Oscillations: Review (Chapter 12)
 Oscillations: motions that are periodic in time (i.e. repetitive)
o Swingingobject(pendulum)
o Vibratingobject(spring,guitarstring,etc.)
o Partofmedium(i.e.string,water)aswavepassesby
 Oscillation requires
o Restoringforce(pushes“system”backtowardequilibrium)
o Inertia(keeps“system”goingpastequilibrium)
 Oscillator example: Mass on a Spring:
o Restoringforcefromspring.Varieswith“extension”
 Hooke’sLaw: Fx  kx (linearrestoringforce)
 Ifmassmdisplacedfromequilibriumthenreleased→willoscillate
 Ifforceislinearandrestoring(andthusconservative)
o Motionissimpleharmonicmotion(SHM)
o Systemisasimpleharmonicoscillator(SHO)
Acceleration in SHM: not constant → depends on x
 Newton’s2ndlawsays  Fx  ma x k
a


x
o ForSHM,implies  kx  ma x sothat x
m k
a


A
v  0
 Atmax.positivedisplacementfromequil.  x  A have x
m and x
 Atequilibrium  x  0 have ax  0 and v x  vmax k
a


A
v  0
 Atmax.negativedisplacementfromequil.  x   A have x
m and x
How do we find an expression for x(t) in SHM?
 First:findanequationforwhichx(t)isthesolution.
dx
dv x d 2 x
 Remember: v x  dt and a x  dt  dt 2 k
a


x
 Butformassonspring: x
m d 2x
k


x Thisisadifferentialequation.
 Soformassonspring: dt 2
m
d 2x
k
2
2



x


 Moregeneralform: dt 2
Soidentify
m formass/spring
 Functionx(t)thatsatisfiesequation:
o OscillateslikeparticledoingSHM→moregeneralthanspringcase
o Whatis  ?
d 2x
2
How do we solve dt 2   x to get x(t)?
 Tosolvedifferentialequation→guesssolutionthentest!
d 2 cos
d 2 sin 
 Noticethat: d 2   cos and d 2   sin   Sotry: xt   A cos t    o Important:argument  t    mustbeinradians
o Constants A ,  ,and  mustbefoundfromdetailsofmotion
Test xt   A cos t    :
dx
 dt  v x   A sin t   
d 2x
2
2



a


A

cos

t





x
x
 dt 2
d 2x
2
So: xt   A cos t    satisfies dt 2   x
Important!
d 2x
2



x mustbesatisfiedbyanySHM(i.e. xt   A cos t    )
2
 dt
 Foramassonaspring(oneexampleofSHM)
k
a


x o x
m
k


o
m (Naturalangularfrequencyforspring/masssystem)
 ForotherexamplesofSHM,  dependsonspecificformofrestoringforceand
inertia
  isnaturalangularfrequencyofSHO.Unitsareradians/second(reallys‐1)
Parameters that describe SHM: xt   A cos t   
  t    isthephaseinradians.
 Period(T):
o Motionat t  T  isthesameasmotionat t  o Meansthatphaseincreasesby 2 intimeT
o So  t  T       t     2 2
T

o Result:
  Frequency:
o  (angularfrequency)isnumberofradiansbywhichphaseincreases
persecond
1 
f


o
T 2 (frequency)isnumberoftimespersecondthatmotion
repeats
 unitsarecycles/secondorHertz(Hz)
o so   2 f  Phaseconstant   o Dependsonwheremotionisincycleatt=0.i.e. x0  A cos   Amplitude(A)
o xt  oscillatesbetween+Aand–A
Relations between Position, Velocity, and Acceleration in SHM
xt   A cos t   
xA
dx
  A sin  t   
dt
(slope of x vs t ), vx  Aω
vx t  
d 2x
a x t   2   A 2 cos t   
dt
(slope of vx vs t ), a x  Aω 2
How do we get parameters  ,  , and A from description of specific motion?
 Angularfrequency   dependsonpropertiesofoscillator
o i.e.forspring/masssystem,   k / m  GetAmplitudeandphaseconstantfromxandvatspecifictime(i.e.t=0)
o Ex.:massm,springk.Startclock(sett=0)whenmassatarbitrary xi  So x0   xi  A cos  and v0  vi   A sin   SolveforA(usetrigonometryidentity):
xi2
vi2
2
2
 sin   cos   A 2  A 2 2  1  gives: A 
x 
2
i
vi2
2  Solveforphaseconstant:
vi
sin 
tan





cos 
xi  Check for two simple cases:
o If vi  0 ,motionstartsfromrestatmaximumorminimumdisplacement
 A  xi2  xi and tan   0 sothat   0 o If xi  0 ,motionstartsfromequilibriumwithmaximumspeed
 A
vi2
2
1
and   tan     
 So x 


cos

t



2
2


vi2

2
EXAMPLES (to finish yourself if not completed on board):
Chapter 12, Problem 14:
 A piston oscillates in an engine with position x  5.00 cm cos2t   / 6 . At
t  0 s , find (a) position, (b) velocity, and (c) acceleration. (d) Find the period
and Amplitude of the motion.
Chapter 12, Problem 9:
 A particle’s motion along the x axis is SHM. It starts from equilibrium at
t  0 s and moves to the right. Amplitude is 2.00 cm and frequency is 1.50
Hz. (a) Find an expression for xt  . (b) Find the maximum speed and
earliest time t  0 at which particle has this speed. (c) Find the maximum
acceleration and earliest time t  0 at which particle has this acceleration.
Energy and SHM
 Mechanicalenergy(E=K+U)conservedifnonon‐conservativeforces
o ForSHM,alternatesbetweenkineticenergyandpotentialenergy
2
 Example:formassmonspringksothat   k / m o x  A cos t    and v   A sin t    1 2 1 2 2
K

mv  kA sin t    o KineticEnergy:
2
2
1 2 1 2
2
 t    U

kx

kA
cos
o PotentialEnergy:
2
2
1 2
1 2
2
2




E

K

U

kA
sin

t



cos

t



kA o MechanicalEnergy:
2
2
1 2
E

kA o SototalmechanicalenergyonlydependsonkandA:
2
o From E  K  U ,alsoget:
1 2 1 2 1 2
 2 kA  2 kx  2 mv k 2
2
2
2
v


A

x



A

x
 Givesmassspeedasfunctionofx:
m





