PhysicsForm 8.0 4/15/03 12:16 PM Page PHYSICAL CONSTANTS Acceleration due to gravity Avogadro’s number ELECTROMAGNETIC CONSTANTS WAVELENGTHS OF LIGHT IN A VACUUM (m) g 9.8 m/s NA 6.022 × 10 23 k × 109 N·m2 /C2 Gravitational constant G 6.67 × 10−11 N·m2 /kg Green Planck’s constant h 6.63 × 10 Blue Ideal gas constant R Permittivity of free space ε0 8.8541 × 10−12 C/(V·m) Permeability of free space µ0 4π × 10−7 Wb/(A·m) J·s 331 m/s 3.00 × 108 m/s Electron charge e 1.60 × 10 Electron volt eV 1.6022 × 10 Atomic mass unit u 1.6606 × 10 kg = 931.5 MeV/c2 Rest mass of electron me 9.11 × 10−31 kg = 0.000549 u = 0.511 MeV/c2 mp 1.6726 × 10−27 kg = 1.00728 u = 938.3 MeV/c2 .of proton −19 J −27 Mass of Earth 5.976 × 1024 kg Radius of Earth 6.378 × 10 m 4.9 – 5.7 × 10−7 1011 1012 microwaves 10-1 10-2 = wavelength (in m) 4.2 – 4.9 × 10−7 10-3 1013 1014 1015 10-4 1016 1017 1018 ultraviolet infrared 10-5 10-6 10-7 10-8 10-9 1020 1019 gamma rays X rays 10-10 R O Y G B I 10-11 10-12 V = 780 nm visible light 4.0 – 4.2 × 10−7 Violet 360 nm INDICES OF REFRACTION FOR COMMON SUBSTANCES ( l = 5.9 X 10 –7 m) Air 1.00 Alcohol 1.36 Corn oil 1.47 1.47 Diamond 2.42 1.33 Glycerol Water incident ray θinciden t = θreflected c n= (v is the speed of light in the medium) v Law of Reflection Index of refraction angle of incidence 01 0' angle of reflection n1 sin θ1 = n2 sin θ2 � � θc = sin −1 nn21 Snell’s Law Critical angle 02 normal angle of refraction refracted ray reflected ray LENSES AND CURVED MIRRORS 1.6750 × 10−27 kg = 1.008665 u = 939.6 MeV/c2 …of neutron 1010 REFLECTION AND REFRACTION C −19 109 radio waves OPTICS c Speed of light in a vacuum 108 5.7 – 5.9 × 10−7 Yellow 8.314 J/(mol·K) = 0.082 atm ·L/(mol·K) Speed of sound at STP ƒ = frequency (in Hz) Orange 5.9 – 6.5 × 10−7 Coulomb’s constant −34 6.5 – 7.0 × 10−7 Red molecules /mol q image size =− p object size 1 + = f q p Optical instrument Lens: Concave Convex Focal distance f Image distance q Type of image negative positive negative (same side) negative (same side) positive (opposite side) virtual, erect virtual, erect real, inverted negative (opposite side) virtual, erect negative (opposite side) positive (same side) virtual, erect real, inverted pf p h Convex negative Concave positive pf DYNAMICS V F Mirror: q NEWTON’S LAWS First Law: An object remains in its state of rest or motion with constant velocity unless acted upon by a net external force dp F = Second Law: Fnet = ma dt Third Law: For every action there is an equal and opposite reaction Weight Fw = mg Normal force FN = mg cos θ (θ is the angle to the horizontal) h h F p V p q Kinetic friction fk = µk FN µs is the coefficient of static friction µk is the coefficient of kinetic friction For a pair of materials, µk < µs W = F · s = F s cos θ � W = F · ds $5.95 CAN $3.95 mv Centripetal force Fc = r ˆ a = axˆi + ayˆi + az k Magnitude a = |a| = Dot product a · b = ax bx + ay by + az yz = ab cos θ � a2x + a2y + a2z Cross product a × b = (ay bz � � ax � = �� ax � ˆi axb Gravitational potential energy Ug = mgh Total mechanical energy E = KE + U Average power Pavg = Instantaneous power MOMENTUM AND IMPULSE Linear momentum p = mv Impulse J = �Ft = ∆p J= F dt = ∆p a COLLISIONS b ˆ − az by ) ˆi + (az bx − ax bz) ˆj + (ax by − ay bx ) k � ay az �� ay bz �� ˆj ˆ � k This downloadable PDF copyright © 2004 by SparkNotes LLC ∆W ∆t P =F·v a b p2 mv = 2m ∆U = −W Potential energy Notation |a × b| = ab sin θ a × b points in the direction given by the right-hand rule: KE = Kinetic energy All collisions m1 v1 + m2 v2 = m1 v1� + m2 v2� Elastic collisions 1 1 2 m1 v12 + m2 v22 = m1 (v1� ) + m2 (v2� ) 2 2 v1 − v2 = − (v1� − v2� ) q V h F V F p h F V F q WORK, ENERGY, POWER Work F p (for conservative forces) VECTOR FORMULAS (θ is the angle between a and b) F Work-Energy Theorem W = ∆KE UNIFORM CIRCULAR MOTION v2 Centripetal acceleration ac = r q FRICTION Static friction fs, max = µs FN h V p q KINEMATICS Average velocity vavg = ∆s ∆t DISTANCE s (m) Instantaneous ds v= velocity dt Displacement ∆s = Average acceleration � aavg = v dt ∆v ∆t Instantaneous dv a= acceleration dt Change in velocity vf = v0 + at vavg = (v0 + vf ) s = s0 + v0 t + at = s0 − vf t + at = s0 + vavg t = v02 VELOCITY v (m/s) + t (s) ∆v = � CONSTANT ACCELERATION vf2 t (s) a dt – ACCELERATION a (m/s2) + t (s) – + 2a(sf − s0 ) CONTINUED ON OTHER SIDE SPARKCHARTS™ Physics Formulas page of PhysicsForm 8.0 4/15/03 12:16 PM Page WAVES ELECTRICITY T WAVE ON STRING Tension in string FT Mass density µ = Length L mass length F =k Electric field E= Potential difference W ∆V = q Fon q q F = Eq CIRCUITS ∆Q ∆t Current I= Resistance R=ρ Ohm’s Law I= SOUND WAVES Power dissipated by resistor P = V I = I 2R Beat frequency Heat energy dissipated by resistor W = P t = I Rt Speed of standing wave v= Wavelength of standing wave λn = FT µ 2L n fbeat = |f1 − f2 | DOPPLER EFFECT Motion of source Stationary Motion of observer Stationary v λ f veff = v + vo λeff = λ � � o feff = f v+v v Towards source at vo Toward observer at vs Away from observer at vs veff = v � � s λeff = λ v−v � v � v feff = f v−v s veff = v � � s λeff = λ v+v � v � v feff = f v+v s veff = v ± vo � � s λeff = λ v±v v � � o feff = f v±v v±vs Away from source at vo veff = v − vo λeff = λ � � o feff = f v−v v ROTATIONAL MOTION Angular position Angular velocity ωavg = ∆θ ∆t ω= v r dθ dt at r dω dt ω= α= Angular acceleration αavg = s r ∆ω ∆t α= a CONSTANT ωf = ω0 + αt T = 2π � v=0 U = max KE = MASS-SPRING SYSTEM R R sphere R MR ring disk Elastic potential energy Period MR L rod TORQUE AND ANGULAR MOMENTUM Torque τ = dL dt F = −k(∆)x ∆x is the distance the spring is stretched or compressed from the equilibrium position, and k is the spring constant ML2 12 R R2 R3 Magnetic force on moving charge F = qvB sin θ F = q (v × B) Magnetic force on current-carrying wire F = BI� sin θ F = I (� × B) MAGNETIC FIELD PRODUCED BY… Magnetic field due to a moving charge B= µ0 qv × ˆr 4π r2 Magnetic field produced by a current-carrying wire B= µ0 I 2π r Magnetic field produced by a solenoid B = µ0 nI Biort-Savart Law dB = Lenz’s Law and Faraday’s Law ε=− MAXWELL’S EQUATIONS � Gauss’s Law �s Gauss’s Law for magnetic fields �s Restoring force MOMENTS OF INERTIA (I ) � I= r dm Moment of inertia MR 2 mg cos v=0 U = max KE = equilibrium position = θ0 + ωavg t MR v = max U = KE = max τ = F r sin θ τ =r×F τ = Iα Angular momentum L = pr sin θ L=r×p L = Iω Rotational KE rot = 12 Iω kinetic energy GAS LAWS Universal Gas Law P V = nRT Combined Gas Law P2 V2 P1 V1 = T2 T1 2π T = � k(∆x)2 � m T = 2π k Ue = x = A sin(ωt) Equation of motion where ω = k m is the angular frequency and A = (∆x)max is the amplitude THERMODYNAMICS First Law ∆ (Internal Energy) = ∆Q + ∆W Second Law: All systems tend spontaneously toward maximum entropy ∆Qout Alternatively, the efficiency e = − ∆Qin of any heat engine always satisfies ≤ e < Boyle’s Law P1 V1 = P2 V2 Charles’s Law P2 P1 = T2 T1 This downloadable PDF copyright © 2004 by SparkNotes LLC R3 R1 mg sin = (ω0 + ωf ) ωf2 = ω02 + 2α(θf − θ0 ) R2 Parallel circuits Ieq = I1 + I2 + I3 + · · · Veq = V1 = V2 = V3 = 1 1 + ··· + + = R2 R2 R1 Req T � g αt particle Period mg ωavg θ = θ0 + ω t + 2g� (1 − cos θmax ) R1 MAGNETISM Velocity at equilibrium position � Series circuits Ieq = I1 = I2 = I3 = Veq = V1 + V2 + V3 + · · · Req = R1 + R2 + R3 + · · · Loop rule: The sum of all the (signed) potential differences around any closed loop is zero Node rule: The total current entering a juncture must equal the total current leaving the juncture PENDULUM v= V R KIRCHHOFF’S RULES SIMPLE HARMONIC MOTION θ= L A Faraday’s Law c � Ampere’s Law �c Ampere-Maxwell Law c E · dA = r) µ0 I (d� × ˆ r2 4π dΦB dt Qenclosed ε0 B · dA = E · ds = − � SPARKCHARTS �� t q1 q2 q1 q2 = 4πε0 r r2 Coulomb’s Law $5.95 CAN Wave speed v = f λ Wave equation � � y(x, t) = A sin(kx − ωt) = A sin 2π λx − ELECTROSTATICS $3.95 2π T ∂ ∂ΦB =− ∂t ∂t � s B · dA B · ds = µ0 Ienclosed B · ds = µ0 Ienclosed + µ0 ε0 ∂ ∂t � s E · dA GRAVITY m1 m2 r2 Newton’s Law of Universal Gravitation F =G Acceleration due to gravity a= Gravitational potential U (r) = − Escape velocity vescap e 20593 36340 ω = 2πf = 2π = f ω Angular frequency ω TM Period T Contributors: Bernell K Downer, Anna Medvedovsky Design: Dan O Williams Illustration: Dan O Williams, Matt Daniels Series Editors: Sarah Friedberg, Justin Kestler T = Wavelength λ Report errors at www.sparknotes.com/errors Frequency f Amplitude A GM Earth rEarth GM m r � GM = r KEPLER’S LAWS OF PLANETARY MOTION Planets revolve around the Sun in an elliptical path with the Sun at one focus The imaginary segment connecting the planet to the Sun sweeps out equal areas in equal time The square of the period of revolution is directly proportional to the cube of the length of the semimajor axis of revolution: T is constant a3 SPARKCHARTS™ Physics Formulas page of