Discover the answer to this question in this chapter.

Transcription

1 In a oto ide such as the one shown in the figue, what is the maximum peiod of otation that the oto ide can hae so that people do not slip down the wall if the coefficient of fiction between the wall and people is 0.7 and if the adius of the oto ide is.5 m? Discoe the answe to this question in this chapte.

2 Application of Newton s Law to Cicula Motion As seen peiously, thee is an acceleation towads the cente in a cicula motion. This conclusion is einfoced if this motion is examined with Newton s laws. If thee is no foce, the object continues in a staight line (figue to the left). If the tajectoy changes, it is because a foce acted on the object. In the cente figue, a foce acted towads the ight fo a shot time in ode to deflect the object towads the ight. In a cicula motion (figue to the ight), it is necessay to deflect the object continuously. This means that thee is always a foce diected towads the cente of the cicle. This confims that thee is an acceleation towads the cente, as seen in chapte. If Newton s second law F = ma is combined with the fact that the acceleation is ²/ towads the cente, the following conclusion is eached. Cicula Motion In a cicula motion, thee must be a net foce diected towads the cente of the cicle whose magnitude is m. Thus, if an object tight at the end of a ope otates, thee must be a net foce diected towads the cente of the cicle. This foce is exeted by the tension foce of the sting acting on the stone. 018 Vesion 6 Cicula Motion

3 astonomy.nmsu.edu/thaiso/ast105/ast105week04.html Centipetal Foce The component of the net foce diected towads the cente of the cicle is called the centipetal foce. This is not a new type of foce; it is just a name gien to foces aleady known, like gaity, nomal foce, tension foce o fiction foce, when thei action esult in a cicula motion. In the example aboe, the tension foce is the centipetal foce. Ca on a Bump Now suppose that a ca passes oe a bump. The foces acting on the ca ae shown in this figue. f.depositphotos.com/577683/stock-illustation-ca.html Since the ca makes a cicula motion, the net foce must be diected towads the cente of the cicle. This foce must then be diected downwads when the ca is in the position shown in the figue. This means that the weight must be geate than nomal foce in ode to hae a esulting foce diected downwads. w > F N In this case, the nomal foce cancels pat of the weight and the emaining pat of the weight makes the downwads foce. Hee, the centipetal foce is thus made by a pat of the gaitational foce. 018 Vesion 6 Cicula Motion 3

4 Ca in a Dip Now suppose that a ca passes in a dip. The foces acting on the ca ae shown in this figue. f.depositphotos.com/577683/stock-illustation-ca.html Since the ca is a cicula motion, the net foce must be pointing towads the cente of the cicle. Hee the net foce must be diected upwads. This means that the nomal foce must be geate than the weight so that thee is a esultant upwads foce. w < F N In this case, a pat of the nomal foce cancels the weight, and the emaining pat of the nomal foce makes the upwads foce. Hee, the centipetal foce is thus made by a pat of the nomal foce. Object on a Tuntable Now imagine an object placed on a tuntable. de.physicslab.og/document.aspx?doctype=5&filename=compilations_cpwokbook_centipetalfoce.xml&asid=5679&siteid=%7 B9CF166F-C64-436A-B5DE-DF386F0FF11%7D In this case, the nomal foce and the weight cannot make the centipetal foce as they ae not diected towads the cente of the cicula motion. The only foce that can be diected towads the cente of the cicle is the fiction foce. The fiction foce is thus the centipetal foce hee. 018 Vesion 6 Cicula Motion 4

5 What Happens If the Centipetal Foce Disappeas? If the centipetal foce suddenly disappeas, then the cicula motion stops and the object continues its motion in a staight path because thee is no longe any foce acting on it. In the case illustated in the figue, the foce made by the ope disappeas when the ope beaks. The object can no longe be in a cicula motion as thee is no longe any foce towads the cente. The object then goes in a staight line in the diection of the elocity that the object had when the ope boke. Since the elocity is always tangent to the cicle in a cicula motion, the staight line motion is tangent to the cicle as shown in the figue. astonomy.nmsu.edu/thaiso/ast105/ast105week04.html This is what happens in this ideo. The peson in the ide can make a cicula motion because the nomal foce exeted by the metal ailing on the back of the peson is diected towads the cente. When the peson loses contact with the metal ailing, the foce disappeas, and the peson can no longe be in a cicula motion. The peson then continued its motion in a staight line and is thown to the gound. A too Geat o too Small Inwads Foce Too Much Inwads Foce If the sum of foces towads the cente cicle is bigge than m²/, then the net foce is geate than the foce equied to make a cicula motion with a constant adius. This excess of foce gies too much acceleation towads the cente of the cicle, and the object then gets close to the cente of the cicle. Hee s the tajectoy obtained if the sum of the foces is always geate than m²/. 018 Vesion 6 Cicula Motion 5

6 dawsketch.about.com/od/doodles/ss/geometic-spials-to-daw-and-doodle_3.htm Not Enough Inwads Foce If the sum of foces towads the cente cicle is less than m²/, then the net foce is smalle than the foce equied to make a cicula motion with a constant adius. This lack of foce does not gie enough acceleation towads the cente of the cicle, and the object then moes away fom the cente of the cicle. Hee s the tajectoy obtained if the sum of the foces is always infeio to m²/. dawsketch.about.com/od/doodles/ss/geometic-spials-to-daw-and-doodle_3.htm 018 Vesion 6 Cicula Motion 6

7 A Moie Mistake The cued bullet tajectoies in the film wanted, eceated hee is, theefoe, impossible. Een if the gun is otated, thee is no centipetal foce acting on the bullet once it has left the gun. Thus, the bullet is supposed to moe in a staight line (while falling to the gound because of gaity). It is, theefoe, impossible fo the path to be cued as seen in the clip. Many people think that thee is some sot a conseation of otational motion, but thee is none. In a unifom cicula motion, the speed of the object is constant. To sole a poblem of unifom cicula motion, it is bette to use an axis paallel to the acceleation. As the acceleation is towads the cente of the cicle, the axis must be in the adial diection, which means that it must be towads the cente of the cicula motion (inwads) o in the opposite diection to the cente of the cicle (outwads). Then, Newton s laws with an inwads acceleation of ²/ ae applied. Unifom Cicula Motion Radial Diection F = m with an axis diected inwads. F = m with an axis diected outwads. Fo all Othe Diections F = Vesion 6 Cicula Motion 7

8 In a unifom cicula motion, the centipetal acceleation can also be calculated with the fomula a = 4π whee T is the peiod, i.e. the time it takes fo the object to go aound the cicle once. This means that unifom cicula motion poblem-soling can also be done with the following fomulas. Unifom Cicula Motion Radial Diection T 4 F = with an axis diected inwads. m T π 4 F = with an axis diected outwads. m T π Fo all Othe Diections F = 0 This fom is mainly used if the peiod of otation is gien instead of the speed o if the peiod has to be found. Example 6..1 A 500 g object attached at the end of a m ope otates in a etical plane at a steady speed of 10 m/s. a) What is the tension of the ope when the object is at the lowest point of the tajectoy? Vesion 6 Cicula Motion 8

9 The foces acting on the stone ae: 1) The weight. ) The tension foce. Depending on the position of the object, the diections of these foces ae shown in the figue. Using an axis pointing upwads, the foce equation at the lowest point is F = m + = mg T m Theefoe, the tension is T = m + mg This equation indicates that the sum of the foces is not zeo since a centipetal foce must be pesent. Hee, only the tension foce is in the ight diection to make the centipetal foce. The tension foce should, theefoe, cancel the foce of gaity and also make the centipetal foce. ( 10 ) = 0.5kg + 0.5kg 9.8 m = 5N + 4.9N = 9.9N m s Of this 9.9 N tension foce, 4.9 N ae thee to cancel gaity, and 5 N ae thee to make the centipetal foce. N kg b) What is the tension of the ope when the object is at the highest point of the tajectoy? Using an axis pointing upwads, the foce equation at the highest point is 018 Vesion 6 Cicula Motion 9

10 Fy = m = mg T1 m Thee is a minus sign in font of m²/ since the centipetal foce is towads the cente of the cicle, which is downwads when the object is at the highest point. (An axis pointing upwads is used.) In this case, the tension foce and the weight ae in the ight diection to make the centipetal foce. The tension should, theefoe, be simply equal to what is missing fom the gaitation foce to make the centipetal foce. The tension is theefoe T1 = m mg m ( 10 ) s = 0.5kg 0.5kg 9.8 m = 5N 4.9N = 0.1N Thee ae aleady 4.9 N in the ight diection coming fom the weight. The tension must then be (0.1 N) to hae the equied 5 N centipetal foce fo this motion. Hee ae two emaks on the peious example: 1) Should the ope beak in such a motion, it would beak at the lowest point because the ope must compensate fo the foce of gaity and make the centipetal foce at this point. It is, theefoe, at this location that the tension is lagest. ) If the speed of the object wee to be lowe, say m/s, then thee would be a poblem at the highest point since a calculation simila to the one done in the peious example would gie us a tension of -3.9 N. As the tension cannot be negatie (that would mean that the ope pushes), this situation is impossible. At this speed, the centipetal foce equied is only 1 N, while the weight is 4.9 N. As thee is nothing to cancel the weight, thee is an excess of centipetal foce. The object would then moe close to the cente of the cicle, and the ope would slacken. N kg 018 Vesion 6 Cicula Motion 10

11 Example 6.. At what maximum speed can a 1000 kg ca take a cue haing a 10 m adius if the coefficient of static fiction between the ties and the oad is 0.8? Thee foces ae acting on the ca: 1) The weight diected downwads. ) The nomal foce diected upwads. 3) The static fiction foce, paallel to the gound. (It is a static fictional foce because the ties do not slip on the asphalt.) Only the fiction foce is in the ight diection to make the centipetal foce. Theefoe, the centipetal foce must be made by the static fictional foce. Theefoe, the equations ae Fx = Ff = m F = mg + F = 0 y N As thee is a maximum static fictional foce, we hae ² m = F µ F m µ sfn f.depositphotos.com/577683/stock-illustation-ca.html f s N Substituting the alue of the nomal foce (found with the sum of the y-component of the foces) into this equation, we obtain max max m µ sfn m µ s mg µ g = m 9.8 m = = 31.9 s s N kg km h Note that this speed is independent of the mass of the ca. 018 Vesion 6 Cicula Motion 11

12 This speed inceases if the oad is banked, as shown in the figue. and libay.thinkquest.og/5844/cicula_motion/banked.htm Let s see why the speed inceases with an example. Example 6..3 At what speed can a 1000 kg ca take cue haing a 10 m adius if the coefficient of static fiction between the ties and the oad is 0.8 and if the oad is banked 30 fom the hoizontal? Thee ae still 3 foces acting on the ca. f.depositphotos.com/577683/stock-illustation-ca.html 1) The weight diected downwads. ) The nomal foce pependicula to the oad. 3) The fiction foce, paallel to the oad. The x-axis is still hoizontal because the solution is geatly simplified if an axis in the diection of the acceleation is used. As the acceleation is towads the cente of the cicle, it is in the diection indicated in the figue, which is hoizontal. The weight is still not in the ight diection to make the centipetal foce, but the nomal foce has a hoizontal component, which now contibutes to the centipetal foce. On the othe hand, the fictional foce is not in the hoizontal diection anymoe. Only the hoizontal component of fiction foce can contibute to the centipetal foce. The centipetal foce is, theefoe, made hee by a component of the nomal foce and a component of the fiction foce. 018 Vesion 6 Cicula Motion 1

13 As the angle between the positie x-axis and the foces ae shown in the figue, the table of foces is Foces x y Weight 0 -mg Nomal foce N cos 60 N sin 60 Fiction foce f cos -30 f sin -30 Theefoe, the equations ae Fx = FN cos 60 + Ff cos ( 30 ) = m F = mg + F sin 60 + F sin 30 = 0 y N f ( ) ( ) As the maximum speed is wanted, the maximum static fictional foce is needed towads the cente. This happens when the fiction is at its maximum F fmax = µ s F N. The equations then become max Fx = F N cos 60 + µ sfn cos ( 30 ) = m F = mg + F sin 60 + µ F sin 30 = 0 ( ) ( ) y N s N The nomal foce can be found with the second equation. ( ( ) s ( )) mg + F sin 60 + µ sin 30 = 0 F N N mg = sin 60 sin 30 F N F ( ) + µ ( ) =.1458 mg N = 1,08N Substituting this nomal foce into the sum of the x-component of foces, the maximum speed is obtained. max.1458 mg cos 60 + µ s.1458 mg cos( 30 ) = m max max max 018 Vesion 6 Cicula Motion 13 s ( s ( )) N ( ) =.1458g cos 60 + µ cos 30 ( ) = m 9.8 cos ,8 cos 30 kg m = = 57 The mass of the ca does not matte since it no longe appeas in the equation of the maximum speed. It can also be noted that the maximum speed has inceased compaed to the peious example. A pat of nomal foce now acts as a centipetal foce, and this allows the ca to moe faste in the cue without slipping. s km h

14 It is possible to take a banked cue een if thee is no fiction. In this case, only the hoizontal component of the nomal foce acts as the centipetal foce. Howee, as the component of the nomal foce has a ey pecise alue, the ca must moe at exactly the ight speed to take such a cue without slipping if thee is no fiction. The oad can be banked een moe in ey tight cues. Thee is no limit to the banking angle gien to the oad, as shown in this ideo. As the wall is almost etical, the centipetal foce is made almost exclusiely by the nomal foce. Hee, the fiction foce peents the ca fom sliding down the wall. Banking the oad is not the only option to incease the maximum speed of a ca in a cue. Fomula 1 ace cas may take non-banked cues at much highe speed than the speed found in example 6.. ( max µ s g). To achiee this, wings ae used to incease the nomal foce between the ca and the gound. When the ai stikes the wings, it exets a downwads foce. The equation of foce is then F = F w F = 0 y N wings F = w + F N wings This shows that nomal foce is geate with wings. In fact, at high speeds, the foce made by the wing is much geate than the weight and the nomal foce is way lage than it would be without wings. This geate nomal foce geatly inceases the maximum static fiction foce (µ s F N ). This allows the dies to take cues at much highe speeds because the maximum centipetal foce is much geate (since the static fiction foce is the only foce acting as the centipetal foce in a non-banked cue). If one of the wings beaks, the nomal foce dops and the die can no longe take a cue at high speed. When the ea wing beaks, the nomal foce on the ea wheels suddenly dops, which causes a sudden dop of the fictional foce and thus of the centipetal foce. Thee is then a lack of centipetal foce on the ea wheels so that the back of the ca cannot take this cue as quickly. Theefoe, the ea of the ca difts towads the outside of the cue, causing the accident. The dag, howee, exets a foce on the wings which opposes the motion of the fomula 1. Fomula 1 dag coefficients ae ey high mainly because of this dag foce on the wings. This highe dag foce is, of couse, a handicap fo a ace ca but ultimately is it bette to hae wings. The cas go a little slowe on the staights, but they can go so much faste in cues that, in the end, they can go aound the tack much moe quickly with wings. 018 Vesion 6 Cicula Motion 14

15 The foces exeted by the wings ae so lage at high speeds that it would be possible fo a Fomula 1 ace ca to die upside down on a ceiling without falling (at speed lage than about 160 km/h) since the foce made by the wings, which ae pushing the ca towads the ceiling, would be geate than the weight. Example 6..4 In a oto ide such as this one (you ll see at 1:10 that the floo is emoed and that the people emain stuck on the wall), what is the maximum peiod of otation that the oto ide can hae so that people do not slip down the wall if the coefficient of fiction between the wall and people is 0.7 and if the adius of the oto ide is.5 m? Thee ae thee foces on the peson. 1) The weight, diected downwads. ) The fiction foce, diected upwads. 3) The nomal foce, diected towads the cente of the cicula motion (inwads). With an x-axis pointing inwads and a y-axis pointing upwads, the equations ae 4π Fx = FN = m T F = mg + F = 0 y f As the otation peiod is sought, the centipetal foce fomula with the otation peiod is used. Accoding to the second equation, the fiction foce is mg = F f Since this fiction foce must be smalle than the maximum fiction foce, we hae mg = F < F f f max The peiod can now be found. 018 Vesion 6 Cicula Motion 15

16 mg < µ F 4π mg < µ s m T s N T 4π µ g s The maximum otation peiod is theefoe T max = 4π.5m 0.7 =.655s 9.8 N kg Sometimes, an object loses contact with anothe duing a cicula motion. In ode to know whethe this happens o not, the magnitude of the nomal foce between the two objects in contact must be found, assuming they ae in contact. A positie alue fo the nomal foce means that the objects do indeed touch each othe wheeas a negatie alue fo the nomal foce means that this situation is impossible and that the objects do not touch each othe. Example 6..5 A ca going at 1 m/s passes oe a bump haing a 5 m adius, as shown in the figue. a) Is the ca still in contact with the oad at the top of the bump? Let s assume that the ca is still in contact with the oad at the top of the bump. The foces on the ca ae: 1) The weight, diected downwads. ) The nomal foce, diected upwads. f.depositphotos.com/577683/stock-illustation-ca.html The sum of etical foces is 018 Vesion 6 Cicula Motion 16

17 Fy = mg + FN = m m²/ is negatie because the centipetal foce is diected downwads since the cente of the cicle is unde the ca. This equation gies FN = mg m = m g N = m 9.8 kg = m N ( 19 kg ) ( 1 ) Regadless of the alue of the mass, the nomal foce is negatie, which is impossible. Theefoe, the ca is not in contact with the oad. In fact, only the foce of gaity was in the ight diection hee to act as the centipetal foce. Howee, this gaitational foce is not sufficiently lage to make the centipetal foce, and thee is, theefoe, a lack of centipetal foce. Thus, the ca moes away fom the cente of the cicle, theeby losing contact with the oad. b) What maximum speed can a ca passing on this bump hae without losing contact with the oad? To hae a positie nomal foce, we must hae m s 5m F N = m g > 0 g > < g < 5m 9.8 < 7 m s N kg Theefoe, the maximum speed is 7 m/s. 018 Vesion 6 Cicula Motion 17

18 Common Mistake: Thinking That Thee Is a Centifugal Foce Diected Outwads in a Cicula Motion. You suely head the tem centifugal foce befoe. When a peson is ejected fom a mey-go-ound, the centifugal foce is often inoked as the cause of this ejection. That is wong. The lack of foce centipetal is eally the cause of the ejection. A Lack of Centipetal Foce Conside the situation shown in this small animation to illustate ou point. Thee is an object on a olling cat and thee is no fiction between the object and the cat. When the cat stats to tun, the small object cannot follow this cicula motion since no foce can act as the centipetal foce. Only the weight and the nomal foce ae acting on the small object, and none is in the ight diection to deflect the tajectoy to allow the object to tun with the cat. Without a centipetal foce, the small object continues in a staight line as the cat tuns. As a esult, the small object falls fom the cat. No centifugal foces ae pushing the object towads the outside of the cue. It is instead the absence of centipetal foce which caused the object to fall fom the cat since it has not been able to follow the cicula motion of the cat. If a ca ties taking a cue with too much speed, it will leae the oad, falling into the outside ditch of the cue. This is not because a centifugal foce pushes the ca outwads, but because thee is a lack of centipetal foce. In a cue, the ehicle tuns because thee is a fictional foce between the ties and the oad acting as centipetal foce. As thee is a maximum to this fictional foce, thee is a maximum to the centipetal foce. If the ca is going too fast in the cue, it is possible that class/cicles/u6l1c.cfm the fiction foce, een at its maximum alue, is smalle than the centipetal foce equied to take this cue. The fiction deflects the path of the ca, but not enough to follow the cuatue of the oad. The ca thus makes a cicula motion, but with a geate adius of cuatue than the oad, causing the ca to fall into the outside ditch. A centifugal foce does not push the ca into the ditch; it fell thee because thee was a lack of centipetal foce. stockcascience.com/sciencetopics/scsracing_centifugalfoce.php 018 Vesion 6 Cicula Motion 18

19 This is what happens to these cas. In this othe situation, the cicula motion becomes so fast that this peson can no longe hold his toso upight. The centifugal foce is not the foce pushing his toso outwads, it is the lack of centipetal foce. The muscles of the peson must make the inwads foce, allowing his toso to make the cicula motion. When the otational speed becomes too geat, the muscles ae no longe sufficient, and thee is then a lack of centipetal foce. The toso of the peson can no longe follow the cicula motion, and it then moes away fom the axis of otation. Once his toso is hoizontal, the tensile esistance of his body now acts as the centipetal foce. Fotunately, he did not lose contact with the ailing fo then the centipetal foce would hae completely disappeaed, and he would hae flown in a staight line motion as in this clip. Thee Ae Foces Diected Outwads When a stone otates at the end of a ope and a peson holds the ope, it is possible to think that thee is a centifugal foce because the peson feels that the ope pulls on its hand towads the outside of the cicle. That is tue, but the foces made by the ope at each end of the ope must be looked at ey caefully. The ope pulls on the stone towads the inside of the cicle. This is the centipetal foce equied fo the stone to make its cicula motion. The ope also pulls with equal foce to the othe end of the ope. So the ope pulls on the hand towads the outside of the cicle. This is indeed a foce diected outwads, but this foce does not act on the object making the cicula motion. The net foce on the object making the cicula motion is always diected towads the cente of the cicula path. The foce can be diected outwads on othe objects. astonomy.nmsu.edu/thaiso/ast105/ast105week04.html In a ca taking a cue, thee seem to be foces diected outwads. Suppose you e sitting on the passenge side, without a seatbelt, on a ey slippey seat. If the ca tuns left, you slide outwads until you come into contact with the doo. You slide towads the doo because thee is no centipetal foce acting on you. You then continue you motion in a staight line while the ca is tuning. If you continue in a staight line and the ca is tuning, 018 Vesion 6 Cicula Motion 19

20 you eentually come into contact with the side doo. With this contact, thee is now a nomal foce pushing you inwads that acts as the centipetal foce to allow you to tun with the ca. As a contact foce is always a epulsie foce, the foce that the doo exets on you is indeed diected towads the cente of the cicula motion. If the doo pushes on you inwads, then, accoding to Newton s thid law, you must exet a foce outwads on the doo. Does this mean that thee is an outwads foce on the ca and that thee is, theefoe, a centifugal foce? No. Thee is indeed this outwads foce on the ca, but the net foce on the ca must be diected inwads. When you exet an outwads foce on the doo, the inwads fiction foce between the oad and the ties must incease slightly to compensate fo this outwads foce. The Centifuge Although the centifugal foce does not exist, centifuges exist. Howee, the centifugal foce does not explain why substances settle at the bottom of test tubes. Hee s a simplified centifuge model, used to undestand the mechanism. A test tube is simply otating as shown in the figue. f.openclassooms.com/sciences/cous/la-physique-chimie-en-seconde/a-centifugation A Test Tube Filled With Wate Only Conside fist a test tube only filled with wate. As the wate makes a cicula motion, thee must be a foce acting on the wate. This foce is made by the wate pessue. When the test tube is otated, a pessue aiation appeas in the wate. The pessue is lagest at the bottom of the tube and deceases towads the suface of the liquid. matoumatheux.ac-ennes.f/num/factions/6/epouette.htm 018 Vesion 6 Cicula Motion 0

21 Let s examine the foce on a small slice of wate in the test tube. The pessue foce exeted below the laye is geate than the pessue foce exeted aboe the laye. The excess pessue foce towads the cente of the cicula path is the foce that makes the centipetal foce of the wate slice. In fact, the pessue as a function of depth could be calculated based on the assumption that the pessue incease with depth should geneate the ight centipetal foce on any slice of wate. A Test Tube Filled with Wate Containing Paticles Now imagine that thee is a paticle in the wate. The wate pessue then acts on the paticle. As the pessue gets lage towads the bottom of the test tube, the pessue foce is geate on this side. The sum of these foces acting on the paticle is a foce towads the axis of otation. This foce is, in fact, a buoyant foce and is in the ight diection to act as a centipetal foce. The magnitude of this foce can easily be found, using the fomula of the buoyant foce. Howee, the alue of the gaitational acceleation must be eplaced by the centipetal acceleation of the wate in this fomula. Theefoe, the foce is F B ρwate Vf = 018 Vesion 6 Cicula Motion 1

22 The sum of the x-component of the foce is then (using a positie x-axis pointing to the left, so towads the axis of otation) x ρwate f F = V = ma Fom this equation, the acceleation can be found ρ ρ wate ρ V wate = wate = a f ρ = ma m a V f paticle ρ wate = ρ paticle a If the paticle density is the same as the wate density, the acceleation of the paticle is exactly ²/. This means that the buoyant foce has just the ight magnitude to make the centipetal foce. The paticle then makes its cicula motion and always emains at the same distance fom the axis of otation. The paticle, theefoe, emains at the same position in the tube. This occus only if the paticle has the same density as wate. If the paticle has a highe density than wate, then a < ²/. The buoyant foce is not lage enough, and thee is a lack of centipetal foce. Thus, the paticle cannot follow the cicula motion with a constant adius. With a lack of centipetal foce, the paticle makes a cicula motion while moing away fom the axis of otation, until it eaches the bottom of the tube. The highe the density of the paticle, the lage the lack of centipetal foce is, and the faste the paticle eaches the bottom of the cylinde. The dense paticles will, theefoe, each the bottom fist, followed by othe less dense paticles, and then by othes less dense paticles and so on. Thee is then a sepaation of the paticle accoding to thei density at the bottom of the test tube. If the paticle has a smalle density than wate, then a > ²/. The buoyant foce is too lage, and thee is an excess of centipetal foce. In this case, the paticle makes a cicula motion while moing towads the axis of otation, until it eaches the suface of the wate in the test tube. In the end, we get a test tube in which the paticles ae deposited in ode of density at the bottom of the tube fo paticles dense than wate and whee the paticles less dense than wate float to the suface of the tube. The faste the otation, the faste this sepaation is done. The centifugal foce does not make this sepaation. 018 Vesion 6 Cicula Motion

23 Sometimes, an object makes a cicula motion, but with a changing speed. This is a nonunifom cicula motion. In this kind of motion, thee ae centipetal and tangential acceleations. These two acceleations can be found with the sum of the foces. This is easie if the axes ae well chosen. 1) Use an x-axis in the tangential diection ( possibilities). ) Use a y-axis in the adial diection (towads the cente of the cicula path o in the diection opposite to the cente of the cicula path). The poblem is, theefoe, soled by following these ules. Non-Unifom Cicula Motion Radial Diection m Fy = if the axis is diected inwads m Fy = if the axis is diected outwads Tangential Diection F x = ma T 018 Vesion 6 Cicula Motion 3

24 Remembe this common mistake. 4π Common Mistake: Using ac = to Calculate the T Centipetal Acceleation in a Non-Unifom Cicula Motion. This fomula was obtained by assuming that the speed is constant. Do not use it in a nonunifom cicula motion, whee the speed changes. Example An object is sliding on a spheical suface. What ae the tangential acceleation and the magnitude of nomal foce when the object is at the position shown in the figue? The foces acting on the object ae: 1) The weight. ) A nomal foce. Using the axes shown in the figue, the foce equations ae ( ) F = mg cos 50 = ma x Fy = mg sin ( 50 ) + FN = m R With the fist equation, the tangential acceleation can be found T a T ( ) = g cos 50 = 6.99 m s² The nomal foce can be found with the second equation. 018 Vesion 6 Cicula Motion 4

25 FN = mg sin ( 50 ) m R ( ) ( 5 ) N = 8kg 9.8 kg sin 50 8kg 5m = 60.06N 40N = 0.06N Note that the total acceleation hee is (een if it was not asked to find it) m s a = a + a c T ( 5 ) m s = + 5m = 8.04 m s² m ( 6.99 ) If the speed had been highe, say 7 m/s, the nomal foce would hae been FN = mg sin ( 50 ) m R ( ) s² N = 8kg 9.8 sin 50 8kg = 60.06N 78.4N = 18.34N kg ( 7 ) which is impossible. This means that to hae this cicula motion at such a speed, the nomal foce must pull on the object, something a nomal foce cannot do. Thee is, theefoe, a lack of centipetal foce. Thus, the object would not able to make a cicula motion with such a small adius, and it would leae the suface of the sphee. m s 5m 018 Vesion 6 Cicula Motion 5

26 Speed and Peiod on a Cicula Obit Objects in a cicula obit aound a planet o a sta make a cicula motion. Theefoe, a centipetal foce must act on the object. As the foce of gaity is the only foce acting on the object, this foce must be the centipetal foce. This means that GM cm = m The mass of the body at the cente of the obit is called M c, fo cental mass. The object in obit must hae a ey pecise obital speed to follow a cicula obit. This speed can be found by soling the peious equation fo. Speed of an Object on a Cicula Obit = GM c The time equied to go aound the obit once is called the peiod T. This peiod is T distance π = = speed GM c Simplified, this is Peiod of an Object on a Cicula Obit T = π 3 GM c This fomula is often called Keple s thid law since Keple discoeed in 1618 that T² α ³ fo planets in obit aound the Sun. 018 Vesion 6 Cicula Motion 6

27 Example Knowing that the Eath eoles aound the Sun with a peiod of days and that the adius of the Eath s obit is 149,600,000 km, calculate the mass of the Sun. The mass is found with T = π s = π M c 3 GM = c ( m) kg 11 Nm kg 3 M c This is how the mass of eey planet of the sola system was found. What Happens if the Object in Obit Does Not Hae the Right Speed? The object must hae a ey pecise speed fo its obit to be cicula. Howee, no ham is done if the speed is not exactly equal to the speed equied to make a cicula obit. If the elocity of the object is too small, we hae GM cm > m This means that thee is an excess of centipetal foce. In this case, the object makes a cicula motion while moing towads the cente of the obit. This excess foce does not necessaily mean that the object will cash on the cental mass because the object will gain speed as it appoaches the cental mass and the gaitational foce will incease, theeby changing the elation between the centipetal and gaitational foces. Then, the gaitational foce could become smalle than the centipetal foce and the object could stat to moe away fom the Eath Vesion 6 Cicula Motion 7

28 If the speed is too high, we hae GM cm < m This means that thee is a lack of centipetal foce. In this case, the object makes a cicula motion while moing away fom the cente of the obit. This lack of foce does not necessaily mean that the object will get lost in space because the object will lose speed as it moes away fom the cental mass and the gaitational foce will decease, theeby changing the elation between the centipetal and gaitational foces. Then, the gaitational foce could become smalle than the centipetal foce and the object could stat to moe back towads the Eath. The esulting tajectoies will be examined with moe details in chapte 14. Objects in Obit Ae in Fee-Fall This is not obious but objects in obit ae fee-falling. To poe this, suppose that balls ae launched hoizontally fom the top of a ey high cliff, but at diffeent speeds. Thee ae no dag foces in these situations. The fist ball is thown with a elatiely small speed. This is a pojectile that falls with a tajectoy bending towads the gound esembling the path shown in this figue. As all pojectiles, this ball is fee-falling. slid.es/tofegegg/gaity-and-fluid-dynamics/fullsceen#/ 018 Vesion 6 Cicula Motion 8

29 If the ball is thown with moe speed, we still hae a fee-falling ball. The path then looks like the path shown in this figue (whee the cuatue of the Eath is much exaggeated). slid.es/tofegegg/gaity-and-fluid-dynamics/fullsceen#/ The ball was thown with so much speed that the cuatue of the Eath begins to be significant. The ball still cued towads the gound, but the gound also cues. In this case, the ball eentually hit the gound. If the speed is futhe inceased up to the speed equied fo a cicula obit, the following situation is achieed. slid.es/tofegegg/gaity-and-fluid-dynamics/fullsceen#/ The tajectoy of this fee-falling object still cues towads the gound because of the foce of gaity. Howee, the gound also cues so that the object is always at the same distance fom the gound. Thus, the object, een if it is fee-falling, nee hit the gound and moes continuously on its cicula obit. So, why doesn t the Moon fall towads the Eath if it is attacted by the foce of gaitation? Just as the ball fom the last example, the Moon is indeed fee-falling, but it nee hits the Eath because gaity only cues the tajectoy of the Moon so that it does not appoach the Eath. The following clip gies you the same explanation Vesion 6 Cicula Motion 9

30 Common Mistake: Thinking That Thee is a Balance between Gaity and a Centifugal foce on an Object in Obit. Often, it is said that thee is a balance between centifugal and gaitational foce fo objects in a cicula obit, as shown in the figue to the left. This cannot be coect since the centifugal foce does not exist. Moeoe, if these two foces wee to cancel each othe, the sum of the foces would be zeo and the object would moe in a staight line. A cicula path cannot be obtained if the sum of the foces is zeo because this would be in obious contadiction with Newton s fist law. I een found this explanation in a magazine in India whose goal is to help futue uniesity students pepae fo uniesity entance exams. This is bad. xp.hauduoy.fee.f/mise_en_obite.html Synchonous Satellites It is possible to place a satellite at just the ight distance fom Eath so that its peiod is 4 hous (actually 3 h 56 min 4 s). The satellite then otates aound the Eath at exactly the same ate as the Eath otates on itself and thus always emains aboe the same spot on Eath. In othe wods, the satellite always sees the same side of the Eath. These satellites ae impotant because thei positions can easily be found since they ae always in the same diection in the sky. Once an antenna is adjusted to point towads the satellite, it is always in the ight diection. If the satellite wee obiting faste o slowe, it would be necessay to constantly change the diection of the antenna to eceie the signal. This little ideo gies you an animated esion of this explanation. Example 6.5. How fa fom the suface of the Eath should synchonous satellites be placed and what should be thei speed knowing that the mass of the Eath is x 10 4 kg? Using the peiod fomula, the distance can be found. 018 Vesion 6 Cicula Motion 30

31 s = π T = π 3 GM c Nm 4 kg 7 = m = 4, 45km kg All synchonous satellites hae this obital adius. As the Eath has a adius of 6378 km, they ae thus 35,866 km away fom the Eath s suface. The obital speed of these satellites is = GM All synchonous satellites hae this speed. c 11 Nm² kg ² kg = m m km = 3,07 = 11,060 s h Thee ae, theefoe, two ey busy locations aound the Eath to place satellites. Most satellites ae placed close to the Eath (which is cheape) o in synchonous obit. The following figue showing the position of satellites aound the Eath illustates this. Most of them ae ey nea o at a distance whee the satellites hae a 4-hou peiod. 018 Vesion 6 Cicula Motion 31

32 Unifom Cicula Motion Radial Diection F = m o 4π F = m if the axis is diected inwads. T F = m o 4π F = m if the axis is diected outwads. T Fo all Othe Diections F = Vesion 6 Cicula Motion 3

33 Non-Unifom Cicula Motion Radial Diection m Fy = if the axis is diected inwads m Fy = if the axis is diected outwads Tangential Diection F x = ma T Speed of an Object on a Cicula Obit = GM c Peiod of an Object on a Cicula Obit T = π 3 GM c 018 Vesion 6 Cicula Motion 33

34 6. Unifom Cicula Motion 1. This 00 kg sled moes along a tack with a steady speed of 34 m/s. a) What is the magnitude of the centipetal foce when the sled is the bend with a 33 m adius? b) What is the magnitude of the centipetal foce when the sled is the bend with a 4 m adius? whs.wsd.wednet.edu/faculty/busse/mathhomepage/busseclasses/apphysics/studyguides/chapte5/apphysicsch5_006.php. A ca going at a steady speed of 10 km/h takes a cue with a 100 m adius. The oad is not banked. What is the minimum alue that the coefficient of static fiction between the ties and the oad must hae so that the ca takes this cue without slipping? Vesion 6 Cicula Motion 34

35 3. A small 00 g block is placed on a tuntable, 10 cm fom the axis of otation. de.physicslab.og/document.aspx?doctype=5&filename=compilations_cpwokbook_centipetalfoce.xml&asid=5679& siteid=%7b9cf166f-c64-436a-b5de-df386f0ff11%7d What is the otation peiod if the block is taelling at the maximum speed that it can hae without slipping off the tuntable if the coefficient of static fiction between the block and the table is 0.6? 4. In the situation shown in the figue, what ae the nomal foces exeted on the 1000 kg ca at points A and B? 5. How fast must this ca go in ode to stay in contact with the tack at the top of its tajectoy (point B)? Vesion 6 Cicula Motion 35

36 6. A ca passes on a banked oad (θ = 0 ) haing a adius of = 80 m. The oad is so icy that thee is absolutely no fiction between the oad and the ties. How fast must this ca go to take this cue without slipping? 7. A 100 kg ca passes on a banked oad (θ = 30 ) haing a adius of = 80 m. a) What is the foce of fiction exeted on the ca if the ca s speed is 100 m/s? b) What is the foce of fiction exeted on the ca if the ca s speed is 10 m/s? 8. Hee is a 100 kg ca unning on a etical wall. conques-the-wall-of-death The coefficient of static fiction between the ties and the wall is 0.8, and the adius of the tack is 5 m. a) What is the minimum speed that the ca must hae so that it does not slip? b) What is the magnitude of the nomal foce exeted on the ca if the ca s speed is equal to the minimum speed? 9. The peson in this ide has a mass of 60 kg. mail.dcd.ab.ca/~smolesky/physics35/1) KinCic/Day1.html a) What is the tension of the ope? b) How long does it take fo this peson to go aound the ide once? 018 Vesion 6 Cicula Motion 36

37 10. What ae the tensions of the stings in the following system? 11. When this cone tuns at the ate of eolutions pe second, the small block does not slide on the cone. If thee is no fiction between the block and the cone, what is the alue of x? Non-Unifom Cicula Motion 1. A small tube takes compessed ai to a 3 kg block so that the compessed ai popels the block and gies it a tangential acceleation. The block is placed on a table, and thee is no fiction between the block and the table. Hee s this situation, iewed fom the top. 018 Vesion 6 Cicula Motion 37

38 Initially, the block is at est and ai steam exeted a 1.5 N foce on the block. Neglect the mass of the tube. a) What is the acceleation of the block seconds afte the stat of the motion? b) What is the tension in the tube seconds afte the stat of the motion? 13. Hee s Gontan, whose mass is 65 kg, playing Tazan. a) What is the magnitude of Gontan s centipetal acceleation? b) What is the magnitude of Gontan s tangential acceleation? c) What is the magnitude of Gontan s acceleation? d) What is the tension of the ope? 018 Vesion 6 Cicula Motion 38

39 6.5 Objects on Cicula Obits 14. Calculate the mass of the Eath knowing that the Moon eoles aound it with a 7.3 day peiod on an obit whose adius is km. 15. Io obits Jupite with a day peiod on an obit whose adius is 41,700 km. Ganymede also obits aound Jupite but on an obit whose adius is 1,070,400 km. a) What is the peiod of Ganymede? b) What is the obital speed of Ganymede? 16. The Apollo capsule was obiting the Moon at an altitude of 100 km aboe the suface. Knowing that the Moon has a mass of 7.35 x 10 kg and a adius of 1737 km, how long did it take fo the Apollo capsule to go aound the Moon once? 17. How fa fom the suface of the Eath should a satellite be placed so that it goes aound the Eath in days? (Mass of the Eath = 5.97 x 10 4 kg, adius of the Eath = 6378 km.) Challenges (Questions moe difficult than the exam questions.) 18. A ope loop haing a 50 cm adius and a 6 kg mass is spinning at a ate of 4 eolutions pe second. What is the tension of the ope? Vesion 6 Cicula Motion 39

40 19. In the situation shown in the figue, the sping constant is 000 N/m and the length of the sping when it is neithe stetched no compessed is 80 cm. What is the length of the sping if this system otates at 10 eolutions pe second and if it is in a spaceship fa away fom all stas and planets (so thee is no gaity)? (Thee is no fiction.) 6. Unifom Cicula Motion 1. a) 7006 N b) 9633 N s 4. Lowest point: 38,600 N highe point: 19,000 N 5. 7 m/s m/s 7. a) 14,03 N downhill b) 4581 N uphill 8. a) 7.86 m/s b) 14,700 N 9. a) 1391 N b) 4.5 s N (lowe sting) and 86,71 N (uppe sting) cm 6.4 Non-Unifom Cicula Motion 1. a) m/s² b).5 N 13. a) 0 m/s² b) 6.99 m/s² c) m/s² d) 1788 N 018 Vesion 6 Cicula Motion 40

41 6.5 Objects on Cicula Obits x 10 4 kg 15. a) 7.63 j b) 10.7 km/s h ,666 km Challenges N cm 018 Vesion 6 Cicula Motion 41