College Physics Formula Sheet by twistr9

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									                   Reference Guide & Formula Sheet for Physics
         Dr. Hoselton & Mr. Price                                                                                   Page 1 of 8
                                                                               #20       Heating a Solid, Liquid or Gas
#3         Components of a Vector                                                                   Q = m•c•∆T (no phase changes!)
                if   V = 34 m/sec ∠48°                                                                              Q = the heat added
                                    then                                                                              c = specific heat.
     Vi = 34 m/sec•(cos 48°); and VJ = 34 m/sec•(sin 48°)                                                   ∆T = temperature change, K
                                                                               #21       Linear Momentum
#4        Weight = m•g                                                                   momentum = p = m•v = mass • velocity
         g = 9.81m/sec² near the surface of the Earth                                            momentum is conserved in collisions
          = 9.795 m/sec² in Fort Worth, TX
                                                                               #23       Center of Mass – point masses on a line
           Density = mass / volume                                                                xcm = Σ(mx) / Mtotal

                     ρ=
                            m
                            V
                                (
                              unit : kg / m 3       )                          #25       Angular Speed vs. Linear Speed
                                                                                             Linear speed = v = r•ω = r • angular speed
#7       Ave speed = distance / time = v = d/t
       Ave velocity = displacement / time = v = d/t                            #26       Pressure under Water
       Ave acceleration = change in velocity / time                                                   P = ρ•g•h
                                                                                                                       h = depth of water
#8         Friction Force                                                                                            ρ = density of water
                         FF = µ•FN                                             #28       Universal Gravitation
       If the object is not moving, you are dealing with static                                             m1 m2
          friction and it can have any value from zero up to µs FN                                  F =G
       If the object is sliding, then you are dealing with kinetic
                                                                                                             r2
          friction and it will be constant and equal to µK FN                                                     G = 6.67 E-11 N m² / kg²

#9         Torque                                                              #29       Mechanical Energy
                                                                                                PEGrav = P = m•g•h
                       τ = F•L•sin θ                                                            KELinear = K = ½•m•v²
           Where θ is the angle between F and L; unit: Nm
                                                                               #30       Impulse = Change in Momentum
#11        Newton's Second Law                                                                      F•∆t = ∆(m•v)
                    Fnet = ΣFExt = m•a
                                                                               #31       Snell's Law
#12        Work = F•D•cos θ                                                                       n1•sin θ1 = n2•sin θ2
                     Where D is the distance moved and                                   Index of Refraction
                        θ is the angle between F and the                                                n=c/v
                                     direction of motion,                                                 c = speed of light = 3 E+8 m/s
                                                  unit : J
                                                                               #32       Ideal Gas Law
#16        Power = rate of work done                                                                 P•V = n•R•T
                                           Work                                                                     n = # of moles of gas
                         Power =                             unit : watt                                            R = gas law constant
                                           time                                                                        = 8.31 J / K mole.
           Efficiency = Workout / Energyin                                     #34       Periodic Waves
      Mechanical Advantage = force out / force in                                                     v = f •λ
                   M.A. = Fout / Fin                                                                   f=1/T         T = period of wave

#19        Constant-Acceleration Linear Motion                                 #35       Constant-Acceleration Circular Motion
                      v = vο + a•t             x                                                  ω = ωο + α•t           θ
                  (x-xο) = vο•t + ½•a•t²       v                                                θ−θο= ωο•t + ½•α•t²      ω
                                                                                                   2     2
                     v ² = vο² + 2•a• (x - xο) t                                                  ω = ωο + 2•α•(θ−θο) t
                  (x-xο) = ½•( vο + v) •t      a                                                θ−θο = ½•(ωο + ω)•t      α
                  (x-xο) = v•t - ½•a•t²        vο                                               θ−θο = ω•t - ½•α•t²      ωο

                                                                     Version 5/12/2005
                Reference Guide & Formula Sheet for Physics
        Dr. Hoselton & Mr. Price                                                                             Page 2 of 8
                                                                #53         Resistor Combinations
#36       Buoyant Force - Buoyancy                                                       SERIES
      FB = ρ•V•g = mDisplaced fluid•g = weightDisplaced fluid                      Req = R1 + R2+ R3+. . .
                              ρ = density of the fluid                                 PARALLEL
                                                                                                                n
                             V = volume of fluid displaced                      1     1   1       1                   1
                                                                               R eq
                                                                                    =   +
                                                                                      R1 R 2
                                                                                             +K +
                                                                                                  Rn
                                                                                                     =         ∑R
                                                                                                               i =1   i
#37       Ohm's Law
                         V = I•R                                #54         Newton's Second Law and
                   V = voltage applied                                           Rotational Inertia
                   I = current                                                          τ = torque = I•α
                   R = resistance                                         I = moment of inertia = m•r² (for a point mass)
                                                                          (See table in Lesson 58 for I of 3D shapes.)
          Resistance of a Wire
                       R = ρ•L / Ax                             #55         Circular Unbanked Tracks
                 ρ = resistivity of wire material
                                                                                          mv 2
                 L = length of the wire                                                        = µmg
                Ax = cross-sectional area of the wire                                      r
                                                                #56         Continuity of Fluid Flow
#39       Heat of a Phase Change                                                       Ain•vin = Aout•vout                   A= Area
                    Q = m•L                                                                                               v = velocity
                    L = Latent Heat of phase change             #58         Moment of Inertia       -       I
                                                                             cylindrical hoop              m•r2
#41       Hooke's Law                                                        solid cylinder or disk    ½ m•r2
                                                                                                       2
                        F = k•x                                              solid sphere                /5 m•r2
          Potential Energy of a spring                                       hollow sphere             ⅔ m•r2
                                                                                                    1
           W = ½•k•x² = Work done on spring                                  thin rod (center)        /12 m•L2
                                                                             thin rod (end)            ⅓ m•L2
#42       Electric Power
                   P = I²•R = V ² / R = I•V                     #59         Capacitors      Q = C•V
                                                                                 Q = charge on the capacitor
#44       Speed of a Wave on a String                                            C = capacitance of the capacitor
                                 mv 2                                            V = voltage applied to the capacitor
                         T=                                                 RC Circuits (Discharging)
                                  L                                                                 − t/RC
                                                                                         Vc = Vo•e
                                        T = tension in string
                                                                                          Vc − I•R = 0
                                        m = mass of string
                                        L = length of string
                                                                #60         Thermal Expansion
#45       Projectile Motion
                                                                                 Linear: ∆L = Lo•α•∆T
                   Horizontal: x-xο= vο•t + 0
                   Vertical: y-yο = vο•t + ½•a•t²                                Volume: ∆V = Vo•β•∆T

#46       Centripetal Force                                     #61         Bernoulli's Equation
                         mv  2                                                  P + ρ•g•h + ½•ρ•v ² = constant
                   F=       = mω 2 r                                       QVolume Flow Rate = A1•v1 = A2•v2 = constant
                          r
                                                                #62         Rotational Kinetic Energy (See LEM, pg 8)
#47       Kirchhoff’s Laws                                                                            2
                                                                              KErotational = ½•I•ω = ½•I• (v / r)2
           Loop Rule: ΣAround any loop ∆Vi = 0                                                                      2
                                                                             KErolling w/o slipping = ½•m•v2 + ½•I•ω
           Node Rule: Σat any node Ii = 0

#51       Minimum Speed at the top of a                               Angular Momentum = L = I•ω = m•v•r•sin θ
                Vertical Circular Loop                                  Angular Impulse equals
                                                                        CHANGE IN Angular Momentum
                         v = rg                                                 ∆L = τorque•∆t = ∆(I•ω)

                                                      Version 5/12/2005
             Reference Guide & Formula Sheet for Physics
      Dr. Hoselton & Mr. Price                                                                                     Page 3 of 8
                                                                    #75        Thin Lens Equation
#63    Period of Simple Harmonic Motion                                                                                       f = focal length
            T = 2π
                      m           where k = spring constant                         1   1    1  1 1  i = image distance
                                                                                      =    +   = +
                      k                                                             f   D o D i o i o = object distance
                                           f = 1 / T = 1 / period
#64    Banked Circular Tracks                                                             Magnification
                  v2 = r•g•tan θ                                                    M = −Di / Do = −i / o = Hi / Ho

#66    First Law of Thermodynamics                                  Helpful reminders for mirrors and lenses
                 ∆U = QNet + WNet                                   Focal Length of:      positive                 negative
       Change in Internal Energy of a system =                        mirror              concave                  convex
                          +Net Heat added to the system              lens               converging                                    diverging
                          +Net Work done on the system              Object distance = o all objects
                                                                    Object height = Ho    all objects
       Flow of Heat through a Solid                                 Image distance = i    real                     virtual
               ∆Q / ∆t = k•A•∆T / L                                 Image height = Hi     virtual, upright         real, inverted
                             k = thermal conductivity               Magnification         virtual, upright         real, inverted
                            A = area of solid
                            L = thickness of solid                  #76        Coulomb's Law
                                                                                                          q1 q 2
#68    Potential Energy stored in a Capacitor                                                    F =k
                   P = ½•C•V²                                                                                r2
                                                                                                 1   N ⋅m2
                                                                                         k=             = 9E9
       RC Circuit formula (Charging)                                                     4πε o         C2
                                  − t / RC
             Vc = Vcell•(1 − e             )                        #77        Capacitor Combinations
                                  R•C = τ = time constant                                 PARALLEL
             Vcell - Vcapacitor − I•R = 0                                            Ceq = C1 + C2+ C3 + …
                                                                                            SERIES
                                                                                                                        n
                                                                                     1     1   1       1                      1
#71    Simple Pendulum                                                              C eq
                                                                                         =   +
                                                                                           C1 C 2
                                                                                                  +K +
                                                                                                       Cn
                                                                                                          =           ∑C
                                                                                                                       i =1       i
                   L and f = 1/ T
           T = 2π
                   g                                                #78        Work done on a gas or by a gas
                                                                                          W = P•∆V
#72    Sinusoidal motion
          x = A•cos(ω•t) = A•cos(2•π•f •t)                          #80        Electric Field around a point charge
                               ω = angular frequency                                                         q
                                                                                                     E=k
                                         f = frequency                                                       r2
#73    Doppler Effect                                                                            1    N ⋅m2
                                                                                         k =            = 9E9
                          343 ±   Toward
                                           vo                                           4πε o          C2
                 f′= f
                                  Away

                          343 m   Toward
                                  Away     vs                       #82        Magnetic Field around a wire
       vo = velocity of observer: vs = velocity of source                                        µ I
                                                                                             B= o
                                                                                                 2π r
#74    2nd Law of Thermodynamics                                                         Magnetic Flux
       The change in internal energy of a system is                                      Φ = B•A•cos θ
       ∆U = QAdded + WDone On – Qlost – WDone By
                                                                               Force caused by a magnetic field
       Maximum Efficiency of a Heat Engine                                             on a moving charge
       (Carnot Cycle) (Temperatures in Kelvin)                                           F = q•v•B•sin θ
                             Tc
             % Eff = (1 −       ) ⋅100%                             #83     Entropy change at constant T
                             Th                                                           ∆S = Q / T
                                                                    (Phase changes only: melting, boiling, freezing, etc)

                                                         Version 5/12/2005
              Reference Guide & Formula Sheet for Physics
      Dr. Hoselton & Mr. Price                                                                        Page 4 of 8
                                                                #95      Relativistic Time Dilation
#84     Capacitance of a Capacitor                                                     ∆t = ∆to / β
                  C = κ•εo•A / d
                    κ = dielectric constant                     #96      Relativistic Length Contraction
                    A = area of plates                                                 ∆x = β•∆xo
                     d = distance between plates
                    εo = 8.85 E(-12) F/m                                 Relativistic Mass Increase
                                                                                       m = mo / β
#85     Induced Voltage                 N = # of loops
                              ∆Φ                                #97       Energy of a Photon or a Particle
                    Emf = N                                                            E = h•f = m•c2
                               ∆t
Lenz’s Law – induced current flows to create a B-field                   h = Planck's constant = 6.63 E(-34) J sec
opposing the change in magnetic flux.                                           f = frequency of the photon

#86     Inductors during an increase in current                 #98      Radioactive Decay Rate Law
                                                                                   −kt
                 VL = Vcell•e
                             − t / (L / R)                                 A = Ao•e    = (1/2n)•A0 (after n half-lives)
                                                                               Where k = (ln 2) / half-life
                                   − t / (L / R)
             I = (Vcell/R)•[ 1 - e            ]                 #99      Blackbody Radiation and
                                    L / R = τ = time constant                    the Photoelectric Effect
#88     Transformers                                                     E= n•h•f where h = Planck's constant
                     N 1 / N 2 = V 1 / V2
                        I1•V1 = I2•V2                           #100     Early Quantum Physics
#89     Decibel Scale                                                    Rutherford-Bohr Hydrogen-like Atoms
        B (Decibel level of sound) = 10 log ( I / Io )
            I = intensity of sound                                             1        1   1 
            Io = intensity of softest audible sound
                                                                                 = R ⋅  2 − 2 meters −1
                                                                                       n      
                                                                               λ        s n 
                                                                                            or
#92     Poiseuille's Law
                 ∆P = 8•η•L•Q/(π•r )
                                      4
                                                                                   c       1   1 
                                                                             f =     = cR  2 − 2  Hz
                                                                                          n      
                         η = coefficient of viscosity                              λ       s n 
                         L = length of pipe                                     R = Rydberg's Constant
                         r = radius of pipe                                        = 1.097373143 E7 m-1
                        Q = flow rate of fluid                                  ns = series integer (2 = Balmer)
        Stress and Strain                                                       n = an integer > ns
             Y or S or B = stress / strain
                      stress = F/A                                       Mass-Energy Equivalence
        Three kinds of strain: unit-less ratios                                        mv = mo / β
                   I. Linear: strain = ∆L / L                             Total Energy = KE + moc2 = moc2 / β
                  II. Shear: strain = ∆x / L                             Usually written simply as   E = m c2
                 III. Volume: strain = ∆V / V
                                                                          de Broglie Matter Waves
#93     Postulates of Special Relativity                                 For light:      Ep = h•f = h•c / λ = p•c
        1. Absolute, uniform motion cannot be
           detected.                                                         Therefore, momentum: p = h / λ
        2. No energy or mass transfer can occur                           Similarly for particles, p = m•v = h / λ,
           at speeds faster than the speed of light.                     so the matter wave's wavelength must be
                                                                                        λ=h/mv
#94     Lorentz Transformation Factor                                     Energy Released by Nuclear
                           v2                                             Fission or Fusion Reaction
                   β = 1− 2                                                             E = ∆mo•c2
                           c

                                                     Version 5/12/2005
              Reference Guide & Formula Sheet for Physics
    Dr. Hoselton & Mr. Price                                                                 Page 5 of 8

MISCELLANEOUS FORMULAS                                 Fundamental SI Units
                                                       Unit        Base Unit                Symbol
                 Quadratic Formula                                        …………………….
                 if a x² + b x + c = 0                 Length             meter  m
                          then
                                                       Mass               kilogram          kg
                    − b ± b − 4ac2
                                                       Time               second            s
               x=
                          2a                           Electric
                                                         Current          ampere            A
           Trigonometric Definitions                   Thermodynamic
     sin θ = opposite / hypotenuse                       Temperature      kelvin            K
     cos θ = adjacent / hypotenuse                     Luminous
     tan θ = opposite / adjacent                         Intensity        candela           cd
                                                       Quantity of
             sec θ = 1 / cos θ = hyp / adj               Substance        moles             mol
             csc θ = 1 / sin θ = hyp / opp
             cot θ = 1 / tan θ = adj / opp             Plane Angle        radian            rad

       Inverse Trigonometric Definitions               Solid Angle        steradian         sr or str
               θ = sin-1 (opp / hyp)
               θ = cos-1 (adj / hyp)
               θ = tan-1 (opp / adj)                   Some Derived SI Units
                                                       Symbol/Unit Quantity                 Base Units
                  Law of Sines                                            …………………….
         a / sin A = b / sin B = c / sin C             C coulomb          Electric Charge A•s
                         or
         sin A / a = sin B / b = sin C / c             F farad            Capacitance       A2•s4/(kg•m2)

                 Law of Cosines                        H henry            Inductance        kg•m2/(A2•s2)
         2      2
        a = b + c2 - 2 b c cos A
        b2 = c2 + a2 - 2 c a cos B                     Hz hertz           Frequency         s-1
        c² = a² + b² - 2 a b cos C
                                                       J   joule          Energy & Work kg•m2/s2 = N•m
                     T-Pots
          For the functional form                      N newton           Force             kg•m/s2
                     1 1 1
                      = +                              Ω ohm              Elec Resistance kg•m2/(A2•s2)
                     A B C
                                                       Pa pascal          Pressure          kg/(m•s2)
  You may use "The Product over the Sum" rule.
                           B ⋅C                        T tesla            Magnetic Field    kg/(A•s2)
                     A=
                           B+C
                                                       V volt             Elec Potential    kg•m2/(A•s3)
        For the Alternate Functional form
                     1 1 1                             W watt             Power             kg•m2/s3
                      = −
                     A B C                             Non-SI Units
                                                       o
                                                        C degrees Celsius            Temperature
             You may substitute T-Pot-d
                    B ⋅C    B ⋅C                       eV electron-volt              Energy, Work
               A=        =−
                    C−B     B−C

                                             Version 5/12/2005
              Reference Guide & Formula Sheet for Physics
       Dr. Hoselton & Mr. Price                                                                Page 6 of 8
                                                           Αα Alpha angular acceleration, coefficient of
Aa acceleration, Area, Ax=Cross-sectional Area,                   linear expansion,
         Amperes, Amplitude of a Wave, Angle,              Ββ Beta coefficient of volume expansion,
Bb Magnetic Field, Decibel Level of Sound,                        Lorentz transformation factor,
         Angle,                                            Χχ Chi
Cc specific heat, speed of light, Capacitance,
         Angle, Coulombs, oCelsius, Celsius
         Degrees, candela,                                 ∆δ Delta ∆=change in a variable,
Dd displacement, differential change in a variable,
         Distance, Distance Moved, distance,               Εε Epsilon εο = permittivity of free space,
Ee base of the natural logarithms, charge on the
         electron, Energy,                                 Φφ Phi Magnetic Flux, angle,
Ff Force, frequency of a wave or periodic motion,
         Farads,                                           Γγ Gamma surface tension = F / L,
Gg Universal Gravitational Constant, acceleration                 1 / γ = Lorentz transformation factor,
         due to gravity, Gauss, grams, Giga-,              Ηη Eta
Hh depth of a fluid, height, vertical distance,
         Henrys, Hz=Hertz,
                                                           Ιι Iota
Ii Current, Moment of Inertia, image distance,
         Intensity of Sound,
                                                           ϑϕ Theta and Phi lower case alternates.
Jj Joules,
Kk K or KE = Kinetic Energy, force constant of             Κκ Kappa dielectric constant,
         a spring, thermal conductivity, coulomb's
         law constant, kg=kilograms, Kelvins,
         kilo-, rate constant for Radioactive
         decay =1/τ=ln2 / half-life,
                                                           Λλ Lambda wavelength of a wave, rate constant
Ll Length, Length of a wire, Latent Heat of
         Fusion or Vaporization, Angular                         for Radioactive decay =1/τ=ln2/half-life,
         Momentum, Thickness, Inductance,
                                                           Μµ Mu friction, µo = permeability of free space,
Mm mass, Total Mass, meters, milli-, Mega-,
                                                                 micro-,
         mo=rest mass, mol=moles,
Nn index of refraction, moles of a gas, Newtons,           Νν Nu alternate symbol for frequency,
         Number of Loops, nano-,
Oo                                                         Οο Omicron
Pp Power, Pressure of a Gas or Fluid, Potential            Ππ Pi 3.1425926536…,
         Energy, momentum, Power, Pa=Pascal,
Qq Heat gained or lost, Maximum Charge on a                Θθ Theta angle between two vectors,
         Capacitor, object distance, Flow Rate,
Rr radius, Ideal Gas Law Constant, Resistance,             Ρρ Rho density of a solid or liquid, resistivity,
         magnitude or length of a vector,
         rad=radians
Ss speed, seconds, Entropy, length along an arc,           Σσ Sigma Summation, standard deviation,
Tt time, Temperature, Period of a Wave, Tension,           Ττ Tau torque, time constant for a exponential
         Teslas, t1/2=half-life,                                  processes; eg τ=RC or τ=L/R or τ=1/k=1/λ,
Uu Potential Energy, Internal Energy,                      Υυ Upsilon
Vv velocity, Velocity, Volume of a Gas, velocity of        ςϖ Zeta and Omega lower case alternates
wave, Volume of Fluid Displaced, Voltage, Volts,           Ωω Omega angular speed or angular velocity,
Ww weight, Work, Watts, Wb=Weber,                                 Ohms
Xx distance, horizontal distance, x-coordinate             Ξξ Xi
         east-and-west coordinate,
Yy vertical distance, y-coordinate,                        Ψψ Psi
         north-and-south coordinate,
Zz z-coordinate, up-and-down coordinate,                   Ζζ Zeta


                                                 Version 5/12/2005
                 Reference Guide & Formula Sheet for Physics
      Dr. Hoselton & Mr. Price                                                                   Page 7 of 8
      Values of Trigonometric Functions
           for 1st Quadrant Angles                                                    Prefixes
          (simple mostly-rational approximations)
     θ           sin θ           cos θ         tan θ                       Factor Prefix Symbol Example
      o
    0           0            1             0
    10o        1/6        65/66          11/65                             1018    exa-     E    38 Es (Age of
    15o        1/4        28/29         29/108                                                    the Universe
                                                                                                  in Seconds)
    20o        1/3        16/17          17/47
                                                                           1015    peta-    P
    29o      151/2/8        7/8         151/2/7
    30o        1/2         31/2/2        1/31/2
       o                                                                   1012    tera-    T    0.3 TW (Peak
    37         3/5          4/5           3/4                                                     power of a
    42o        2/3          3/4           8/9                                                     1 ps pulse
    45o       21/2/2       21/2/2          1                                                      from a typical
       o
    49         3/4          2/3           9/8                                                     Nd-glass laser)
    53o        4/5          3/5           4/3
    60        31/2/2        1/2           31/2                             109     giga-    G    22 G$ (Size of
    61o        7/8        151/2/8       7/151/2                                                  Bill & Melissa
       o
    70        16/17         1/3          47/17                                                   Gates’ Trust)
       o
    75        28/29         1/4         108/29
    80o       65/66         1/6          65/11                             106     mega-    M    6.37 Mm (The
    90 o
                1            0             ∞                                                      radius of the
                                                                                                  Earth)
    (Memorize the Bold rows for future reference.)
                                                                           103     kilo-    k    1 kg (SI unit
Derivatives of Polynomials                                                                        of mass)

For polynomials, with individual terms of the form Axn,                    10-1    deci-    d    10 cm
we define the derivative of each term as
                                                                           10-2    centi-   c    2.54 cm (=1 in)
            d
            dx
                 ( )
               Ax n = nAx n −1                                             10-3    milli-   m    1 mm (The
                                                                                                  smallest
To find the derivative of the polynomial, simply add the                                          division on a
derivatives for the individual terms:                                                             meter stick)

          d
          dx
             (              )
             3x 2 + 6 x − 3 = 6 x + 6                                      10-6    micro-   µ

                                                                           10-9    nano-    n    510 nm (Wave-
Integrals of Polynomials                                                                          length of green
                                                                                                  light)
For polynomials, with individual terms of the form Axn,
we define the indefinite integral of each term as                          10-12   pico-    p    1 pg (Typical
                                                                                                  mass of a DNA
          ∫ (Ax )dx = n + 1 Ax
                 n      1           n +1
                                                                                                  sample used in
                                                                                                  genome
To                                     find the indefinite                                        studies)
integral of the polynomial, simply add the integrals for                   10-15   femto-   f
the individual terms and the constant of integration, C.
                                                                           10-18   atto-    a    600 as (Time

      ∫ (6 x + 6)dx = [3x       + 6x + C   ]                                                     duration of the
                            2
                                                                                                 shortest laser
                                                                                                 pulses)


                                                       Version 5/12/2005
                Reference Guide & Formula Sheet for Physics
       Dr. Hoselton & Mr. Price                                                                    Page 8 of 8

Linear Equivalent Mass                                        The only external force on this system is the weight of
                                                              the hanging mass. The mass of the system consists of
Rotating systems can be handled using the linear forms        the hanging mass plus the linear equivalent mass of the
of the equations of motion. To do so, however, you must       fly-wheel. From Newton’s 2nd Law we have
use a mass equivalent to the mass of a non-rotating
object. We call this the Linear Equivalent Mass (LEM).        F = ma, therefore,         mg = [m + (LEM=½M)]a
(See Example I)
                                                                                         mg = [m + ½M] a
For objects that are both rotating and moving linearly,
you must include them twice; once as a linearly moving                                   (mg – ma) = ½M a
object (using m) and once more as a rotating object
(using LEM). (See Example II)                                                            m(g − a) = ½Ma

The LEM of a rotating mass is easily defined in terms of                                 m = ½•M•a / (g − a)
its moment of inertia, I.
                                                                                         m = ½• 4.8 • 1.00 / (9.81 − 1)
                       LEM = I/r2
                                                                                         m = 0.27 kg
For example, using a standard table of Moments of
Inertia, we can calculate the LEM of simple objects           If a = g/2 = 4.905 m/s2,   m = 2.4 kg
rotating on axes through their centers of mass:
                                                              If a = ¾g = 7.3575 m/s2,   m = 7.2 kg
                              I             LEM
                                                              Note, too, that we do not need to know the radius unless
Cylindrical hoop            mr2                 m             the angular acceleration of the fly-wheel is requested. If
                                                              you need α, and you have r, then α = a/r.
Solid disk                ½mr2              ½m
                                                              Example II
Hollow sphere             2
                           ⁄5mr2            2
                                             ⁄5m
                                                              Find the kinetic energy of a disk, m = 6.7 kg, that is
Solid sphere              ⅔mr2              ⅔m                moving at 3.2 m/s while rolling without slipping along a
                                                              flat, horizontal surface. (IDISK = ½mr2; LEM = ½m)

Example I                                                     The total kinetic energy consists of the linear kinetic
                                                              energy, KL = ½mv2, plus the rotational kinetic energy,
A flywheel, M = 4.80 kg and r = 0.44 m, is wrapped            KR = ½(I)(ω)2 = ½(I)(v/r)2 = ½(I/r2)v2 = ½(LEM)v2.
with a string. A hanging mass, m, is attached to the end
of the string.                                                          KE = ½mv2 + ½•(LEM=½m)•v2

When the                                                                KE = ½•6.7•3.22 + ½•(½•6.7)•3.22
hanging mass is
released, it                                                            KE = 34.304 + 17.152 = 51 J
accelerates
downward at                                                   Final Note:
1.00 m/s2. Find
the hanging                                                   This method of incorporating rotating objects into the
mass.                                                         linear equations of motion works in every situation I’ve
                                                              tried; even very complex problems. Work your problem
To handle this problem using the linear form of               the classic way and this way to compare the two. Once
Newton’s Second Law of Motion, all we have to do is           you’ve verified that the LEM method works for a
use the LEM of the flywheel. We will assume, here, that       particular type of problem, you can confidently use it for
it can be treated as a uniform solid disk.                    solving any other problem of the same type.



                                                    Version 5/12/2005

								
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