Uniform Circular Motion. Banked and Unbanked Curves Circular Orbits Nonuniform Circular Motion Tangential and Angular Acceleration Artificial Gravity

Transcription

1 Chapte 5: Cicula Motion Unifom Cicula Motion Radial Acceleation Banked and Unbanked Cues Cicula Obits Nonunifom Cicula Motion Tangential and Angula Acceleation Atificial Gaity 1

2 Unifom Cicula Motion y θ f Δθ θ i x θ is the angula position. Angula displacement: Δθ θ f θ i Note: angles measued CW ae negatie and angles measued CCW ae positie. θ is measued in adians. π adians eolution

3 The aeage and instantaneous angula elocities ae: ω a Δθ Δθ and ω lim Δt Δt 0 Δt ω is measued in ads/sec. 3

4 y θ f aclength s Δθ Δθ θ i x Δθ s Δθ is a atio of two lengths; it is a dimensionless atio! 4

5 y θ f An object moes along a cicula path of adius ; what is its aeage speed? Δθ θ i x total distance Δ θ Δ θ a total time Δt Δt ω a Also, ω (instantaneous alues). 5

6 The time it takes to go one time aound a closed path is called the peiod (T). total distance total time a π T Compaing to ω: π ω πf T f is called the fequency, the numbe of eolutions (o cycles) pe second. 6

7 Centipetal Acceleation Conside an object moing in a cicula path of adius at constant speed. y Hee, Δ 0. The diection of is changing. x If Δ 0, then a 0. The net foce cannot be zeo. 7

8 Conclusion: to moe in a cicula path, an object must hae a nonzeo net foce acting on it. It is still tue that ΣF ma, but what acceleation do we use? 8

9 The elocity of a paticle is tangent to its path. Fo an object moing in unifom cicula motion, the acceleation is adially inwad. 9

10 The magnitude of the centipedal (o adial) acceleation is: a ω ω 10

11 Unifom Cicula Motion Slide 6-13

12 Examples The disk in a had die in a desktop compute otates at 700 pm. The disk has a diamete of 5.1 in (13 cm.) What is the angula speed of the disk? The had die disk in the peious example otates at 700 pm. The disk has a diamete of 5.1 in (13 cm.) What is the speed of a point 6.0 cm fom the cente axle? What is the acceleation of this point on the disk? Slide 6-14

13 Quiz 1. Fo unifom cicula motion, the acceleation A. is paallel to the elocity. B. is diected towad the cente of the cicle. C. is lage fo a lage obit at the same speed. D. is always due to gaity. E. is always negatie. Slide 6-

14 Answe 1. Fo unifom cicula motion, the acceleation B. is diected towad the cente of the cicle. Slide 6-3

15 Undestanding When a ball on the end of a sting is swung in a etical cicle: We know that the ball is acceleating because A. the speed is changing. B. the diection is changing. C. the speed and the diection ae changing. Slide 6-9

16 Answe When a ball on the end of a sting is swung in a etical cicle: We know that the ball is acceleating because B. the diection is changing. Slide 6-10

17 Undestanding When a ball on the end of a sting is swung in a etical cicle: What is the diection of the acceleation of the ball? A. Tangent to the cicle, in the diection of the ball s motion B. Towad the cente of the cicle Slide 6-11

18 Answe When a ball on the end of a sting is swung in a etical cicle: What is the diection of the acceleation of the ball? B. Towad the cente of the cicle Slide 6-1

19 Cicula Motion Dynamics When the ball eaches the beak in the cicle, which path will it follow? Slide 6-19

20 Answe When the ball eaches the beak in the cicle, which path will it follow? Slide 6-0

21 Foces in Cicula Motion ω a ωω { m F { net ma, towad cente of cicle} Slide 6-1

22 Example A leel cue on a county oad has a adius of 150 m. What is the maximum speed at which this cue can be safely negotiated on a ainy day when the coefficient of fiction between the ties on a ca and the oad is 0.40? Slide 6-4

23 Diing oe a Rise A ca of mass 1500 kg goes oe a hill at a speed of 0 m/s. The shape of the hill is appoximately cicula, with a adius of 60 m, as in the figue at ight. When the ca is at the highest point of the hill, a. What is the foce of gaity on the ca? b. What is the nomal foce of the oad on the ca at this point? Slide 6-6

24 Example: The oto is an amusement pak ide whee people stand against the inside id of a cylinde. Once the cylinde is spinning fast enough, the floo dops out. (a) What foce keeps the people fom falling out the bottom of the cylinde? y Daw an FBD fo a peson with thei back to the wall: x N f s w It is the foce of static fiction. 4

25 Example continued: (b) If μ s 0.40 and the cylinde has.5 m, what is the minimum angula speed of the cylinde so that the people don t fall out? Apply Newton s nd Law: ( 1) Fx N ma ( ) F f w 0 y s mω Fom (): f s w Fom (1) μ N s μ ( m ) s ω mg ω g μ s 9.8 m/s m ( )( ) 3.13 ad/s 5

26 Example (text poblem 5.79): A coin is placed on a ecod that is otating at pm. If μ s , how fafomthe fom cente of the ecod can the coin be placed without haing it slip off? Daw an FBD fo the coin: y N x f s Apply Newton s nd Law: () 1 Fx fs ma ( ) F N w 0 F y mω w 6

27 Example continued: () Fom 1 f s : f s mω ( mg ) m ω μ N μ s s Fom () μs g Soling fo : What is ω? ω ω e π ad 1min 33.3 min 1e 60 sec 3.5 ad/s μ g ω ( )( m/s ) 0.08 m s ( 3.50 ad/s) 7

28 Unbanked and Banked Cues Example (text poblem 5.0): A highway cue has a adius of 85 m. At what angle should the oad be banked so that a ca taeling at 6.8 m/s has no tendency to skid sideways on the oad? (Hint: No tendency to skid means the fictional foce is zeo.) θ Take the ca s motion to be into the page. 8

29 Example continued: y FBD fo the ca: θ N x w Apply Newton s Second Law: ( 1 ) F N sin x ( ) F N cosθ w 0 y θ ma m 9

30 Example continued: Rewite (1) and (): ( 1) N sinθ m ( ) N cosθθ mg Diide (1) by (): ( 6.8 m/s) ( 9.8 m/s )( 85 m) tanθ g θ

31 The Foce of Gaity Slide 6-31

32 Cicula Obits Conside an object of mass m in a cicula obit about the Eath. Eath The only foce on the satellite is the foce of gaity: F F g GmsM e msa m s Sole fo the speed of the satellite: Gm M s e GM e m s 3

33 Example: How high aboe the suface of the Eath does a satellite need to be so that it has an obit peiod of 4 hous? Fom peious slide: GM e Also need, π T Combine these expessions and sole fo : GM e T 4π 1 3 ( 11 )( Nm /kg kg) ( s) 4π m 1 3 Re + h h Re 35,000 km 33

34 GM e T 4π 1 3 is Keple s Thid Law. It can be genealized to: GM T 4π 1 3 Whee M is the mass of the cental body. Fo example, it would be M sun if speaking of the planets in the sola system. 34

35 Example: What is the minimum speed fo the ca so that it maintains i contact t with the loop when it is in the pictued position? FBD fo the ca at the top of the loop: y Apply Newton s nd Law: N w x F y N w N + w ma m m 35

36 Example continued: The appaent weight at the top of loop is: N N + w m m g N 0 when N m g 0 g This is the minimum speed needed to make it aound the loop. 36

37 Example continued: Conside the ca at the bottom of the loop; how does the appaent weight compae to the tue weight? FBD fo the ca at the bottom of the loop: y N x w Apply Newton s nd Law: F y N w ma N w m N m + g c m Hee, N > mg 37

38 Nonunifom Cicula Motion Hee, the speed is not constant. a a a t Thee is now an acceleation tangent to the path of the paticle. The net acceleation of the body is a t a a + a 38

39 a t a a t changes the magnitude of. a a changes the diection of. Can wite: F F t ma ma t 39

40 Atificial Gaity A lage otating cylinde in deep space (g 0). 40

41 FBD fo peson at the bottom position FBD fo peson at the top position N y y x x Apply Newton s nd Law to each: Fy N ma mω Fy N ma mω N 41

42 Example (text poblem 5.56): A space station is shaped like a ing and otates t to simulate gaity. If the adius of the space station is 10m, at what fequency must it otate so that it simulates Eath s gaity? Using the esult fom the peious slide: ω F y N m N ma mg m mω g 0.8 ad/sec The fequency is f (ω/π) Hz (o.7 pm). 4

43 Tangential and Angula Acceleation The aeage and instantaneous angula acceleation ae: α a Δω Δω and α lim Δ t Δ t 0 Δt α is measued in ads/sec. 43

44 Recalling that the tangential elocity is t ω means the tangential acceleation is Δt Δω α t Δt Δt α 44

45 The kinematic equations: Linea Angula t a Δ α ω ω Δ t t a t x x Δ Δ + Δ θ α ω θ θ Δ Δ + Δ t t x a Δ + 0 θ α ω ω Δ + 0 With α ω a t t and 45

46 Example (text poblem 5.66): A high speed dental dill is otating ti at ads/sec. d/ Though hhow many degees does the dill otate in 1.00 sec? Gien: ω ads/sec; Δt 1 sec; α 0 Want Δθ. 1 θ θ0 + ω0δt + αδt θ θ + ω t Δθ ω 0 0Δ ( ads/sec)( 1.0 sec) 0Δt ads degees 46