11. Rotational Motion 11. Rotasiebeweging
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1 Rotation Transparencies.doc:. Rotational Kinematics 3/05/0 :57:00. Rotational Motion. Rotasiebeweging. Rotational Kinematics (HRW 0- to 0-5) Rotational variables. Rotasiekinematika (HRW 0- tot 0-5) Rotasieveranderlikes Angular position (always in radians!) θ θ ( t) θ s r Hoekposisie (altyd in radiaal!) Angular displacement [rad] θ θ θ Hoekverplasing [rad] Angular velocity [rad/s] dθ ( t) ω ( t) Hoeksnelheid [rad/s] Angular acceleration [rad/s ] dω( t) α ( t) Hoekversnelling [rad/s ] Rotation Transparencies.doc:. Rotational Kinematics 3/05/0 :57:00
2 Rotation Transparencies.doc:. Rotasiekinematika 3/05/0 :57:00 Relating linear and angular variables Verband tussen liniëre en hoekveranderlikes Position s rθ Posisie Speed ds dθ v r rω Spoed Tangential acceleration dv dω at r α r Raaklynige versnelling Radial acceleration (centripetal acceleration) a a v r r centripetal / Radiale versnelling (sentripetale versnelling) Rotation Transparencies.doc:. Rotasiekinematika 3/05/0 :57:00
3 Rotation Transparencies.doc:. Kinetic energy and moment of intertia 3 3/05/0 :57:00. Kinetic energy and moment of intertia (HRW 0-6, 0-7). Kinetiese energie en traagheidsmoment (HRW 0-6, 0-7) Kinetic energy of a rotating body Kinetiese energie van n roterende liggaam K m v + m v + m v m v i i m ( ωr ) i miri ω i K Iω, I m r i i Rotational inertia (Moment of inertia) Point masses Continuous body I I Traagheidsmoment m r Puntmassas i i r dm Kontinue liggaam Parallel-axis theorem I I Mh CM + Parallel-as-stelling Rotation Transparencies.doc:. Kinetic energy and moment of intertia 3 3/05/0 :57:00
4 Rotation Transparencies.doc:. Kinetiese energie en traagheidsmoment 4 3/05/0 :57:00 Some rotational inertia n Aantal traagheidsmomente Rotation Transparencies.doc:. Kinetiese energie en traagheidsmoment 4 3/05/0 :57:00
5 Rotation Transparencies.doc:.3 Torque and Newton s 5 3/05/0 :57:00.3 Torque and Newton s second law (HRW 0-8, 0-9).3 Draaimoment en Newton se tweede wet (HRW 0-8, 0-9) Torque Draaimoment (wringkrag) Torque τ rf sinφ Draaimoment / Wringkrag Newton s second law τ Iα Newton se tweede wet Rotation Transparencies.doc:.3 Torque and Newton s 5 3/05/0 :57:00
6 Rotation Transparencies.doc:.4 Work and kinetic energy 6 3/05/0 :57:00.4 Work and kinetic energy (HRW 0-0).4 Arbeid en kinetiese energie (HRW 0-0) Work done by a torque Arbeid deur n wringkrag verrig Work done by a torque W θ τ θ f d Arbeid deur n wringkrag verrig θi For a constant torque W τ ( θ θ ) Vir n konstante wringkrag f Work kinetic energy theorem K W Arbeid kinetiese energie-stelling i Power dw P τω Drywing Rotation Transparencies.doc:.4 Work and kinetic energy 6 3/05/0 :57:00
7 Rotation Transparencies.doc:.5 Rolling 7 3/05/0 :57:00.5 Rolling (HRW - to -5).5 Rolbeweging (HRW - tot -5) vcom ωr Kinetic energy of rolling Kinetiese energie van n rollende voorwerp K I ω + Mv com com Rotation Transparencies.doc:.5 Rolling 7 3/05/0 :57:00
8 Rotation Transparencies.doc:.6 Conservation of angular momentum 8 3/05/0 :57:00.6 Conservation of angular momentum (HRW -6 to -).6 Behoud van hoekmomentum (HRW -6 tot -) Vector product (cross product) (HRW 3-7) Vektorproduk (kruisproduk) (HRW 3-7) c a b c absinθ, direction given by right hand rule note that a b b a Cross product is usually calculated using determinants. Angular variables as vectors Hoekveranderlikes as vektore ω α Torque / Draaimoment: τ r F Newton II: τ Iα [ τ rf sin φ] (Note: θ can not be represented by a vector.) Rotation Transparencies.doc:.6 Conservation of angular momentum 8 3/05/0 :57:00
9 Rotation Transparencies.doc:.6 Behoud van hoekmomentum 9 3/05/0 :57:00 Angular momentum Hoekmomentum (draaimomentum) Angular momentum defined for a particle l r p mr v Hoekmomentum van n enkele deeltjie Angular momentum for a system of particles L l Hoekmomentum van n stelsel van deeltjies Angular momentum for a rigid body L Iω Hoekmomentum vir n starre liggaam Newton s law in terms of angular momentum τ d L Newton se wet i.t.v. hoekmomentum Conservation of angular momentum If the net torque acting on a system is zero, the angular momentum of the system remains constant, i.e. L f Li. Behoud van hoekmomentum As daar geen netto wringkrag op n stelsel is nie, bly die hoekmomentum van die stelsel konstant, d.w.s. L f Li. Rotation Transparencies.doc:.6 Behoud van hoekmomentum 9 3/05/0 :57:00
10 Rotation Transparencies.doc: Summary: Comparison between linear and rotational motion 0 3/05/0 :57:00 Summary: Comparison between linear and rotational motion Opsomming: Vergelyking tussen lineêre en rotasiebeweging Linear kinematics Lineêre kinematika x θ s rθ v a dx dv dθ ω v rω dω α at rα, [ α s rω v / r ] Energy and momentum K W mv K Iω F x W τ θ Energie en momentum I miri r dm [ τ rf sinφ ] F ma τ Iα p mv L Iω F dp τ dl Rotation Transparencies.doc: Summary: Comparison between linear and rotational motion 0 3/05/0 :57:00
11 Rotation Transparencies.doc:.7 Equilibrium (HRW - to -5) 3/05/0 :57:00.7 Equilibrium (HRW - to -5).7 Ewewig (HRW - tot -5) Requirements for equilibrium Voorwaardes vir ewewig F net 0 τ 0 Rotation Transparencies.doc:.7 Equilibrium (HRW - to -5) 3/05/0 :57:00
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