General Physics (PHY 2130)

Transcription

1 Geneal Physics (PHY 130) Lectue 1 Rotational kinematics Angula speed and acceleation Unifom and non-unifom cicula motion Obits and Keple s laws

2 Lightning Reiew Last lectue: 1. Rotational kinematics angula displacement, angula elocity and angula acceleation elations between angula and linea quantities Reiew Poblem: A high speed dental dill is otating at ads/sec. Though how many degees does the dill otate in 1.00 sec? How many eolutions does that constitute?

3 Example: A high speed dental dill is otating at ads/sec. Though how many degees does the dill otate in 1.00 sec? How many eolutions does that constitute? Gien: ω ads/sec Δt 1 sec α 0 Find: Δθ? Solution: 1. Angula acceleation is zeo, so θ θ + ω Δt 0 0 Δθ ω Δt 0 0 θ θ + ω Δt ( ads/sec)( 1.0 sec) αδt ads degees. One eolution is 360 degees, so N ( ) o / 360 o 5000 eolutions 3

4 Analogies Between Linea and Rotational Motion Rotational Motion About a Fixed Axis with Constant Acceleation Linea Motion with Constant Acceleation ω ω + αt i i + at 1 Δθ ω it + α t Δ x i t + 1 at ω i ω + αδθ + aδx i

5 Relationship Between Angula and Linea Quantities Displacements Δθ Δs Speeds Acceleations Δθ 1 Δt o ω 1 Δs Δt a α

6 Centipetal Acceleation An object taeling in a cicle, een though it moes with a constant speed, will hae an acceleation (since elocity changes diection) This acceleation is called centipetal ( cente-seeking ). The acceleation is diected towad the cente of the cicle of motion

7 Centipetal Acceleation and Angula Velocity The angula elocity and the linea elocity ae elated ( ω) The centipetal acceleation can also be elated to the angula elocity Simila tiangles! Δ Δs Δ Δs, but a Δ Δt a Δs Δt Thus: a C o a C ω

8 Total Acceleation What happens if linea elocity also changes? Two-component acceleation: the centipetal component of the acceleation is due to changing diection the tangential component of the acceleation is due to changing speed Total acceleation can be found fom these components: slowing-down ca a a + a t C

9 Vecto Natue of Angula Quantities As in the linea case, displacement, elocity and acceleation ae ectos: Assign a positie o negatie diection A moe complete way is by using the ight hand ule Gasp the axis of otation with you ight hand Wap you finges in the diection of otation You thumb points in the diection of ω

10 Foces Causing Centipetal Acceleation Newton s Second Law says that the centipetal acceleation is accompanied by a foce F ma C m F stands fo any foce that keeps an object following a cicula path Foce of fiction (leel and banked cues) Tension in a sting Gaity

11 Example: leel cues Conside a ca diing at 0 m/s (~45 mph) on a leel cicula tun of adius 40.0 m. Assume the ca s mass is 1000 kg. 1. What is the magnitude of fictional foce expeienced by ca s ties?. What is the minimum coefficient of fiction in ode fo the ca to safely negotiate the tun?

12 Example: Gien: masses: m1000 kg elocity: 0 m/s adius: 40.0m Find: 1. f?. µ? f µ mg m N µ 1000 kg 9.8m s 1. Daw a fee body diagam, intoduce coodinate fame and conside etical and hoizontal pojections F y 0 N mg N mg F x ma f f ma m. Use definition of fiction foce: 4 N, 1.0 thus ( 0 m s) 1000 kg m Lesson: µ fo ubbe on dy concete is 1.00! ubbe on wet concete is 0.! 4 N diing too fast

13 ConcepQuestion Is it static o kinetic fiction that is esponsible fo the fact that the ca does not slide o skid? 1. Static. Kinetic

14 Example: banked cues Conside a ca diing at 0 m/s (~45 mph) on a 30 banked cicula cue of adius 40.0 m. Assume the ca s mass is 1000 kg. 1. What is the magnitude of fictional foce expeienced by ca s ties?. What is the minimum coefficient of fiction in ode fo the ca to safely negotiate the tun? A component of the nomal foce adds to the fictional foce to allow highe speeds tan θ g

15 Example: Gien: masses: m1000 kg elocity: 0 m/s adius: 40.0m angle: α 30 Find: 1. f?. µ? 1. Daw a fee body diagam, intoduce coodinate fame and conside etical and hoizontal pojections F x m cos30 f mg sin30 f m cos30 mg sin N F y m sin30 N mg cos30 N m sin30 + mg cos Use definition of fiction foce: 4 N f µ N, thus minimal µ is µ s s fs 3760 N N N s Lesson: by inceasing angle of banking, one deceases minimal µ o fiction with which one can take cue!

16 Tension as a souce of centipetal acceleation The hoizontal component of the tension causes the centipetal acceleation a C g tanθ

17 Foces in Acceleating Refeence Fames Distinguish eal foces fom fictitious foces Centifugal foce is a fictitious foce Real foces always epesent inteactions between objects

18 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 Sole fo the speed of the satellite: g GmsM e msa GmsM e GM e m m s s 18

19 Escape Speed The escape speed is the speed needed fo an object to soa off into space and not etun esc GM R E E Fo the eath, esc is about 11. km/s Note, is independent of the mass of the object

20 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 0

21 Keple s Laws All planets moe in elliptical obits with the Sun at one of the focal points. A line dawn fom the Sun to any planet sweeps out equal aeas in equal time inteals. The squae of the obital peiod of any planet is popotional to cube of the aeage distance fom the Sun to the planet.

22 Keple s Laws, cont. Based on obseations made by Bahe Newton late demonstated that these laws wee consequences of the gaitational foce between any two objects togethe with Newton s laws of motion

23 Keple s Fist Law All planets moe in elliptical obits with the Sun at one focus. Any object bound to anothe by an inese squae law will moe in an elliptical path Second focus is empty

24 Keple s Second Law A line dawn fom the Sun to any planet will sweep out equal aeas in equal times Aea fom A to B and C to D ae the same

25 Keple s Thid Law The squae of the obital peiod of any planet is popotional to cube of the aeage distance fom the Sun to the planet. T K Fo obit aound the Sun, K S.97x10-19 s /m 3 K is independent of the mass of the planet 3 o GM e T 4π 1 3