Circular Mo+on. Chapter 5

Transcription

1 Cicula Mo+on Chapte 5

2 Cicula mo+on Rota+ons Descibe the mo-on: 1) Linea speed: the distance (metes, km, feet,..) tavelled pe second (o minute o ) ) Rota-onal (o angula) speed: the numbe of ota+ons pe second (o minute o ) To define cicula mo-on we will concentate on angles instead of distances.

3 Radians Measuing θ in degees (deg in you calculato) tuns out to be a poo choice. Radians ae a moe natual choice of angula unit. π adians evolution 1otation adian θ (ad) s s ac length - adius 3

4 Conceptual ques+on Radians Q1 A wheel tuns though five complete evolu+ons and then one quate of a evolu+on. Though what total angle in adians has the wheel tuned? A. 33 adians. B. 50 adians. C. 5 adians. D. 66 adians. 1 evolu+on π adians 5 + ¼ evolu+ons (5 + ¼ )xπ adians 10π+0.5π 10.5π 10.5x adians 4

5 Angula posi+on and displacement y! f θ f θ i! i x Just like we did in Chapte 3 fo linea mo+on, we will define: Angula posi+on θ (adians) measued counteclockwise (CCW) fom the x-axis posi+ve angle θ i ini+al angula posi+on θ f final angula posi+on Angula displacement Δθ (adians) Δθ θ f θ i When object moves along a cicula path, posi+on vectos i and f have magnitudes equal to the adius of the cicle. 5

6 Angula velocity Just like we did in Chapte 3 fo linea mo+on, we will define: Aveage angula velocity angula displacement pe +me ω av Δθ θ f Δt t f θ t i i Instantaneous angula velocity ω Δθ lim Δ t 0 Δt Just like linea mo+on but with θ Units of angula velocity: adians pe second [ad/s] Counteclockwise (CCW) ota+on epesents posi+ve ota+on. Clockwise ota+on (CW) epesents nega+ve ota+on. 6

7 Conceptual ques+on Ladybug Q A ladybug sits at the oute edge of a mey-go-ound, and a gentleman bug sits halfway between he and the axis of ota+on. The mey-go-ound makes a complete evolu+on on each second. The gentleman bug s angula velocity is A. Half the ladybug s. B. The same as ladybug s. C. Twice the ladybug s. D. It can t be detemined. Both bugs cove the same angle (Δθ) ove the same amount of +me angula velocity is the same fo both bugs 7

8 Rela+on between linea and angula speed y θ f s An object moving along a cicula path duing angula displacement (Δθ) will tavel the distance (s) equal to the ac length: s Δθ θ i Aveage linea speed (Chapte ): v av total distance total time s Δt v av Δθ Δt Δθ Δt v av ω av adius of a cicle 8

9 Conceptual ques+on Ladybug Q3 A ladybug sits at the oute edge of a mey-go-ound, and a gentleman bug sits halfway between he and the axis of ota+on. The mey-go-ound makes a complete evolu+on on each second. The gentleman bug s linea speed is A. Half as the ladybug s. B. The same as the ladybug s. C. Twice the ladybug s. D. It can t be detemined. Both bugs have the same angula speed (ω) but the linea speeds ae diffeent ladybug: v L ωr gentleman bug: v G ωr/ ½ v L 9

10 Unifom cicula mo+on Unifom cicula mo+on è when the speed of a point moving in a cicle is constant. Peiod and Fequency The fequency (f) is the numbe of complete evolu+ons pe second. Units: Hetz [Hz (ev/s)] The peiod (T) is the +me it takes a point to make one evolu+on. Linea speed: T 1 f v distance taveled time π T πf ω v π f π T 10

11 Execise: Speed in centifuge A centifuge is spinning at 5400 pm. a) Find the peiod and fequency of mo+on. 1 pm 1 evolu+ons pe minute this is fequency Fist, convet pm to ev/s Hz:! f 5400 ev $! # & 1min $ # & 90 ev " min % " 60s % s 90 Hz Peiod: T 1 f T 1 90Hz 0.011s b) Find angula speed. π ω πf T ω π ad 90Hz 180π ad s 565. ad s 11

12 Radial accelea+on Conside an object moving in a cicula path of adius at constant speed. v! i v! f Δ v! Linea velocity is always a tangent to the mo+on path. Diec+on of vecto velocity is changing velocity is changing object has non zeo accelea+on! Δv! v f! v i aδt a is in the same diec+on as Δv When Δt 0, instantaneous accelea+on points towads the cente of the cicle (adially inwad) and is called adial accelea-on a. 1

13 Magnitude of adial accelea+on Similaly, to: s Δθ Δθ s v! f v! i Δ! v Δθ We can also say: Δv Δ vδθ v vωδt v! i v! f v Magnitude of adial accelea+on: a Δv v a v Δt ω ω Units: [m/s ] 13

14 Conceptual ques+on Radial accelea+on Q4 An LP ecod, 10 cm in adius, spins with angula velocity of 0 ad/s. If the angula speed of the ecod is doubled, what happens to its adial accelea+on at the ims? A. Radial accelea+on doubles. B. Radial accelea+on quaduples. C. Radial accelea+on quates. D. Radial accelea+on halves. v vω ω a ω doubles a quaduples O mathema+cally: a 1 ω 1 (0ad / s) 0.1m 40 m/s a ω (40ad / s) 0.1m 160 m/s a a

15 Applying Newton s second law Radial accelea+on has constant magnitude and is diected towad the cicle s cente. Newton s second law:!! F net ma Something must povide the foce Centipetal foce Centipetal foce is NOT a foce of natue. Any of the foces we discussed in Chapte 4 can play a ole of centipetal foce when object s moving along a cicle. To solve poblems in unifom cicula mo+on apply the same set of ules povided in Chapte 4 set x-axis along adial accelea+on thee will be no accelea+on y diec+on fo unifom cicula mo+on Fx ma mω and F 0 y 15

16 Execise: Amusement pak ide The oto is an amusement pak ide whee people stand against the inside of a cylinde. Once the cylinde is spinning fast enough, the floo dops out. (a) What foce keeps the people fom falling out the botom of the cylinde? Fist, daw a fee body diagam (FBD) of one peson standing against the wall. y x a N f s Make sue that: Weight points staight down Nomal foce is pependicula to the suface Fic+on foce is paallel to the suface and opposes the mo+on w The foce of sta+c fic+on keeps people fom falling out the botom. X diec+on: f sx 0 N x N W x 0 Y diec+on: f sy f s N y 0 W y - W - mg 16

17 Execise: Amusement pak ide The oto is an amusement pak ide whee people stand against the inside of a cylinde. Once the cylinde is spinning fast enough, the floo dops out. (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? y Apply Newton s nd Law: f s F x ma N mω x N w F y 0 f s W 0 f s W N m µ s µ s ω As long as foce of sta+c fic+on is lage o equal to foce of gavity (weight) è people will not fall out mg ω g µ s 9.8 m/s 0.40 ( )(.5 m) 3.13 ad/s 17

18 Taking exams Fom now on, I will povide eveyone with equa+ons necessay fo the exam. I will post the equa+on sheet on Canvas a week befoe the exam so eveyone can see it an go ove it. You ae NOT allowed to bing anything to the exam (except fo pen/pencil and calculato). The equa+on sheet will be atached to you exam. 18

19 Execise: Slipping coin A 0-g coin is placed on a ecod that is ota+ng at 33.3 pm. If μ s 0.1, how fa fom the cente of the ecod can the coin be placed without having it slip off? Daw an FBD fo the coin. y x a f s N X diec+on: f sx f s N x 0 W x 0 Apply Newton s nd Law: Y diec+on: f sy 0 N y N W y - W - mg Foce of sta+c fic+on keeps the coin fom slipping off. w F x ma f s mω We need to find f s fist and use it to find the adius. f s µ s N F y 0 a y 0 (Thee is no accelea+on in ve+cal diec+on). N W 0 N W mg N kg 9.8 m/s N 19

20 Execise: Slipping coin A 0-g coin is placed on a ecod that is ota+ng at 33.3 pm. If μ s 0.1, how fa fom the cente of the ecod can the coin be placed without having it slip off? y f s µ s N N N x a f s N f s mω f s mω You ae given fequency f 33.3 pm convet it to Hz and then calculate angula speed ω πf w ev π ad 1min ω πf ad/s min 1ev 60 sec kg m/s kg 3.50 ad/s ( ) 0.08 m Note that the same pinciple applies when you dive a ca in a cicula path along an unbanked oad. 0

21 Banked cuves To pevent cas fom going into skid è oads ae banked (+lted at a slight angle) N y Apply Newton s nd Law: a F y 0 N cosθ mg x F x ma x v g tanθ N sinθ ma m v mgtanθ m v tanθ v g Fic+on foce negligible compaed hoizontal component of the nomal foce. 1

22 Cicula obits Conside a satellite in a cicula obit about the Eath. But thee is no sting joining the satellite to the Eath no is thee anything to have fic+on against Gavity What foce is holding the satellite in a cicula obit? It s gavity: F g GmsM e Apply Newton s nd law F m g s a GmsM e m s v v GM e The speed of a satellite in a cicula obit does not depend on mass of the satellite

23 Keple s 1 st Law of Planetay Mo+on The law of obits: All planets move in ellip+cal obits, with the Sun at one focus Luckily fo us, we will only wok with pefect cicles in this class. 3

24 Keple s nd Law of Planetay Mo+on The law of aeas: A line that connects a planet to the Sun sweeps out equal aeas in the plane of the planet s obit in equal +mes. Eath aound Sum - closest distance è v 30.3 km/s - futhest distance è v 9.3 km/s GSCI 101, Pof. M. Nikolic Planets ae moving faste when they ae close to the Sun 4

25 Keple s 3 d Law of Planetay Mo+on The law of peiods: The squae of the peiod of any planet is popo+onal to the cube of the semi majo axis of its obit. O in English: Oute planets have futhe to go and move moe slowly in thei obits aound Sun T π GM 3/ o T 4 π GM 3 GSCI 101, Pof. M. Nikolic 5

26 Execise: Keple s laws The Hubble Space Telescope obits Eath 613 km above Eath s suface. What is the peiod of the telescope s obit? T π GM 4 E 3 What is given: Mass of the Eath: M E 5.98x10 4 kg Radius of the Eath: R E 6371 km h 613 km G 6.67x10-11 N m /kg Distance is measued fom the cente of the Eath to the telescope: R E + h and has to be conveted to metes: 6371 km km 6984 km 6984x10 3 m T 4π GM E ( R e + h) 3 T ( m) Nm /kg kg s 6

27 Nonunifom cicula mo+on What happens when the speed in cicula mo+on is not constant? Then, when Δt 0, Δv does not point towads the cente of the cicle Thee is now anothe component to accelea+on tangent to the path of the cicle tangen-al accelea-on a t a a a t v The net accelea+on is: a a + a t a t changes the magnitude of v a changes the diec-on of v And Newton s nd law s+ll applies: F ma F t ma t 7

28 Execise: Child on a swing A 35-kg child swings on a ope with a length of 6.5 m that is hanging fom a tee. At the botom of the swing, the child is moving at a speed of 4. m/s. What is the tension in the ope? Daw an FBD fo the child at the botom of the swing. x X diec+on: T x 0 W x 0 y T w a t a Y diec+on: T y T W y - W - mg adial accelea+on points towads the cente of the cicle (along y axis) tangen+al accelea+on is tangent to the mo+on Apply Newton s nd Law: F ma T 35kg 9.8m / s v T W mω m v T mg + m + 35kg (4.m / s) 6.5m 438 N 8

29 Tangen+al and angula accelea+on Duing nonunifom cicula mo+on angula velocity (ω) is changing thee should be an angula accelea-on (α) Aveage angula accelea-on α av Δω ω f Δt t f ω t Units of angula accelea+on: adians pe second [ad/s ] i i Instantaneous angula accelea-on α Δω lim Δ t 0 Δt Just like linea mo+on but with v ω 9

30 30 Angula mo+on vs. linea mo+on Tangen+al accelea+on: a t Δv Δt Δω Δt α ω v Linea mo-on Angula mo-on t a v v v x ix fx x Δ Δ 1 t a t v x x x x ix i f Δ + Δ Δ v fx v ix a x Δx ( ) t v v x fx ix Δ + Δ 1 t i f Δ Δ α ω ω ω 1 t t i i f Δ + Δ Δ α ω θ θ θ ω f ω i αδθ ( ) t f i Δ + Δ ω ω θ 1

31 Execise: Stopping the Eath Suppose the Eath stated to undego an angula accelea+on of ad/s in the opposite diec+on to its cuent ota+on. a) Aze how long would it come to a stop (in ota+on)? What is given: α ad/s The Eath is slowing down accelea+on is nega+ve ω f 0 the Eath stopped Δω ω f ω αδt i We do not know ini+al (cuent) angula velocity but, hopefully, eveyone knows the peiod of Eath s ota+on: T 1 day 4 h s ω i π T ω i s ad s Δt ω f ωi α Δt 0 ω i α ad / s ad / s s 7 days 31

32 Execise: Stopping the Eath Suppose the Eath stated to undego an angula accelea+on of ad/s in the opposite diec+on to its cuent ota+on. b) How many evolu+ons would the Eath make befoe it stopped? What is given o aleady found: α ad/s The Eath is slowing down accelea+on is nega+ve ω f 0 the Eath stopped ω i ad/s Δt 6.1x10 5 s Revolu+ons ae expessed in angle θ 1 evolu+on π adians Δθ 1 ( ω + ω ) Δt i f Δθ 5 ( ad/s + 0) s. ad 1 5 Δθ 1evolution. ad 3.54 evolutions π ad 3

33 Appaent weight and a+ficial gavity Ty to ecall poblems with the appaent and tue weight (peson in an elevato) peson feels weightless when a g Simila situa+on fo astonauts in space We can simulate gavity by ota+ng the space sta+on astonaut hits the walls nomal foce makes him otate with the space sta+on with adial accelea+on a mω Fo a space sta+on we want: a g N v F ma mω m mg 33

34 Execise: A+ficial gavity If a washing machine s dum with adius of 5 cm poduces a+ficial gavity on the clothes of 16g, what is the fequency in evolu+ons pe minute of the dum? What is given: 5 cm 0.5 m a 16g 16 x 9.8 m/s a ω ω 16g ω m / s 0.5m 5 ad/s To find the fequency: f ω π f 5 ad/s Hz 60s 4 ev/s 1min 40 pm 34

35 Hollywood movies busted Sta Tek: USS Entepise not possible 001: A space Odyssey possible Sta Was: Millennium falcon not possible 35

36 Weightless in the ISS The Intena+onal Space Sta+on obits the Eath evey 91 minutes at a distance of 353 km above the suface of the Eath Intena+onal space sta+on does not otate è no appaent gavity 36