6.3.1 applications of simple harmonic motion applications of simple harmonic motion the spring...
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6.3.1Applicationsofsimpleharmonicmotion
Thespringpendulum
Therearetwofactorsthataffecttheperiodofoscillationofamass-springsystem;the‘stiffness’ofthespring(springconstantk)andthemassm.https://phet.colorado.edu/sims/mass-spring-lab/mass-spring-lab_en.html
(1)!Howwouldtheperiodchangeifyouincreasedthespringconstant?
(2)!Howwouldtheperiodchangeifyouincreasedthemass?
ProvidedthespringisnotstretchedbeyonditselasticlimitiswillobeyHooke’sLaw:
𝐹 = −𝑘𝑥
TheminussignistherebecausetherestoringforceFisalwaysintheoppositedirectiontothedisplacement(fromequilibrium).
(3)!UsingF=ma,substituteforFintheequation,above,andrearrangetomakexthesubject.
Weknowthatthemass-springsystemwilloscillatewithSHM,andadefiningfeatureofSHMis:
𝑎 = −(2𝜋𝑓),𝑥,
whereAistheamplitude,𝑥isthedisplacementand𝑓isthefrequency.
(4)!Substitutefor𝑥,usingthepreviousequationandrearrangethisequationtomake𝑓thesubject.
(5)!Theperiod𝑇 = /0.SubstituteforfandrewritetheequationtomakeTthesubject.
Youshouldhavefoundtheperiodofthespringpendulumisgivenby:
𝑇 = 2𝜋𝑚𝑘
k
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(6)!Howwouldyoushowexperimentallythatthisrelationshipiscorrect?
TheSimplependulum
Therearetwofactorsthataffecttheperiodofapendulum;thelengthofthependulum𝑙andthegravitationalfieldstrengthg.
Theperiodofapendulumisgivenby:
𝑇 = 2𝜋𝑙𝑔
SystemswilloscillatewithSHMiftherestoringforceisproportionaltothedisplacementfromtheequilibriumposition.
(7)!Forapendulum,wheredoestherestoringforcecomefrom?
(Note:Thisistricky.)
(8)!Showthatforsmallangles(Hint:usesmallangleapproximation),therestoringforceisproportionaltothedisplacements.
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