Unifom Cicula Motion Have you eve idden on the amusement pak ide shown below? As it spins you feel as though you ae being pessed tightly against the wall. The ide then begins to tilt but you emain glued to the wall. What is unique about moving in a cicle that allows you to appaently defy gavity? What causes people on the ide to stick to the wall?
Unifom Cicula Motion Amusement pak ides ae excellent examples of cicula motion. When an object is moving in a cicle of constant adius and its speed is constant, it is moving with unifom cicula motion. When objects ae moving in a cicula path, speed is constant but diection continuously changes as it moves along the cicle. Theefoe they ae expeiencing centipetal acceleation towads the cente of the path. Centipetal is Latin fo cente-seeking.
Unifom Cicula Motion Fo example, conside an object as it moves fom point P to point Q as shown. If its velocity changes fom v1 to v2 then: v = v 2 v 1 Using tiangle conguencies and the equations v = d/ t fo the distance tavelled and a = v/ t then as t appoaches zeo yields: a c = v2 Note that since v1 and v2 ae pependicula to the adii of the cicle, the change in velocity and acceleation vecto points diectly towads the cente of the cicle. Acceleation that is diected towads the cente of a cicula path is called centipetal acceleation. This is the instantaneous acceleation towads the cicle cente.
Centipetal Acceleation SUMMARY of UNIFORM CIRCULAR MOTION occus when an object moves in a cicle of constant adius and its speed is constant since diection changes the object expeiences acceleation which is always diected towad the cente of the cicle a c = v2 whee a c is the centipetal acceleation (m/s 2 ) v is the velocity (m/s) is adius of the cicula path (m) Note: a c is the instantaneous acceleation ( t is vey small)
Centipetal Acceleation Example #1: A child ides a caousel with a adius of 5.1 m that otates with a constant speed of 2.2 m/s. Calculate the magnitude of the centipetal acceleation of the child. a c = v2 a c = (2.2m s )2 5.1 m a c = 0.95 m/s 2
Centipetal Acceleation Sometimes the speed of an object moving with unifom cicula motion is unknown. Often we can measue the time it takes fo the object to move once aound the cicle, o the peiod (T). If the object is moving too quickly, you would measue the numbe of evolutions pe unit time, o the fequency (f) whee f = 1/T. How ae the following fomulas deived? (Hint: use the cicumfeence of the cicle) a c = 4π2 T 2 and a c = 4π 2 f 2
Centipetal Acceleation Since v = d whee d is the cicumfeence of a cicle (C=2π) t then v = 2π whee T is the peiod of one cicle cicumfeence. T Substituting into a c = v2 T 2 gives: 2π = a c = 4π2 T 2 Since peiod and fequency ae elated as T = 1 f then: a c = 4π 2 f 2 whee a c is centipetal acceleation (m/s 2 ) is the adius of the cicula path (m) T is the peiod of otation (s) f is the fequency of otation (Hz o s -1 )
Centipetal Acceleation Example #2: a) A salad spinne with a adius of 9.7 cm otates clockwise with a fequency of 12 Hz. At a given instant, a piece of lettuce is moving in the westwad diection. Detemine the magnitude and diection of the centipetal acceleation of the lettuce in the spinne at the moment shown. a c = 4π 2 f 2 a c = 4π 2 (0. 097m)(12Hz) 2 a c = 550 m/s 2 [N] b) How does the salad spinne wok to emove wate fom the lettuce? The wate is able to pass though a sceen to the outside of the spinne whee it is collected, leaving the lettuce dy.
Centipetal Acceleation Example #3: The planet Mecuy moves in an appoximately cicula path aound the sun at an aveage distance of 5.8 x 10 10 m, acceleating centipetally at 0.04 m/s 2. What is its peiod of evolution aound the sun? a c = 4π2 T 2 T = 4π 2 (5.8 x 10 10 m) 0.04 m/s 2 T = 7.6 x 10 6 s o appx. 88 days on Eath
Centipetal Foce Accoding to Newton s laws of motion, an object will acceleate only if a net foce is exeted on it. Since objects moving with unifom cicula motion ae always acceleating, thee must always be a foce exeted on it in the same diection as the acceleation as shown. The foce pointing to the cente of a cicula path is called a centipetal foce (Fc). Without this foce, objects would not be able to move in a cicula path.
Centipetal Foce Using Newton s second law and a c = v2 deived as follows: the fomula fo Fc is Substitute a c = v2 into F net = ma : F net = ma c F net = m v2 F c = F net = m v2
Centipetal Foce SUMMARY of CENTRIPETAL FORCE is like net foce that causes centipetal acceleation (Fc =Fnet) always choose motion towads the cente of the cicle as the +ve diection F c = mv2 = 4π2 mf 2 = 4π2 m T 2 whee Fc is the centipetal foce that acts towads the cente of cicle (N) m is the mass (kg)
Centipetal Foce A centipetal foce can be supplied by any type of foce. Fo example, gavity povides the centipetal foce that keeps the Moon in a oughly cicula path aound Eath, fiction povides a centipetal foce that causes a ca to move in a cicula path on a flat oad, and the tension in a sting tied to a ball will cause the ball to move in a cicula path when you twil it aound.
Centipetal Foce Example #4: An astonaut in deep space twils a yo-yo on a sting. a) What type of foce causes the yo-yo to tavel in a cicle? Tension causes F c b) What would happen if the sting suddenly boke? The yo-yo would continue along a staight line accoding to Newton s Fist Law of Inetia.
Centipetal Foce Example #5: A ca with a mass of 2200 kg is ounding a cuve on a level oad. If the adius of the cuvatue of the oad is 52 m and the coefficient of static fiction between the ties and the oad is 0.70, what is the maximum speed at which the ca can make the cuve without skidding off the oad? Since Fc = F fs = μsf N = μmg and F c = mv2 0. 70 2200kg (9. 81 m s 2) = s (2200kg)v 2 52m v = 19 m/s o 68 km/h
Centipetal Foce and Banked Cuves Cas and tucks can use fiction as a centipetal foce. Howeve, the amount of fiction vaies with oad conditions and can become vey small when oads ae wet o icy. As well, fiction causes wea and tea on ties causing them to wea out faste. Fo these easons, enginees who design highways whee speeds ae high with lage centipetal foces ae equied to incopoate anothe souce of centipetal foce banked cuves. Aiplanes also geneate a centipetal foce when they bank o tun.
Centipetal Foce and Banked Cuves Example #6: A ca (m = 1.1 x 10 3 kg) tavels aound a fictionless banked cuve of adius 85 m. The bank is 19 to the hoizontal. a) What foce povides the centipetal acceleation? The hoizontal component of F N acts towads the cente of the cicle and esults in Fc and theefoe a c. Note that F f is not needed to ceate Fc; only F Nx ceates Fc. b) What constant speed must the ca maintain to tavel safely aound the cuve? Since Fx: F c = F N sinθ ; ma c = F N sinθ #1 Fy: F g = F N cosθ; mg = F N cosθ; F N = mg cosθ #2 Sub 2 into 1: ma c = mg sinθ whee mass cancels cosθ v 2 = gtanθ v = gtanθ v = 17 m/s c) What happens if v > 17 m s? If v < 17 m s? Ca does not maintain position; slides up o down oad. F c
Centipetal Foce and Banked Cuves Example #7: Copy the scenaio fom Sample Poblem 3 on pg. 122 and attempt to solve. Note that this poblem incopoates fiction between the ties and oad to detemine the maximum speed at which a ca can maintain unifom cicula motion.
Centifugal Foce Sometimes when an object expeiences unifom cicula motion, an obseve moving elative to the object may feel as though thee ae othe foces acting on them. Ex: On a mey-go-ound you feel as though you ae being pushed to the outside of the ide o while tuning a cone shaply in a ca o on a bike you body feels as though it leans away; fom an inetial efeence fame you body wants to keep moving in a staight line elative to Eath (due to inetia) This is explained by the centifugal foce (Latin fo centefleeing) which is a fictitious foce in a non-inetial otating fame of efeence ; Newton s Laws do not apply in an acceleating efeence fame. Centifugal foces help to explain the peceived motion of objects in an acceleating efeence fame
Unifom Cicula Motion Read Section 3.4 on Rotating Fames of Refeence and Centifugal Foces pgs. 125-130 What challenges do long space jouneys pose? How could atificial gavity be ceated? www.tcm.com/mediaoom/video/11279/2001-a-space- Odyssey-Wide-Release-Taile-.html
Centipetal Foce and Vetical Motion Example #8: You ae playing with a yo-yo of mass 225 g by swinging it vetically. The full length of the sting is 1.2 m. a) Calculate the minimum speed at which you can swing the yo-yo while keeping it in a cicula path. Fc is caused by tension of sting. At the top of the swing if F T = 0, then v is min. To keep the yo-yo in a cicula path Fc Fg. F c Since F c = F T + F g and if F T = 0 Then F C = F g mv 2 v = = mg whee mass cancels 9. 81m/s 2 1. 2m v = 3.4 m/s
Centipetal Foce and Vetical Motion Example #8 Continued: You ae playing with a yo-yo (m = 225 g) by swinging it vetically. The full length of the sting is 1.2 m. b) At the speed just detemined, what is the tension in the sting at the bottom of the swing? Since F: Fc = F T - Fg mv 2 = F T mg F T = 0.225g(3.431035)2 + (0. 225kg 9. 81 m ) 1.2m s 2 F T = 2. 20725 N + 2. 20725 N F c F T = 4.4 N
Centipetal Foce and Vetical Motion Example #9: A olle coaste ca is at the lowest point on its cicula tack. The adius of cuvatue is 22 m. The appaent weight of one of the passenges is 3.0 times he tue weight (F N = 3.0Fg). Detemine the speed of the olle coaste. Since F c = F N F g mv 2 mv 2 = 3.0mg mg = 2. 0mg v = 2g whee mass cancels F N F c v = 2 22m 9. 81m/s 2 F g v = 21 m/s
Centipetal Foce and Vetical Motion Rolle coastes have evolved ove time. The cicula loop that was used almost a centuy ago has been eplaced by the clothoid loop found in moden looping coastes. Compae these 2 designs. Refe to Section 3.5 in you text Then and now..
Rolle Coaste Physics Check out these cool links: http://physicsbuzz.physicscental.com/2013/04/oll e-coaste-g-foces-weve-got-data.html http://physics.gu.se/liseberg/eng/pendill_loop _2013.pdf https://www.youtube.com/watch?v=stoff4_opkm