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Spring2016Physics7ALec001(Yildiz)MidtermII
1. (15points)AnobjectisreleasedfromrestatanaltitudehabovethesurfaceoftheEarth.his
comparabletotheradiusofEarth(RE),sogravitationalacceleration(g)isnotconstant.
a) WhatisthevelocityoftheobjectwhenithitsthesurfaceofEarth?
b) WhatisthegravitationalaccelerationoftheobjectatdistancerfromtheEarth’scenter,where
𝑅! < 𝑟 < 𝑅! + ℎ?
c) Whatistherateofchangeofthegravitationalacceleration𝑔 𝑟 asafunctionofthedistancerfrom
theEarth’scenter,where𝑅! < 𝑟 < 𝑅! + ℎ?
Solution:
a) −∆𝑈 = ∆𝐾
𝐺𝑀𝑚
𝑈! (𝑟) = −
𝑟
1
1
1
𝑚𝑣 ! = −𝐺𝑀𝑚
−
2
𝑅! + ℎ 𝑅!
!
𝑣 = 2𝐺𝑀
!!
−
!
!! !!
b)𝐹! 𝑟 = −
𝑔 𝑟 =−
c)
!"
!"
=
!"#
!!
= 𝑚. 𝑔(𝑟)
𝐺𝑀
𝑟!
!!"
!!
2.(20 points) Assume a cyclist of weight mg can exert a
force on the pedals equal to 0.80 mg on the average. The
pedals rotate in a circle of radius 18 cm, the wheels have a
radius of 34 cm, and the front and back sprockets on which
the chain runs have 42 and 19 teeth respectively. The mass
of the bike is 14 kg and that of the rider is 63 kg. Assume
there is no slipping between the ground and the wheel. Use
2
g = 10 m/s for calculations.
a)Howistheangularvelocityoftherearwheelofabicycle
relatedtotheangularvelocityofthefrontsprocketandpedals?Theteetharespacedthesameonboth
sprocketsandtherearsprocketisfirmlyattachedtotherearwheel.
b)Determinethemaximumsteepnessofhillthecyclistcanclimbatconstantspeed.Assumethecyclist's
averageforceisalwaystangentialtopedalmotion.
c)Determinethemaximumsteepnessofhillthecyclistcanclimbatconstantspeed.Assumethecyclist's
averageforceisalwaysdownward.
Iftheriderisridingataconstantspeed,thenthepositiveworkinputbytheridertothe(bicycle+
rider)combinationmustbeequaltothenegativeworkdonebygravityashemovesuptheincline.
Thenetworkmustbe0ifthereisnochangeinkineticenergy.
(a) Iftherider’sforceisdirecteddownwards,thentheriderwilldoanamountofworkequalto
theforcetimesthedistanceparalleltotheforce.Thedistanceparalleltothedownwardforce
wouldbethediameterofthecircleinwhichthepedalsmove.Thenconsiderthatbyusing2
feet,theriderdoestwicethatamountofworkwhenthepedalsmakeonecomplete
revolution.Soinonerevolutionofthepedals,theriderdoestheworkcalculatedbelow.
Wrider = 2 ( 0.90mrider g ) d pedal motion
Inonerevolutionofthefrontsprocket,therearsprocketwillmake 42 19 revolutions,andso
thebackwheel(andtheentirebicycleandrideraswell)willmoveadistanceof
( 42 19) (2π r ) .Thatisadistancealongtheplane,andsotheheightthatthebicycleand
wheel
riderwillmoveis h = ( 42 19 )( 2π rwheel ) sin θ . Finally,theworkdonebygravityinmovingthat
heightiscalculated.
WG = ( mrider + mbike ) gh cos180° = − ( mrider + mbike ) gh = − ( mrider + mbike ) g ( 42 19 )( 2π rwheel ) sin θ
Setthetotalworkequalto0,andsolvefortheangleoftheincline.
Wrider + WG = 0 → 2 [0.90mrider g ] d pedal − ( mrider + mbike ) g ( 42 19 )( 2π rwheel ) sin θ = 0 →
motion
( 0.90mrider ) d pedal
θ = sin
motion
−1
( mrider + mbike )( 42 19 )(π rwheel )
= sin
−1
0.90 ( 65 kg )( 0.36 m )
( 77 kg )( 42 19 ) π ( 0.34 m )
= 6.7°
(b)Iftheforceistangentialtothepedalmotion,thenthedistancethatonefootmoveswhile
exertingaforceisnowhalfofthecircumferenceofthecircleinwhichthepedalsmove.The
restoftheanalysisisthesame.
⎛
⎞
Wrider = 2 ( 0.90mrider g ) ⎜ π rpedal ⎟ ; Wrider + WG = 0 →
⎝
motion
⎠
( 0.90mrider ) π rpedal
θ = sin −1
motion
( mrider + mbike )( 42 19 )(π rwheel )
= sin −1
0.90 ( 65 kg )( 0.18 m )
( 77 kg )( 42 19 )( 0.34 m )
3.(20points)TwoblockofmassesMand2Mareconnectedtoaspringof
springconstantkthathasoneendfixed,asshown.Thehorizontalsurface
= 10.5° ≈ 10°
andthepulleyarefrictionlessandthepulleyhasnegligiblemass.Theblocksarereleasedfromrestwith
thespringrelaxed.
a) Whatisthevelocityoftheblockswhenthehangingblockhasfallenadistanceh?
b) Whatmaximumdistancehmaxdoesthehangingblockfallbeforemomentarilystopping?
c) Intheabsenceoffriction,thesystemisexpectedtooscillatebackandforthinfinitely.However,if
thereiskineticfrictionbetweenblockandthehorizontalsurface(coefficientofkineticfrictionisµk),the
systemwilleventuallycometoacompletestop.Whatistheequilibriumpositionofthespring’s
extensionwhenthesystemcomestoacompletestop?Whatisthetotaldistancetraveledbythe
hangingblockbeforethesystemcomestoacompletestop?(Assumethatthefrictionalforceonthe
massMonthehorizontalsurfaceisnegligiblewhenthesystemcomestoacompletestop.)
Solution
a) 𝑈! + 𝐾! = 𝑈! + 𝐾! 0=
1 !
1
1
𝑘ℎ − 2𝑀𝑔ℎ + 𝑀𝑣 ! + 2𝑀𝑣 ! 2
2
2
𝑣=
b) 𝑈! + 𝐾! = 𝑈! + 𝐾! and
4𝑀𝑔ℎ − 𝑘ℎ!
3𝑀
𝑣! = 0 𝑤ℎ𝑒𝑛 ℎ = ℎ!"# 0=
1
!
𝑘ℎ
− 2𝑀𝑔ℎ!"# + 0
2 !"#
ℎ!"# =
4𝑀𝑔
𝑘
c) Inthepresenceofanon-conservativeforce,𝑈! + 𝐾! + 𝑊!" = 𝑈! + 𝐾! 𝑊!" = −𝐹!" 𝑑!"! 𝐹!" = 𝜇! . 𝑁 = 𝜇! 𝑚𝑔
𝐾! = 0 𝑠𝑦𝑠𝑡𝑒𝑚 𝑖𝑠 𝑠𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑟𝑦 , 𝑡𝑜 𝑓𝑖𝑛𝑑 𝑈! , 𝑤𝑒 𝑛𝑒𝑒𝑑 𝑡𝑜 𝑓𝑖𝑛𝑑 𝑡ℎ𝑒 𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛.
𝐴𝑡 𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚, 𝐹! = 0,
𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝑇 − 𝑘𝑥!" = 0
𝐵𝑒𝑐𝑎𝑢𝑠𝑒 𝑡ℎ𝑒 ℎ𝑎𝑛𝑔𝑖𝑛𝑔 𝑚𝑎𝑠𝑠 𝑖𝑠 𝑎𝑙𝑠𝑜 𝑠𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑟𝑦,
𝑇 𝑖𝑠 𝑡ℎ𝑒 𝑡𝑒𝑛𝑠𝑖𝑜𝑛 𝑜𝑛 𝑡ℎ𝑒 𝑟𝑜𝑝𝑒 𝐹! = 0,
𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝑇 − 2𝑀𝑔 = 0
2𝑀𝑔 = 𝑘𝑥!" 𝑈! + 𝐾! + 𝑊!" = 𝑈! + 𝐾! 0 − 𝜇! 𝑚𝑔𝑑!"! =
1
𝑘𝑥 ! − 2𝑀𝑔𝑥!" + 0
2 !"
−𝜇! 𝑚𝑔𝑑!"!
1 2𝑀𝑔
= 𝑘
2
𝑘
𝑑!"! =
!
!!"
!!!
− 2𝑀𝑔
2𝑀𝑔
𝑘
4.(25points)Supposea
spacecraftwithinitialmassof
mi.Withoutitspropellant,the
spacecrafthasamassofmf=
mi/3.Therocketthatpowers
thespacecraftisdesignedto
ejectthepropellantwitha
speedofurelativetothe
rocketataconstantrateofR.Thespacecraftisinitiallyatrestinspaceandtravelsinastraightline.
a) Howlongwouldittaketherockettoreleaseallofitspropellant?
b) Whatism(t),themassoftherocketasafunctionoftime?
c) Whatisv(t),thespeedoftherocketasafunctionoftime?
d) Howfarwillthespacecrafttravelbeforeitsrocketusesallthepropellantandshutsdown?
Hint: ln 1 − 𝑎𝑥 𝑑𝑥 =
!"!!
!
ln 1 − 𝑎𝑥 − 𝑥
Solution:
a) 𝑡! =
!! !!!
!
=
!
!! ! !
!
!
! !!
=!
!
b) 𝑚 𝑡 = 𝑚! − 𝑅𝑡
c) Because 𝑭!"# = 0
!"
!"
𝑚
!"
!"
= 𝑣!"#
!!
!"
= −𝑅𝑣!"# = −𝑢 (𝑟𝑒𝑔𝑎𝑟𝑑𝑙𝑒𝑠𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑝𝑒𝑒𝑑 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑜𝑐𝑘𝑒𝑡)
𝑚! − 𝑅𝑡
!
𝑣=
!
𝑢𝑅
𝑑𝑡 𝑚! − 𝑅𝑡
𝑣 = 𝑢𝑙𝑛
d) 𝑥! =
𝑡𝑓
!
𝑣𝑑𝑡 = −𝑢
𝑡𝑓
!
𝑙𝑛 1 −
!"
𝑚𝑖
𝑑𝑣
= 𝑢𝑅
𝑑𝑡
𝑚𝑖
𝑚! − 𝑅𝑡
𝑑𝑡 UsingtheHintgiveninthequestion,𝑥! = −𝑢
𝑡−
𝑚𝑖
!
𝑙𝑛 1 −
!"
𝑚𝑖
−𝑡
𝑡𝑓
!
and𝑡! =
! !!
! !
𝑥! = −𝑢
2 𝑚𝑖 𝑚𝑖
𝑅 2 𝑚𝑖
2 𝑚𝑖
𝑚𝑖
𝑅. 0
−
𝑙𝑛 1 −
−
− −
𝑙𝑛 1 −
−0 3𝑅
𝑅
𝑚𝑖 3 𝑅
3𝑅
𝑅
𝑚𝑖
𝑥! = −𝑢
1 𝑚𝑖
2
2 𝑚𝑖
𝑙𝑛 1 −
−
3𝑅
3
3𝑅
𝑥! =
𝑢 𝑚𝑖 2
𝑅 3
−
1
𝑙𝑛3 3
5.(20points)Acylindricallysymmetricspoolofmassm
andradiusRsitsatrestonahorizontaltablewith
friction.Withyourhandonamasslessstringwrapped
aroundtheaxleofradiusr,youpullonthespoolwitha
constanthorizontalforceofmagnitudeTtotheright.As
aresult,thespoolrollswithoutslippingadistanceL
alongthetable.Assumethatthespoolisasoliduniform
cylinder.
a)Whatisthedistancethatyourhandtravelsasthe
spoolmovesadistanceL?
b)Findthefinaltranslationalspeedofthecenterofmassofthespoolusingthework-energyprinciple?
c)Findthevalueofthefrictionforcefandaccelerationofthespoolusingtheequationsfordynamicsof
themotion(e.g.forceandtorque)?
c.Inlectures,weusedF=maandτ=Iαtofindfrictionalforceandacceleration.Hereisanalternative
solutionusingImpulse-momentumtheorem.