΉ = ππ π£π£ 2 ππ momentum Centripetal force Equation of motion Mechanical Energy πΈπΈππ = 1 πππ£π£ 2 2 Kinetic energy kinetic energy ππ = πΉπΉβ ⋅ ππβ Golden rule of mechnanics W: work F: force r: distance πΈπΈππ = πΈπΈππ + ππ Change of energy Ef: final energy Ei: initial energy W: work πΉπΉ = −ππ ⋅ π₯π₯ Hook's law k: spring constant ππ = πΉπΉππ ⋅ π₯π₯ = ππ ⋅ ππ ⋅ π₯π₯ Friction work ππ1,2 = ∫ πΉπΉβ ⋅ οΏ½οΏ½οΏ½οΏ½β ππππ Generall math formulation πΈπΈ = 1 2 πππ₯π₯ 2 ππππ ππππ ππ ππΜ = = πΉπΉ ⋅ π£π£ π‘π‘ ππ = P: Momentum M: mass Total momentum Spring energy Instantaneous power [W] Mean power Potential energy π₯π₯π₯π₯ = ππππ(π¦π¦ππ − π¦π¦ππ )πΈπΈππππππ = ππππβ 1 ππ(π₯π₯) = πππ₯π₯ 2 2 πΉπΉ(π₯π₯) = −ππππ(π₯π₯) ππππ Potential energy Elastic energy Force dermined from a potiental Collisions J-Integral Fm: mean force J-Integral == change of momentum Impulse p or π½π½ = πΉπΉππ ⋅ π₯π₯π₯π₯ π½π½β = π₯π₯ππβ ππβ = ππ ⋅ π£π£β 2 πΉπΉ = πΈπΈππππππ = οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½β πππ‘π‘π‘π‘π‘π‘ = ∑ ππ οΏ½οΏ½οΏ½βπ€π€ π₯π₯π₯π₯ π₯π₯π₯π₯ ππ2 2 ⋅ ππ To save paper rubbish do not write into the formulas document. After the exam, please return the document. BFH/MNG, Dr. F. Löwenthal Version 4.2 Elektrotechnics Sources ππππ Open circuit voltage πππΎπΎ terminal voltage π π ππ Internal resistance Electrical charge 1 β£ππ1 β£β£ππ2 β£ πΉπΉ = 4ππππ0 ππ 2 Electrical field (static) πΈπΈοΏ½β = πΈπΈ = πΈπΈ = Current Coulombs law q charges r distance between charges Coulomb-Gesetz ππ0 = 8.85 ⋅ 10−12 πΆπΆ 2 ⁄ππ ⋅ ππ2 ππππ Power of the consumer RV: Resistance of the consumer Ri: internal resistance of the source F Force πΉπΉβ ππ0 1 β£ππβ£β£ 4ππππ0 ππ 2 E-Field of a point charge 1 ππ ππ = ππππ 2ππππ0 π§π§ 3 E-Field of a dipole ππ = π π β πΌπΌ Ohms law U tension [V] R Ohm’s resistance I Current πΉπΉβ = πππΈπΈοΏ½β π π = ππ β Unloaded Voltage divider πΏπΏ π΄π΄ Force of a charge q in the EField Specific resistance ππ L Length A Area 1 π π π‘π‘ = οΏ½οΏ½ οΏ½ π π π π −1 ππ π π π‘π‘ = οΏ½ π π ππ Parallel resistances Serial resistances ππ Condensor Capacity πͺπͺππ = οΏ½ πͺπͺππ ππ Ct = οΏ½οΏ½ ππ 1 οΏ½ πΆπΆππ −1 Parallel condensors Serial condensors 3 πππΎπΎ = ππππ − πΌπΌ ⋅ π π ππ ππ1 π π 1 = ππ2 π π 2 ππππ = π π ππ ⋅ ππππ2 (π π ππ + π π ππ )2 To save paper rubbish do not write into the formulas document. After the exam, please return the document. BFH/MNG, Dr. F. Löwenthal Version 4.2 Oscillations πΎπΎ = Oscillations without friction π₯π₯Μ + ππ02 π₯π₯ = 0 2ππ ππ = ππ ππ = 1 ππ Harmonic oscillations πΈπΈ(π‘π‘) = πΈπΈ0 ⋅ ππ −2πΎπΎπΎπΎ vmax = xm ππ ππ = οΏ½ ππ ππ ππ ππ = 2πποΏ½ ππ ππ = 2πποΏ½ πΏπΏ ππ ππ = 2πποΏ½ πΏπΏππ = πΌπΌππππ + ππππ 2 ππππππ πΌπΌππππ + ππππ 2 ππππ πππ‘π‘ = οΏ½ Maximal speed Maximal acceleration Damping term M: mass b: Ns/m: Fr = - b v Damped frequency Energy loss Doppler shift Periode T ππ′ = ππ ⋅ Frequence [Hz] ππππππππ = ππ2 π₯π₯ππ t ππ2 = ππ02 − πΎπΎ 2 Equation of oscillations π₯π₯(π‘π‘) = π₯π₯ππ sin(ππππ − ππ0 ) ππ 2ππ π£π£ Resonance = 1 ± π£π£ ⁄ππ 1 β π£π£ ⁄ππ f, f': Frequency v: velocity Sender / recipient c: velocity of sound Resonance Equation π₯π₯Μ + 2πΎπΎπ₯π₯Μ + ππ02 = ππππ cos(ππππ) Spring oscillations ππ Angular velocity Stationary solution Periode of a spring oscillator Periode Mathematical pedulum Periode Physical pendulum Amplitude π΄π΄(ππ) = Reduced length of pendulum Lr Phase πΏπΏ(ππ) πΉπΉππ ⁄ππ οΏ½(ππ02 −ππ2 )2 +ππ 2 ππ2 οΏ½ππ2 = arctan οΏ½ Radius of inertia r πΌπΌππππ ππ ππ = Damped oscillations Equation of damped harmonic oscillations π₯π₯Μ + + = Μ + 2 πΎπΎ xΜ + ππ02 x = 0 x(t) 4 ππππ οΏ½ ππ(ππ02 − ππ 2 ) πππ π = οΏ½ππ02 − ππ 2 2m2 πππππππππ‘π‘ = √4ππππ π₯π₯(t) = π΄π΄(ππ)cos(ππππ − πΏπΏ(ππ)) ππ πΎπΎ = 2ππ ππ2 = ππ02 − πΎπΎ 2 πΈπΈ(π‘π‘) = πΈπΈ0 ⋅ ππ −2πΎπΎπΎπΎ Resonance frequency Critical damping To save paper rubbish do not write into the formulas document. After the exam, please return the document. BFH/MNG, Dr. F. Löwenthal Version 4.2 Waves ππ = 2ππ ππ Wave number ππ = 2ππ ππ Periode ππ = 2ππππ Angular frequency ππ ππ Speed in the rope π£π£ = ππ ⋅ ππ π£π£ = οΏ½ ππππ = 1 2 ππππππ2 ⋅ π¦π¦ππ 2 Speed of phase Power Stationary waves ππ = 2ππ ππ ππππ = ππ ⋅ ππ 2L wave number Eigenfrequenzen n = 1,2,3, … 5 To save paper rubbish do not write into the formulas document. After the exam, please return the document. BFH/MNG, Dr. F. Löwenthal Version 4.2 [P] = W Geometrical Optics Besselsche Methode Mirrors für Sammellinse p: distance of object i: distance of image f: focal length R: Radius of curvature a: Abstand Objekt – Bild e: Abstand der beiden Linsenpositionen f: Brennweite Refraction n = Für Streulinse Refraction index n c: speed of light in vacuum cm: speed of light in medium c cm f1: Brennweiter Streulinse a: Distanz ObjektStreulinse d: Distanz StreulinseSammellinse b: Distanz: SammellinseBild Law of refaction Sign convention r : radius of curvature r < 0 for concave surface r > 0 für convex surface Dioptry D Korrektur Kurzsichtig smax: maximale Sehdistanz Lens maker formula 1 = (n-1)οΏ½r 1 1 r2 οΏ½ Abbildungsgleichung dünner Linsen Intensität bei isotropem Leuchtkörper [I] = W/m2 ππ ⋅ sin(πΌπΌ) = ππππππππππ ππ = −π΅π΅ −ππ = πΊπΊ ππ 1 1 1 + = ππ ππ ππ Korrektur Weitsichtig πΌπΌ(ππ) = ππ 2 ri < 0 for convave ri > 0 for convex f ππ0 ππππ Linse in Serie Dünne Linsen in Serie ππ1 ππ2 ππ2 − ππ1 + = ππ ππ ππ Refraction on a spherical surface 1 ππππ = s0: normale Sehdistanz smin: minimale Sehdistanz 4ππππ smin > s0 für Weitsichtigkeit 1 1 1 + = ππ ππ ππ π π = 2 ⋅ ππ Optische Instrumente Konvex f < 0 konkav f > 0 6 ππππ = ππ0 ππππ To save paper rubbish do not write into the formulas document. After the exam, please return the document. BFH/MNG, Dr. F. Löwenthal Version 4.2 Lupe s0 = 25 cm f Brennweite der Lupe Beamexpander D: Ausgangsdurchmesse r d: Eingangsdurchmesser f1: Eingangslinse f2: Ausgangslinse ππ = 2οΏ½ ππ2 − ππ ⋅ ππ 4 ππ2 − ππ 2 4 ⋅ ππ 1 1 1 = + ππ1 ππ ππ − 1 1 1 − ππ2 ππ ππ = ππππππππ Fernrohr ππ = ππππππ m: Winkelvergrösserung fok: Okularlinse fobj: Objektivlinse Mikroskop s0 = 25 cm s = Tubuslänge fok = Okularbrennweite fobj = Objektivbrennweite ππ = π₯π₯π₯π₯ = −π΅π΅ −ππ = πΊπΊ ππ π·π· = 1 1 ππ = −1 π π ππππππ ππππππππππππππ π₯π₯π₯π₯ < 0 1 1 π₯π₯π₯π₯ = − π π 0 π π ππππππ π₯π₯π₯π₯ > 0 ππ = 2οΏ½ ππ = ππ2 − ππ ⋅ ππ 4 ππ2 − ππ 2 4 ⋅ ππ 7 To save paper rubbish do not write into the formulas document. After the exam, please return the document. BFH/MNG, Dr. F. Löwenthal Version 4.2 Thermodynamics π₯π₯πΏπΏ = πΌπΌ ⋅ πΏπΏ ⋅ π₯π₯ππ π₯π₯π΄π΄ = π½π½ ⋅ π΄π΄ ⋅ π₯π₯π₯π₯ π½π½ = 2 ⋅ πΌπΌ π₯π₯π₯π₯ = πΎπΎ ⋅ ππ ⋅ π₯π₯π₯π₯ πΎπΎ = 3 ⋅ πΌπΌ [Rs] = J/kg/K [v] = m3/kg M: mol volume v: spezific volume vm: molar volume 1.Dimensions 2 dimensions Equation of real gas 3 dimensions ππ ⋅ ππ = ππ ⋅ π π π π ⋅ ππ π£π£ππ = ππ ⋅ π£π£ π π = π π π π ⋅ ππ Q = m C βT Specific heat capacity [C] = J kg/K NA = 6.022·1023/mol ππ = ππ ⋅ ππππ ⋅ π₯π₯ππ Molar heat [cm] = J/mol/K [n] mol-1 Work Wx πΆπΆΜ = ππ2 ∫ππ1 πΆπΆ(ππ)ππππ ππ2 − ππ1 ππ = πΏπΏ ⋅ ππ πππππ£π£ = ππ(ππ) ⋅ ππππ πππππ£π£ = ππ(π£π£)ππππ ππππππ = −ππ(ππ) ⋅ ππππ ππππππ = −π£π£(ππ) ⋅ ππππ Mean value ππππ ππππ Melting enthalpy [L] = J/kg ππ ⋅ π£π£ = π π π π ππ ππ ⋅ π£π£ππ = π π ⋅ ππ Avogadro number V-work Wv p-work Heat [Q] = J [q] = J/kg (Spec.) Inner Energy ππ [U] = J ⋅ πππ΄π΄ ⋅ πππ΅π΅ ⋅ ππ [u] = J/kg 2 π π = πππ΄π΄ ⋅ πππ΅π΅ Perfect gas: f = 3 ππ = Perfect gas ππ ⋅ ππ = ππ ⋅ π π π π ⋅ ππ Real gas factor Z πΏπΏπΏπΏ ππ πΏπΏππ ππππ = ππ ππππ = Equation of equilibrium of the perfect gas. [V] = m3 [p] = Pa [m] = kg Rs = spez. Gaskonstante R = universelle Gaskontante R = 8.314 J/mol/K (spez.) Entropy S, s [S] = J/K [s] = J/K/kg 4 Principles of thermodynamics 8 To save paper rubbish do not write into the formulas document. After the exam, please return the document. BFH/MNG, Dr. F. Löwenthal Version 4.2 Four elementary Processes in closed systems Closed systems ππππππ = −ππ(ππ) ⋅ ππππ Volume work ππππππ = −π£π£(ππ) ⋅ ππππ Isochor process 1a. closed systems ππππ = πΏπΏπΏπΏ − πΏπΏππππ ππππ = πΏπΏπΏπΏ − πΏπΏπ€π€π£π£ V = const πππππ£π£ = ππ(ππ) ⋅ ππππ πππ€π€π£π£ = ππ(π£π£)ππππ Work Conversion of energy ππ = ππππππππππ ππ Equation of state 1b. open Systems heat ππππ π₯π₯ = πΏπΏπΏπΏ − πΏπΏππππ ππβ π₯π₯ = πΏπΏπΏπΏ − πΏπΏπ€π€ππ v-work isobar process 2. natural processes ππππ = πΏπΏπΏπΏ ππ takes the direction to πΏπΏπΏπΏ ππππ = increase the total ππ entropy. ππ12 = π’π’2 − π’π’1 π€π€π£π£ = 0 p = const π£π£ = ππππππππππ ππ Equation of state heat ππππ > 0 ππππ > 0 v-work ππ12 = β2 − β1 π€π€π£π£12 = π π ⋅ (ππ2 − ππ1 ) isotherm process T = const lim π₯π₯π₯π₯ = 0 3. It is not possible to ππ→0 lim π₯π₯π₯π₯ = 0 hit the absolut zero ππ→0 temperature by a natural process. Equation of state heat v-work 0. Two systems in a thermodynamical process have the same temperature. adiabatic process ππ = ππππππππππ ππ ππ ⋅ π£π£ = πππππππππ‘π‘ π€π€π£π£12 = ππ1 ⋅ π£π£1 ⋅ ln ππ12 = π π ⋅ ππ ⋅ ln π£π£2 π£π£1 k = adiabatic exponent k = cp/cv = 1.4 Equation of state ππ ⋅ π£π£ ππ = ππππππππππ ππ ⋅ π£π£ ππ = πππππππππ‘π‘ ππ ⋅ π£π£ ππ = πππππππππ‘π‘ open systems 9 π£π£2 π£π£1 ππ ⋅ π£π£ ππ = πππππππππ‘π‘ π»π» = ππ + ππ ⋅ ππ β = π’π’ + ππ ⋅ π£π£ To save paper rubbish do not write into the formulas document. After the exam, please return the document. BFH/MNG, Dr. F. Löwenthal Version 4.2 Enthalpy H, h [H] = J [h] = J/kg Totale Enthalpy H* [H*] = J [h*] = J/kg 1 π»π» π₯π₯ = π»π» + ππππ 2 2 1 π»π» π₯π₯ = ππ + ππππ + ππππ 2 2 1 β π₯π₯ = β + ππ 2 2 1 β π₯π₯ = π’π’ + ππ ⋅ π£π£ + ππ 2 2 10 To save paper rubbish do not write into the formulas document. After the exam, please return the document. BFH/MNG, Dr. F. Löwenthal Version 4.2 Quantum mechanics Quantisierung der Energieniveaus h: Planckkonstante f: Photonenfrequenz Eu: oberes Energieniveau El: unteres Energieniveau Projektion des Spindipolmoment Photoelektrischer Effekt β ⋅ ππ = πΈπΈπ’π’ − πΈπΈππ Photonenenergie f: Frequenz Wellenlänge Frequenz Photonenimpuls Photonenmasse Atomarer Drehimpuls (Bahndrehimpuls) Klassischer Drehimpuls Atomarer Bahndrehimpuls L Bahndrehimpuls l Quantenzahl, ππππππ, ππ < ππ Photelektrische Gleichung πΏπΏ = ππ ⋅ ππ ⋅ π£π£ πΏπΏ = οΏ½ππ(ππ + 1) ⋅ β K: Kinetische Energie der Austrittselektronen ππππ Austrittsarbeit V: Stoppotential e: Elementarladung n: Hauptquantenzahl Z-Projektion des Drehimpulses, β£πππ§π§ β£ ≤ ππ, πππ§π§ ππππ Bohrsches Magnetonπππ΅π΅ Magnetisches Dipolmoment Z-Projektion des magnetischen Dipolmoments πΏπΏπ§π§ = ππππ ⋅ β πππ΅π΅ = πππ΅π΅ = οΏ½οΏ½οΏ½οΏ½β Materiewellen – Wahrscheinlichkeit ππ β 2ππππ De Broglie Wellenlänge −ππ οΏ½β πΏπΏ 2ππ πππ΅π΅,π§π§ = −ππππ ⋅ πππ΅π΅ ππππ ππππ Elektronenspin (Eigendrehimpuls) Spin des Elektrons Z-Projektion des Spins Magnetisches Dipolmoment des Spins ππ = οΏ½π π (π π + 1) ⋅ β 1 π π = 2 πππ§π§ = πππ π ⋅ β 1 πππ π = ± οΏ½ οΏ½ 2 ππβ πππ π = − οΏ½ οΏ½ ππβ οΏ½οΏ½οΏ½β ππππ οΏ½οΏ½οΏ½β πππ π = −2πππ΅π΅ ππβ 11 πππ π ,π§π§ = ±πππ΅π΅ πππ΅π΅ = οΏ½οΏ½οΏ½οΏ½β −ππ πΏπΏοΏ½β 2ππ πΈπΈππβ = βππ ππ ⋅ ππ = ππ β ⋅ ππ β = ππ ππ β ⋅ ππ ππππβ = 2 ππ ππ = πΈπΈππβ = πΎπΎ + ππππ β ⋅ ππ = πππ π π π π π π π ⋅ ππ + ππππ β ππππ πππ π π π π π π π = οΏ½ οΏ½ ⋅ ππ − ππ ππ πππ π ,π§π§ = ±πππ΅π΅ ππ = β ππ To save paper rubbish do not write into the formulas document. After the exam, please return the document. BFH/MNG, Dr. F. Löwenthal Version 4.2 Constants Speed of light Elementary charge Electron mass Proton mass Avogadro number Boltzmann constant Planck constant h, β oder β = 2ππ Bohr magneton ππ = 2.998 ⋅ 108 ππ⁄π π ππ = 1.60210 ⋅ 10−19 πΆπΆ ππππ = 9.11 ⋅ 10−31 ππππ ππππ = 1.67 ⋅ 10−27 ππππ ππ = 1.38 ⋅ 10−23 π½π½⁄πΎπΎ β = 6.626 ⋅ 10−34 π½π½π½π½ β = 1.054 ⋅ 10−34 π½π½π½π½ ππ = ⋅ πππ΅π΅ = 9.274 ⋅ 10−24 π½π½⁄ππ πππ΄π΄ = 6.0225 ⋅ 1023 ππππππ −1 12 To save paper rubbish do not write into the formulas document. After the exam, please return the document. BFH/MNG, Dr. F. Löwenthal Version 4.2 Updates Version updated 2.3 - Formatting, some minor corrections upload: 27.10.17 2.4 - Update sign in lens maker formula - Added: Work of loss upload: 31.10.17 2.5 - added: cinematics of rotation upload 7.11.17 2.6 - update: mechanics formula for air resistance 2.7 Added: formulas for physics of atoms Added: stern – triangle – formula minor corrections upload 14.11.17 2.8 Minor corrections 3.0 - added: some formulas for oscillations upload 17.4.18 3.2 - added: some formulas of atom physics - added: some formulas of electronics 22.2.18 3.3 - added: Some formulas rotations - added: some formulas standing waves 11.6.18 4.0 - Change to english - minor corrections 4.1 upload 14.01.18 11.9.18 05.06.20 - added: some formulas of electrotechnics 5.6.2020 4.2 - translated: thermodynamics -> english 13