Spring

3PotentialenergyofaspringTheforcekxexertedbyaspringisaconservativeforce.Theworkdoneincompressingorstretchingaspringisstoredaspotentialenergyandcanbeusedlatertodowork.Theworkdoneinstretchingorcompressingaspringofspringconstantkbyadistancexis12kx2.Thusthepotentialenergy(U)ofaspringisU=12kx2.(60)Example:Ifamassmrestingonafrictionlesshorizontaltableisattachedtotheendofaspringofspringconstantk.Theotherendofthespringisfixed.ThemassisdisplacedadistanceAfromitsequilibriumpositionandreleasedfromrest.Whatisthemaximumspeedofthemassasitoscillates?Note:Ifamassmisattachedtotheendofaspringandthenallowedtooscillatebackandforth,theenergyofthesystemwillremainconstant.12kx2+12mv2=E=constant.Solution:Thetotalenergyofthesystemisconstant,sothekineticenergywillbegreatestwhenthepotentialenergyisaminimum,andthisoccurswhenx=0.ThusE=U1+K.E1=U2+K.E212kA2+0=0+12mv2v=kmAExample:AnarcherybowexertsaHooke’slawforcekxonanarrowwhenthestringispulledbackadistancex.Supposethatanarcherexertsaforceof220Nindrawingbackanarrowadistanceof64cm.Whatisthespringconstantofthebow?Withwhatspeedwillanarrowofmass24kleavethebow?Solution:k=Fx=2200.64=343.75N/mP.E1+K.E1=P.E2+K.E212kx2+0=0+12mv2v=kmxv=343.750.024(0.64)

3PotentialenergyofaspringTheforcekxexertedbyaspringisaconservativeforce.Theworkdoneincompressingorstretchingaspringisstoredaspotentialenergyandcanbeusedlatertodowork.Theworkdoneinstretchingorcompressingaspringofspringconstantkbyadistancexis12kx2.Thusthepotentialenergy(U)ofaspringisU=12kx2.(60)Example:Ifamassmrestingonafrictionlesshorizontaltableisattachedtotheendofaspringofspringconstantk.Theotherendofthespringisfixed.ThemassisdisplacedadistanceAfromitsequilibriumpositionandreleasedfromrest.WhatisthemaximumspeedofthemassasitoscillatesAde ?Note:Ifamassmisattachedtotheendofaspringandthenallowedtooscillatebackandforth,theenergyofthesystemwillremainconstant.12kx2+12mv2=E=constant.Solution:Thetotalenergyofthesystemisconstant,sothekineticenergywillbegreatestwhenthepotentialenergyisaminimum,andthisoccurswhenx=0.ThusE=U1+K.E1=U2+K.E212kA2+0=0+12mv2v=kmAExample:AnarcherybowexertsaHooke’slawforcekxonanarrowwhenthestringispulledbackadistancex.Supposethatanarcherexertsaforceof220Nindrawingbackanarrowadistanceof64cm.Whatisthespringconstantofthebow?Withwhatspeedwillanarrowofmass24kleavethebow?Solution:k=Fx=2200.64=343.75N/mP.E1+K.E1=P.E2+K.E212kx2+0=0+12mv2v=kmxv=343.750.024(0.64)