ap simple harmonic motion

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  • 8/9/2019 AP Simple Harmonic Motion

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    Simple

    Harmonic

    Motion

    AP Physics C

    Mrs. Coyle

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    Periodic MotionPeriodic Motion A motion of an object that repeats with a

    constant period.

    http://www.sccs.swarthmore.edu/users/!/ajb/e"#/lab#/

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    SimpleSimpleHarmonicHarmonicMotionMotion

    $t is aperiodic motion.A%&

    $t has arestoringforce thatacts to restore

    the oscillator to e'uilibrium. (he restorin)force is )i*en by:

    Hookes Law F=-kx

    x is the displacement from e'uilibrium and kis theforce constant +sprin) constant,.

    (he period of SHM oscillator does not depend

    on the amplitude.

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    Simple Harmonic Motion

    Simulationshttp://bcs.wiley.com/he-bcs/oos0

    action1minina*2bcs$d13442item$d15"#

    "6!#72asset$d1###"2resource$d1#8

    ##

    -SHM

    -Particle oscillatin) in SHM

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    Calculatin) from a 9 *s ;raph is the slope of a 9 *s )raph

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    (he acceleration in SHM is not

    constant:

    Hooke Newton

    x

    x

    F F

    kx ma

    ka xm

    =

    =

    =

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    Acceleration in Simple

    Harmonic Motion Acceleration

    >et

    a1 -8x

    2

    2

    d x ka x

    dt m= =

    2 k

    m =

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    Characteristic ?uantities of Simple

    Harmonic Motion &isplacement

    Amplitude: maimum displacement

    9re'uency

    Period

    f1#/( (1#/f

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    SimpleSimpleHarmonicHarmonicMotionMotion Velocity:

    @ maimumas it

    passes throu)h

    e'uilibrium

    @ eroas it passes

    throu)h the etremepositions in its

    oscillation.

    cceleration!a=F"m = -kx"m

    -maimumat etremepoints

    -eroat e'uilibrium

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    B'uations of Motion-

    &isplacement

    1Asin+t+, or 1Acos+t +)

    1-Asin+t+, 1-Acos+t+,

    =an)ular fre'uency rad/s is the phase constant

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    Bample

    a, Dhat is the amplitude0

    b, Dhat is the period0

    c, Dhat total distance does the particle tra*el in

    one period0

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    Bample

    Drite the e'uation of motion for the abo*e oscillator.

    Answer:

    1-.6sin+Et,

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    %ote #$e acceleration is not constant and

    t$erefore t$e kinematics e%&ations cannot'e &sed.

    #$ere are two options!#.Fse conser*ation of mechanical ener)y to

    find * at a )i*en position.

    B1 G m*8= G 8 1 constant or8. (ae the first deri(ati(eof +e'uation of

    motion, to find (and the second deri(ati(e

    to find a.

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    Bamples of Motion B'uations

    for Simple Harmonic Motion

    22

    2

    ( ) cos ( )

    sin ( t )

    cos( t )

    x t A t

    dxv A

    dt

    d x

    a Adt

    = +

    = = +

    = = +

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    max

    2

    max

    kv A A

    mk

    a A A

    m

    = =

    = =

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    ;raphs of

    SHM

    (he *elocity is 7o

    out of phase withthe displacement

    (he acceleration is

    #!oout of phasewith thedisplacement

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    Bample #

    t1

    x +,1A

    v +, 1 1

    ama 1 8A

    *ma 1 A

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    Bample 8

    t1

    x +,1

    v +, 1 vi

    1 /8

    (he )raph is shiftedone-'uarter cycle to theri)ht compared to the)raph ofx +, 1A

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    Period and 9re'uency

    22

    T

    = =

    12 , 2

    m kTk m

    = =

    2 k

    m

    =

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    Period of A Sprin) Mass scillator

    IIII

    (18)m/

    m mass

    sprin) constant

    ( does not depend on ) (he period is smaller for a stiffer sprin)

    +lar)e *alues of k,.

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    Harmonic Motion of a Pendulum

    http://www3.interscience.wiley.com:!#/le)acy/colle)e/halliday/5"#386/simulations4e/inde.htm0newwindow1true

    Period

    III

    (18)>/)

    >1len)th of strin) ( depends on )

    >

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    ?uestion

    $f you had a sprin)-mass system on the

    moon would the period be the same or

    different than that of this system on the

    earth0

    Dhat if it were a pendulum system0