Objects can have translational energy Objects can have rotational energy Objects can have both K = ½ m v 2 + ½ I ω 2

Transcription

1 Physics 07 Lectue 17 Goals: Lectue 17 Chapte 1 Define cente of mass Analyze olling motion Use Wok Enegy elationships Intoduce toque Equilibium of objects in esponse to foces & toques Assignment: HW7 due tomoow Wednesday, Exam eview Physics 07: Lectue 17, Pg 1 Combining tanslation and otation Objects can have tanslational enegy Objects can have otational enegy Objects can have both K = ½ m v + ½ I ω Physics 07: Lectue 17, Pg Page 1

2 Physics 07 Lectue 17 1 st : A special point fo otation Cente of ass (C) A fee object will otate about its cente of mass. Cente of mass: Whee the system is balanced! A mobile exploits this centes of mass. + m 1 m + m m 1 mobile Physics 07: Lectue 17, Pg 3 System of Paticles: Cente of ass How do we descibe the position of a system made up of many pats? Define the Cente of ass (aveage position): o a collection of N individual point like paticles whose masses and positions we know: C N i= 1 m i i 1 y C m 1 m (In this case, N = ) x Physics 07: Lectue 17, Pg 4 Page

3 Physics 07 Lectue 17 Sample calculation: Conside the following mass distibution: m at ( 0, 0) N m at (1,1) C m = i i i= 1 = X C î + Y X C = (m x 0 + m x 1 + m x 4 )/4m metes C = (1,6) Y C = (m x 0 + m x 1 + m x 0 )/4m metes C ĵ + Z C (1,1) m X C = 1 metes Y C = 6 metes m m kˆ m at (4, 0) (0,0) (4,0) Physics 07: Lectue 17, Pg 5 System of Paticles: Cente of ass o a continuous solid, one can convet sums to an integal. N y dm x C C= i= 1 dm = dm whee dm is an infinitesimal mass element but thee is no new physics. m i i dm Physics 07: Lectue 17, Pg 6 Page 3

4 Physics 07 Lectue 17 Connection with motion... An unconstained igid object with otation and tanslation otates about its cente of mass! KTOTAL TOTAL =K + otation K =K + otation Any point p otating: K p K 1 = Tanslation mv C N i= 1 C ω m 1 1 m ) pvp = mp( p i i p p p V C p ω p p p Physics 07: Lectue 17, Pg 7 Wok & Kinetic Enegy: Wok Kinetic-Enegy Theoem: K = W NET Applies to both otational as well as linea motion. K = ( ω ) 1 f ωi + m(vcf vci = WNET 1 ) I What if thee is olling without slipping? Physics 07: Lectue 17, Pg 8 Page 4

5 Physics 07 Lectue 17 Same Example : olling, without slipping, otion A solid disk is about to oll down an inclined plane. What is its speed at the bottom of the plane? h θ v? Physics 07: Lectue 17, Pg 9 olling without slipping motion Again conside a cylinde olling at a constant speed. V C V C C Physics 07: Lectue 17, Pg 10 Page 5

6 Physics 07 Lectue 17 otion Again conside a cylinde olling at a constant speed. otation only V Tang = ω C V C Both with V Tang = V C V C C Sliding only C V C If acceleation a cente of mass = -α Physics 07: Lectue 17, Pg 11 Example : olling otion A solid cylinde is about to oll down an inclined plane. What is its speed at the bottom of the plane? Use Wok-Enegy theoem Disk has adius h θ v? gh = ½ v + ½ I C ω and v =ω gh = ½ v + ½ (½ )(v/) = ¾ v v = (gh/3) ½ Physics 07: Lectue 17, Pg 1 Page 6

7 Physics 07 Lectue 17 How do we econcile foce, angula velocity and angula acceleation? Physics 07: Lectue 17, Pg 13 om foce to spin (i.e., ω)? A foce applied at a distance fom the otation axis gives a toque θ Tangential = Tang sin θ adial If a foce points at the axis of otation the wheel won t tun Thus, only the tangential component of the foce mattes With toque the position & angle of the foce mattes τ NET = Tang sin θ a Tangential adial Physics 07: Lectue 17, Pg 14 Page 7

8 Physics 07 Lectue 17 otational Dynamics: What makes it spin? a Tangential τ NET = Tang sin θ adial Toque is the otational equivalent of foce Toque has units of kg m /s = (kg m/s ) m = N m τ NET = Tang = m a Tang = m α = (m ) α o evey little pat of the wheel Physics 07: Lectue 17, Pg 15 Toque The futhe a mass is away fom this axis the geate the inetia (esistance) to otation a Tangential andial τ NET = Iα This is the otational vesion of NET = ma oment of inetia, I Σ i m i i, is the otational equivalent of mass. If I is big, moe toque is equied to achieve a given angula acceleation. Physics 07: Lectue 17, Pg 16 Page 8

9 Physics 07 Lectue 17 otational Dynamics a Tangential τ NET = Iα sin θ adial A constant toque gives constant angula acceleation if and only if the mass distibution and the axis of otation emain constant. Physics 07: Lectue 17, Pg 17 Toque, like ω, has pos./neg. values agnitude is given by (1) sin θ () tangential (3) pependicula to line of action Diection is paallel to the axis of otation with espect to the ight hand ule cos(90 θ) = Tang. line of action sin θ a 90 θ θ adial And fo a igid object τ = I α Physics 07: Lectue 17, Pg 18 Page 9

10 Physics 07 Lectue 17 Statics Equilibium is established when Tanslational motion Σ Net = 0 0 otational motion Στ Net = In 3D this implies SIX expessions (x, y & z) Physics 07: Lectue 17, Pg 19 Example Two childen (30 kg & 60 kg) sit on a hoizontal teete-totte. The lage child is at the end of the ba and 1.0 m fom the pivot point. The smalle child is tying to figue out whee to sit so that the teete-totte emains hoizontal and motionless. The teete-totte is a unifom ba of length 3.0 m and mass 30 kg. Assuming you can teat both childen as point like paticles, what is the initial angula acceleation of the teete-totte when the lage child lifts up thei legs off the gound (the smalle child can t each)? The moment of inetia of the ba about the pivot is 30 kg m. o the static case: otational motion Στ Net = 0 Physics 07: Lectue 17, Pg 0 Page 10

11 Physics 07 Lectue 17 Example: Soln. Use Στ Net = 0 30 kg N 30 kg 60 kg 0.5 m 1 m 300 N 300 N 600 N Daw a ee Body diagam (assume g = 10 m/s ) 0 = 300 d x N x x 1.0 0= d d = 1.5 m fom pivot point Physics 07: Lectue 17, Pg 1 ecap Assignment: HW7 due tomoow Wednesday: eview session Physics 07: Lectue 17, Pg Page 11