Chapter12 Ch.12 Rota/onofarigidbody
Uptotonow,wehavebasicallymodeledallobjectsas pointobjects. Inthischapter webeginamorerealis/cdescrip/onofanobjectasa distributedobject. ThisrequiresustoexpandouruseofNewton slawstoincorporate simultaneoustransla/onlinearmo/on)androta/onangularmo/on).
Reviewofrota/on: 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
Reviewofrota/on:
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. Slide12N28
The fan blade is speeding up. What are the signs of ω and α? A. ωisposi/veandαisposi/ve. B. ωisposi/veandαisnega/ve. C. ωisnega/veandαisposi/ve. D. ωisnega/veandαisnega/ve. Slide12N30
For distributed objectsunderrota/on, theywillfreerotateaboutapointcalled the centerofmass CM). TheCMdoesNOThavetoresideinsideof theobject.
Centerofmassofacollec/onofobjects: 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 Na 0a2a Ex.12N1
CMofacollec/onofobjects: 2kgand3kgmassesareseparatedby40cm. Theyareconnectedbya1kgrod.Whatisthe centerofmassofthissystem? Ex.12N2 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
Example12N3: GivenarodoflengthLwithamassdensityof: λ x ) = " M %" $! # L ' 1+2 x % $ ' &# L & Findthemassandcenterofmass. Ex.12N3 x=0x=l
MomentofIner/a: Considertobeaformof rota/onalmass Moment!of!inertia:!! I = r 2 dm I = 2 m! i r i
Example12N4a: Findthemomentofiner/aabout axisa 1.Assumetheobjectsare eachseparatedbyadistance a. 2m 3m m m A 1
Example12N4b: Findthemomentofiner/aaboutaxisA 2. 2m 3m m m A 2 a)16ma 2 b)7ma 2 c)5ma 2 d)9ma 2
ParallelNAxisTheorem:I axis =I CM +md 2 + m 3m m Ex.12N5 Compute:I CM andi A A CM
Mathema/csbreak:ThecrossNproductvector,vectormul/plica/on) NAcrossproductresultsinanewvectorthatisperpendiculartothe direc/onofbothoftheoriginalvectors. NThe righthandrule canbeusedtoevaluatethenewvectordirec/on. 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
Wehaveconsideredforcestogiverisetothemo/onofanobject. Inpar/cular:linearmo/on,circularmo/on Forarigidobject,theapplica/onofaTORQUEwillgiverisetoa rota/onoftheobjectaboutanaxis.
DefineTorque as pushorpull thatgivesrisetorota/on.! τ =! r! F τ = rf sinθ CCWrota/on posi/ve CWrota/onNnega/ve θ
Someexamples findthenumberofappliedtorques;arethey posi/veornega/ve? Pushingonadoorviewedfromabove)
Torquecalcula/ons: a) Let b) Right c) Intoboard d) Outofboard Torquecalcula/ons: a) Up b) Down c) Let d) Right
Examples findthenumberofappliedtorques;aretheyposi/ve ornega/ve? Hangingsign, supportedbywires wall sign
Newton s2 nd Lawforrota/on:! τ = I! α
Newton s2 nd Lawforrota/on:! τ = I! α m 3m m A F" Ex.12N6:Findtheangularaccelera/on
Ex.12N7:Solvefortheangularaccelera/onofa realpulley! τ = I! α F=100N m=10kg m p =5kg r p =0.1m Idisk)=½mr 2 m m p F"
Example12N8: Solveforpowerusageofagrindingwheel: PushdownwithaforceF=20Nfor5secat angleof110.iftheini/alangularvelocityis 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 Sumoftransla/onal+rota/onalmo/on v CM v ROT NRω Rω v=2v cm =2Rω v=0 v=v cm =Rω
Transla/onalKine/cEnergy: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
Example12N9: Ahoopandaballofthesamemass,m=1.2kg, andradius,r=0.3m,slidedowninclinedplanes withh=0.5m.whichonearrivesattheborom withahighervelocity? I hoop =MR 2 I sphere =2/5MR 2
Angularmomentum: L=IωkgNm 2 /s) ) τ = dl dt = d Iω dt = I dω dt = Iα N N Hassimilarpropertytolinearmomentum Ifτ net =0,angularmomentumisconserved L ini/al =L final )
Ex.12N10:Angularmomentum1) Athindiskm=1kg,r=0.1m)is rota/ngat10rad/sec.amassm= 0.4kgisplacedattheouteredgeof thedisk.whatisthenewangular velocity?
Ex.12N11Angularmomentum A1500kg,0.8mdia.cylindricalsatellite hasapairof50kgvariablesolarpanels. Thesatelliteisini/allyrota/ngat2rev/ secwhenthepanelsat1m.ifthepanels moveoutto1.3m,whatisthenew rota/onalspeed. Panel:L=0.4mxH=0.2m
N L h θ f fric mg N# f fric # mg# L" h" θ