Chapter10&11 Ch.10 Rota0onofarigidbody Ch.11 TorqueandAngular Momentum
s = rθ v = ds dt = r dθ dt = rω = v tan ω = 2π f = 2π T α = dω! dt CCW:ωis+ CW:ωisD a centripetal = v2 r v tangential = ωr a! tangential = αr
Linear!to!Circular!Motion x θ v ω! a α θ f =θ i +ω i ω f = ω i +α t t)+ 1 2 α t ) 2 ) ) ω 2! f = ω 2 i +2α θ f θ i
ExampleCirc1: A0.5mdiameterdiskisspunupfromrestto500rpm over30seconds.findtheangularaccelera0on,final centripetalaccelera0on,andtotalangular displacementduringthismo0on. θ f =θ i +ω i ω f = ω i +α t t)+ 1 2 α t ) 2 ) ) ω 2! f = ω 2 i +2α θ f θ i
Reviewofrota0on: θ f =θ i +ω i t + 1 2 αt2 ω f =ω i +αt ω 2! f =ω 2 i +2α θ ω = dθ s = rθ!!!!!!!!!!!f = 1 dt T!!!!!!!!!!!v t α = dω! dt = d 2 θ ω =2π f!!!!!!!a dt 2 r = a c = v2! =2πrf = 2πr T =ωr r!!!!!!a tan = v2 r =ω2 r
iclickerques0ons
Reviewofrota0on:
Two coins rotate on a turntable. Coin B is twice as far from the axis as coin A. A. The angular velocity of A is twice that of B. B. The angular velocity of A equals that of B. C. The angular velocity of A is half that of B. Slide10D28
The fan blade is speeding up. What are the signs of ω and α? A. ωisposi0veandαisposi0ve. B. ωisposi0veandαisnega0ve. C. ωisnega0veandαisposi0ve. D. ωisnega0veandαisnega0ve. Slide10D30
Uptotonow,wehavebasicallymodeledallobjectsas pointobjects. Inthischapter webeginamorerealis0cdescrip0onofanobjectasa distributedobject. ThisrequiresustoexpandouruseofNewton slawstoincorporate simultaneoustransla0onlinearmo0on)androta0onangularmo0on).
For distributed objectsunderrota0on, theywillfreerotateaboutapointcalled the centerofmass CM). TheCMdoesNOThavetoresideinsideof theobject.
Centerofmassofacollec0onofobjects: x cm = 1! M m x = m x +m x +m x +... 1 1 2 2 3 3 i i m 1 +m 2 +m 3 +... y cm = 1! M m y = m y +m y +m y +... 1 1 2 2 3 3 i i m 1 +m 2 +m 3 +... a 0 Da 0a2a Ex.10D1
CMofacollec0onofobjects: 2kgand3kgmassesareseparatedby40cm. Theyareconnectedbya1kgrod.Whatisthe centerofmassofthissystem? Ex.10D2 2kg 3kg x cm = 1! M m x = m x +m x +m x +... 1 1 2 2 3 3 i i m 1 +m 2 +m 3 +...
Mass:!!!!M = x cm = 1 M y cm = 1! M xdm ydm dm Linear:!!!!dm= λ!dx Area:!!!!!!!dm=σ!dA! Volume:!!dm= ρ!dv λ=mass/length σ=mass/area ρ=mass/volume x=x 1 x=x 2 dm
Example10D3: GivenarodoflengthLwithamassdensityof: λ x ) = " M %" $! # L ' 1+2 x % $ ' &# L & Findthemassandcenterofmass. Ex.10D3 x=0x=l
MomentofIner0a: Considertobeaformof rota0onalmass Moment!of!inertia:!! I = r 2 dm I = 2 m! i r i
Example10D4a: Findthemomentofiner0aabout axisa 1.Assumetheobjectsare eachseparatedbyadistance a. 2m 3m m m A 1
Example10D4b: Findthemomentofiner0aaboutaxisA 2. 2m 3m m m A 2 a)16ma 2 b)7ma 2 c)5ma 2 d)9ma 2
ParallelDAxisTheorem:I axis =I CM +md 2 + m 3m m Ex.10D5 Compute:I CM andi A A CM
Mathema0csbreak:ThecrossDproductvector,vectormul0plica0on) DAcrossproductresultsinanewvectorthatisperpendiculartothe direc0onofbothoftheoriginalvectors. DThe righthandrule canbeusedtoevaluatethenewvectordirec0on. C B A A B = C C = ABsinθ ab Key property: B A = C = A B ) î ĵ ˆk +#.# î ĵ = ˆk ĵ ˆk = î ˆk î = ĵ +) ĵ î = k ˆk ĵ = î î ˆk = ĵ ) î î = ĵ ĵ = ˆk ˆk = 0
Someexamples: Vectoroutofpage Vectorintopage A+ B+ A+ B+ A+ B+ A+ B+ A+ B+ î ĵ ˆk +#.# î ĵ = ˆk ĵ ˆk = î ˆk î = ĵ +) ĵ î = k ˆk ĵ = î î ˆk = ĵ ) î î = ĵ ĵ = ˆk ˆk = 0
Someexamples: Vectoroutofpage Vectorintopage A = 3î + 4 ˆk B = 2î ĵ Find : A B A = 2 ĵ + 3 ˆk B = 2î 3 ˆk Find : A B î ĵ ˆk +#.# î ĵ = ˆk ĵ ˆk = î ˆk î = ĵ +) ĵ î = k ˆk ĵ = î î ˆk = ĵ ) î î = ĵ ĵ = ˆk ˆk = 0
Wehaveconsideredforcestogiverisetothemo0onofanobject. Inpar0cular:linearmo0on,circularmo0on Forarigidobject,theapplica0onofaTORQUEwillgiverisetoa rota0onoftheobjectaboutanaxis.
DefineTorque as pushorpull thatgivesrisetorota0on.! τ =! r! F τ = rf sinθ CCWrota0on posi0ve CWrota0onDnega0ve θ
Someexamples findthenumberofappliedtorques;arethey posi0veornega0ve? Pushingonadoorviewedfromabove)
Examples findthenumberofappliedtorques;aretheyposi0ve ornega0ve? Hangingsign, supportedbywires wall sign
Newton s2 nd Lawforrota0on:! τ = I! α Considertwocases: D Sta0c:α=0 D Dynamic:α 0
Sta0cequilibrium1: GivenaplankofmassMandlengthLwith apivotat2/3)l.whatisthemaximum massofaboxthatcanbeplacedatone endandallowthesystemtoremain balanced?
Sta0cequilibrium2: A20kgsignishungfromtheendofa50 kg,1mlongrod.ifthesystemis supportedbyawireat30,whatisthe tensioninthewire? wall sign
Newton s2 nd Lawforrota0on:! τ = I! α A F" Ex.10D6a:Findtheangularaccelera0on:Rodpushedatoneend
Newton s2 nd Lawforrota0on:! τ = I! α A F" Ex.10D6a:Findtheangularaccelera0on:Platepushedatoneend Assume:PlatehasmassM,LxL;RodhasmassM,lengthL.
Newton s2 nd Lawforrota0on:! τ = I! α m 3m m A F" Ex.10D6c:Findtheangularaccelera0on
Ex.10D7:Solvefortheangularaccelera0onofa realpulley! τ = I! α F=100N m=10kg m p =5kg r p =0.1m Idisk)=½mr 2 m m p F"
Example10D8: Solveforpowerusageofagrindingwheel: PushdownwithaforceF=20Nfor5secat angleof110.iftheini0alangularvelocityis 200rad/sec,findfinalvelocityandpower usageover5sec. F=20N m=10kg m w =60kg r p =0.2m Idisk)=½mr 2 F" m p
Rollingw/oslipping rollingconstraint:v cm =ωr Sumoftransla0onal+rota0onalmo0on v CM v ROT DRω Rω v=2v cm =2Rω v=0 v=v cm =Rω
Transla0onalKine0cEnergy:K rot =½Iω 2 Forrolling:K=K trans +K rot K = 1 2 mv 2 CM + 1 2 I cm ω 2 = K CM + K rot
Example10D9: Ahoopandaballofthesamemass,m=1.2kg, andradius,r=0.3m,slidedowninclinedplanes withh=0.5m.whichonearrivesatthebouom withahighervelocity? I hoop =MR 2 I sphere =2/5MR 2
Angularmomentum: L=IωkgDm 2 /s) ) τ = dl dt = d Iω dt = I dω dt = Iα D D Hassimilarpropertytolinearmomentum Ifτ net =0,angularmomentumisconserved L ini0al =L final )
Ex.10D10:Angularmomentum1) Athindiskm=1kg,r=0.1m)is rota0ngat10rad/sec.amassm= 0.4kgisplacedattheouteredgeof thedisk.whatisthenewangular velocity?
Ex.10D11Angularmomentum A1500kg,0.8mdia.cylindricalsatellite hasapairof50kgvariablesolarpanels. Thesatelliteisini0allyrota0ngat2rev/ secwhenthepanelsat1m.ifthepanels moveoutto1.3m,whatisthenew rota0onalspeed. Panel:L=0.4mxH=0.2m