Phsics 0 ectue 8 ectue 8 Goals: Define and anale toque ntoduce the coss poduct Relate otational dnamics to toque Discuss wok and wok eneg theoem with espect to otational motion Specif olling motion (cente of mass velocit to angula velocit So what causes otation? Phsics 0: ectue 8, Pg So what foces make things otate? l Which of these scenaios will cause the ba to spin? ied otation ais. A B C D E Phsics 0: ectue 8, Pg Net etenal toques cause objects to spin l An etenal foce (o foces) popel placed induced changes in the angula velocit. This action is defined to be a toque l A foce vecto o a component of a foce vecto whose line of action passes though the ais of otation povides no toque l A foce vecto o the component of a foce vecto whose line of action does NOT pass though the ais of otation povides toque l The eact position whee the foce is applied mattes. Alwas make sue the foce vecto s line of action contacts the point at which the foce is applied. Phsics 0: ectue 8, Pg 3 Net etenal toques cause objects to spin l Toque inceases popotionatel with inceasing foce l Onl components pependicula to vecto ields toque l Toque inceases popotionatel with inceasing distance fom the ais of otation. τ sin l This is the magnitude of the toque l is the angle between the adius and the foce vecto l Use Right Hand Rule fo sign Phsics 0: ectue 8, Pg 4 oce vecto line of action must pass though contact point l oce vecto cannot be moved anwhee l Just along line of action Resolving foce vecto into components is also valid l e point: Vecto line of action must pass though contact point (point to which foce is applied) = t line of action line of action t t = sin Phsics 0: ectue 8, Pg 5 Phsics 0: ectue 8, Pg 6 Page
Phsics 0 ectue 8 A. Case B. Case C. Same Eecise Toque l n which of the cases shown below is the toque povided b the applied foce about the otation ais biggest? n both cases the magnitude and diection of the applied foce is the same. l Remembe toque equies, and sin o the tangential foce component times pependicula distance ais case case Phsics 0: ectue 8, Pg 7 Toque is constant along the line of action l Even though case has a much lage adius vecto the toque emains constant. l Case : τ = l Case : τ = sin = sin l Notice that the sin = sin (π ) = - cos π sin = sin l Case : τ = sin = sin = sin (π ) = Hee = 35 and = ½ so sin = -/ ( ½ ) = ais case case π π Phsics 0: ectue 8, Pg 8 Toque, like ω, is a vecto quantit l Magnitude is given b () sin () tangential (3) pependicula to line of action l Diection is paallel to the ais of otation with espect to the ight hand ule tangential line of action sin adial l Toque is the otational equivalent of foce Toque has units of kg m /s = (kg m/s ) m = N m Phsics 0: ectue 8, Pg 9 v τ = Toque can also be calculated with the vecto coss poduct l The vecto coss poduct is just a definition v = î j ˆ kˆ î ĵ kˆ v = v = î ˆj = ( ( )î ( ) ĵ )kˆ Phsics 0: ectue 8, Pg 0 kˆ Toque and Newton s nd aw v = ma = l Appling Newton s second law Tangential ma Tangential t = mat at t = m = m α = α t = τ = α Phsics 0: ectue 8, Pg l A solid 6.0 kg wheel with adius = 0.50 m otates feel about a fied ale. Thee is a ope wound aound the wheel. Stating fom est, the ope is pulled such that it has a constant tangential foce of = 8 N. How man evolutions has the wheel made afte 0 seconds? Phsics 0: ectue 8, Pg Page
Phsics 0 ectue 8 l m = 6.0 kg adius = 0.50 m l = 8 N fo 0 seconds Constant Constant τ constant α solid disk = ½ m = kg m α = τ = α = 4 Nm/ kg m = ad/s = 0 ω 0 t ½ α t 0 0 t ½ α t Rev = ( 0 ) / π =( 0 ½ α t )/ π Rev = (0.5 00) / 6.8 = 6 Wok l Conside the wok done b a foce acting on an object constained to move aound a fied ais. o an angula displacement d then ds = d dw = Tangential d = t ds dw = ( t ) d ais of otation dw = τ d (and with a constant toque) R d l We can integate this to find: W = τ = τ ( f i ) φ d =Rd Phsics 0: ectue 8, Pg 3 Phsics 0: ectue 8, Pg 4 Rotation & inetic Eneg... l The kinetic eneg of a otating sstem looks simila to that of a point paticle: Wok & inetic Eneg: l Recall the Wok inetic-eneg Theoem: = W NET o W EXT Point Paticle = mv v is linea velocit m is the mass. Rotating Sstem = ω = m i i i ω is angula velocit is the moment of inetia about the otation ais. l This is tue in geneal, and hence applies to otational motion as well as linea motion. l So fo an object that otates about a fied ais: = ( ω f ωi ) = WNET Phsics 0: ectue 8, Pg 5 Phsics 0: ectue 8, Pg 6 l A solid 6 kg wheel with adius = 0.50 m otates feel about a fied ale. Thee is a ope wound aound the wheel. Stating fom est, the ope is pulled such that it has a constant tangential foce of = 8 N. What is the angula velocit afte 6 evolutions? l Mass 6 kg adius = 0.50 m solid disk =½m = kg m l Constant tangential foce of = 8 N. l Angula velocit afte ou pull fo 3π ad? W = ( f - i ) = =0.5 8.0 3π J = 40 J = ( f - i ) = f = ½ ω = 40 J ω = 0 ad/s Phsics 0: ectue 8, Pg 7 Phsics 0: ectue 8, Pg 8 Page 3
Phsics 0 ectue 8 Eecise Wok & Eneg l Stings ae wapped aound the cicumfeence of two solid disks and pulled with identical foces fo the same linea distance. l Disk, on the left, has a bigge adius, but both have the same mass. Both disks otate feel aound aes though thei centes, and stat at est. v Which disk has the biggest angula velocit afte the pull? ω ω W = τ = d = ½ ω Smalle bigge ω (A) Disk (B) Disk (C) Same stat finish d Phsics 0: ectue 8, Pg 9 Home Eample: Rotating Rod l A unifom od of length =0.5 m and mass m= kg is fee to otate on a fictionless pin passing though one end as in the igue. The od is eleased fom est in the hoiontal position. What is (A) its angula speed when it eaches the lowest point? (B) its initial angula acceleation? (C) initial linea acceleation of its fee end? m Phsics 0: ectue 8, Pg 0 Eample: Rotating Rod l A unifom od of length =0.5 m and mass m= kg is fee to otate on a fictionless hinge passing though one end as shown. The od is eleased fom est in the hoiontal position. What is (B) its initial angula acceleation?. o foces ou need to locate the Cente of Mass is at / ( halfwa ) and put in the oce on a BD. The hinge changes evething! m Σ = 0 occus onl at the hinge but τ = α = sin 90 at the cente of mass and ( m(/) ) α = (/) Eample: Rotating Rod l A unifom od of length =0.5 m and mass m= kg is fee to otate on a fictionless hinge passing though one end as shown. The od is eleased fom est in the hoiontal position. What is (C) initial linea acceleation of its fee end?. o foces ou need to locate the Cente of Mass is at / ( halfwa ) and put in the oce on a BD. The hinge changes evething! m a = α and solve fo α Phsics 0: ectue 8, Pg Phsics 0: ectue 8, Pg / Eample: Rotating Rod l A unifom od of length =0.5 m and mass m= kg is fee to otate on a fictionless hinge passing though one end as shown. The od is eleased fom est in the hoiontal position. What is (A) its angula speed when it eaches the lowest point?. o foces ou need to locate the Cente of Mass is at / ( halfwa ) and use the Wok-Eneg Theoem. The hinge changes evething! m W = h = ½ ω W = / = ½ ( m (/) ) ω and solve fo ω Connection with motion l f an object of mass M is moving lineal at velocit V without otating then its kinetic eneg is = l f an object of moment of inetia is otating in place about its cente of mass at angula velocit ω then its kinetic eneg is R = ω l What if the object is both moving lineal and otating? V M T Phsics 0: ectue 8, Pg 3 Phsics 0: ectue 8, Pg 4 Page 4
Phsics 0 ectue 8 Connection with motion... l So fo a solid object which otates about its cente of mass and whose is moving: l Now conside a clinde olling at a constant speed. V V ω The clinde is otating about and its is moving at constant speed (V ). Thus its total kinetic eneg is given b : Phsics 0: ectue 8, Pg 5 Phsics 0: ectue 8, Pg 6 l Again conside a clinde olling at a constant speed. l Now conside a clinde olling at a constant speed. V V V The clinde is otating about and its is moving at constant speed (V ). Thus its total kinetic eneg is given b : Phsics 0: ectue 8, Pg 7 Phsics 0: ectue 8, Pg 8 Motion l Again conside a clinde olling at a constant speed. Rotation onl v t R Both with v t = v V V Sliding onl V l Read though.3 o Tuesda Phsics 0: ectue 8, Pg 9 Phsics 0: ectue 8, Pg 30 Page 5