Rotational Motion and the Law of Gravity

Transcription

1 Chape 7 7 Roaional Moion and he Law of Gaiy PROBLEM SOLUTIONS 7.1 (a) Eah oaes adians (360 ) on is axis in 1 day. Thus, ad 1 day ad s 4 1 day s Because of is oaion abou is axis, Eah bulges a he equao. 7. The disance aeled is s = θ, whee θ is in adians. Fo 30, ad s Fo 30 adians, s ad Fo 30 eoluions, ad s e e 7.3 (a) s i 580 f 1.0 f 1 i ad Page 7.1

2 Chape 7 1 e ad ad e 7.4 (a) 1.00 e s 0 e ad s s 1 e 0.09 ad s Yes. When an objec sas fo es, is angula speed is elaed o he angula acceleaion and ie by he equaion = (). Thus, he angula speed is diecly popoional o boh he angula acceleaion and he ie ineal. I he ie ineal is held consan, doubling he angula acceleaion will double he angula speed aained duing he ineal e in 0 ad 1 in 3.0 s 1 e 60.0 s (a) 81 ad s 1 1 ad s ad i s 7.6 e ad 1 in i ad s in 1 e 60.0 s ad 50.0 e 314 ad 1 e Thus, w ad s i 6 ad s 314 ad 7.7 (a) Fo, he angula displaceen is 0. ad s 0.06 ad s ad 0.70 ad s Page 7.

3 Chape 7 Fo he equaion gien aboe fo, obsee ha when he angula acceleaion is consan, he displaceen is popoional o he diffeence in he squaes of he final and iniial angula speeds. Thus, he angula displaceen would incease by a faco of 4 if boh of hese speeds wee doubled. 7.8 (a) The axiu heigh h depends on he dop s eical speed a he insan i leaes he ie and becoes a pojecile. The eical speed a his insan is he sae as he angenial speed, =, of poins on he ie. Since he second dop ose o a lesse heigh, he angenial speed deceased duing he ineening oaion of he ie. Fo a y, wih 0 = a y = g, and = 0 when y = h, he elaion beween he angenial 0 y speed of he ie and he axiu heigh h is found o be 0 g h o gh Thus, he angula speed of he ie when he fis dop lef was gh1 1 1 and when he second dop lef, he angula speed was gh Fo 0, wih =, he angula acceleaion is found o be 1 gh gh1 g h h1 o 9.80 s ad s ad 7.9 Main Roo: e ad 1 in s in 1 e 60 s Page 7.3

4 Chape 7 sound = 179 = 0.5 s 343 s sound Tail Roo: e ad 1 in s in 1 e 60 s sound = 1 = s 343 s sound 7.10 We will beak he oion ino wo sages: (1) an acceleaion peiod and () a deceleaion peiod. The angula displaceen duing he acceleaion peiod is f i 5.0 e s ad 1 e 0 1 a 8.0 s 16 ad and while deceleaing, f i e s ad 1 e 1 s 188 ad 1 e ad 50 e. ad The oal displaceen is 7.11 (a) The linea disance he ca aels in coing o es is gien by 0 f a x as f s x a 1.75 s 40 Since he ca does no skid, he linea displaceen of he ca and he angula displaceen of he ies ae elaed by x = (). Thus, he angula displaceen of he ies is Page 7.4

5 Chape 7 x ad 1 e 116 e ad When he ca has aeled 10 (one half of he oal disance), he linea speed of he ca is x 0 a 9.0 s 1.75 s s and he angula speed of he ies is 0.5 s ad s 7.1 (a) The angula speed is ad s.30 s 5.75 ad s. Since he disk has a diaee of 45.0 c, is adius is = ( )/ = 0.5. Thus, ad s 1.9 s and a ad s s The angula displaceen of he disk is f ad s f 6.61 ad ad s ad and he final angula posiion of he adius line hough poin P is f o i is a 76 couneclockwise fo he + x-axis afe uning 19 beyond one full eoluion. i 7.13 Fo a, we find he iniial angula speed o be ad 37.0 e 1 e i 98.0 ad s 57.0 ad s 3.00 s Page 7.5

6 Chape 7 The angula acceleaion is hen i 98.0 ad s 57.0 ad s 3.00 s 13.7 ad s 7.14 (a) The iniial angula speed is e in ad 1 in 1 e 10.5 ad s 60.0 s The ie o sop (i.e., each a speed of = 0) wih =.00 ad/s is ad s.00 ad s 5.5 s ad s a 5.5 s 7.6 ad The cenipeal acceleaion is a whee adius of he cicula pah followed by he objec in quesion. The angula speed of he oaing Eah is c ad 1 day day s ad s 4 (a) Fo a peson on he equao, = R E = , so a ad s s c 6 5 Fo a peson a he Noh Pole, 0 a c 0. The cenipeal acceleaion of an objec is dieced owad he cene of he cicula pah he objec is following. Thus, he foces inoled in poducing his acceleaion ae all foces acing on he objec which hae a coponen along he adius line of he cicula pah. These foces ae he gaiaional foce and he noal foce The adius of he cylinde is elociy is i.5 i Thus, fo a c =, he equied angula Page 7.6

7 Chape 7 a c 9.80 s ad s 7.17 The final angula elociy is f e 1 in ad ad s in 60 s 1 e and he adius of he disk is.54 c 5.0 in 1.7 c in (a) The angenial acceleaion of he bug as he disk speeds up is 8.17 ad s a 3.0 s s The final angenial speed of he bug is f ad s 1.0 s 8.17 ad s i s.7 ad s 3.0 s A = 1.0 s Thus, a 0.35 s as aboe, while he adial acceleaion is ac ad s 0.94 s The oal acceleaion is a a a 1.0 s, and he angle his acceleaion akes wih he diecion of a c is c a an an 0 a 0.94 c 7.18 The noal foce exeed by he wall behind he pe- Page 7.7

8 Chape 7 son s back will supply he necessay cenipeal acceleaion, o Figue P7.18 n a c whee = 9 f is he adius of he cicula pah followed by he peson. If i is desied o hae n = 0 weigh = 0g, hen i is necessay ha 0 g, o 0g s 9 f f 4.7 ad s 1 e ad 60 s 1 in 45 e in 7.19 The oal foce, dieced owad he cene of he cicula pah, acing on he ide a he op of he loop is he su of he noal foce and he gaiaion foce. If he agniude of he noal foce (exeed on he ide by he sea) is o hae a agniude equal o he ide s weigh, he oal cenipeal foce is hen F n F g g g c g Also, F so we sole fo he needed speed a he op of he loop as c op op g o g op Ignoing any ficion and using conseaion of enegy fo when he coase sas fo es ( i = 0) a heigh h unil i eaches he op of he loop gies 1 i gh 1 op g o 0 gh g g 1 and educes o h = 3 = 3(4.00 ) = (a) The naual endency of he coin is o oe in a saigh line (angen o he cicula pah of adius 15.0 c), Page 7.8

9 Chape 7 and hence, go fahe fo he cene of he unable. To peen his, he foce of saic ficion us ac owad he cene of he unable and supply he needed cenipeal foce. When he necessay cenipeal foce exceeds he axiu alue of he saic ficion foce, ( f ) n g, he coin begins o slip. s ax s s When he unable has angula speed, he equied cenipeal foce is F c =. Thus, if he coin is no o slip, i is necessay ha g, o s s s g ad s Wih a consan angula acceleaion of = ad/s, he ie equied o each he ciical angula speed is ad s ad s 6.55 s 7.1 (a) Fo F = a c, we hae 55.0 kg 4.00 s T N 1.10 kn The ension is lage han he weigh by a faco of T g N 55.0 kg 9.80 s.04 ies 7. (a) The cenipeal acceleaion is a c. Thus, when a c = a = /s, we hae ac s 00 s 14.1 s A his ie, i 00 s 0 a s 8.3 s Page 7.9

10 Chape 7 and he linea displaceen is i 00 s 0 s 8.3 s 00 a The ie is 8.3 s as found in pa aboe. 7.3 Ficion beween he ies and he oadway is capable of giing he uck a axiu cenipeal acceleaion of a c,ax 3.0 s 6.83 s 150,ax If he adius of he cue changes o 75.0, he axiu safe speed will be,ax ac,ax s.6 s 7.4 Since F c, he needed angula elociy is F c = = N kg a 1 e ad = ad s = e s ad s 18.0 s 7.5 (a) c F c ac 50.0 kg 18.0 s 900 N We know he cenipeal acceleaion is poduced by he foce of ficion. Theefoe, he needed saic ficion foce is f s = 900 N. Also, he noal foce is n = g = 490 N. Thus, he iniu coefficien of ficion equied is fs ax 900 N = 1.84 s n 490 N So lage a coefficien of ficion is uneasonable, and she will no be able o say on he ey-go-ound. 7.6 (a) The only foce acing on he asonau is he noal foce exeed on hi by he floo of he cabin. Page 7.10

11 Chape 7 Figue P7.6 F c n If, 1 n g E hen 1 n 60.0 kg 9.80 s 94 N (d) Fo he equaion in Pa, 94 N 10.0 n 60.0 kg 7.00 s (e) Since 1 =, we hae 7.00 s ad s (f) The peiod of oaion is T ad s 8.98 s (g) Upon sanding, he asonau s head is oing slowe han his fee because his head is close o he axis of oaion. When sanding, he adius of he cicula pah followed by he head is head = = 8.0, and he angenial speed of he head is ad s 5.74 s head head 7.7 (a) Since he 1.0-kg ass is in equilibiu, he ension in he sing is T g 1.0 kg 9.8 s 9.8 N Page 7.11

12 Chape 7 The ension in he sing us poduce he cenipeal acceleaion of he puck. Hence, F c = T = 9.8 N. Fo Fc puck, we find puck N Fc 0.5 kg 6.3 s. 7.8 (a) Since he ass hangs in equilibiu on he end of he sing, Fy T g 0 o T g The puck oes in a cicula pah of adius R and us hae an acceleaion dieced owad he cene equal o a R. The only foce acing on he puck and dieced owad he cene is he ension in he sing. c Newon s second law equies F a c owad 1 cene giing T 1 R Cobing he esuls fo (a) and gies R 1 g o gr 1 (d) Subsiuion of he nueic daa fo poble 7.7 ino he esuls fo (a) and shown aboe will yield he answes gien fo ha poble. 7.9 (a) The foce of saic ficion acing owad he oad s cene of cuaue us supply he biefcase s equied cenipeal acceleaion. The condiion ha i be able o ee his need is ha c s ax s F f g, o s /g. When he angenial speed becoes lage enough ha s g he biefcase will begin o slide. As discussed aboe, he biefcase sas o slide when he coefficien of saic ficion us be s g. If his occus a he speed, = 15.0 /s, 15.0 s s s Page 7.1

13 Chape (a) The exenal foces acing on he wae ae he gaiaional foce and he conac foce exeed on he wae by he pail. The conac foce exeed by he pail is he os ipoan in causing he wae o oe in a cicle. If he gaiaional foce aced alone, he wae would follow he paabolic pah of a pojecile. When he pail is ineed a he op of he cicula pah, i canno hold he wae up o peen i fo falling ou. If he wae is no o spill, he pail us be oing fas enough ha he equied cenipeal foce is a leas as lage as he gaiaional foce. Tha is, we us hae g o g s 3.13 s (d) If he pail wee o suddenly disappea when is i a he op of he cicle and oing a 3.13 /s, he wae would follow he paabolic ae of a pojecile launched wih iniial elociy coponens of 3.13 s, 0 0x 0y 7.31 (a) The cenipeal acceleaion is ac e ad 1 in s in 1 e 60 s A he boo of he cicula pah, he noal foce exeed by he sea us suppo he weigh and also poduce he cenipeal acceleaion. Thus, n g ac 40.0 kg s 455 N upwad A he op of he pah, he weigh us offse he noal foce of he sea plus supply he needed cenipeal acceleaion. Theefoe, g = n+ a c, o n g ac 40.0 kg s 39 N upwad (d) A a poin halfway up, he sea exes an upwad eical coponen equal o he child s weigh (39 N) and a coponen owad he cene haing agniude F c ac 40.0 kg 1.58 s 63. N. The oal foce exeed by he sea is F 39 N 63. N 397 N dieced inwad and a R Page 7.13

14 Chape N an 80.8 aboe he hoizonal 63. N 7.3 (a) A A, he ack suppos he weigh and supplies he cenipeal acceleaion. Thus, 0.0 s n g 500 kg 9.80 s 5 kn 10 A B, he weigh us offse he noal foce exeed by he ack and poduce he needed cenipeal acceleaion, o g n. If he ca is on he ege of leaing he ack, hen n = 0 and g. Hence, g s 1 s 7.33 A he half-way poin he spaceship is fo boh bodies. The foce exeed on he ship by he Eah is dieced owad he Eah and has agniude F E G E s N kg kg kg N The foce exeed on he ship by he Moon is dieced owad he Moon and has a agniude of F M G M s N kg kg kg N The esulan foce is (35 N 4.00 N) = 31 N dieced owad Eah The adius of he saellie s obi is RE h Page 7.14

15 Chape 7 (a) PE g GM E kg 100 kg 11 N J kg F kg 100 kg 6 GM E 11 N N kg a 7.35 The foces exeed on he.0-kg ass by he ohe bodies ae F x and F y shown in he diaga a he igh. The agniudes of hese foces ae F x N kg.0 kg 4.0 kg N and F y N kg.0 kg 3.0 kg N 10 The esulan foce exeed on he.0-kg ass is F F F N dieced a an 1( F ) an 1 y Fx aboe he x axis. x y 7.36 (a) The densiy of he whie dwaf would be M M M 3M V V R R sun sun sun Eah 4 3 E E and using daa fo Table 7.3, kg kg F g GM g, so he acceleaion of gaiy on he suface of he whie dwaf would be Page 7.15

16 Chape 7 g N kg kg GM R sun s E The geneal expession fo he gaiaional poenial enegy of an objec of ass a disance fo he cene of a spheical ass M is PE of he whie dwaf would be GM. Thus, he poenial enegy of a 1.00-kg ass on he suface PE GM sun R 1.00 kg E N kg kg 1.00 kg J 7.37 (a) A he idpoin beween he wo asses, he foces exeed by he 00-kg and 500-kg asses ae opposiely dieced, so fo F GM and 1, we hae GM GM GM F o F N kg 50.0 kg 500 kg 00 kg N owad he 500-kg ass A a poin beween he wo asses and disance d fo he 500-kg ass, he ne foce will be zeo when d G 50.0 kg 00 kg G 50.0 kg 500 kg d o d 0.45 Noe ha he aboe equaion yields a second soluion d = A ha poin, he wo gaiaional foces do hae equal agniudes, bu ae in he sae diecion and canno add o zeo The equilibiu posiion lies beween he Eah and he Sun on he line connecing hei cenes. A his poin, he gaiaional foces exeed on he objec by he Eah and Sun hae equal agniudes and opposie diecions. Le his poin be locaed disance fo he cene of he Eah. Then, is disance fo he Sun is Page 7.16

17 Chape 7 ( ), and we ay deeine he alue of by equiing ha G G E S whee E and S ae he asses of he Eah and Sun especiely. This educes o S E 577 o = 578, which yields = fo cene of he Eah (a) When he ocke engine shus off a an aliude of 50 k, we ay conside he ocke o be beyond Eah s aosphee. Then, is echanical enegy will eain consan fo ha insan unil i coes o es oenaily a he axiu aliude. Tha is, KE f + PE f = KE i + PE i o 0 GME ax 1 i GM E i o 1 i 1 GM ax E i Wih R 50 k and i E i 6.00 k s s, his gies ax s N kg kg 7 1 o ax = The axiu aliude aboe Eah s suface is hen h R E ax ax k If he ocke wee fied fo a launch sie on he equao, i would hae a significan easwad coponen of elociy because of he Eah s oaion abou is axis. Hence, copaed o being fied fo he Souh Pole, he ocke s iniial speed would be geae, and he ocke would ael fahe fo Eah We know ha kg, o 5.00 kg 1 Page 7.17

18 Chape 7 F G N kg N kg N kg 6.00 kg N kg Thus, kg kg 0, o 3.00 kg, so.00 kg kg.00 kg 0 giing 1 1 The answe 1.00 kg and = 3.00 kg is physically equialen (a) The gaiaional foce us supply he equied cenipeal acceleaion, so G E This educes o G E which gies 4 11 N kg kg s The aliude aboe he suface of he Eah is hen h R E The ie equied o coplee one obi is T cicufeence of obi obial speed s s 5.57 h 7.4 Fo an objec in obi abou Eah, Keple s hid law gies he elaion beween he obial peiod T and he aeage Page 7.18

19 Chape 7 adius of he obi ( sei-ajo axis ) as T 4 GM 3 E Thus, if he aeage adius is in ax k k k he peiod (ie fo a ound ip fo Eah o he Moon) would be N kg kg T s GME The ie fo a one way ip fo Eah o he Moon is hen s 1 day T s 4.99 d 7.43 The gaiaional foce exeed on Io by Jupie poides he cenipeal acceleaion, so G M, o M G The obial speed of Io is days s day T s Thus, M s N kg kg 7.44 (a) The saellie oes in an obi of adius = R E and he gaiaional foce supplies he equied cenipeal acceleaion. Hence, R G R E E E, o Page 7.19

20 Chape 7 4 G kg E 11 N s R kg E The peiod of he saellie s oion is T s s 3.98 h The gaiaional foce acing on he saellie is F = G E /, o F kg 600 kg N N kg 7.45 The adius of he saellie s obi is RE h (a) Since he gaiaional foce poides he cenipeal acceleaion, G E o 4 G kg E 11 N s kg Hence, he peiod of he obial oion is T s s 1.48 h The obial speed is s as copued aboe. Assuing he saellie is launched fo a poin on he equao of he Eah, is iniial speed is he Page 7.0

21 Chape 7 oaional speed of he launch poin, o i RE 1 day s 464 s The wok kineic enegy heoe gies he enegy inpu equied o place he saellie in obi as W KE PE KE PE, o nc g f g i 1 GM E 1 GM E i 1 1 Wnc i GME R E RE Subsiuion of appopiae nueic alues ino his esul gies he iniu enegy inpu as W nc J A synchonous saellie will hae an obial peiod equal o Jupie s oaion peiod, so he saellie can hae he ed spo in sigh a all ies. Thus, he desied obial peiod is T s 1 h 9.84 h s Keple s hid law gies he peiod of a saellie in obi aound Jupie as T 4 GM 3 Jupie The equied adius of he cicula obi is heefoe JupieT N kg kg s 8 GM and he aliude of he saellie aboe Jupie s suface should be h R Jupie The gaiaional foce on ass locaed a disance fo he cene of he Eah is F g GM g E. Thus, he acceleaion of gaiy a his locaion is g GM. If g = 9.00 /s a he locaion of he saellie, he a- E Page 7.1

22 Chape 7 dius of is obi us be N kg kg GM E g 9.00 s Fo Keple s hid law fo Eah saellies, T 3 G M S, he peiod is found o be 4 E T N kg kg s GME o T 3 1 h s 1.50 h 90.0 in s 7.48 The gaiaional foce on a sall pacel of aeial a he sa s equao supplies he cenipeal acceleaion, o G M R s s R s R s Hence, G M R3 s s N kg kg ad s 7.49 (a) s 98.0 i h 1 i h ad 1 e e s 0.74 s ad i 9.40 e s 0 1 e 44. e s Page 7.

23 Chape 7 a c s 98.0 i h 1 i h s a e ad s 1 e s In he adial diecion a he elease poin, he hand suppos he weigh of he ball and also supplies he cen ipeal acceleaion. Thus, F g a g a, o 3 F kg 9.80 s s 514 N In he angenial diecion, he hand supplies only he angenial acceleaion, so F a kg 06 s 40.8 N 1.30 s 7.50 (a) i i 56.5 ad s 1.30 s f f.4 ad s The duaion of he ecoding is Thus, 74 in 60 s in 33 s s a ad s 3 f i ad s s (d).4 ad s 56.5 ad s 3 f i ad s ad (e) The ack oes pas he lens a a consan speed of = 1.30 /s fo seconds. Theefoe, he lengh of Page 7.3

24 Chape 7 he spial ack is 3 s 1.30 s s k 7.51 The angula elociy of he ball is e s ad s. (a) ad s.51 s ad s 7.90 s a c We iagine ha he weigh of he ball is suppoed by a ficionless plafo. Then, he ope ension need only poduce he cenipeal acceleaion. The foce equied o poduce he needed cenipeal acceleaion is F. Thus, if he axiu foce he ope can exe is 100 N, he axiu angenial speed of he ball is ax N Fax 5.00 kg 4.00 s 7.5 (a) When he ca is abou o slip down he incline, he ficion foce, f, is dieced up he incline as shown and has he agniude f n. Thus, F n cos n sin g 0 y o n g cos sin [1] Also, F nsin n cos ( R), o x in nr in sin cos [] Page 7.4

25 Chape 7 Subsiuing equaion [1] ino [] gies in sin cos an R g R g cos sin 1 an If he ca is abou o slip up he incline, f = n is dieced down he slope (opposie o wha is shown in he skech). Then, F n cos n sin g 0, o y n g cos sin [3] Also, F x n sin n cos ax R o nr ax sin cos [4] Cobining equaions [3] and [4] gies ax sin cos an R g R g cos sin 1 an If R = 100, = 10, and = 0.10, he lowe and uppe liis of safe speeds ae in an s an 10 s and ax an s 17 s an The adius of he saellie s obi is Page 7.5

26 Chape 7 RE h i i (a) The equied cenipeal acceleaion is poduced by he gaiaional foce, so G M E, which gies GM E 4 11 N kg s kg The ie fo one coplee eoluion is T s s 89.3 in 7.54 (a) A he lowes poin on he pah, he ne upwad foce (i.e., he foce dieced owad he cene of he pah and supplying he cenipeal acceleaion) is F up T g, so he ension in he cable is T 3.00 s g kg 9.80 s 8.4 N Using conseaion of echanical enegy, KE PEg KE PEg o he highes poin on he pah gies, as he bob goes fo he lowes f i 1 0 g L 1 cosax i 0, o i cosax 1 gl 3.00 s cos 1 cos gl 9.80 s i 1 ax Page 7.6

27 Chape 7 A he highes poin on he pah, he bob is a es and he ne adial foce is F T g cos ax 0 Theefoe, ax T g cos kg 9.80 s cos N 7.55 (a) When he ca is a he op of he ac, he noal foce is upwad and he weigh downwad. The ne foce dieced downwad, owad he cene of he cicula pah and hence supplying he cenipeal acceleaion, is F g n. down Thus, he noal foce is n g. If 30.0 and n 0, hen g 0 o he speed of he ca us be g s 17.1 s 7.56 The escape speed fo he suface of a plane of adius R and ass M is gien by e GM R If he plane has unifo densiy, ρ, he ass is gien by M olue R R The expession fo he escape speed hen becoes e G 4 R3 8 G R consan R R 3 3 o he escape speed is diecly popoional o he adius of he plane. Page 7.7

28 Chape The speed he peson has due o he oaion of he Eah is = whee is he disance fo he oaion axis and ω is he angula elociy of oaion. The peson s appaen weigh, (F g ) appaen, equals he agniude of he upwad noal foce exeed on hi by he scales. The ue weigh, (F g ) ue = g, is dieced downwad. The ne downwad foce poduces he needed cenipeal acceleaion, o g g g F down n F F F ue appaen ue (a) A he equao, RE, so F g Fg RE Fg ue appaen appaen. A he equao, i is gien ha s, so he appaen weigh is g g F F appaen ue 75.0 kg s 73 N A eihe pole, = 0 (he peson is on he oaion axis) and F g Fg g appaen 75.0 kg 9.80 s 735 N ue 7.58 Choosing y = 0 and PE g = 0 a he leel of poin B, applying he wok enegy heoe o he block s oion gies Wnc gy g ( R), o W nc 0 g R y [1] (a) A poin A, y = R and W nc = 0 (no nonconseaie foce has done wok on he block ye). Thus, A 0 gr. The noal foce exeed on he block by he ack us supply he cenipeal acceleaion a poin A, so A 0 na g R R s 0.50 kg 9.8 s 15 N Page 7.8

29 Chape 7 A poin B, y = 0 and W nc is sill zeo. Thus, 0 4 gr. Hee, he noal foce us supply he cenipeal acceleaion and suppo he weigh of he block. Theefoe, B B 0 nb g 5 g R R s 0.50 kg s 30 N When he block eaches poin C, y = R and W f L g L. A his poin, he noal foce is nc k k o be zeo, so he weigh alone us supply he cenipeal acceleaion. Thus, c R g, o he equied speed a poin C is c Rg. Subsiuing his ino equaion [1] yields R g 0 kgl 0, o k 4.0 s s 0 Rg gl 9.8 s Define he following sybols: M = ass of oon, M e = ass of he Eah, R = adius of oon, R e = adius of he Eah, and = adius of he Moon s obi aound he Eah. We inepe luna escape speed o be he escape speed fo he suface of a saionay oon alone in he uniese. Then, GM launch escape o R launch 8 GM R Applying conseaion of echanical enegy fo launch o ipac gies g launch g 1 1 ipac PE PE, o f i PEg PEg ipac launch i f The needed poenial enegies ae G M G Me PE and PE e g i R g f Re G M G M Page 7.9

30 Chape 7 Using hese poenial enegies and he expession fo launch fo aboe, he equaion fo he ipac speed educes o ipac 3 M M M e e M G R Re Wih nueic alues of G N kg, M kgs, and R , R e, s we find 4 ipac s 11.8 k s 7.60 (a) When he passenge is a he op, he adial foces poducing he cenipeal acceleaion ae he upwad foce of he sea and he downwad foce of gaiy. The downwad foce us exceed he upwad foce o yield a ne foce owad he cene of he cicula pah. A he lowes poin on he pah, he adial foces conibuing o he cenipeal acceleaion ae again he upwad foce of he sea and he downwad foce of gaiy. Howee, he upwad foce us now exceed he downwad foce o yield a ne foce dieced owad he cene of he cicula pah. The sea us exe he geaes foce on he passenge a he lowes poin on he cicula pah. (d) A he op of he loop, o F Fg n 4.00 s n Fg g kg 9.80 s 546 N A he boo of he loop, F ( ) n Fg o Page 7.30

31 Chape s n Fg g kg 9.80 s 86 N 7.61 (a) In ode o launch youself ino obi by unning, you unning speed us be such ha he gaiaional foce acing on you exacly equals he foce needed o poduce he cenipeal acceleaion. Tha is, GM, whee M is he ass of he aseoid and is is adius. Since M densiy olue [(4 3) 3], his equieen becoes 4 3 G 3 o 3. 4G The adius of he aseoid would hen be s N kg kg o 15.3 k. The ass of he aseoid is gien by M kg kg 3 3 You peiod will be T s s 7.6 (a) Page 7.31

32 Chape 7 The elociy eco a A is shoe han ha a B. The gaiaional foce acing on he spacecaf is a conse aie foce, so he oal echanical enegy of he caf is consan. The gaiaional poenial enegy a A is lage han a B. Hence, he kineic enegy (and heefoe he elociy) a A us be less han a B. The acceleaion eco a A is shoe han ha a B. Fo Newon s second law, he acceleaion of he spacecaf is diecly popoional o he foce acing on i. Since he gaiaional foce a A is weake han ha a B, he acceleaion a A us be less han he acceleaion a B Choosing PE s = 0 a he op of he hill, he speed of he skie afe dopping disance h is found using conseaion of echanical enegy as 1 g h 0 0, o gh The ne foce dieced owad he cene of he cicula pah, and poiding he cenipeal acceleaion, is F g cos n R Soling fo he noal foce, afe aking he subsiuions R h h gh and cos 1 R R gies h g h 3h n g 1 g 1 R R R The skie leaes he hill when n 0 This occus when 3h 1 0 o R h R The cenipeal acceleaion of a paicle a disance fo he axis is a If we ae o hae a c = 100g, hen i is necessay ha 100g 100 g o The equied oaion ae inceases as deceases. In ode o ainain he equied acceleaion fo all paicles in he casing, we use he iniu alue of and find c Page 7.3

33 Chape 7 in 100g s ad 1 e 60.0 s e s ad 1 in in 7.65 The skech a he igh shows he ca as i passes he highes poin on he bup. Taking upwad as posiie, we hae Fy ay n g o n g (a) If 8.94 s, he noal foce exeed by he oad is n 8.94 s kg N 10.6 kn s 0.4 When he ca is on he ege of losing conac wih he oad, n = 0. This gies g = / and he speed us be g s 14.1 s 7.66 When he ope akes angle θ wih he eical, he ne foce dieced owad he cene of he cicula pah is F T g cos as shown in he skech. This foce supplies he needed cenipeal acceleaion, so T g cos, o T g cos Using conseaion of echanical enegy, wih KE = 0 a = 90 and PE g = 0 a he boo of he ac, he speed when 1 he ope is a angle θ fo he eical is gien by cos 0 g g, o gcos. The expession fo he ension in he ope a angle θ hen educes o T 3 g cos. (a) A he beginning of he oion, = 90 and T = 0 Page 7.33

34 Chape 7 A 1.5 fo he boo of he ac,.5.5 cos and he ension is T 3 70 kg 9.8 s N 1.3 kn A he boo of he ac, = 0 and cos = 1.0, so he ension is T 3 70 kg 9.8 s N.1 kn 7.67 (a) The desied pah is an ellipical ajecoy wih he Sun a one of he foci, he depaue plane a he peihelion, and he age plane a he aphelion. The peihelion disance D is he adius of he depaue plane s obi, while he aphelion disance is he adius of he age plane s obi. The sei-ajo axis of he desied ajecoy is hen a. D T If Eah is he depaue plane, AU. D Wih Mas as he age plane, T 1 AU AU 11 Thus, he sei-ajo axis of he iniu enegy ajecoy is a D T 1.00 AU 1.5 AU 1.6 AU Keple s hid law, T = a 3, hen gies he ie fo a full ip aound his pah as T a AU y so he ie fo a one-way ip fo Eah o Mas is y T 0.71 y Page 7.34

35 Chape 7 This ip canno be aken a jus any ie. The depaue us be ied so ha he spacecaf aies a he aphelion when he age plane is locaed hee (a) Conside he skech a he igh. A he boo of he loop, he ne foce owad he cene (i.e., he cenipeal foce) is Fc n F R g so he pilo s appaen weigh (noal foce) is F g g n F 1 g Fg Fg R R gr o n 71 N N s s s A he op of he loop, he cenipeal foce is F c R n Fg, so he appaen weigh is Fg g n Fg Fg Fg 1 R R gr s 71 N N s s Wih he igh speed, he needed cenipeal foce a he op of he loop can be ade exacly equal o he gaiaional foce. A his speed, he noal foce exeed on he pilo by he sea (his appaen weigh) will be zeo, and he pilo will hae he sensaion of weighlessness. (d) When n = 0 a he op of he loop, F c R g Fg, and he speed will be g Rg s 177 s R Page 7.35

36 Chape (a) A he insan he ud leaes he ie and becoes a pojecile, is elociy coponens ae. Fo 0x 0, 0y R y a 0y y wih ay g, he ie equied fo he ud o eun o is saing poin (wih y = 0) is gien by 0 R g fo which he nonzeo soluion is R g The angula displaceen of he wheel (uning a consan angula speed ) in ie is =. If he displaceen is 1 e ad a R g, hen R ad g o g and R g R 7.70 (a) A each poin on he eical cicula pah, wo foces ae acing on he ball: (1) The downwad gaiaional foce wih consan agniude Fg g () The ension foce in he sing, always dieced owad he cene of he pah The skech a he igh shows he foces acing on he ball when i is a he boo of he cicula pah and when i is a he highes poin on he pah. Noe ha he gaiaional foce has he sae agniude and diecion a each poin on he cicula pah. The ension foce aies in agniude a diffeen poins and is always dieced owad he cene of he pah. A he op of he cicle, F c T Fg, o T Fg g g s 0.75 kg 9.80 s 6.05 N Page 7.36

37 Chape 7 (d) A he boo of he cicle, F c T Fg T g, and soling fo he speed gies T g T g and T g If he sing is a he beaking poin a he boo of he cicle, hen T =.5 N, and he speed of he objec a his poin us be.5 N s 7.8 s 0.75 kg 7.71 Fo Figue (a) a he igh, obsee ha he angle he sings ake wih he eical is cos Also, he adius of he cicula pah is Figue gies a fee-body diaga of he objec wih he +y-axis eical and he +x-axis dieced owad he cene of he cicula pah. (a) Since he objec has zeo eical acceleaion, Newon s second law gies Fy T1cos T cos g 0 o T1 T g [1] cos In he hoizonal diecion, he objec has he cenipeal acceleaion a dieced in he +x-diecion (owad he cene of he cicula pah). Thus, c F T sin T sin x 1 o T T 1 sin [] Adding equaions [1] and [] gies Page 7.37

38 Chape 7 T 1 g cos sin so he ension in he uppe sing is T 1 cos sin kg 6.00 s 9.80 s 109 N To copue he ension T in he lowe sing, subac equaion [1] aboe fo equaion [] o obain g cos T sin Thus, T 4.00 kg 9.80 s 1.3 sin 41.4 cos s 56.4 N 7.7 The axiu lif foce is F L ax C, whee C N s and is he flying speed. Fo he ba o say alof, he eical coponen of he lif foce us equal he weigh, o FL cos g whee is he banking angle. The hoizonal coponen of his foce supplies he cenipeal acceleaion needed o ake a un, o F whee is he adius of he un. sin ( L ) (a) To say alof while flying a iniu speed, he ba us hae 0 also use he axiu lif foce possible a ha speed. Tha is, we need cos cos 1 ) and (o gie ax FL cos ax g, o C in 1 ax g Thus, we see ha iniu flying speed is kg 9.8 s g C N s in 4.1 s To ainain hoizonal fligh while banking a he axiu possible angle, we us hae FL ax ax cos g, o C cos g. Fo 10 s, his yields ax Page 7.38

39 Chape kg 9.8 s N s 10 s cos ax g 0.17 C o ax 80 The hoizonal coponen of he lif foce supplies he cenipeal acceleaion in a un, F sin L.Thus, he iniu adius un possible is gien by in F sin L ax ax C sin ax C sin ax whee we hae ecognized ha sin has is axiu alue a he lages allowable alue of. Fo a flying speed of = 10 /s, he axiu allowable bank angle is ax = 80as found in pa. The iniu adius un possible a his flying speed is hen kg in N s sin (d) No. Flying slowe acually inceases he iniu adius of he achieable uns. As found in pa, in C sin ax eical coponen of he lif foce us equal he weigh o. To see how his depends on he flying speed, ecall ha he bank angle, cos will be a iniu. This occus when and L FL cos g. A he axiu allowable F F C. Thus, cos g C L ax ax g sin ax 1 cos ax 1 C This gies he iniu adius un possible a flying speed as in C g 1 C Deceasing he flying speed will decease he denoinao of his expession, yielding a lage alue fo he iniu adius of achieable uns. Page 7.39

40 Chape The angula speed of he luggage is = /T whee T is he ie fo one coplee oaion of he caousel. The esulan foce acing on he luggage us be dieced owad he cene of he hoizonal cicula pah (ha is, in he +x diecion). The agniude of his esulan foce us be a c Thus, and o F a f cos nsin a [1] x x s c F a f sin n cos g 0 y y s n g f s sin [] cos Subsiuing equaion [] ino equaion [1] gies sin f cos g an f a cos s s c o f s a g an c cos sin cos [3] (a) Wih T = 38.0 s and = 7.46, we find ha ad s and a 30.0 kg ad s 6.09 N c Equaion [3] hen gies he ficion foce as Page 7.40

41 Chape 7 f s 6.09 N 30.0 kg 9.80 s an N sin cos 0.0 cos N If T = 34.0 s and = 7.46, hen = ad/s and ac 30.0 kg 7.94 s ad s 8.15 N Fo equaion [1], f s 8.15 N 30.0 kg 9.80 s an N 108 N sin cos 0.0 cos 0.0 while equaion [] yields n 30.0 kg 9.80 s 108 N sin 0.0 cos N Since he luggage is on he ege of slipping, ax f f n and he coefficien of saic ficion us be s s s fs 108 N s n 73 N The hoizonal coponen of he ension in he cod is he only foce dieced owad he cene of he cicula pah, so i us supply he cenipeal acceleaion. Thus, T sin Lsin o sin T L [1] Also, he eical coponen of he ension us suppo he weigh of he ball, o T cos g [] Page 7.41

42 Chape 7 (a) Diiding equaion [1] by [] gies sin cos Lg o Lg sin [3] cos Wih L = 1.5 /s and = 30, s sin 30 cos 30.1 s Fo equaion [3], wih sin 1 cos, we find 1 cos cos Lg o Lg cos cos 1 0 Soling his quadaic equaion fo cos gies cos 1 L g L g If L = 1.5 and = 4.0 /s, his yields soluions: cos 1.7 (which is ipossible), and cos = +1.7 (which is possible). Thus, = cos -1 (0.59) = 54. Fo equaion [], when T = 9.8 N and he cod is abou o beak, he angle is 0.50 kg 9.8 s g T 9.8 N cos 1 cos 1 60 Then equaion [3] gies s Lg sin sin s cos cos 60 Page 7.4

43 Chape The noal foce exeed on he peson by he cylindical wall us poide he cenipeal acceleaion, so n = ( ). If he iniu accepable coefficien of ficion is pesen, he peson is on he ege of slipping and he axiu saic ficion foce equals he peson s weigh, o Thus, f n g. s ax s in s in g g 9.80 s n ad s If he block will jus ake i hough he op of he loop, he foce equied o poduce he cenipeal acceleaion a poin C us equal he block s weigh, o. ( R) g c This gies Rg, as he equied speed of he block a poin C. c We apply he wok enegy heoe in he fo W nc = ( KE + PE g PE s ) f ( KE + PE g PE s. ) i fo when he block is fis eleased unil i eaches poin C o obain c 1 1 fk AB cos180 g R kd The ficion foce is f k = u k (g), and fo iniu iniial copession of he sping, c as found aboe. Thus, he wok enegy equaion educes o Rg d in k kg AB Rg g R g AB 5 R k k d in 0.50 kg 9.8 s N 0.75 Page 7.43