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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.

s