AP * PHYSICS B. Circular Motion, Gravity, & Orbits. Teacher Packet

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1 AP * PHYSICS B Cicula Motion, Gavity, & Obits Teache Packet AP* is a tademak of the College Entance Examination Boad. The College Entance Examination Boad was not involved in the poduction of this mateial. Copyight 008 Laying the oundation, Inc., Dallas, TX. All ights eseved. Visit:

2 Cicula Motion, Gavity, & Obits Objective To eview the student on the concepts, pocesses and poblem solving stategies necessay to successfully answe questions on cicula motion, gavity, and obits. Standads Cicula motion, gavity, and obits ae addessed in the topic outline of the College Boad AP* Physics Couse Desciption Guide as descibed below. I. Newtonian Mechanics E. Cicula motion and otation 1. Unifom cicula motion. Toque and otational statics. Oscillations and Gavitation 4. Newton s law of gavity 5. Obits of planets and satellites (cicula) AP Physics Exam Connections Topics elating to cicula motion, gavity, and obits ae tested evey yea on the multiple choice and in most yeas on the fee esponse potion of the exam. The list below identifies fee esponse questions that have been peviously asked ove cicula motion, gavity, and obits. These questions ae available fom the College Boad and can be downloaded fee of chage fom AP Cental. ee Response Questions 004 Question om B Question Question 1 00 om B Question 1999 Question Question 1 (a) AP* is a tademak of the College Entance Examination Boad. The College Entance Examination Boad was not involved in the poduction of this mateial. Copyight 008 Laying the oundation, Inc., Dallas, TX. All ights eseved. Visit:

3 Cicula Motion, Gavity, & Obits What I Absolutely Have to Know to Suvive the AP* Exam oce is any push o pull. It is a vecto. Accoding to Newton s Second Law, bodies in motion continue to move in a staight line unless acted upon by a foce. oces that change the diection of a body s velocity vecto (vesus changing the magnitude of the body s velocity vecto) ae called cente seeking foces o centipetal foces. A centipetal foce is NOT a new kind of foce, but is any natually occuing foce in a system that changes the diection of a body s motion, like the nomal, fictional, weight, tension, and gavitational foce. Centipetal foces poduce centipetal acceleations. The centipetal acceleation a body expeiences is elated to the adius of the body s path and the magnitude of the body s velocity. When a body is being pushed o pulled out of staight line motion, thee exists a single foce, the component of a single foce, o the combination of foces and components that act centipetally. The NET centipetal foce is always diected towad the cente of the cicula motion. If a body is following a cuved path, thee must be a natually occuing foce like tension, component of the nomal foce, o gavitation that is diected towad the cente of the cicula path. Centipetal acceleation is always pependicula to the velocity vecto o line of motion. Key omulas and Relationships v ac whee vvelocity o speed of an object in UCM o obiting satellite ac centipetal acceleation π v π f adius of cicula motion o adius of obit T T peiod of evolution f fequency c mac c centipetal foce π m T 4π m c mmass T GM -11 Nm v G6.67x10 univesal gavitational constant kg M mass of object being obited AP* is a tademak of the College Entance Examination Boad. The College Entance Examination Boad was not involved in the poduction of this mateial. Copyight 008 Laying the oundation, Inc., Dallas, TX. All ights eseved. Visit:

4 Cicula Motion, Gavity, & Obits Centipetal oce is NOT a new foce, but one o moe of these nomally occuing foces o thei components Weight Gavitational Nomal g, W G N, N g mg Always diected towad the Eath s cente. oce on a fee falling body, if we neglect ai fiction. mm 1 G G A foce of attaction between any two massive objects. When the Eath is one of the two bodies involved, then the foce felt by the second body while positioned on the Eath s suface will always be diected towad the Eath s cente. A foce of suppot, povided to an object by a suface in which the object is in contact. Always diected pependicula to and away fom the suface poviding the suppot. iction is poduced by the atomic inteaction between two bodies as they eithe slide ove one anothe (kinetic fiction) o sit motionless in contact with one anothe (static fiction). f smax μ (static) s N μ fk k N (kinetic) iction, f f WARNING! The two quantities f s and f k may look the same, but they tell us diffeent things. Kinetic fiction is typically less than static fiction fo the same two sufaces in contact. Note that the nomal foce sometimes equals the weight but not always. When you daw a fee body diagam of foces acting on an object o system of objects, be sue to include the fictional foce as opposing the elative motion (o potential fo elative motion) of the two sufaces in contact. Tension Applied T, T Subscipt T is a foce that is applied to a body by a ope, sting, o cable. T is applied along the line of the sting and away fom the body in question. Push me, pull you foce that does not fall into one of the above categoies, fo example, a fiend shoves you. The magnitude of the foce is chaacteized by an, with an appopiate subscipt. Copyight 008 Laying the oundation, Inc., Dallas, TX. All ights eseved. Visit:

5 Cicula Motion, Gavity, & Obits Stategy on Centipetal oce Poblems 1. Take one body in the system and daw a ee Body Diagam (BD) fo it. Remembe that the centipetal foce is not a NEW foce, so do NOT include it on you BD. You BD will have you old fiends like the weight foce, nomal foce, fictional foce, tension in a sting, the gavitational foce, etc. poviding the cente seeking foce.. Choose x and y axes and place them beside you BD. One axis must be in the diection of the centipetal acceleation you ae tying to find. Remembe that the centipetal acceleation and centipetal foce always point towad the cente of the cicula path. The velocity vecto is pependicula to both and tangential to the path of the object. 3. If thee ae foces on the BD that ae not along the x and y diections, find thei espective x and y components. Remembe the x o y component might be the centipetal o cente seeking foce. 4. Using Newton s nd Law, sum the foces in the diection of the centipetal foce and set them equal to. Impotant Concepts An object that moves in a cicle at constant speed, v, is said to expeience unifom cicula motion (UCM). The magnitude of the velocity emains constant, but the diection of the velocity is continuously changing, as shown in the diagam below. Notice that the velocity vectos ae not pointing in the same diection. The velocity vecto is always tangent to the cicula path. So fa, the only acceleations we have dealt with have been those that change the magnitude of an object s velocity, i.e. that make an object speed up o slow down. This type of acceleation is called tangential acceleation, in which the object speeds up if the acceleation and the velocity ae in the same diection. The object slows down if the acceleation and the velocity ae in opposite diections. Thee is anothe way that an acceleation can change an object s motion: it can change the diection of the object s velocity Copyight 008 Laying the oundation, Inc., Dallas, TX. All ights eseved. Visit:

6 Cicula Motion, Gavity, & Obits vecto. An object evolving in a cicle is continuously acceleating, even when the speed emains constant. The acceleation vecto is always pointing towad the cente of the cicula path and it is always pependicula to the velocity vecto, as shown below. Centipetal (cente-seeking) acceleation, a c, is defined as: a c v If we measue the peiod, T, the time needed fo the object to make a complete evolution, we can calculate the speed of the object in UCM. Duing this time, it tavels a distance equal to the cicumfeence of the π cicle, π, whee is the adius of the cicula path. The object s speed, then, is epesented by v. T Accoding to Newton s Second Law (Σ ma), an object that is acceleating must have a net foce acting upon it and we call this foce, the centipetal foce. This foce acts in the same diection as the acceleation which is towad the cente of the cicle. The equation fo centipetal foce can be expessed as π v T 4π c ma mac m m T Copyight 008 Laying the oundation, Inc., Dallas, TX. All ights eseved. Visit:

7 Cicula Motion, Gavity, & Obits Centipetal foce is not a new kind of foce, but the net foce which points towad the cente of the cicle. It may be a single foce, a component of a foce, o a combination of foces that ae cente seeking. We ae not talking about a new kind of foce, but a new kind of situation in which the same old foces such as, the gavitational foce, fictional foce, nomal foce, the tension in a sting, o a combination theeof constain the object to follow a cuved path. The centipetal foce does NO wok on an object, since wok is defined as W cosθ since the angle between the centipetal foce and is 90º and the cosine of 90º0. In addition, since thee is no change in the speed of the object, by the wok kinetic enegy theoem, the enegy of the system emains constant unless wok is done on the system and hence the wok done is zeo. Centipetal foce is a tem used to identify foces that motivate an object out of staight line motion and into cuved motion. These cente seeking foces must exist natually within a system. Do not confuse the centipetal foce with the centifugal foce which is a fictitious foce. Centifugal foce is a tem used to identify foces that must be assumed to exist if one is to use Newton s Second Law in analyzing poblems fom a non-inetial fame of efeence. Do NOT use the tem, centifugal foce on the AP Exam since it is a foce that does not exist. Example UCM 1. A sphee on the end of a sting is swung in a vetical cicle. The mass of the sphee is m and the adius of the cicle is R. The position of the ball is maked evey quate of a evolution in the diagam below. 1 4 R 3 Copyight 008 Laying the oundation, Inc., Dallas, TX. All ights eseved. Visit:

8 Cicula Motion, Gavity, & Obits (a) On the diagam below, daw all of the foces acting on the sphee when it is at position 4. T W The only foces acting on the sphee in position 4 ae the weight foce and the Tension in the sting. The tension povides the centipetal foce. Do not include the centipetal foce on you BD since in this case, the tension is the centipetal foce. (b) When the sphee is at position 4, the tension foce in the sting is twice as geat as the weight of the sphee. Detemine the speed of the sphee at position 4 in tems of the given quantities and any fundamental constants. T W mg R v gr v gr (c) On the diagams below, daw and label the foces acting on the sphee i. at the top of the cicle (position 1) and ii. at the bottom of the cicle (position 3). Position 1 Position 3 T T W W Copyight 008 Laying the oundation, Inc., Dallas, TX. All ights eseved. Visit:

9 Cicula Motion, Gavity, & Obits (d) Wite NSL fo the sphee when in position 1. Assume speed is v at position 1. Note that the centipetal foce is the combination of the sum of the tension and the weight of the sphee which both point towad the cente. c Σ mac T + W R (e) What is the minimum speed of the sphee at position 1 such that the sphee follows a cicula path at the top. v min min occus when T 0 T + mg 0 + mg v gr min min (f) Wite NSL fo the sphee when in position 3. Assume the speed of the sphee is v at the bottom. Note that the centipetal foce is the combination of the diffeence between the tension and the weight of the sphee since they point in opposite diections. c Σ mac T mg R (e) Suppose the sting beaks just as the sphee is at position 1. i. Descibe the subsequent motion of the sphee. The sphee will tavel tangent to the path and follow a paabolic tajectoy. ii. Assuming the speed at position 1 is v min, the minimum speed necessay to maintain the cicula path, find the hoizontal distance taveled by the sphee using the gound level as being located at position 1. Copyight 008 Laying the oundation, Inc., Dallas, TX. All ights eseved. Visit:

10 Cicula Motion, Gavity, & Obits time in the ai depends upon vetical displacement y ( R) R t g g g hoizontal distance depends on hoizontal velocity R gr x vxt ( gr) R g g If the foce causing the object to move in a cicle magically disappeas as it did hee, then emembe the poblem becomes a D pojectile poblem with the initial velocity being the tangential velocity of the object which was undegoing. Copyight 008 Laying the oundation, Inc., Dallas, TX. All ights eseved. Visit:

11 Cicula Motion, Gavity, & Obits Question 1 (15 pts) ee Response A student swings a stoppe of mass 0.40 kg in a hoizontal cicle (paallel to the floo), as shown in the figue above. The sting has a length L 0.50 m, and is swung so that it emains at an angle of θ 60 fom the vetical. A constant tension in the sting is maintained as the stoppe is swung in a cicula path at this angle. A. On the diagam below, daw all of the foces acting on the stoppe as it swings in a cicula path at this angle. Teaching Note: students should not daw components on thei BD as the latest ubics count the components as extaneous vectos. θ T +y W points One point fo each coectly dawn and labeled foce vecto (tension and weight). 1 point fo no extaneous foces, such as the centipetal foce o even the x and y components of the tension. B. Detemine the centipetal foce on the stoppe which causes it to tavel in a cicula path. y 0 m Ty W mg ( kg) N s x ma ( )( ) T T tanθ 0.40 N tan N x y 1 point o indicating that the sum of the vetical components of the foces is zeo 1 point o indicating that the sum of the hoizontal components of the foces is not zeo 1 point o detemining that the centipetal foce acting on the stoppe is the x component of the tension 1 point o the coect answe with units Copyight 008 Laying the oundation, Inc., Dallas, TX. All ights eseved. Visit:

12 Cicula Motion, Gavity, & Obits C. Detemine the linea speed v necessay to keep the stoppe evolving in this cicula path. x ma R Tx, R whee the adius R Lsinθ ( )( ) R 0.50 m sin m ( )( ) TR 0.69 N 0.43 m x m v.7 m kg s 1 point o setting the x component of the tension in the sting equal to the centipetal foce 1 point o attempting to find R and not using the length of the sting 1 point o substituting the mass of the stoppe in the equation fo centipetal foce 1 point o the coect answe with units consistent with the ealie answe fo the centipetal foce D. Detemine the peiod (time fo one complete evolution) of the stoppe as it is swung in this cicula path. T T π 1 point o a coect equation fo the v peiod of evolution π ( 0.43m) 1.0 s 1 point o the coect answe with m.7 units consistent with the ealie answe s fo speed and adius E. Is it possible to keep the stoppe whiling in a pefectly hoizontal cicle such that the stoppe does not doop and θ90º fom the vetical as shown in the diagam below? Check Yes o No and explain you answe. θ Copyight 008 Laying the oundation, Inc., Dallas, TX. All ights eseved. Visit:

13 Cicula Motion, Gavity, & Obits Yes No x 1 point o checking NO It is not possible to keep the mass evolving in a hoizontal cicle with θ90º, since the foce of gavitational attaction ( W ) must pull the stoppe down to an equilibium angle. The stoppe could only evolve in a hoizontal cicle if it had no mass and no weight to pull the sting down away fom the hoizontal. As θ appoaches 90º the sting would beak as the tension in the sting appoaches infinity. 1 point o a coect explanation involving the weight foce and the tension in the sting Copyight 008 Laying the oundation, Inc., Dallas, TX. All ights eseved. Visit:

14 Cicula Motion, Gavity, & Obits Question (10 pts) A satellite of mass m is moving in a cicula obit with linea speed v, aound a planet of mass M, obiting at a paticula distance fom the cente of the planet. v M A. Detemine the adius of evolution, of the satellite, in tems of the given quantities and any fundamental constants? c G GM v GmM GmM 1 point o indicating that the gavitational foce plays the ole of the centipetal foce 1 point o a coect application of Newton s law of Univesal Gavitation 1 point o a coect application of msatellitev 1 point o a coect expession fo the adius of evolution in tems of the given quantities and the fundamental constant G Copyight 008 Laying the oundation, Inc., Dallas, TX. All ights eseved. Visit:

15 Cicula Motion, Gavity, & Obits B. What is the elationship between the adius of evolution and the mass of the satellite which is undegoing unifom cicula motion? Justify you esponse. Radius of evolution is independent of the mass of the satellite. Radius of evolution depends upon the mass of the planet the satellite is obiting, not the mass of the satellite. 1 point o indicating that thee is no elationship between the adius and the mass of the satellite 1 point o a pope justification, i.e. the mass of the satellite divides out of both sides of the equation C. Assume the acceleation due to gavity g at a distance fom the cente of the planet of mass M is 9 m/s. In tems of the adius of evolution, what would the speed of the satellite have to be to emain in a cicula obit aound this planet at this distance? 1 point o indicating that the weight w c foce plays the ole of the centipetal mg foce g v 1 point o indicating that the mass of the satellite divides out of the equation v g point o substituting 9 m/s fo g 1 point o the coect expession in tems of the adius of evolution, Copyight 008 Laying the oundation, Inc., Dallas, TX. All ights eseved. Visit:

16 Cicula Motion, Gavity, & Obits Multiple Choice I II Questions 1 4 A ball on the end of a sting is being swung in a vetical cicle, otating clockwise as shown above. 1. Which of the following vectos epesents the centipetal foce acting on the ball at position I? A) B) III C) D) E) Tension povides the centipetal foce that causes the ball to follow a cicula path. The velocity vecto is always tangent to the path and in the diection of motion, and the centipetal foce and acceleation ae towad the cente of the cicle. D. If the sting wee suddenly cut when the ball is at position II shown in the figue above, the subsequent motion of the ball would be A) to move to the ight. B) to move to the left. C) to move to the top of the page. D) to move down and to the ight E) to move up and to the left. Copyight 008 Laying the oundation, Inc., Dallas, TX. All ights eseved. Visit:

17 Cicula Motion, Gavity, & Obits Unifom Cicula Motion At the instant the ball is in the position shown, it is moving tangent to the path, so up towad the top of the page C 3. The tension in the sting is I. geate at position I than at position III. II. geatest at position II. III. geate at position III than at position I. A) I only B) II only C) III only D) II and III only E) I and II only Tension povides the centipetal foce that causes the ball to follow a cicula path. At position I, the sting goes slack due to the weight of the ball pulling downwad. At position II, the weight of the ball pulls downwad, but the tension in the sting pulls towad the cente. At position III, the weight is diected downwad, but the tension is diected upwad, causing the sting to be tighte than it is at I o II. C 4. The ball has a mass m and a speed v as it moves aound the vetical cicle of adius. Which of the following expessions can be used to find the minimum speed of the ball at position I such that the cicula path is maintained? (A) v (B) v g g (C) v ( T ) mg m (D) v ( T + ) mg m (E) v ( T ) mg m Copyight 008 Laying the oundation, Inc., Dallas, TX. All ights eseved. Visit:

18 Cicula Motion, Gavity, & Obits At top of cicle the tension appoaches zeo at minimum velocity T + mg T 0 at minimum velocity at top mg v g A 5. A 30 kg child sits on the edge of a mey go ound at a adius of 6 m. The tangential speed of the child is 1 m/s. The wok done by the centipetal foce duing one complete evolution is A) 0 J B) 1440 J C) 5760 J D) J E) J Centipetal foces do no wok W dcosθ W ΔKE Angle between and d is 0 degees. Enegy emains constant unless thee is wok done on the system. A Copyight 008 Laying the oundation, Inc., Dallas, TX. All ights eseved. Visit:

19 Cicula Motion, Gavity, & Obits 6. An object evolves in unifom cicula motion. Which of the gaphs below best epesents the centipetal foce on the vetical axis vs. the speed of the object on the hoizontal axis? (A) (B) (C) (D) (E) Centipetal foce is diectly popotional to the squae of the velocity c C Copyight 008 Laying the oundation, Inc., Dallas, TX. All ights eseved. Visit:

20 Cicula Motion, Gavity, & Obits 7. The lage the obital adius of a satellite I. the geate the gavitational foce acting on it. II. the lage its acceleation. III. the longe its obital peiod. A) I only B) II only C) III only D) II and III only E) I, II, and III Keple s thid law Accoding to Keple s thid law, the peiod of evolution squaed is popotional to the aveage obital adius cubed, causing a lage obit to have a longe peiod. C Questions A planet obits the sun and is moving fom point A to point E as shown in the figue below. A E B C D Copyight 008 Laying the oundation, Inc., Dallas, TX. All ights eseved. Visit:

21 Cicula Motion, Gavity, & Obits 8. The speed of the planet is geatest at point A) A B) B C) C D) D E) E Speed of a satellite The speed is geatest when the planet is closest to the sun. E 9. The centipetal foce exeted on the planet is at its lowest value at point A) A B) B C) C D) D E) E c The speed is least at point A, is the geatest, so the centipetal foce exeted on the planet is at its lowest value at point A. A 10. The centipetal acceleation of the planet is geatest at point A) A B) B C) C D) D E) E v ac The acceleation of the planet is geatest at the point whee the gavitational foce between the sun and the planet, point E. E Copyight 008 Laying the oundation, Inc., Dallas, TX. All ights eseved. Visit:

22 Cicula Motion, Gavity, & Obits 11. A coin of mass m is placed on a vinyl steeo ecod of adius R and moves at a constant tangential speed v. If the fictional foce between the coin and the vinyl ecod is at its maximum value, then which of the following expessions can be used to find the value of the coefficient of fiction between the coin and the vinyl ecod? A) B) C) D) E) μ μ gr v gv R μ vgr μ μ v gr R ictional foce plays the ole of the centipetal foce f μn, and N mg, so R v μmg, and μ R gr D 1. A student holds one end of a 1.5 m sting and swings a 1.0 kg ball on the othe end of the sting in a hoizontal cicula path with a constant speed and a centipetal acceleation of 1.5 m/s. The centipetal foce is most nealy A) 0 N B) 0.7 N C) 1.5 N D) 3.0 N E) 4.5 N c c c ma c m ( 1.0kg) N s C Copyight 008 Laying the oundation, Inc., Dallas, TX. All ights eseved. Visit:

23 Cicula Motion, Gavity, & Obits 13. A canival ide has a.0 m adius and otates once each 0.90 s. The centipetal acceleation of the ide is most nealy A) 1.8 m/s B).5 m/s C) 14 m/s D) 56 m/s E) 98 m/s Peiod to find speed, then find centipetal acceleation T 0.90s d π π(.0 m) m v 14 t T 0.90 s s a c m 14 v s m 98.0 m s E 14. A line dawn fom the sun to a planet sweeps out equal aeas in equal A) distances B) displacements C) speeds D) acceleations E) times Keple s second law This is a statement of Keple s second law of planetay motion and implies that a planet tavels at a highe speed when it is close to the sun. E Copyight 008 Laying the oundation, Inc., Dallas, TX. All ights eseved. Visit:

DYNAMICS OF UNIFORM CIRCULAR MOTION

DYNAMICS OF UNIFORM CIRCULAR MOTION Chapte 5 Dynamics of Unifom Cicula Motion Chapte 5 DYNAMICS OF UNIFOM CICULA MOTION PEVIEW An object which is moing in a cicula path with a constant speed is said to be in unifom cicula motion. Fo an object