Classical Mechanics Lecture 16
Today’sConcepts:
a)RollingKine6cEnergy
b)AngularAccelera6on
Physics211Lecture16,Slide1
Thoughts from the Past
Justge=ngdizzywithrota6onalversionsofnewton'slaws.-.Toomanyequa6onstorememberandcopewithIt seems weird that the ball would roll slower than a block sliding
down a ramp. My intuition says the ball would go faster, but alas...
What unmerciful doom doth rot upon the error of my soul?Whatdoesthecenterofmasshavetodowiththissec6on?AliJlebrushuponlastlecturesmaterialwouldhelp,justtoreinforcewhathas
beentaughtmomentsofiner6as6llthrowmeittakesmeacoupletriestothinkthrough
themandwithreallyrota6onperiodIamfindingitmorechallengingdespitethefactIhadtroublelearningitlastyearaswell
juststopteachingthesethingsplz-Itseemsweirdthattheballwouldrollslowerthanablockslidingdownaramp.
Myintui6onsaystheballwouldgofaster,butalas...
Startedoffterrible,butnowthatwe'vegottheballrollingI'mstar6ngtogetit.
MechanicsLecture16,Slide2
Ifeltlikeeveryslidejusthadatonofequa>onsjustbeingusedtofindnewequa>ons.Otherthanthattheactualuseoftheequa>onsdoesn'tseemsobad,butIs>lldon'treallyknowwhattodofortheCheckPoints.
EnergyConserva6on
F = ma
α
Mg
f a
MechanicsLecture16,Slide4
H
Rolling Motion
Objectsofdifferent I rollingdownaninclinedplane:
v = 0 ω = 0 K = 0
Δ K = − ΔU = M g hR
M
h
v = ωR
MechanicsLecture16,Slide6
Rolling
Ifthereisnoslipping:
Inthelabreferenceframe IntheCM referenceframe
v vv
Wherev = ωR
ω
MechanicsLecture16,Slide7
Rolling
Usev = ωR andI = cMR2 .
Doesn’tdependonMorR,justonc(theshape)
Hoop:c = 1
Disk:c = 1/2
Sphere:c = 2/5
etc...
c c
Rampdemo
v
So: c c
MechanicsLecture16,Slide8
A) B) C)
Clicker Question
Ahula-hooprollsalongthefloorwithoutslipping.Whatisthera6oofitsrota6onalkine6cenergytoitstransla6onalkine6cenergy?
RecallthatI = MR2 forahoopabout
anaxisthroughitsCM.
MechanicsLecture16,Slide9
Ablockandaballhavethesamemassandmovewiththesameini6alvelocityacrossafloorandthenencounteriden6calramps.Theblockslideswithoutfric6onandtheballrollswithoutslipping.Whichonemakesitfurthestuptheramp?
A)Block
B)Ball
C)Bothreachthesameheight.
CheckPoint
v
v ω
MechanicsLecture16,Slide10
Theblockslideswithoutfric6onandtheballrollswithoutslipping.Whichonemakesitfurthestuptheramp?
A)BlockB)Ball
C)Same
v
v ω
A)Theballlossesenergytorota6on
B)Theballhasmoretotalkine6cenergysinceitalsohasrota6onalkine6cenergy.Therefore,itmakesithigheruptheramp.
C)Iftheyhavethesamevelocitythentheyshouldgothesameheight.Therota6onalenergyshouldnotaffecttheball.
MechanicsLecture16,Slide11
CheckPoint
CheckPoint
Acylinderandahoophavethesamemassandradius.Theyarereleasedatthesame6meandrolldownarampwithoutslipping.WhichonereachestheboJomfirst?
A)CylinderB)HoopC)BothreachtheboJomatthesame6me
MechanicsLecture16,Slide12
A)CylinderB)HoopC)BothreachtheboJomatthesame6me
A)samePEbutthehoophasalargerrota6onaliner6asomoreenergywillturnintorota6onalkine6cenergy,thuscylinderreachesitfirst.
B)Ithasmorekine6cenergybecauseithasabiggermomentofiner6a.
C)BothhavethesamePEsotheywillhavethesameKEandreachthegroundatthesame6me.
Which one reaches the bottom first?
MechanicsLecture16,Slide13
CheckPoint
Asmalllightcylinderandalargeheavycylinderarereleasedatthesame6meandrolldownarampwithoutslipping.WhichonereachestheboJomfirst?
A)Smallcylinder
B)Largecylinder
C)BothreachtheboJomatthesame6me
MechanicsLecture16,Slide14
Asmalllightcylinderandalargeheavycylinderarereleasedatthesame6meandrolldownarampwithoutslipping.WhichonereachestheboJomfirst?
A)Smallcylinder
B)Largecylinder
C)BothreachtheboJomatthesame6me
A)Becausethesmalleronehasasmallermomentofiner6a.B)Thelargecylinderhasalargermomentofiner6a,andtherefore,endsupwithmoreenergy.C)Themassiscanceledoutinthevelocityequa6onandtheyarethesameshapesotheymoveatthesamespeed.Therefore,theyreachtheboJomatthesame6me.
MechanicsLecture16,Slide15
CheckPoint
Accelera>ondependsonlyontheshape,notonmassorradius.
What you saw in your Prelecture:
MechanicsLecture16,Slide16
Clicker Question
A)Smallcylinder
B)Largecylinder
C)BothreachtheboJomatthesame6me
Asmalllightcylinderandalargeheavycylinderarereleasedatthesame6meandslidedowntherampwithoutfric6on.WhichonereachestheboJomfirst?
MechanicsLecture16,Slide17
vω
a RM
We can try this…
Interesting aside: how v is related to v0 :
MechanicsLecture16,Slide22
Doesn’tdependonµ
v = v0 � µg
2v0
7µg
!
Position vs time for Bowling Ball
Quadra6ccurvefittoregion8.66sto9.05s:
v0=2.67m/s
Linearfittoregion9.1sto10.3s
v=1.78m/s
Linear velocity vs time
thevelocityvs6megraphshowsthatclearlyvisnearly(5/7)v0th
row
ing
slipping rolling
Supposeacylinder(radiusR,massM)isusedasapulley.Twomasses(m1 > m2)areaJachedtoeitherendofastringthathangsoverthepulley,andwhenthesystemisreleaseditmovesasshown.Thestringdoesnotsliponthepulley.
Comparethemagnitudesoftheaccelera6onofthetwomasses:
A)a1 > a2B)a1 = a2
C)a1 < a2
Clicker Question
a2T1
a1
T2
αR
M
m1m2
MechanicsLecture16,Slide24
Clicker Question
Supposeacylinder(radiusR,massM)isusedasapulley.Twomasses(m1 > m2)areaJachedtoeitherendofastringthathangsoverthepulley,andwhenthesystemisreleaseditmovesasshown.Thestringdoesnotsliponthepulley.
Howistheangularaccelera6onofthewheelrelatedtothelinearaccelera6onofthemasses?
A)α = RaB)α = a/RC)α = R/a
MechanicsLecture16,Slide25
a2T1
a1
T2
αR
M
m1m2
Clicker Question
Supposeacylinder(radiusR,massM)isusedasapulley.Twomasses(m1 > m2)areaJachedtoeitherendofastringthathangsoverthepulley,andwhenthesystemisreleaseditmovesasshown.Thestringdoesnotsliponthepulley.
Comparethetensioninthestringoneithersideofthepulley:
A)T1 > T2
B)T1 = T2C)T1 < T2
MechanicsLecture16,Slide26
a2T1
a1
T2
αR
M
m1m2
Atwood's Machine with Massive Pulley:
Apairofmassesarehungoveramassivedisk-shapedpulleyasshown.! Findtheaccelera>onoftheblocks.
Forthehangingmassesuse F = ma −m1g + T1 = −m1a −m2g + T2 = m2a
Forthepulleyuseτ = Iα
T1R − T2R
(Since foradisk)
MechanicsLecture16,Slide27
y
x
m2gm1g
aT1
a
T2
αR
M
m1m2
Wehavethreeequa6onsandthreeunknowns(T1,T2,a).
Solvefora.
−m1g + T1 = −m1a (1)
−m2g + T2 = m2a (2)
T1 − T2 (3)
MechanicsLecture16,Slide28
y
x
m2gm1g
aT1
a
T2
αR
M
m1m2
Atwood's Machine with Massive Pulley:
a =
m1 � m2
m1 + m2 + M/2
!g
UNIT 12 HOMEWORK BEFORE SESSION ONE
➡SmartPhysics: View the Prelecture 14 “Rotational Kinematics and Moment of Inertia” before the class and answer the checkpoint questions. (The online Homework is due the next day.)
UNIT 12 HOMEWORK AFTER SESSION ONE
➡SmartPhysics: View the Prelecture 15 “Parallel Axis Theorem” before the class of next session and answer the checkpoint questions. (The online Homework is due the next day.)
• Finish Activity Guide Entries for Session 1
•Work supplemental problems SP12-1, SP12-2 and SP12-3SP12-1) A racing car travels on a circular track of radius 250 m. If the car moves with a constant speed of 45 m/s, find (a) the angular speed of the car and (b) the magnitude and direction of the car's acceleration.
SP12-2) A tractor is travelling at 20 km/h through a row of corn. The radii of the rear tires are 0.75 m. Find the angular speed of one of the rear tires with its axle taken as the axis of rotation.
SP12-3) Sharon pulls on a rod mounted on a frictionless pivot with a force of 125 N at a distance of 87 cm from the pivot. Roger is trying to stop the rod from turning by exerting a force in the opposite direction at a distance of 53 cm from the pivot. What is the magnitude of the force he must exert?
UNIT 12 HOMEWORK AFTER SESSION TWO
➡SmartPhysics: View the Prelecture 16 “Rotational Dynamics” before the class of next session and answer the checkpoint questions. (The online Homework is due the next day.)
•Work supplemental problem SP12-4 (Problem SP12-4 will help you prepare for the ex-periment to be done during the next class session.)
SP12-4) (20 pts) A small spool of radius rs and a large Lucite disk of radius rd are connected by an axle that is free to rotate in an almost frictionless manner inside of a bearing as shown in the dia-gram below.
torque to the object and measuring the resultant angular
acceleration.
✍ Activity 12-14: Theoretical Calculations
(a) Calculate the theoretical value of the rotational inertia of the
stack of CDs using basic measurements of its radius and mass.
Be sure to state units! If there’s a significant size hole in the
centre, try to estimate the error involved in ignoring it.
For the CDs, ignoring the hole:
rd = Md =
Id =
For the hole in the centre:
rh = Mh =
Ih =
ICD = Id – Ih =
(b) Calculate the theoretical value of the rotational inertia of
the spool using basic measurements of its radius and mass.
Note: You'll have to do a bit of estimation here. Be sure to
state units.
rs = Ms =
Is =
(c) Calculate the theoretical value of the rotational inertia, I, of
the whole system. Don't forget to include the units. Note: By
noting how small the rotational inertia of the spool is compared
to that of the disk, you should be able to convince yourself that
you can neglect the rotational inertia of the rotating axle in your
calculations.
I =
(c) In preparation for calculating the torque on your system,
summarize the measurements for the falling mass, m, and the
radius of the spool in the space below. Don't forget the units!
Page 12-26 Workshop Physics II Activity Guide SFU 1067
© 1990-93 Dept. of Physics and Astronomy, Dickinson College Supported by FIPSE (U.S. Dept. of Ed.) and NSF. Modified for SFU by N. Alberding, 2005.
Assembled Apparatus
bo
lt
me
tal w
ash
er
CD stack
rub
be
r w
ash
er
rotational motion!
sensor
use 12” LPrubber washer maybe not needed.
Recommended