PHYSICS 220. Lecture 08. Textbook Sections Lecture 8 Purdue University, Physics 220 1

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1 PHYSICS 0 Lectue 08 Cicula Motion Textbook Sections Lectue 8 Pudue Univesity, Physics 0 1

2 Oveview Last Lectue Cicula Motion θ angula position adians ω angula velocity adians/second α angula acceleation adians/s Linea to Cicula convesions s θ Unifom Cicula Motion Speed is constant t Diection is changing Acceleation towad cente a v / Unifom Cicula Acceleation Kinematics Today Simila to linea Banked cuves Cicula obits Lectue 8 Pudue Univesity, Physics 0

3 Cicula Motion Angula velocity Linea velocity ω av Δθ Δtt v ω Δ Linea velocity is tangent to cicle Centipetal acceleation R a c v ω a v Diection: towad the cente Centipetal Acceleation is adially inwad Lectue 8 Pudue Univesity, Physics 0 3

4 Cicula vs. Linea Motion with compaison to 1-D kinematics Angula Linea α constant ω ω0 + αt 1 θ θ0 + ω0t+ αt a constant v v0 + at 1 x x0 + v0t+ at And fo a point at a distance R fom the otation axis: xrθ v ωr R a αrr Lectue 8 Pudue Univesity, Physics 0 4

5 Kinematics fo Cicula Motion Linea Vaiables x,v,a. Constant a with constant acceleation Angula Vaiables θ,ω,α. Constant α Δ x x x0 vavgδt Δ θ θ θ0 ωavgδt v v0 at ω ω α 0 t v0 + v ω0 + ω vavg ωavg 1 x x0 v0t+ at 1 θ θ0 ω0t+ αt v v aδx 0 ω ω αδθ 0 Lectue 8 Pudue Univesity, Physics 0 5

6 Quiz 1) Two childen ide on a mey-go-ound. Bob is m fom the axis of otation and May is 4 m fom the axis. Which is tue: A) May has lage speed and acceleation but the same angula speed B) Bob has lage speed, acceleation, and angula speed C) May has lage speed, acceleation, and angula speed Lectue 8 Pudue Univesity, Physics 0 6

7 Quiz ) Which statement is tue fo a ca in unifom cicula motion? A) The ca is moving with a constant speed, theefoe the ca has a net foce of zeo B) Fo the ca to stay taveling aound a hoizontal cicula tack, thee must be a net foce pointing adially inwad, towad the cente of the cicle. If thee wasn't, the ca would dive in a staight line. C) If you tun shap, you tend to be pulled in the opposite diection of the tun. F N W f R ΣF ma mv /R Lectue 8 Pudue Univesity, Physics 0 7

x-diection Fiction is diected towad the cente of the cicle Since fiction is the only foce acting

8 Unbanked Cuve A ca ounding a cuve tavels in an appoximate cicle The adius of this cicle is called the adius of cuvatue Foces in the y-diection Gavity and the nomal foce Foces in the x-diection Fiction is diected towad the cente of the cicle Since fiction is the only foce acting in the x-diection, it supplies the centipetal foce F fiction mv μs mg Lectue 8 Pudue Univesity, Physics 0 8

9 Unbanked Cuve What is the maximum velocity a ca can go aound an unbanked cuve? y : N mg 0 N mg x F f N μ mg v : c s μs s μ g s mv f s μ N The maximum velocity to go aound an un- banked cuve depends only on μ s (g and ae fixed) Dy oad: μ s 0.9 Icy oad: μ s Lectue 8 Pudue Univesity, Physics 0 9 s

Banked Cuve The maximum speed can be inceased

foces due to gavity and the nomal foce acting

10 Banked Cuve The maximum speed can be inceased by banking the cuve Assume no fiction between the ties and the oad The ca tavels in a cicle, so the net foce is a centipetal foce Thee ae foces due to gavity and the nomal foce acting on the ca Lectue 8 Pudue Univesity, Physics 0 10

11 Banked Cuve Thee is a hoizontal component of the nomal foce Letting the hoizontal be the x-diection The speed at which the ca will just be able to negotiate the tun without sliding up o down the banked oad is v gtanθ When θ 0, v 0 and you cannot tun on a vey icy oad without slipping Lectue 8 Pudue Univesity, Physics 0 11

A) What is the fictional foce on the ca?

12 Banked Cuve A ca dives aound a cuve with adius 410 m at a speed of 3 m/s. The oad is banked at 5.0. The mass of the ca is 1400 kg. A) What is the fictional foce on the ca? B) At what speed could you dive aound this cuve so that the foce of fiction is zeo? Lectue 8 Pudue Univesity, Physics 0 1

13 Banked Cuve y θ x y-diection Σ F ma y y N cosθ mg f sinθ 0 x-diection Σ x x 0 F ma ma v N sin θ + f cos θ ma m + () θ m (1) v 3 m / s W N f Lectue 8 Pudue Univesity, Physics 0 13

14 Banked Cuve equations and unknown we can solve fo N in (1) and substitute in () N f f sinθ + mg cosθ f sin θ + mg sinθ + f cosθ mv cosθ (sin + cos ) cos mgsin v N sin θ + f cos θ m θ θ mv θ θ v f m cosθ gsinθ 300N Lectue 8 Pudue Univesity, Physics 0 14

15 Banked Cuve A ca dives aound a cuve with adius 410 m at a speed of 3 m/s. The oad is banked at 5.0. The mass of the ca is 1400 kg. A) What is the fictional foce on the ca? B) At what speed could you dive aound this cuve so that the foce of fiction is zeo? f 0 v cos θ g sin θ v g tan θ 19 m / s Like an aiplane Lectue 8 Pudue Univesity, Physics 0 15

16 Cicula Motion and Foces Newton s Second Law can be applied to cicula motion: mv F m a F ma C The foce must be diected towad the cente of the cicle The centipetal t foce can be supplied by a vaiety of physical objects o foces The cicle does not need to be a complete cicle Lectue 8 Pudue Univesity, Physics 0 16

17 Centipetal Foce Example The centipetal acceleation is poduced by the tension in the sting If the sting beaks, the object would move in a diection tangent to the cicle at a constant speed Lectue 8 Pudue Univesity, Physics 0 17

18 Conical Pendulum Net foce must point towads the cente of the cicle x : Fx T sin φ max m ω Lsinφ T sin φ mω L sin φ T mω L y: F Tcosφ mg x mω Lcosφ mg ω g Lcosφ ω L g cos φ Lectue 8 Pudue Univesity, Physics 0 18

19 ILQ: Caousel Motion Java applet What is the diection of the acceleation? A) Along the tension in the cable B) In a hoizontal diection towad the cente C) In a hoizontal diection away fom the cente Lectue 8 Pudue Univesity, Physics 0 19

20 Cicula Obits of Planets Ancient Geeks: The Geocentic Model implies Eath-Centeed Univese Lectue 8 Pudue Univesity, Physics 0 0

21 Cicula Obits of Planets Copenicus (1500's) Developed a mathematical ti model fo a Sun-centeed sola system Published in De Revolutionibus Dedicated the book to Paul III The De Revolutionibus was placed on the Index of Pohibited Books in 1616 as a esult of the Galileo affai It was taken off the Index in 1758 Lectue 8 Pudue Univesity, Physics 0 1

22 Cicula Obits of Planets Lectue 8 Pudue Univesity, Physics 0

23 Tycho Bahe ( ) Obits of Planets Johannes Keple ( ) Made pecise Descibed the shape of measuements of the planetay obits positions of the planets as well as thei obital speeds Lectue 8 Pudue Univesity, Physics 0 3

24 Keple s Laws Fist law: The obit of a planet about the Sun is an ellipse with the Sun at one focus. Second law: A line joining a planet and the Sun sweeps out equal aeas in equal intevals of time. Lectue 8 Pudue Univesity, Physics 0 4

25 Keple s Thid Law The squae of a planet's obital peiod is popotional to the cube of the length of its obit's semimajo axis. O simply T R 3 if T is measued in yeas and R is measued in astonomical units. An AU is the aveage distance of the Eath fom the Sun. 1 AU 93,000,000 miles 8.3 light minutes Lectue 8 Pudue Univesity, Physics 0 5

26 Keple s Laws Elliptical obits Equal aeas in equal time T R 3 These wee empiical laws Lectue 8 Pudue Univesity, Physics 0 6

27 Newton s Law of Univesal Gavitation Fom Keple's 3d Law, Newton deduced d d invese squae law of attaction. F Gm m 1 d G Nm /kg You book deives Keple s thidlaw fom Newton s law fo a cicula obit T 4π GM 3 sun Lectue 8 Pudue Univesity, Physics 0 7

obits: A cicula obit in Eath s equatoial plane whose peiod is equal to Eath

28 Satellites and Planets F GmM eath ma mv v GM eath -Speed is independent of mass of satellite - Satellites in lowe obits have geate speeds Geostationay obits: A cicula obit in Eath s equatoial plane whose peiod is equal to Eath s otational peiod. ~35,786 km above gound Lectue 8 Pudue Univesity, Physics 0 8

29 Satellites ACT Two identical (except fo colo) satellites ae in cicula obits aound the Eath. The ed satellite is fathe fom the eath than the geen one. Which satellite has the geate centipetal acceleation? A) Red B) Geen C) They ae the same ac a g 3 ω GM eath GM eath ω F ma c GMm Lectue 8 Pudue Univesity, Physics 0 9

30 Satellites ACT Two identical (except fo colo) satellites ae in cicula obits aound the Eath. The ed satellite is fathe fom the eath than the geen one. Which satellite has the geate velocity? A) Red B) Geen C) They ae the same GmM mv GM v Lectue 8 Pudue Univesity, Physics 0 30

31 Weightlessness Inside the Space Shuttle, the astonauts expeience appaent weightlessness. The foce of gavity on the Eath pulls on the astonauts to keep them in a cicula obit aound the Eath, so we can t accuately say thee is no gavity thee. Both the astonauts and the Space Shuttle ae in unifom cicula motion, and ae continually acceleating towads the Eath. The ae both falling at the same ate. The appaent weight (how heavy you feel) is defined as the magnitude of the nomal foce acting on the body: W m g a whee g is the local gavitational field stength: g GM Lectue 8 Pudue Univesity, Physics 0 31

32 Atificial Gavity Lectue 8 Pudue Univesity, Physics 0 3

33 Summay of Concepts Unifom Cicula Motion Speed is constant, but velocity is not Linea to Cicula convesions s θ Acceleation towad cente a v / Keple s Laws The obit of a planet about the Sun is an ellipse with the Sun at one focus A line joining a planet and the Sun sweeps out equal aeas in equal intevals of time The squae of a planet's obital peiod is popotional to the cube of the length of its obit's semimajo axis T R 3 Lectue 8 Pudue Univesity, Physics 0 33

Chap 5. Circular Motion: Gravitation

Chap 5. Circular Motion: Gravitation Chap 5. Cicula Motion: Gavitation Sec. 5.1 - Unifom Cicula Motion A body moves in unifom cicula motion, if the magnitude of the velocity vecto is constant and the diection changes at evey point and is