Chapter 5: Uniform Circular Motion

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Augustus O’Connor’
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1 Chapte 5: Unifom Cicula Motion Motion at constant speed in a cicle Centipetal acceleation Banked cuves Obital motion Weightlessness, atificial gavity Vetical cicula motion

2 Centipetal Foce Acceleation towad cente of a cicula path of adius : v a c Foce needed to maintain the centipetal acceleation = centipetal foce: F c v m a c m ˆ Foce povided by tension in a sting, fiction, hoizontal component of aiplane s lift, gavity... Note: Centifugal foce is the foce you feel towad the outside of a cuve when going aound a cone. It s not eally a foce, but a consequence of Newton s fist law that says that things tavel at constant velocity unless a foce is applied.

3 Ca A uses ties with coefficient of static fiction 1.1 with the oad on an unbanked cuve. The maximum speed at which ca A can go aound this cuve is 5 m/s. Ca B has ties with fiction coefficient What is the maximum speed at which ca B can negotiate the cuve?

4 Flying aound in cicles m g The Plane banks to tun in a hoizontal cicula path of adius m g

5 Poblem 5.5 A jet (m = 00,000 kg), flying at 13 m/s, banks to make a hoizontal tun of adius 3810 m. Calculate the necessay lifting foce. m g

6 Diving aound in cicles banked oad (no fiction) As fo plane but with lift foce eplaced by nomal foce:

7 Diving aound in cicles banked oad If you dive slowly, you slide down the slope. If you dive fast, you skid up the slope. If = 31 and = 316 m and thee is no fiction, what is the best speed to dive aound the banked cuve? = 316 m no fiction

8 Poblem 5.0 Two banked cuves have the same adius. Cuve A is banked at 13, cuve B at 19. A ca can tavel aound cuve A without elying on fiction at a speed of 18 m/s. At what speed can this ca tavel aound cuve B without elying on fiction?

9 Poblem 5.18 A ca tavels at 8 m/s aound a cuve of adius 150 m. A mass is suspended fom a sting fom inside the oof.

Obiting Eath The secet to flying is to thow youself at the eath and miss. Hitch Hike s Guide to the Galaxy The centipetal foce on the satellite is povided by the gavitational foce fom the eath.

10 Obiting Eath The secet to flying is to thow youself at the eath and miss. Hitch Hike s Guide to the Galaxy The centipetal foce on the satellite is povided by the gavitational foce fom the eath. F c m a c v m ˆ GME m ˆ So: v GM E The smalle the adius, the geate the speed. Synchonous obit: peiod = 4 hous satellite stays above same pat of eath (above the equato) used by communications satellites what is the adius of the obit?

11 Synchonous obit what is the adius? The peiod of an obit is: Fom pevious page: So: T GM E v T GM 3/ E cicumfee nce speed v 1 GM E 3 T GME Keple s 3d law of planetay motion: T 3 With T = s, = m = 4,300 km fom cente of eath, The speed of the satellite is: v 3070 m/s 11, 000km/h T

12 Poblem 5.3 The eath obits the sun once pe yea at a distance of m. Venus obits the sun at a distance of m. What is the length of the yea on Venus?

13 Poblem 5.3 A satellite has a mass of 5850 kg and is in cicula obit m above the suface of a planet. The peiod of the obit is hous. The adius of the planet is m. What is the tue weight of the satellite when it is at est on the planet s suface?

Eveything in the satellite is acceleated towad the cente of the eath at the same ate.

14 Fee Fall Weightlessness An obiting satellite is in fee fall thee s nothing hold it up. Only its fowad speed lets it miss the eath and keep obiting. Eveything in the satellite is acceleated towad the cente of the eath at the same ate. An object exets no foce on the bathoom scales as the scales ae also being acceleated towad the cente of the eath.

15 Atificial Gavity A space station is otating about its axis to povide an atificial gavity. At what speed must the suface of the space station move so that the astonaut expeiences a push on his feet equal to his weight on eath? The adius is 1700 m. F c v m mg Solve fo the velocity v g m 9 80m s 19 m s

16 Atificial Gavity Poblem 5.8: Poblems of motion sickness stat to appea in a otating envionment when the otation ate is geate than evolutions / minute. Find the minimum adius of the station to allow an atificial gavity of one g (a c = 9.8 m/s) while avoiding motion sickness. v m mg v g a c mete v t 60sec ac 3m 4 F c a c

17 v1 F N 1 mg m v m F N v3 F N 3 mg m v4 m F N 4 Vetical Cicula Motion 1) foce towad cente weight lage than mg ) foce towad cente 3) foce towad cente ide falls off if 4) foce towad cente v3 mg m The ide falls off if v 3 F N 3 0 g

18 Poblem 5.40: A motocycle is taveling up one side of a hill and down the othe side. The cest is a cicula ac with a adius of 45 m. Detemine the maximum speed that the motocycle can have while moving ove the cest without losing contact with the oad. The net downwad foce on the bike at the cest of the hill allows the motobike to emain in contact with the gound. Then F N 0. That is: net downwad foce = centipetal foce, mv /.

PS113 Chapter 5 Dynamics of Uniform Circular Motion

PS113 Chapter 5 Dynamics of Uniform Circular Motion PS113 Chapte 5 Dynamics of Unifom Cicula Motion 1 Unifom cicula motion Unifom cicula motion is the motion of an object taveling at a constant (unifom) speed on a cicula path. The peiod T is the time equied