• 107328  Infos

List of elementary physics formulae

    A list of elementary physics formulae commonly appearing in high-school and college introductory physics courses
    Where possible this list has avoided using any specific system of units Axioms are in green definitions are in blue and theorems are black

    SI Prefixes

    significandtimes 1000^n

    n SI prefix name etymology
    8 Y yotta- [1]octo
    7 Z zetta- septem
    6 E exa- hexa
    5 P peta- penta
    4 T tera- tetra
    3 G giga- giant
    2 M mega- great
    1 k kilo- thousand (Greek)
    -1 m milli- thousand (Latin)
    -2 µ micro- small
    -3 n nano- dwarf
    -4 p pico- small
    -5 f femto- (fifteen/3 = 5) (Danish)
    -6 a atto- (eighteen/3 = 6) (Danish)
    -7 z zepto- septem
    -8 y yocto- [2]octo

    Translation and rotation

    positionx theta
    durationDelta tDelta t
    displacementDelta xDelta theta
    conservation of massDelta m Delta m
    velocity v = dx/dt omega = dtheta/dt
    acceleration a = dv/dt alpha = domega/dt
    jerkj = da/dt j = dalpha/dt
    inertia{color{blue} m = int dm = Sigma m_i }{color{blue} I = int r^2 dm = Sigma r^2m_i }
    kinetic energyE_k = mv^2/2 E_k = Iomega^2/2
    momentumP = mv L = Iomega
    conservation of massDelta m = 0 Delta m = 0
    conservation of energyDelta E = 0 Delta E = 0
    conservation of momentumDelta P = 0 Delta L = 0
    force{color{blue}f = dP/dt } = ma = -dU/dx {color{blue}tau = dL/dt } = Ialpha = mathbf{r} times mathbf{f}
    impulse{color{blue}J=int f dt}{color{blue}J=int tau dt}
    work{color{blue}W = int f dx } = mathbf{f} cdot mathbf{d}{color{blue}W = int tau dtheta }
    power{color{blue} P = dW/dt } = fv {color{blue} P = dW/dt } = tauomega
    Newton's Third Law f_{ab} = - f_{ba} tau_{ab} = -tau_{ba}

    Equations for constant acceleration

    displacementDelta v = atDelta omega = alpha t
    timeDelta(v^2) = 2aDelta xDelta(omega^2) = 2alphaDelta theta
    accelerationDelta x = tDelta v/2Delta theta = tDelta omega/2
    initial velocityDelta x = -at^2/2 + v_2tDelta theta = -alpha t^2/2 + omega_2t
    final velocityDelta x = +at^2/2 + v_1tDelta theta = +alpha t^2/2 + omega_1t

    Common forces

    spring force lies parallel to springf_k = -kd
    weight points toward the centre of gravityf_g = -f_n = mg
    tension lies within the cordf_t = f
    static friction maxiumum lies tangent to the surfacef_{s}=mu_sf_n
    kinetic friction lies tangent to the surfacef_{k}=mu_kf_n
    drag force tangent to the pathf_{d} =mu_drho a v^2/2

    Mapping rotation to translation

    centripetal force points to the axis of rotationf_c = mv^2/r
    angular to linear displacementx = theta r
    angular to linear speedv = theta omega
    angular to linear acceleration tangential componenta_t = alpha r
    angular to linear acceleration radial componenta_r = omega^2r
    angular to linear acceleration centripetal accelerationalpha=v^2/r


    work done by a spring positive when relaxesW_s = -kDelta(x^2)/2
    inertial framesx_{PA} = x_{PB} + x_{AB}
    v_{PA} = v_{PB} + v_{AB}
    a_{PA} = a_{PB} + 0
    greatest distance1/8 rotations
    flight distancev_0^2sin{2theta}/g
    flight time?T=2pi r/v
    terminal velocityv_t=sqrt{2fg/(mu_drho A)}
    elastic potential energyu(x)=kx^2/2
    mechanical energy{color{blue} E_{mec}=K + U}
    work by type of energyW = Delta E_{mec} + Delta E_{th} + Delta E_{int}
    friction creates heatDelta E_{th} = f_kd
    center of mass COMmathbf{r}_{com}=M^{-1}Sigma m_i mathbf{r}_i
    x_{com}=M^{-1}int x dm, cdots
    for constant density:x_{com}=V^{-1}int x dV, cdots
    COM is in all planes of symmetry
    elastic collisionDelta E_k = 0
    inelastic collisionDelta E_k = maximum
    system COM remains inertmathbf{v}_{com}={(mathbf{P}_{1i}+mathbf{P}_{2i})over(M_1+M_2)} = const
    elastic collision 1D, M2 stationaryv_{1f}={(m_1 - m_2)over(m_1 + m_2)}v_{1i}
    v_{2f}={(2m_1)over(m_1 + m_2)}v_{1i}
    first rocket equationthrustt = Rv_{rel}=Ma
    second rocket equation-v_i + v_f = v_{rel}ln(M_i/M_f)
    parallel axis theoremI = I_{com} + Mh^2
    smooth rollingv_{com}=Romega
    down a ramp axis xa_{comx}=-frac{gsintheta}{I_{com}/MR^2}


    adiabaticDelta Q = 0
    degrees of freedomf
    linear expansiondL/dt = alpha L
    volume expansiondV/dt = 3 alpha V
    heatDelta E_Q due to Delta T
    heat capacity C_{th} = Delta T/Q
    specific heatc_{th} = mDelta T/Q
    heat of vaporizationL_v = Q/m
    heat of fusionL_f = Q/m
    work due to gas expansionW = int_{i}^{f}pdV
    adiabaticDelta E_{int} = W
    constant volumeDelta E_{int} = Q
    free expansionDelta E_{int} = 0
    closed cycleQ + W = 0
    thermal conductivitykappa
    thermal resistanceR=L/ kappa
    conduction rateP_{cond} = A(T_H - T_C)/(L/kappa) = Q/t
    steady state conduction through a composite slabP_{cond} = A(T_H - T_C)/Sigma(L/kappa)
    Stefan-Boltzmann constantsigma = 567*10^{-8}?
    thermal radiationP_{rad} = sigma epsilon A T ^4_{sys}
    thermal absorptionP_{rad} = sigma epsilon A T ^4_{env}
    Boltzmann Constantk = 138 times 10^{-23}J/K
    ideal gas lawPV = kTN
    work constant temperatureW=kTNln(V_f/V_i)
    work constant volumeW=0
    work constant pressureW=pDelta V
    translational energyE_{kavg} = kTf/2
    internal energyE_{int} = NkTf/2
    mean speedv_{avg}= sqrt{(kT/m)(8/pi)}
    mode speedv_{prb} = sqrt{(kT/m)2}
    root mean square speedv_{rms} = sqrt{(kT/m)3}
    mean free pathlambda = 1/( sqrt{2} pi d^2 N / V)?
    Maxwell's Distribution]P(v)=4pi(m/(2pi kT))^{3/2}V^2e^{-(mv^2/(2kT))}
    molecular specific heat at a constant volumeC_V = Q/(NDelta T)
    ?Delta E_{int} = NC_V Delta T
    molecular specific heat at a constant pressureC_p = Q/(NDelta T)
    ?W = p Delta V = Nk Delta T
    ?k = C_p - C_V
    adiabatic expansionpV^{gamma} = constant
    adiabatic expansionTV^{gamma - 1} = constant
    four special processes on page 529
    multiplicity of configurationsW = N!/n_1!n_2!
    microstate in one half of the boxn_1 n_2
    Boltzmann's equation]S = klnW
    entropy{color{blue}S = - ksum_i P_i ln P_i !}
    change in entropyDelta S = int_i^f(1/T)dQ approx Q/T_{avg}
    change in entropyDelta S = kNln(V_f/V_i) + NC_Vln(T_f/T_i)
    force due to entropyf = -Tds/dx
    engine efficiencyW|/|Q_H|
    Carnot engine efficiencyQ_H|-|Q_L|)/|Q_H| = (T_H-T_L)/T_H
    refrigerator performanceQ_L|/|W|
    Carnot refrigerator performanceQ_L|/(|Q_H|-|Q_L|) = T_L/(T_H-T_L)
    Zeroth Law of Thermodynamicstemperature is an equivalence relation
    First Law of ThermodynamicsDelta E_{int} = Q + W
    Second Law of ThermodynamicsDelta S ge 0
    Third Law of ThermodynamicsS = S_{structural} + CT


    torsion constantkappa = -tau / theta
    frequencyf = 1/T = omega /(2pi)
    angular frequencyomega = 2pi f = 2pi / T
    phase anglephi
    phase(omega t + phi)
    damping constantb
    damping forcef_d = -bv
    phaseky -omega t
    angular number]k
    phase constantphi
    linear densitymu
    harmonic numbern
    harmonic seriesf = v/lambda = nv/(2L)
    wavelengthlambda = k/(2pi)
    bulk modulusB = Delta p /(Delta V / V)
    path length differenceDelta L
    resonanceomega_d = omega
    phase differencephi = 2 pi Delta L / lambda
    fully constructive interferenceDelta L/lambda = n
    fully destructive interferenceDelta L/lambda = n+05
    intensityI = P/A = rho v omega^2 s^2_m/2
    source powerP_s
    sound intensity by distanceI = P_s/(4pi r^2)
    standard reference intensityI_0
    sound levelBeta = (10 dB)log(I/I_0)
    pipe two open endsf=v/lambda = nv/(2L)
    pipe one open endf = v/lambda = nv/(4L) for n odd
    beatss(t) = ' t cos omega t
    beat frequencyf_{beat} = f_1 - f_2
    Doppler effectf' = f(v+-v_D)/(v+-v_S)
    Mach cone anglesin theta = v/v_s
    average wave powerP_{avg}=mu v omega^2 x_m^2/2
    pressure amplitudeDelta p_m = (vrho omega )x_m
    wave equationfrac{partial y}{partial x^2} = frac{1}{v^2} frac{partial ^2 y}{partial t^2}
    wave superpositionx'(yt) = x_1(yt) + x_2(yt)
    wave speedv = omega/k = lambda/T = lambda f
    wave speed on a stretched stringv=sqrt{f_t/mu}
    angular frequency of a linear simple harmonic oscillatoromega = sqrt{k/m}
    angular frequency of an angular simple harmonic oscillatoromega = sqrt{I/kappa}
    angular frequency of a low amplitude simple pendulumomega = sqrt{L/g}
    angular frequency of a low amplitude physical pendulumomega = sqrt{I/mgh}
    speed of soundv = sqrt{B/ rho }
    damped angular frequencyomega ' = sqrt{(k/m)-(b^2/4m^2)}
    displacementx(t)=cos(omega t + phi)x_m
    damped displacementx(t)=cos(omega 't+phi)x_m(e^{-bt/2m})
    velocityv(t)=sin(omega t + phi)x_m ( - omega)
    accelerationa(t)=cos(omega t + phi)x_m( - omega^2 )
    transverse sinusoidal wavex(yt) = sin(ky-omega t)x_m
    wave travelling backwardsx(yt) = sin(ky+omega t)x_m
    resultant wavex'(yt) = sin(ky-omega t + phi/2)x_m(2cosphi/2)
    standing wavex'(yt) = cos(omega t)(2ysin ky)
    sound displacement functionx(yt) = cos(ky-omega t)x_m
    sound pressure-variation functionDelta p(yt) = sin(ky-omega t)Delta p_m
    potential harmonic energyE_U(t) = kx^2/2 = kx_m^2cos^2(omega t + phi)/2
    kinetic harmonic energyE_K(t) = kx^2/2 = kx_m^2sin^2(omega t + phi)/2
    total harmonic energyE(t) = kx_m^2/2 = E_U + E_K
    damped mechanical energyE_{mec}(t) = ke^{-bt/m}x^2_m/2


    gravityf_G = Gm_1m_2/r^2
    potential energyE_G = -Gm_1m_2/r
    potential energyE_g = ma_{g}y
    standard gravity{color{blue} a_g = Gm_{Earth}/r_{Earth}^2 } approx 981m/s^2
    Wg=mgdcos theta
    ?Delta k = W_{lift} + W_g
    potential workDelta u = -W
    understand path independenceW_{ab1}=W_{ab2}=cdots
    ?W_{pull} = - W_s


    electric constantepsilon_0 F/m
    electrostatic constantk = 1/(4piepsilon_0) m/F
    Coulomb's lawq_1/r^2
    elementary chargee in Coloumbs
    electric chargeq = ne for integer n
    conservation of chargeDelta e = 0
    test chargeq_0
    electric fieldmathbf{E} =mathbf{F}/q_0
    electric field of a point chargemathbf{E}q =mathbf{F}
    electric field linesend at a negative charge
    r hat?hat{r}
    electric field of a point chargemathbf{E} = mathbf{F}/q_0 = (1/(epsilon_04pi))q/r^2hat{r}
    electric moment]mathbf{p}
    electric field of a moment]E = (1/(epsilon_0 2pi))(p/z^3)
    electric field of a charged ringE = qz/(epsilon_04pi(z^2 + R^2)^(3/2)
    surface charge densitysigma = Q/A
    electric field of a charged diskE = (sigma/epsilon_0 2)(1 - z/sqrt{z^2 + R^2})
    torque on a dipolemathbf{tau}=mathbf{p}timesmathbf{E}
    potential of a dipole]U = -mathbf{p}cdotmathbf{E}
    work done on a dipoleW_a=-W=Delta U
    Gaussian surface?
    electric fluxPhi = ointmathbf{E}cdot d mathbf{A}
    Gauss' lawq_{enc} = epsilon_0Phi
    Gauss' lawq_{enc} = epsilon_0ointmathbf{E}cdot d mathbf{A}
    volume charge densityrho = Q/V
    linear charge densitylambda = Q/L
    acceleration due to chargea = qE/m
    conducting surfaceE = sigma / epsilon_0
    electric field of a line of chargeE = lambda / epsilon_0 2pi r
    non-conducting sheet of chargeE = sigma /epsilon_0 2
    electric field outside spherical shell r>=RE = q/epsilon_0 4 pi r^2
    electric field inside spherical shell rE = 0
    electric field of uniform charge r<=RE = qr/epsilon_0 4 pi R^3
    electric potential energyU
    Work done by electric potential energyDelta U=-W
    finite energy of the system]U - W_infty
    electric potentialV = U/q
    electric potential differenceDelta V = Delta U/q = -W/q
    potential definedV = -W_{infty}/q
    potential from the electric fieldDelta V = -int_i^f mathbf{E}cdot dmathbf{s}
    potential of a point chargeV = q/epsilon_0 4 pi r
    potential of a charge group]V = Sigma V_i = (1/epsilon_0 4 pi) Sigma q_i/r_i
    potential of a dipoleV = pcostheta/epsilon_0 4 pi r^2
    potential of continuous chargeV = int dV = (1/epsilon_0 4 pi)int dq/r
    field from potentialmathbf{E} = nabla V
    potential of a pair of point chargesU = W = q_2V = q_1q_2/epsilon_04pi r
    capacitanceC = q/V
    parallel plate capacitorC = epsilon_0A/d
    cylindrical capacitorC = epsilon_0 2 pi L/ln(b/a)
    spherical capacitorC = epsilon_0 4 pi ba/(b-a)
    isolated spherical capacitorC = epsilon_0 4 pi R
    parallel capacitorsC_{eq} = Sigma C_i
    series capacitors1/C_{eq} = Sigma 1/C_i
    potential stored in a capacitor]U=q^2/2C = CV^2/2
    energy densityu = epsilon_0 E^2/2
    dielectric constantkappa ge 1
    dimensions of capacitanceC = epsilon_0mathcal{L}
    dielectricepsilon_0 to epsilon_0kappa
    Gauss' law with dialectricq = epsilon_0 oint kappamathbf{E}cdot d mathbf{A}
    electric displacementmathbf{D} = epsilon_0 kappa mathbf{E}
    currenti = dq/dt
    charge densitymathbf{J} = i/A
    ?i = int JdA
    drift speedmathbf{v}_d
    ?mathbf{J} = nemathbf{v}_d/m^3
    resistanceR = V/i
    resistivity in ohm-metersrho = mathbf{E}/mathbf{J}
    conductivitysigma = mathbf{J}/mathbf{E} = 1/rho
    ?R/rho = L/A
    variation of with temperature]rho - rho_0 = rho_0alpha(T-T_0)
    temperature of resistivity]alpha
    Ohm's lawV=iR
    electrical powerP=iV
    resistive dissipationP = i^2R = V^2/R
    emfmathcal{E} = dW/dq = iR
    rules for calculating emfloop resistance emf
    internal resistancei = mathcal{E}/(R+r)
    resistors in seriesR_{eq}=Sigma R_i
    resistors in parallel1/R_{eq} = 1/ Sigma R_i
    potential difference across a real batteryp = mathcal{E} - iR
    power of an emf deviceP_{emf} = imathcal{E}
    Kirchoff's junction rulei_{in} = i_{out}
    RC charging a capacitorq = Cmathcal{E}(1-e^{-t/RC}
    RC charging a capacitori = (mathcal{E}/R)e^{-t/RC}
    RC charging a capcitorV_C = mathcal{E}(1-e^{-t/RC}
    capacitive constant]tau = RC
    magnetic fieldmathbf{F}_B = qmathbf{v}timesmathbf{B}
    Hall effectn = Bi/Vle
    circulating charged particleq|vB=mv^2/r
    cyclotron resonance conditionf = f_{osc}
    force on a currentmathbf{F}_B=imathbf{L}timesmathbf{B}
    magnetic momentmu=NiA
    magnetic torque]mathbf{tau}=mathbf{mu}timesmathbf{B}
    magnetic potential energyU(theta)=-mathbf{mu}cdotmathbf{B}
    magnetism constantmu_0 in Tm/A
    Biot-Savart lawdmathbf{B} = (mu_0/4pi)(idmathbf{s}timeshat{r}/r^2)
    magnetic field due to a long straight wireB = mu_0i/2pi R
    magnetic field due to a semi-infinite straight wireB=mu_0i/4pi R
    magnetic field at the center of a circular arcB=mu_0iphi/4pi R
    Ampere's lawoint mathbf{B}cdot dmathbf{s} = mu_0i_{enc}
    ideal solenoidB=mu_0in
    toroidB=mu_0iN/2pi r
    current carrying coilmathbf{B}=mu_0mathbf{mu}/2pi z^3
    magnetic flux through APhi _B = int mathbf{B}cdot dmathbf{A}
    ?Phi_B = BA
    Faraday's lawmathcal{E}=dPhi_B/dt
    Lenz's law?
    Faraday's lawointmathbf{E}cdot dmathbf{s} = -dPhi_B/dt
    self-induced emfmathcal{E}_L = -Ldi/dt
    RL circuitLdi/dt+Ri=mathcal{E}
    RL circuit rise of currenti = mathcal{E}/R(1-e^{-t/tau_L})
    RL circuit constant]tau_L=L/R
    RL circuit decay of currenti=mathcal{E}e^{-t/tau_L}/R=i_0e^{-t/tau_L}
    magnetic energyU_B=Li^2/2
    magnetic density]u_B=B^2/2mu_0
    mutual inductionmathcal{E}_1=-Mdi_2/dtmathcal{E}_2=-Mdi_1/dt
    LC circuitomega = 1/sqrt{LC}
    LC oscillationsLd^2q/dt^2+q/C = 0
    LC chargeq = Qcos(omega t + phi)
    LC currenti=-omega Q sin(omega t + phi)
    LC electrical energyU_E=q^2/2C=Q^2cos^2(omega t + phi)/2C
    U_B=Q^2sin^2(omega t + phi)/2C
    RLC circuit ODELd^2/q/dt^2 + Rdq/dt +q/C = 0
    RLC circuit ODE solutionq = QeT^{-Rt/2L}cos(omega't+phi)
    resistive loadV_R=I_RR
    capacitive reactanceX_C = 1/omega_d C
    capacitive loadV_C = I_C X_C
    inductive reactanceX_L = omega_d L
    inductive loadV_L = I_L X_L
    phase constanttanphi=X_L - X_C /R
    electromagnetic resonanceomega_d = omega = 1/sqrt{LC}
    rms currentI_{rms}=I/sqrt{2}
    rms voltageV_{rms}=V/sqrt{2}
    rms emfmathcal{E}_{rms}=mathcal{E}_m/sqrt{2}
    average powerP_{avg}=mathcal{E}I_{rms}cosphi
    transformation of voltageV_s N_p = V_p N_s
    transformation of currentsI_s N_s = I_p N_p
    transformer reistanceR_eq = (Np/Ns)^2R
    Gauss' law for magnetic fieldsPhi_B = oint mathbf{B} cdot d mathbf{A} = 0
    Maxwell's law of inductionoint mathbf{B} /cdot d mathbf{s} = mu_0epsilon_0dPhi_E/dt
    Ampere-Maxwell lawoint mathbf{B} cdot d mathbf{s} = mu_0epsilon_0dPhi_E/dt + mu_0i_{enc}
    displacement currenti_d = epsilon_0 dPsi_E/dt
    Ampere-Maxwell lawoint mathbf{B} cdot dmathbf{s} = mu_0i_{denc} + mu_0i_{enc}
    induced field inside a circular capacitor]B = (mu_0i_d/2pi R^2)r
    induced field outside a circular capacitor]B = mu_0i_d/2pi rr
    spin magnetic moment]mathbf{mu_s} = -emathbf{S}/m
    Bohr magnetonmu_B = eh/4pi m
    ?U = -mathbf{mu}_scdotmathbf{B}_{ext} = -mu_{sz}B_{ext}
    orbital magnetic moment]mathbf{mu}_{orb}=-emathbf{L}_orb/2m
    ?U = -mathbf{mu}_{orb}cdotmathbf{B}_{ext} = -mu_{orbz}B_{ext}


    electric component]E = E_m sin(kx-omega t)
    magnetic component]B = B_m sin(kx-omega t)
    celerityc = 1/sqrt{mu_0epsilon_0}
    magnitude ratioE/B = c
    Poynting vectormathbf{S} = mu_0^{-1}mathbf{E}timesmathbf{B}
    instantaneous flow rate]S = E^2/cmu_0
    intensityI = E^2_{rms}/cmu_0
    intensity at the sphereI = P_s/4pi r^2
    total absorptionDelta p = Delta U/c
    total reflection back along pathDelta p = 2 Delta U/c
    total absorptionp_r = Ic
    total reflection back along pathp_r = 2I/c
    one-half ruleI = I_0/2
    cosine-squared ruleI = I_0cos^2theta
    law of reflectiontheta_1'=theta_1
    law of refraction (snell's)n_1sintheta_1 = n_2sintheta_2
    critical angletheta_c = sin^{-1}n_2/n_1
    Brewster angletheta_B = tan^{-1}n_2/n_1
    plane mirrori = -p
    spherical mirrorf =r/2
    spherical mirror1/p + 1/i = 1/f
    lateral magnificationm|=h'/h
    lateral magnificationm=-i/p
    ?n_1/p + n_2/i = (n_2 - n_1)/r
    thin lens1/f = 1/p +1/i
    thin lens in air1/f = (n-1)(1/r_1 - 1/r_2)
    index of refractionn=c/v
    path length differenceDelta L = d sintheta
    maxima - bright fringesd sintheta = mlambda for m natural
    minima - dark fringesd sintheta = (m + 1/2)lambda for m natural
    double-slit interference intensityI = 4I_0cos^2(theta/2)
    ?theta = 2pi d sintheta / lambda
    bright film in air maxima2L = (m + 1/2)lambda/n_2
    bright film in air minima(m)lambda/n_2
    single-slit minimaa sin theta = mlambda
    single-slit intensityI(theta)=I_m(sinalpha/alpha)^2
    ?alpha = phi/2 = pi a sintheta/lambda
    circular aperture first minimumsintheta = 122lambda/d
    Rayleigh's criteriontheta_R = 122lambda/d
    double slit intensityI(theta) = I_m(cos^2Beta)(sinalpha/alpha)^2
    diffraction grating maxima linesdsintheta = mlambda for natural m
    diffraction grating half-widthDeltatheta_{hw} = lambda/Ndcostheta
    diffraction grating dispersionD=m/d costheta
    diffraction grating resolving powerR=Nm
    Bragg's law2dsintheta = mlambda


    The Relativity Postulate
    The of Light Postulate]
    Lorentz factorgamma = 1/sqrt{1-(v/c)^2}
    speed parameterBeta = v/c
    time dilationDelta t = gamma Delta t_0
    length contractionL = L_0/gamma
    Lorentz transformationx'=gamma(x-vt)
    y' = y
    z' = z
    t' = gamma(t-xv/c^2)
    Doppler effect for lightf=f_0sqrt{1-Beta/1+Beta}
    wavelength Doppler shiftDeltalambda|c/lambda_0
    momentummathbf{p}=gamma mmathbf{v}
    rest energyE_0 = mc^2
    total energyE = E_0 + K = mc^2 + K = gamma mc^2
    Q = -Delta mc^2
    kinetic energyK = E - mc^2 = gamma mc^2 - mc^2 = mc^2(gamma -1)
    momentum and energy triangle]E = sqrt{(pc)^2 + (mc^2)^2}


    quantum constanth in energy/frequency
    photon energyE = hf
    photoelectric equationhf = K_{max} + Phi
    photon momentump = hf/c = h/lambda
    de Broglie wavelengthlambda = h/p
    Schrodinger's equation one dimensional motiond^2psi/dx^2 + 8pi^2m[3]psi/h^2 = 0
    Schrodinger's equation free particled^2psi/dx^2 + k^2psi = 0
    Heisenberg's principle]Delta x cdot Delta p_x ge hbar
    infinite well]E_n = h^2n^2/8mL^2
    energy changesDelta E = hf
    wave function of a trapped electronpsi_n(x) = A sin(npi x/L) for positive int n
    probability of detectionp(x) = psi^2_n(x)dx
    normalizationint psi^2_n(x)dx = 1
    hydrogen orbital energyE_n = 1361eV/n^2 for positive int n
    hydrogen changes]1/lambda = R(1/n^2_{low} - 1/n^2_{high}
    hydrogen radial density]P(r) = 4r^2/a^3e^{2r/a}
    spin magnetic quantum numberm_s in {-1/2+1/2}
    orbital magnetic moment]mathbf{mu}_{orb} = -emathbf{L}/2m
    orbital magnetic moment components]mathbf{mu}_{orbz} = -m_mathcal{L}mu_B
    Bohr magnetonmu_B = ehbar/2m
    angular components]L_z = mmathcal{L}hbar
    spin momentum magnitude]S = hbarsqrt{s(s+1)}
    cutoff wavelengthlambda_{min} = hc/K_0
    density of statesN(E) = 8sqrt{2}pi m^{3/2}E^{1/2}/h^3
    occupancy probabilityP(E) = 1/(e^{(E-E_F)/kT}+1)
    Fermi energyE_F = (3/16sqrt{2}pi)^{2/3}h^2n^{2/3}m
    mass numberA = Z+N
    nuclear radiir=r_0A^{1/3}
    mass excessDelta = M - A
    radioactive decayN = N_0e^{-lambda t}
    conservation of number]
    conservation of number]
    conservation of strangeness
    the eightfold way
    weak force
    strong force
    Hubble constantH = 710km/s
    Hubble's lawv=Hr

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