1D10-002
Caroncirculartrack-rotationalvectors
STATEMENTOFTHEPROBLEM
Acarisdrivingclockwisearoundacircularracetrack.Thetiresonthecarrotate50times
everysecond.a)Whatisthetires’angularvelocityasitasittravelsduenorthanddueeast?
b)Whatistheangularaccelerationofthetiresduringthe10secondsittakestogofrom
travelingduenorthtotravelingdueeast.
STRATEGY
Theproblemasksaboutvectorquantities,thustheanswershavebothamagnitudeanda
directioncomponentwhichcanbeconsideredseparately.First,wefindthemagnitudeof
theangularvelocityandusetherighthandruletofindthedirectionoftheangularvelocity
vector.Tofindthemagnitudeoftheacceleration,weneedtofindthechangeinangular
velocityandthendividebytime.Tofindthedirectionoftheangularaccelerationwefind
theinversetangentofthey-componentdividebythex-component.
IMPLEMENTATION
Tofindthemagnitudeoftheangularvelocity,weapplyunitanalysis.Tofindthedirection
oftheangularvelocity,weapplytheright-handrule.Wefindtheangularacceleration
usingtherelation:𝛼 =
∆!
∆!
=
!! !!!
!! !!!
.
CALCULATION
f
a)Magnitudeofangularvelocity:
𝜔 = 50
!"#$#%"&'
!"#$%&
×2𝜋
!"#$"%&
!"#$#%"&
= 314
!"#
!
v
Usingtheright-handrule,whenthecaristravelling
north,itstires’angularvelocityvectorpointswest,andwhen
thecaristravellingeast,itstires’angularvelocityvector
pointsnorth.
b)Usingnorthasthe+𝚥hatdirectionandeastasthe+𝚤 direction,theangularaccelerationis:
∆𝜔 (314𝚥 − (−314 𝚤)) rad
1
𝑟𝑎𝑑
𝛼=
=
×
= 31.4 𝚤 + 31.4𝚥 ! ∆𝑡
s
10 s
𝑠
Themagnitudeoftheangularaccelerationis:
𝛼=
Theangleis:
31.4
𝜃 = 𝑡𝑎𝑛!!
= 45°
31.4
31.4! + 31.4!
v
= 44.4
𝑟𝑎𝑑
𝑠!
W
N
S
o
O
= ?
= ?
f
= ?
E
1D10-002
Caroncirculartrack-rotationalvectors
Tofindthedirectionoftheaccelerationvector,wecanalsodrawavectorgoingfromthetip
ofthewestarrowtothetipofthenorthangularvelocityvector.Thustheacceleration
vectoristothenortheast.
SELF-EXPLANATIONPROMPTS
1. Vectorscanbeaddedpictoriallybydrawingthevectorssuchthatthetailofonevector
connectstothetailofanothervector.Showhowthisisconsistentwiththeabove
statementthattheaverageangularaccelerationvectorgoesfromthetipofinitialvector
tothetipofthefinalvector.
2. Whichdirectionistheangularvelocityvectorwhenthecaristravelingwest?South?
3. Describetheangularaccelerationvectorifthecarhadcontinuedinastraightline,but
appliedthebrakes,slowingthecar.