Unifom Cicula Motion constant speed Pick a point in the objects motion... What diection is the velocity? HINT Think about what diection the object would tavel if the sting wee cut Unifom Cicula Motion constant speed v v always tangential to the cicle Now pick anothe point in the objects motion. What diection is the velocity? Compae it to the pevious point Is the object acceleating?
stat hee What diection is the acceleation? v 1 v 2 vecto pat of a a = Δv = v f v i = v 2 v 1 t t t acceleation points in the diection of Δv!!! Δv v 1 v 2 Δv v 2 v 1 *emembe constant speed > v 1 = v 2 = 45 o Δv and theefoe a point towad the cente of the cicle!!! This is called centipetal acceleation, a c d = vδt These ae 2 simila isosceles tiangles Can compae coesponding sides!!! Δv v 1 v 2 Δv = vδt v Δv v = 2 Δt a c = v2
Definitions fo Cicula Motion Cycle in geneal: an object coming back to its initial position afte leaving cicula motion one evolution Peiod the time fo an object to make one cycle T = time cycle Fequency the # of cycle made in a cetain amount of time (invese of the peiod) f = # cycles time **** T = 1 f = 1 **** f T Now we know thee is an acceleation associated with cicula motion which means?... Thee must be a FORCE associated with cicula motion!! F net = ma
Centipetal Foce, F c F net = ma F c = ma c a c = v 2 F c = mv 2 v c = d = 2π F c = 4π 2 m t T T 2 **** F c is always the net foce!! **** must be povided by some othe foce Tension, Gavity, Nomal etc. Centipetal Foce, F c What diection is F c? ***Remembe, F net and acceleation always point in the same diection! a c F c Δv v 2 v 1 **F c always points towad the cente of the cicle!
Find v Find v
Rounding a Cone A 1200 kg ca ounds a FLAT cicula cone o adius, = 45m. If the μs between the ties and the oad is, μs = 0.82. What is the geatest speed the ca can have without skidding? (not slipping off of its cicula path) Top View Side View No matte how massive the ca they all have the same maximum speed!! What would the adius of the cicle be in which the maximum speed, without skidding, is doubled? (same fictional foce exeted) f = m v 2 α v 2 f = m v 2 constants 4 α (2v) 2 quadupled? α v 2 16 α (4v) 2
AIRPLANES An aiplane is able to get off the gound because of a foce called "lift". The wings ae designed to push ai down and back, "pushing" the plane fowad and up. (3 d Law) Lift Lift is always pependicula to the wing suface Because of Newton's 3d Law!!! AIRPLANES Once in the ai, How do planes tun? The plane must tilt o "bank" its wings and in the pocess ceate some F c What povides that F c?
AIRPLANES F L sin F L cos F L What povides that F c? mg AIRPLANES The hoizontal component of the lift!! (in geneal the pat of the lift that is in the same plane as the cicle) F L sin F c F L F L cos mg Find an expession fo the speed of the plane as function of g, and
AIRPLANES F L sin F L F L cos F c mg The hoizontal component of the lift!! (in geneal the pat of the lift that is in the same plane as the cicle) AIRPLANES F L sin F L F L cos F c mg
Banked Cuves The Nomal Foce povides F c!!! Nsin N Ncos notice cicula path is hoizontal NOT along incline mg Find an expession fo the speed of the ca as function of g, and Banked Cuves Nsin N Ncos mg
At what angle should a cuve of adius 150 m be banked so cas can tavel safely at 25 m/s without elying on fiction? Univesal Law of Gavitation At the suface of the Eath: F g = mg in geneal F g depends on how fa away objects ae fom one anothe F Gm 1 m **** 2 g = **** Newton's Constant G = 6.67 x 10 * do NOT have to memoize! 2 11 Nm2 kg 2
Univesal Law of Gavitation At the suface of the Eath: F g = mg in geneal F g depends on how fa away objects ae fom one anothe F 1 F 2 d F 1 = F 2 F Gm 1 m **** 2 g = **** 2 F g α 1 2 F g e What is the gavitational acceleation at the suface of the Eath? (conside an object in feefall) G = 6.67 x 10 11 Nm 2 /kg 2 M e = 5.98 x 10 24 kg e = 6.38 x 10 6 m
e What is the gavitational acceleation at the suface of the Eath? mg (conside an object in feefall) ƩF = ma G M e m e 2 = ma G M e (6.67 x 10 = a = 11 )(5.98 x 10 24 ) = 9.7991 m 2 e (6.38 x 10 6 ) 2 s 2 Keple's Laws: Johannes Keple (1571-1630) Using geomety & mathematics developed " The Kinematics of Obits" I. Law of Obits All planets move in elliptical obits with the Sun at one of the focal points. * 1 AU = distance Eath to Sun (93 million miles- aveage) * Cicula obits ae just a special case of an elliptical obit
II. Law of Equal Aeas A line dawn fom the Sun to any planet sweeps out equal aeas in equal time intevals. t 1 2 t3 4 t 1 2 t 3 4 A 1 2 A 3 4 *close->faste! III Law of Peiods The squae of the obital peiod of any planet is popotional to the cube of the aveage distance fom the planet to the Sun! CONSTANT
Obit and Gavity Fo an object in obit, gavity povides the centipetal foce! ƩF = F g = F c An object of mass m obiting a cental mass M o at a sepeation of Let's compae thei sepaation,, to the peiod of the objects obit, T. GM o m mv 2 ecall: = 2 v = 2π T GM o ( 2π T ) 2 GM = ==> o 4π 2 2 2 = T 2 4π T 2 = 2 GM 3 o 4π 2 GM o = k (constant)!! T 2 3 = k Keple's 3 d Law Obit and Gavity Let's get fomulas fo the obital speed, v, and the obital peiod, T. GM o m = mv 2 2 ƩF = F g = F c v 2 = GM o ==> Using ou deivation of Keple's 3 d : GM o 2 = 4π 2 T T 2 2 = 4π2 3 v = GM o ==> T = 2π 3 GM o GMo **Neithe obital peiod o speed depend on the mass of the object, only the cental mass.**
Definition: Astonomical Unit (AU) aveage distance fom the Eath to the Sun R E = 1 AU 1 AU 1 AU = 1.5 x 10 11 m 93 x 10 6 miles Jupite obits the Sun at 5.2 times the distance Eath does, what is it's peiod?
Astonomes have discoveed a supemassive black hole in the cente of the galaxy M87. They have obseved matte at distance of 5.7x10 17 m fom the cente obiting at a speed of 7.5x10 5 m/s. Find the mass of the supemassive black hole. How much moe massive is it than Eath? The Sun? F g = weight = mg we also know F g deceases as you get futhe away fom the Eath So... as you get futhe away fom the suface of the Eath you actually weigh less!!!