Physics formula sheet, Lecture notes of Physics

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Carmichael Position, velocity and acceleration ∆x/∆t = vav ∆v/∆t = aav dx/dt = v(t) dv/dt = a(t)∫ a(t)dt = ∆v ∫ v(t)dt = ∆x Uniformly accelerated motion v = v0 + at x = x0 + v0t+ 1 2at 2 v2 = v20 + 2a(x− x0) x = x0 + ( v0 + v 2 ) t Projectile motion 2D (uniform field g=const.) x′(0) = ẋ(0) = vx(0) = v(0) cos θ = v0 cos θ y′(0) = ẏ(0) = vy(0) = v(0) sin θ = v0 sin θ y′′ = −g = const. x′′ = 0 = const. y′(t) = y′(0)− gt x′(t) = x′(0) = const. y(t) = y(0) + y′(0)t− 12gt 2 x(t) = x(0) + x′(0)t Alternative form ay = −g = const. ax = 0 = const. vy(t) = vy(0)− gt vx(t) = vx(0) = const. y(t) = y(0) + vy(0)t− 12gt 2 x(t) = x(0) + vx(0)t Trajectory equation for x(0) = 0 y(x) = [ vy(0) vx(0) ] x− [ 1 2 g v2x(0) ] x2 + y(0) y(x) = x tan θ − [ 1 2 g v2(0) cos2 θ ] x2 + y(0) Velocity-position equations v2(y) = v2(0)− 2gy v2y(y) = v 2 y(0)− 2gy Special points: Range R, height h, flight time T h = v2y(0)/2g h = v 2(0) sin2 θ/2g R = 2vx(0)vy(0)/g R = v 2(0) sin 2θ/g T = 2vy(0)/g T = 2v(0) sin θ/g R = 4h cot θ Circular motion Centripetal acceleration ar = v 2/r = rω2 Arc length s = rθ Tangential speed v = rω = 2πr/T Tangetial acceleration at = rα Angular frequency ω = 2πf = 2π/T Frequency and time period f = 1/T Uniform circular motion at = 0, α = 0 Forces and Momentum Newton’s second law (general) ~F = d~p/dt Potential energy and force (1D) F = −dU/dx Potential energy and force (3D) ~F = −∇U Linear Momentum ~p = m~v Friction (static) fs ≤ fs,max = µsN Friction (kinetic) fk = µkN Grav. fields due to point or spherical sources Force between masses F = Gmm′/r2 Gravity field of mass m g = Gm/r2 G.P.E. two masses U = −Gmm′/r Grav. potential of m V = −Gm/r Orbital motion Kepler’s 2nd Law T 2 = (4π2/GM)r3 Orbit (circular) v2 = GM/r Escape velocity v2 = 2GM/r Constants related to gravity Universal const. of gravitation G = 6.67× 10−11 N ·m2/kg2 Earth surface gravity g = 9.81 m/s2 Earth mass & G GME = 3.98× 1014 m3/s2 Solar mass & G GM = 1.33× 1020 m3/s2 Moon mass & G GM$ = 4.91× 1012 m3/s2 Work and energy Kinetic energy K = 12mv 2 Work W = ∫ ~F · d~r Power P = dE/dt = dW/dt Average Power Pav = ∆E/∆t = W/∆t Instantaneous Power P = ~F · ~v = F‖v Work-energy theorem Wnet = Wc +Wnc = ∆K Work done by con. forces Wc = −∆U Mechanical energy Emech = K + U Conservation of mech. energy Ki + Ui +Wnc = Kf + Uf Work done by non-con. forces Wnc = −∆Emech GPE uniform field U(y) = mgy + U(0) Potential uniform grav. field V (y) = gy + V (0) GPE uniform field ∆Ugrav. = mg∆y = mgh Mechanical energy Emech. = K + Utotal Centre of mass ~Rcm = 1 M ∑ mi~ri ~Rcm = 1 M ∫ ~rdm = 1 M ∫ ~rρdV Theorems for variable forces Impulse-momentum ~J = ∆~p = ~Fav∆t = ∫ ~Fnet(t)dt version: Wednesday 3rd October, 2018 12:06 Page 1 SFSU Department of Physics Formula Sheet: Physics 220 A. Carmichael Work-energy Wnet = ∫ ~Fnet · d~r = ∆K Types of collision • totally elastic: No loss of K.E. , e = 1 • inelastic: Some loss of K.E., 0 < e < 1 • completely inelastic: v1 = v2 = v, e = 0 Max K.E. loss Collision conservation laws (1D & 2D) Momentum m1~u1 +m2~u2 = m1~v1 +m2~v2 K.E. (elastic only) 12m1u 2 1 + 1 2m2u 2 2 = 1 2m1v 2 1 + 1 2m2v 2 2 Newton’s collision law (1D only) Newton’s collision law (1D) (v2 − v1) = −e(u2 − u1) Collisions 1D Elastic (derived from the above) v1 = m1 −m2 m1 +m2 u1 + 2m2 m1 +m2 u2 v2 = 2m1 m1 +m2 u1 + m2 −m1 m1 +m2 u2 Collisions 1D Inelastic (derived from the above) v1 = m1 − em2 m1 +m2 u1 + (1 + e)m2 m1 +m2 u2 v2 = (1 + e)m1 m1 +m2 u1 + m2 − em1 m1 +m2 u2 Rotational motion Anguar velocity, acceleration ω = dθ/dt, α = dω/dt Angular displacement ∆θ = ∫ ωdt Linear and angular connection vt = Rω, at = Rα Torque ~Γ = ~r × ~F Magnitude of torque Γ = rF sinϕ = rF⊥ Angular momentum (particle) ~L = ~r × ~p Angular momentum (solid) ~L = I~ω Moment of inertia (particles) I = Σmr2axis Moment of inertia (solid) I = ∫ r2axisdm N2 for rotation (general form) ~Γ = d~L/dt N2 for rotation (I=const.) ~Γ = I~α Rotational K.E. Kr = 1 2Iω 2 Work done by a torque W = ∫ Γ · dθ = ∆Kr Work done by const. or av. torque W = Γ ·∆θ = ∆Kr Rotational power P = Γω Conservation of ~L Iiωi = Ifωf Rolling without slipping vcm = Rω, acm = Rα Parallel axis theorem I = Icm +MD 2 Total kinetic energy I = 12Icmω 2 + 12Mv 2 cm Rotational motion with (α = const.) ω = ω0 + αt ∆θ = ω0t+ 1 2αt 2 ω2 = ω20 + 2α∆θ ∆θ = ω0 + ω 2 t Substitutions for rotational dynamics s =⇒ ∆θ ~F =⇒ ~Γ u =⇒ ω0 m =⇒ I v =⇒ ω K = 12mv 2 =⇒ Kr = 12Iω 2 a =⇒ α ~p = m~v =⇒ ~L = I~ω Moments of inertia Moment Object Axis I = MR2 Uniform ring/tube Through C.M. I = 12MR 2 Uniform disk/cylinder Through C.M. I = 112ML 2 Uniform rod Through C.M. I = 13ML 2 Uniform rod Through end I = 25MR 2 Uniform sphere Through C.M. I = 23MR 2 Hollow sphere Through C.M. I = 13Ma 2 Slab width a Along edge (door) Simple harmonic motion (SHM) Hooke’s Law F (x) = −kx acceleration a(x) = −ω2x = −n2x Velocity v(x) = ±ω √ A2 − x2 SPE or EPE for a spring U(x) = 12kx 2 Total energy E = 12kA 2 = 12mω 2A2 Position x(t) x(t) = A cos(ωt+ ϕ) Velocity v(t) v(t) = −Aω sin(ωt+ ϕ) Acceleration a(t) a(t) = −Aω2 cos(ωt+ ϕ) Period, mass-spring T = 1 f = 2π ω = 2π √ m k Period, simple pendulum T = 1 f = 2π n = 2π √ l g Period, physical pendulum T = 1 f = 2π n = 2π √ I mgr Period, torsional pendulum T = 1 f = 2π n = 2π √ I κ version: Wednesday 3rd October, 2018 12:06 Page 2 SFSU Department of Physics Formula Sheet: Physics 220 A. Carmichael Mathematical constants e = 2.71828... 1o = 1.745× 10−2 rad π = 3.14159... 1′ = 2.9089× 10−4 rad log10 e = 0.434... 1 ′′ = 4.8481× 10−6 rad ln 10 = 2.3025... 1 rad = 57.296o ln 2 = 0.693... π/6 rad = 30o e−1 = 0.368... π/3 rad = 60o (1− e−1) = 0.632... π/4 rad = 45o √ 3/2 = 0.866... 1 rpm = 0.1047 rad/s 1/ √ 2 = 0.707... 1 rad/s = 9.549 rpm Greek alphabet Letter Upper case Lower case Alpha A α Beta B β Gamma Γ γ Delta ∆ δ Epsilon E , ε Zeta Z ζ Eta H η Theta Θ θ Iota I ι Kappa K κ Lambda Λ λ Mu M µ Nu N ν Xi Ξ ξ Omicron O o Pi Π π Rho P ρ Sigma Σ σ Tau T τ Upsilon Y υ Phi Φ φ, ϕ Chi X χ Psi Ψ ψ Omega Ω ω SI units and derived units Quantity Symbol Unit Name Basic Units Mass m kg kilogram kg Length l m meter m Time t s second s Force F N Newton kg ms−2 Energy E J Joule kg m2s−2 Power P W = Js−1 Watt kg m2s−3 Pressure p Pa = N.m2 Pascal kg/ms2 Abbreviations used: atm.=atmosphere (pressure) con. = conservative (force) AC = Alternating Current BVP = Boundary Value Problem CM = Centre of Mass DC = Direct Current (or Detective Comics) EM or E&M = ElectroMagnetism EMF = ElectroMotive Force (voltage) EPE = Elastic Potential Energy GR = General Relativity GPE = Gravitational Potential Energy G.T. = Galilean Transformation IC = Initial Condition IVP = Initial Value Problem ODE = Ordinary Differential Equation PD = Potential Difference PDE = Partial Differential Equation PE = Potential Energy L.T. = Lorentz Transformation SHM = Simple Harmonic Motion SHO = Simple Harmonic Oscillator SPE = Strain/Spring Potential Energy SR = Special Relativity STP = Standard Temperature and Pressure (20o C, 1 atm) TIR = Total Internal Reflection N1,N2,N3= Newton’s laws of motion T0,T1,T2,T3= the laws of thermal physics K1,K2,K3= Kepler’s laws of planetary motion Metric Prefixes exa E 1018 peta P 1015 tera T 1012 giga G 109 mega M 106 kilo k 103 hecto h 102 deci d 10−1 centi c 10−2 milli m 10−3 micro µ 10−6 nano n 10−9 pico p 10−12 femto f 10−15 atto a 10−18 version: Wednesday 3rd October, 2018 12:06 Page 5 SFSU Department of Physics Formula Sheet: Physics 220 A. Carmichael Unit Conversions Quantity Units Conversion or value Length inch, cm 1 in. = 2.54 cm Length foot, cm 1 ft = 30.48 cm Length mile, km 1 mile = 1.609 km Energy electron-volt, Joule 1 eV = 1.602× 10−19 J Energy calorie, Joule 1 cal = 4.1868 J Energy British thermal unit, Joule 1 Btu = 1055 J Energy foot-pound, Joule 1 ft · lb = 1.356 J Energy kilowatt-hour, Joule 1 kW · h = 3.600 MJ Power horsepower, Watt 1 hp = 746 Watt Mass atomic unit, kg 1 u = 1.6605× 10−27 kg Force pound, Newton 1 lb = 4.442 N Density g/cm3 → kg/m3 1 g/cm3 = 1000 kg/m3 Pressure Pascal, psi 1 Pa = 1 N/m2 = 1.450× 10−4 psi Pressure atmosphere, Pascal 1 atm = 101, 325 Pa = 760 Torr = 14.7 psi Pressure psi, Pascal 1 psi = 6.895× 103 Pa Pressure mm Hg 1 torr = 1 mm Hg = 0.0394 in Hg = 1.333× 102 Pa Pressure bar 1 bar = 105 Pa Volume litre 1 l = 103 cm3 = 10−3 m3 = 1.057 qt (US) Volume quart (US) 1 qt (US) = 946 ml Volume gallon (US) 1 gal.(US) = 3.758 l Angle rev, rad,deg 1 rev = 360o = 2π rad Astrophysical Data Body surface g Mass GM Radius Orbit Radius Orbit Period Symbol (m/s2) kg (m3/s2) m m Earth years Sun −− 1.99× 1030 1.33× 1020 6.96× 108 −− −− Earth 9.81 5.97× 1024 3.98× 1014 6.37× 106 1.50× 1011 1.00 ♁ Moon 1.62 7.36× 1022 4.91× 1012 1.74× 106 −− −− $ version: Wednesday 3rd October, 2018 12:06 Page 6 SFSU Department of Physics Formula Sheet: Physics 220 A. Carmichael Symbols used in mechanics: A Amplitude for SHM A, A1, A2 Cross sectional area of pipe a Acceleration at Tangential component of acceleration ar Radial component of acceleration e Coefficient of resitution E Total energy F , Fav Force, average force f Frequency (rev/second or cycles/second) f Friction (force) G Universal gravitation constant g Gravitational field strength h depth or height I Moment of inertia ~J Impulse (change in momentum ~J = ∆~p) K Kinetic energy Kr Rotational kinetic energy k Spring constant k wavenumber 2π/λ ~L Angular momentum l Length M , m Mass n Normal force P Power Pav Average power p Momentum r radius s Displacement T Time period/ time of flight T tension U Potential energy u velocity at time t = 0 v velocity at time t W Work Wc Work done by a con. force(s) Wnc Work done by non-con. force(s) Wnet Work done by net force Y Young’s modulus α Angular acceleration (rad/s2) ∆ change in... µk Coefficient of kinetic friction µs Coefficient of static friction ω Angular speed at time t (rad/s) ω0 Angular speed at time t = 0 (rad/s) ∆θ angular displacement ∆θ = θ − θ0 θ0 Angular position at time t = 0 Γ Torque ρ density (mass/volume) version: Wednesday 3rd October, 2018 12:06 Page 7 SFSU Department of Physics