GAVIAION IDNIFY nd UP: Use the lw of ittion, q(), to detemine F XCU: F G ( sun, moon); F G ( eth) on on F on m G F on G m the dius of the moon s obit ound the eth is ien in Appendi F s 0 m he moon is much, close to the eth thn it is to the sun, so tke the distnce of the moon fom the sun to be, the dius of the eth s obit ound the sun 0 99 0 k 0 m 59 0 k 50 0 m F on F on VALUA: he foce eeted by the sun is le thn the foce eeted by the eth he moon s motion is combintion of obitin the sun nd obitin the eth G IDNIFY: he ity foce between spheiclly syetic sphees is F, whee is the seption between thei centes UP: G 0 N m /k he moment m fo the toque due to ech foce is 050 m XCU: () Fo ech pi of sphees, F ( 0 N m /k )(0 k)(50 k) 0 N Fom (00 m) Fiue in the tetbook we see tht the foces fo ech pi e in opposite diections, so F net 0 (b) he net toque is τ net Fl ( 0 N)(050 m) 0 N m (c) he toque is ey smll nd the pptus must be ey sensitie he toque could be incesed by incesin the mss of the sphees o by decesin thei seption VALUA: he qutz fibe must twist thouh mesuble nle when smll toque is pplied to it IDNIFY: he foce eeted on the pticle by the eth is w m, whee m is the mss of the pticle he G foce eeted by the 00 k bll is F, whee is the distnce of the pticle fom the cente of the bll UP: G 0 N m /k, 90 m/s XCU: F Gbll wies m nd Gmbll ( 0 N m /k )(00 k) 5 0 m 00 It is not fesible to do this; 90 m/s 00 k bll would he dius much le thn 00 VALUA: he ittionl foce between odiny objects is ey smll he ittionl foce eeted by the eth on objects ne its sufce is le enouh to be impotnt becuse the mss of the eth is ey le IDNIFY: Apply q(), enelized to ny pi of spheiclly syetic objects UP: he seption of the centes of the sphees is XCU: he mnitude of the ittionl ttction is G /( ) G / VALUA: q() pplies to ny pi of spheiclly syetic objects; one of the objects doesn't he to be the eth -
- Chpte 5 IDNIFY: Use q() to clculte F eeted by the eth nd by the sun nd dd these foces s ectos () UP: he foces nd distnces e shown in Fiue 5 Let F! nd F! be the ittionl foces eeted on the spceship by the eth nd by the sun Fiue 5 XCU: he distnce fom the eth to the sun is eth; it is then distnce fom the sun F F sys tht G / G /( ) ( ) m / ms/ nd ( ) ( m/ m) m m / nd + m m ( / ) 50 0 m Let the ship be distnce fom the 50 0 m 59 0 m (fom cente of eth) 0 + m/ m + 99 0 k/59 0 k (b) VALUA: At the instnt when the spceship psses thouh this point its cceletion is zeo ince m " m this equl-foce point is much close to the eth thn to the sun IDNIFY: Apply q() to clculte the mnitude of the ittionl foce eeted by ech sphee ch foce is ttctie he net foce is the ecto sum of the indiidul foces UP: Let + be to the iht ( 500 k) ( 00 k) XCU: () F ( 0 N m /k )( 000 k) + 0 Ν, with the ( 000 m) ( 000 m) minus sin indictin net foce to the left (b) No, the foce found in pt () is the net foce due to the othe two sphees VALUA: he foce fom the 500 k sphee is ete thn fo the 00 k sphee een thouh its mss is less, becuse is smlle fo this mss G IDNIFY: he foce eeted by the moon is the ittionl foce, F he foce eeted on the peson by the eth is w m UP: he mss of the moon is m 5 0 k G 0 N m /k XCU: () (b) F F (5 0 k)(0 k) ( 0 N m /k ) 0 N ( 0 m) moon F eth w (0 k)(90 m/s ) 90 N F moon / F 5 0 eth VALUA: he foce eeted by the eth is much ete thn the foce eeted by the moon he mss of the moon is less thn the mss of the eth nd the cente of the eth is much close to the peson thn is the cente of the moon IDNIFY: Use q() to find the foce ech point mss eets on the pticle, find the net foce, nd use Newton s second lw to clculte the cceletion UP: ch foce is ttctie he pticle (mss m) is distnce 000 m fom m 00 k nd theefoe distnce 000 m fom m 50 k Let + be towd the 50 k mss XCU: Gm m (00 k) m F ( 0 N m /k ) ( 0 N/k) m, in the -diection (000 m) Gm m (50 k) m F ( 0 N m /k ) ( 0 N/k) m, in the + -diection he net foce is (000 m) 9 F F F + F ( 0 N/k + 0 N/k) m ( 0 N/k) m m 9 cceletion is 0 m/s, towd the 00 k mss VALUA: he smlle mss eets the ete foce, becuse the pticle is close to the smlle mss 9 0 m/s he
Gittion - 9 IDNIFY: Apply q() to clculte the mnitude of ech ittionl foce ch foce is ttctie 0 UP: he msses e m 5 0 k, m 99 0 k nd m 59 0 k Denote the eth-sun seption s nd the eth-moon seption s m m 0 XCU: () ( Gm ) + 0 0, Ν towd the sun ( + ) (b) he eth-moon distnce is sufficiently smll comped to the eth-sun distnce ( << ) tht the ecto fom the eth to the moon cn be tken to be pependicul to the ecto fom the sun to the moon he ittionl G 0 G 0 0 foces e then 0 Ν nd 99 0 Ν, nd so the foce hs mnitude 0 Ν nd is diected fom the diection towd the sun m m 0 (c) ( Gm ) 0, Ν towd the sun ( ) VALUA: he net foce is ey diffeent in ech of these thee positions, een thouh the mnitudes of the foces fom the sun nd eth chne ey little 0 IDNIFY: Apply q() to clculte the mnitude of ech ittionl foce ch foce is ttctie UP: he foces on one of the msses e sketched in Fiue 0 he fiue shows tht the ecto sum of the thee foces is towd the cente of the sque GmAmB cos 5 GmAmD XCU: FonA FBcos 5 + FD + AB ( 0 N m /k )(00 k) cos 5 ( 0 N m /k )(00 k) FonA + 0 N towd the (00 m) (00 m) cente of the sque VALUA: We he ssumed ech mss cn be teted s unifom sphee ch mss must he n unusully le density in ode to he mss 00 k nd still fit into sque of side lenth 00 cm AD Fiue 0 IDNIFY: Use q() to clculte the ittionl foce ech pticle eets on the thid mss he equilibium is stble when fo displcement fom equilibium the net foce is diected towd the equilibium position nd it is unstble when the net foce is diected wy fom the equilibium position UP: Fo the net foce to be zeo, the two foces on must be in opposite diections his is the cse only when is on the line connectin the two pticles nd between them he fee-body dim fo is ien in Fiue m m nd m m If is distnce fom m, it is distnce 00 m fom m m XCU: () F F + F G G 0 + (00 m ) (00 m ) 00 m ± / ince 00 m is between the two pticles, must be less thn 00 m nd 0 m must be plced t + / point tht is 0 m fom the pticle of mss m nd 0 m fom the pticle of mss m (b) (i) If is displced slihtly to the iht in Fiue, the ttctie foce fom m is le thn the foce fom m nd the net foce is to the iht If is displced slihtly to the left in Fiue, the ttctie foce fom m is le thn the foce fom m nd the net foce is to the left In ech cse the net foce is wy fom equilibium nd the equilibium is unstble (ii) If is displced ey smll distnce lon the y is in Fiue, the net foce is diected opposite to the diection of the displcement nd theefoe the equilibium is stble
- Chpte VALUA: he point whee the net foce on is zeo is close to the smlle mss Fiue IDNIFY: he foce F! eeted by m on nd the foce F! eeted by m on e ech ien by q() nd the net foce is the ecto sum of these two foces UP: ch foce is ttctie he foces on in ech eion e sketched in Fiue Let be t coodinte on the -is XCU: () Fo the net foce to be zeo, F! nd F! must be in opposite diections nd this is the cse only fo!! 0 < < L F +F 0 then equies F F Gm G ( m ) ( L ) nd L ± must be less ( L ) L thn L, so 0L + (b) Fo < 0, F > 0 F 0 s nd F + s 0 Fo > L, F < 0 F 0 s nd F s L Fo 0 < < 0L, F < 0 nd F inceses fom to 0 s oes fom 0 to 0L Fo 0L < < L, F > 0 nd F inceses fom 0 to + s oes fom 0L to L he ph of F esus is sketched in Fiue b VALUA: Any el object is not ectly point so it is not possible to he both m nd ectly t 0 o m nd both ectly t L But the mnitude of the ittionl foce between two objects ppoches infinity s the objects et ey close toethe Fiue
Gittion -5 IDNIFY: Use q() to find the foce eeted by ech le sphee Add these foces s ectos to et the net foce nd then use Newton s nd lw to clculte the cceletion UP: he foces e shown in Fiue sinθ 00 cosθ 00 ke the oiin of coodinte t point P XCU: B Fiue A (0 k)(000 k) FA G G (000 m) F G 5 0 N B F F θ F F θ A A sin (5 0 N)(00) 9 0 N Ay A cos + (5 0 N)(00) + 0 0 N B + B sin + 9 0 N By + B cos + 0 0 N 5 0 N F F θ F F θ F m ies F + A F B m 0 m so 0 Fy my ies F + Ay F By my (0 0 N) (000 k) y 9 0 m/s, diected downwd midwy between A nd B y VALUA: Fo odiny size objects the ittionl foce is ey smll, so the initil cceletion is ey smll By syety thee is no -component of net foce nd the y-component is in the diection of the two le sphees, since they ttct the smll sphee IDNIFY: Apply q() to Pluto UP: Pluto hs mss m 5 0 k nd dius 5 0 m XCU: qution () ies ( 0 N m /k )( 5 0 k) ( 5 0 m) 05 m /s VALUA: t the sufce of Pluto is much less thn t the sufce of th q() pplies to ny spheiclly syetic object m F G, so G, whee is the distnce of the object fom the cente of the eth UP: h+, whee h is the distnce of the object boe the sufce of the eth nd 0 m is the dius of the eth XCU: o decese the cceletion due to ity by one-tenth, the distnce fom the cente of the eth must be incesed by fcto of 0, nd so the distnce boe the sufce of the eth is 5 IDNIFY: ( ) 0 0 m VALUA: his heiht is bout twice the dius of the eth IDNIFY: Apply q() to the eth nd to Venus w m Gm UP: 90 m/s m V 05m nd V 099 w m 50 N GmV G(05 m) Gm XCU: () V 0905 0905 V (099 ) w m 0905 m (0905)(50 N) 9 N (b) V V
- Chpte VALUA: he mss of the ock is independent of its loction but its weiht equls the ittionl foce on it nd tht depends on its loction () IDNIFY nd UP: Apply q() to the eth nd to itni he cceletion due to ity t the sufce of itni is ien by Gm/, whee m is its mss nd is its dius Fo the eth, Gm / XCU: Fo itni, m m /00 nd /, so Gm G( m/00) Gm 00 ( /) 00 ince 90 m/s, (00)(90 m/s ) 0 m/s VALUA: on itni is much smlle thn on eth he smlle mss educes nd is ete effect thn the smlle dius, which inceses (b) IDNIFY nd UP: Use density mss/olume Assume itni is sphee XCU: Fom ection we know tht the ee density of the eth is m m /00 5 5 ρ ρ (5500 k/m ) 00 k/m ( /) 00 00 π π 5500 k/m Fo itni VALUA: he ee density of itni is bout fcto of smlle thn fo eth We cn wite q() fo itni s πg ρ < both becuse ρ < ρ nd < IDNIFY: Apply q() to he UP: ρ mv / he olume of sphee is V π XCU: G 0 k nd ρ ( π /) 0 0 k/m VALUA: he ee density of he is bout one-fouth tht of the eth 9 IDNIFY: Apply q() to the stonut UP: m 59 0 k nd 0 m XCU: F G 00 0 m + so F 0 N At the sufce of the eth, w m 5 N he ity foce is not zeo in obit he stellite nd the stonut he the sme cceletion so the stonut s ppent weiht is zeo VALUA: In q(), is the distnce of the object fom the cente of the eth mn 0 IDNIFY: G, whee the subscipt n efes to the neuton st w m UP: XCU: n n n 00 0 m w 5 N 9 k 90 m/s 0 m n 99 0 k You mss is m 0 99 0 k n ( 0 N m /k ) 0 m/s (00 0 m) You weiht on the neuton st would be wn mn (9 k)( 0 m/s ) 9 0 N VALUA: ince n is much less thn the dius of the sun, the ittionl foce eeted by the neuton st on n object t its sufce is iense IDNIFY nd UP: Use the mesued ittionl foce to clculte the ittionl constnt G, usin q() hen use q() to clculte the mss of the eth: 0 F (00 0 N)(0000 m) XCU: F G so G 0 N m /k (000 k)(00 0 k) Gm ( 0 m) (90 m/s ) ies m 59 0 k G 0 N m /k VALUA: Ou esult ees with the lue ien in Appendi F IDNIFY: Use q() to clculte fo uop he cceletion of pticle moin in cicul pth is ω d UP: In d ω, ω must be in d/s Fo uop, 59 0 m
XCU: Gm ( 0 N m /k )( 0 k) 0 m/s (59 0 m) ies d Gittion - 0 m/s 0 s e ω (055 d/s) 5 pm 5 m min π d VALUA: he dius of uop is bout one-fouth tht of the eth nd its mss is bout one-hundedth tht of eth, so on uop is much less thn on eth he lnde would he some sptil etent so diffeent points on it would be diffeent distnces fom the ottion is nd d would he diffeent lues Fo the ω we clculted, d t point tht is pecisely 5 m fom the ottion is IDNIFY nd UP: mple 5 ies the escpe speed s G/, whee nd e the mss nd dius of the stonomicl object XCU: ( 0 N m /k )( 0 k)/00 m 0 m/s VALUA: At this speed peson cn wlk 00 m in 0 s; esily chieed fo the ee peson We cn wite the escpe speed s whee ρ is the ee density of Dctyl Its dius is much smlle πρg, thn eth s nd its density is bout the sme, so the escpe speed is much less on Dctyl thn on eth IDNIFY: In pt () use the epession fo the escpe speed tht is deied in mple 5 In pt (b) pply consetion of eney UP: 5 0 m In pt (b) let point be t the sufce of the comet G (5 0 m)(0 m/s) XCU: () he escpe speed is so 0 k G ( 0 N m /k ) Gm Gm (b) (i) K m K 000K U ; U K+ U K + Uies Gm Gm m (000)( m ) olin fo ies 050 050(0 m/s) nd 5 km (ii) he debis nee G 5 0 m ( 0 N m /k )( 0 k) loses ll of its initil kinetic eney, but K 0 s he fthe the debis e fom the comet s cente, the smlle is thei kinetic eney VALUA: he debis will he lost 900% of thei initil kinetic eney when they e t distnce fom the comet s cente of bout ten times the dius of the comet 5 IDNIFY: he escpe speed, fom the esults of mple 5, is G/ UP: Fo s, 9 0 m XCU: () 0 k nd 0 0 m Fo Jupite, 90 0 k nd ( 0 N m /k )( 0 k)/(0 0 m) 50 0 m/s (b) ( 0 N m /k (90 0 k)/(9 0 m) 0 0 m/s (c) Both the kinetic eney nd the ittionl potentil eney e popotionl to the mss of the spcecft VALUA: mple 5 clcultes the escpe speed fo eth to be 0 m/s his is le thn ou esult fo s nd less thn ou esult fo Jupite Gm IDNIFY: he kinetic eney is K m nd the potentil eney is U UP: he mss of the eth is 59 0 k XCU: () K 9 (9 k)( 0 m/s) 9 0 J G m ( 0 N m /k )(59 0 k)(9 k) (b) U 0 J 9 0 m VALUA: he totl eney K + U is positie
- Chpte IDNIFY: Apply Newton s nd lw to the motion of the stellite nd obtin nd eqution tht eltes the obitl speed to the obitl dius UP: he distnces e shown in Fiue he dius of the obit is h+ 5 0 0 m + 0 m 0 m Fiue he fee-body dim fo the stellite is ien in Fiue b Fiue b F m () XCU: y y F m G d m Gm ( 0 N m /k )(59 0 k) 0 m/s 0 m π π ( 0 m) (b) 00 s h 0 m/s VALUA: Note tht h+ is the dius of the obit, mesued fom the cente of the eth Fo this stellite is ete thn fo the stellite in mple, so its obitl speed is less IDNIFY: he time to complete one obit is the peiod, ien by q() he speed of the stellite is π ien by UP: If h is the heiht of the obit boe the eth s sufce, the dius of the obit is h+ 0 m nd XCU: () m 59 0 k / 5 / π π(05 0 m+ 0 m) Gm ( 0 N m /k )(59 0 k) 59 0 s 990 min 5 π (05 0 m + 0 m) (b) 9 0 m/s 9 km/s 59 0 s VALUA: he stellite in mple is t lowe ltitude nd theefoe hs smlle obit dius thn the stellite in this poblem heefoe, the stellite in this poblem hs le peiod nd smlle obitl speed But le pecente chne in h coesponds to smll pecente chne in nd the lues of nd fo the two stellites do not diffe ey much!! 9 IDNIFY: Apply F m to the motion of the eth ound the sun π UP: Fo the eth, 5 dys 5 0 s nd 50 0 m π π (50 0 m) XCU: 99 0 s F md ies G m 5 0 s (99 0 s) (50 0 m) 0 m 0 0 k G 0 N m /k 0 VALUA: Appendi F ies m 99 0 k, in ood eement with ou clcultion 0 IDNIFY: We cn clculte the obitl peiod fom the numbe of eolutions pe dy hen the peiod nd the obit dius e elted by q() UP: m 59 0 k nd 0 m he heiht h of the obit boe the sufce of the eth is elted to the obit dius by h+ dy 0 s
XCU: he stellite moes 55 eolutions in 0 s 55 0 s 55 / π ies Gm / / [ 0 N m /k ][59 0 k][55 0 s] Gm π π 5 h 0 m 0 km 0 s, so the time fo 00 eolution is 5 0 m nd Gittion -9 VALUA: he peiod of this stellite is slihtly le thn the peiod fo the stellite in mple nd the ltitude of this stellite is theefoe somewht ete!! π IDNIFY: Apply F m to the motion of the bsebll UP: 0 m D F XCU: () d 5 D GmD ( 0 N m /k )(0 0 k) m ies G m m/s 0 m m/s mph, which is esy to chiee D D π π ( 0 m) (b) 00 s min he me would lst lon time m/s VALUA: he speed is eltie to the cente of Deimos he bsebll would ledy he some speed befoe we thow it, becuse of the ottionl motion of Deimos π IDNIFY: nd F md 0 0 UP: he sun hs mss m 99 0 k he dius of ecuy s obit is 59 0 m, so the dius of Vulcn s obit is F XCU: d 0 0 m m ies G m nd D Gm π π( 0 m) / 0 / π 0 s dys Gm 0 Gm ( 0 N m /k )(99 0 k) VALUA: he obitl peiod of ecuy is 0 d, so we could clculte fo Vulcn s / (0 d)(/) 9 dys IDNIFY: he obitl speed is ien by Gm/, whee m is the mss of the st he obitl peiod is ien π by 0 UP: he sun hs mss m 99 0 k he obit dius of the eth is 50 0 m XCU: () Gm/ 0 ( 0 N m /k )(05 99 0 k)/((50 0 m)(0)) 0 m/s (b) π / 5 0 s 5 dys (bout two weeks) VALUA: he obitl peiod is less thn the dy obitl peiod of ecuy; this plnet is obitin ey close to its st, comped to the obitl dius of ecuy IDNIFY: he peiod of ech stellite is ien by q() et up tio inolin nd / π π UP: ies constnt, so / / / Gm Gm XCU: p / /,000 km 9,00 km p (9 dys) 5 dys Fo the othe stellite, /,000 km (9 dys) dys 9,00 km VALUA: inceses when inceses 5 IDNIFY: In pt (b) pply the esults fom pt () UP: Fo Pluto, e 0 nd 59 0 m Fo Neptune, e 000 nd peiod fo Pluto is 9 y 50 0 m he obitl
-0 Chpte XCU: () he esult follows diectly fom Fiue 9 in the tetbook (b) he closest distnce fo Pluto is ( 0)(59 0 m) 5 0 m he etest distnce fo Neptune is ( + 000)(50 0 m) 55 0 m (c) he time is the obitl peiod of Pluto, y VALUA: Pluto's closest distnce clculted in pt () is 00 0 m 0 0 km, so Pluto is bout 00 million km close to the sun thn Neptune, s is stted in the poblem he eccenticity of Neptune's obit is smll, so its distnce fom the sun is ppoimtely constnt / π π, whee mst is the mss of the st Gm IDNIFY: UP: 0 99 0 k XCU: () m st 5 0 09 dys 0 s he obit dius of ecuy is 59 0 m he mss of ou sun is 5 0 s 0 9 (59 0 m)/9 0 m 9 π π ( 0 m) G ( 0 s) ( 0 N m /k ) st 5 0 st 0 k / π ies Gm sun st m m, so mst msun 9 π π ( 0 m) 5 (b) 5 0 m/s 5 0 s VALUA: he obitl peiod of ecuy is 0 d he peiod fo this plnet is much less pimily becuse the obit dius is much less nd lso becuse the mss of the st is ete thn the mss of ou sun () IDNIFY: If the obit is cicul, Newton s nd lw equies pticul eltion between its obit dius nd obitl speed UP: he ittionl foce eedted on the spcecft by the sun is F Gm m /, whee H m is the mss of the sun nd m H is the mss of the Helios B spcecft Fo cicul obit, d / nd F mh / If we nelect ll foces on the spcecft ecept fo the foce eeted by the sun, F F mh /, so GH/ mh / XCU: Gm 0 9 / ( 0 N m /k )(99 0 k)/ 0 m 5 0 m/s 5 km/s VALUA: he ctul speed is km/s, so the obit cnnot be cicul (b) IDNIFY nd UP: he obit is cicle o n ellipse if it is closed, pbol o hypebol if open he obit is closed if the totl eney (kinetic + potentil) is netie, so tht the object cnnot ech XCU: Fo Helios B, 9 K m H m H( 0 m/s) (5 0 m /s ) mh 0 H/ H( ( 0 N m /k )(99 0 k)/( U Gm m m K + U (5 0 m /s ) m (09 0 m /s ) m 9 9 H H 9 9 0 m)) (09 0 m /s )m H (5 0 m /s )m H VALUA: he totl eney is netie, so the obit is closed We know fom pt () tht it is not cicul, so it must be ellipticl IDNIFY: ection sttes tht fo point mss outside spheicl shell the ittionl foce is the sme s if ll the mss of the shell wee concentted t its cente It lso sttes tht fo point inside spheicl shell the foce is zeo UP: Fo 50 m the point mss is outside the shell nd fo 99 m nd m the point mss is inside the shell G (0000 k)(00 k) 9 XCU: () (i) F ( 0 N m /k ) 5 0 N (ii) F 0 (iii) (50 m) F 0 (b) Fo < 500 m the foce is zeo nd fo > 500 m the foce is popotionl to / he ph of F esus is sketched in Fiue
Gittion - VALUA: Inside the shell the ittionl potentil eney is constnt nd the foce on point mss inside the shell is zeo Fiue 9 IDNIFY: ection sttes tht fo point mss outside unifom sphee the ittionl foce is the sme s if ll the mss of the sphee wee concentted t its cente It lso sttes tht fo point mss distnce fom the cente of unifom sphee, whee is less thn the dius of the sphee, the ittionl foce on the point mss is the sme s thouh we emoed ll the mss t points fthe thn fom the cente nd concentted ll the eminin mss t the cente UP: he density of the sphee is ρ, whee is the mss of the sphee nd is its dius he mss π inside olume of dius < is ρv ( π ) 50 m is outside the sphee nd π 50 m is inside the sphee Gm (0000 k)(00 k) 9 XCU: () (i) F ( 0 N m /k ) 5 0 N (50 m) G m 50 m (ii) F (0000 k) 5 k 500 m (5 k)(00 k) 9 F ( 0 N m /k ) 0 N (50 m) G ( / ) m Gm Gm (b) F fo < nd F fo > he ph of F esus is sketched in Fiue 9 VALUA: At points outside the sphee the foce on point mss is the sme s fo shell of the sme mss nd dius Fo < the foce is diffeent in the two cses of unifom sphee esus hollow shell Fiue 9 0 IDNIFY: he ittionl potentil eney of point of point msses is U G Diide the od into infinitesiml pieces nd intete to find U UP: Diide the od into diffeentil msses dm t position l, mesued fom the iht end of the od dm dl( / L) XCU: () U Gm dm Gm dl l+ L l+ Gm L dl Gm L Intetin, U ln L 0 + Fo L l+ L U Gm/ >>, the ntul loithm is ~ ( / ) L, nd
- Chpte (b) he -component of the ittionl foce on the sphee is F ( / ) ( ( / )) ( ) U Gm L Gm L + L + L the minus sin indictin n ttctie foce As >> L, the denominto in the boe epession ppoches nd F Gm/, s epected VALUA: When is much le thn L the od cn be teted s point mss, nd ou esults fo U nd F do educe to the coect epession when >> L IDNIFY: Find the potentil due to smll sement of the in nd intete oe the entie in to find the totl U () UP: Fiue Diide the in up into smll sements d, s indicted in Fiue XCU: he ittionl potentil eney of d nd m is du Gmd/ he totl ittionl potentil eney of the in nd pticle is U du Gm d/ But + is the sme fo ll sements of the in, so Gm Gm Gm U d + (b) VALUA: When >>, + nd U Gm/ his is the ittionl potentil eney of two point msses septed by distnce his is the epected esult (c) IDNIFY nd UP: Use F du/ d with U( ) fom pt () to clculte F du d Gm XCU: F d d + d F + Gm ( + ) Gm ( ( )( + ) ) d F Gm/( + ) ; the minus sin mens the foce is ttctie VALUA: (d) Fo >>, ( + ) ( ) hen F Gm/ Gm/ his is the foce between two point msses septed by distnce nd is the epected esult (e) Fo 0, U Gm/ ch smll sement of the in is the sme distnce fom the cente nd the potentil is the sme s tht due to point che of mss locted t distnce Fo 0, F 0 When the pticle is t the cente of the in, syeticlly plced sements of the in eet equl nd opposite foces nd the totl foce eeted by the in is zeo IDNIFY: At the equto the object hs inwd cceletion nd the edin w of the blnce is elted to the m tue weiht w 0 (the ittionl foce eeted by the eth) by w0 w At the Noth Pole, d 0 nd w w 0 UP: As shown in ection, 5 m/s 0 m w0 m (5 m/s) XCU: w 0 5 N nd m 99 k w w0 5 N (99 k) N 0 m VALUA: he ottion of the eth cuses the scle edin to be slihtly less thn the tue weiht, since thee must be net inwd foce on the object IDNIFY nd UP: Ate the noth pole, F w m, whee 0 0 0 is ien by q() pplied to Neptune At the equto, the ppent weiht is ien by q() he obitl speed is obtined fom the ottionl peiod usin q(), with,
Gittion - XCU: () 0 Gm/ ( 0 N m /k )(0 0 k)/(5 0 m) 0 m/s his ees with the lue of ien in the poblem F w0 m0 (50 k)(0 m/s ) 5 N; this is the tue weiht of the object (b) Fom q(), w w m 0 / π π π (5 0 m) ies 0 m/s ( h)(00 s/ h) / ( 0 s) /5 0 m 09 m/s hen w 5 N (50 k)(09 m/s ) 5 N VALUA: he ppent weiht is less thn the tue weiht his effect is le on Neptune thn on eth G IDNIFY: he dius of blck hole nd its mss e elted by c 5 UP: 050 0 m, G 0 N m /k nd c 00 0 m/s 5 c (00 0 m/s) (050 0 m) XCU: 0 k G ( 0 N m /k ) VALUA: he ee density of the blck hole would be 0 k 5 G ρ 9 0 k/m We cn combine ρ nd 5 to ie π π(050 0 m) π c c ρ he ee density of blck hole inceses when its mss deceses he ee density π G of this mini blck hole is much ete thn the ee density of the much moe mssie blck hole in mple 5 IDNIFY nd UP: A blck hole with the eth s mss hs the chwzschild dius ien by q(0) XCU: G/ c ( 0 N m /k )(59 0 k)/(99 0 m/s) 5 0 m he tio of to the cuent dius is / 9 5 0 m/ 0 m 9 0 VALUA: A blck hole with the eth s dius is ey smll IDNIFY: Apply q() to clculte the ittionl foce Fo blck hole, the mss nd chwzschild dius e elted by q(0) UP: he speed of liht is c 00 0 m/s Gm ( c /) mc XCU: () ( )( ) ( 500 k 00 0 m/s 0 m ) (b) 50 N 00 0 m ( ) ( ) ( ) c 00 0 m 00 0 m/s (c) olin q(0) fo, 9 0 k G ( 0 N m /k ) VALUA: he mss of the blck hole is bout twice the mss of the eth G IDNIFY: he obitl speed fo n object distnce fom n object of mss is he mss of blck hole nd its chwzschild dius e elted by q(0) 5 UP: c 00 0 m/s ly 9 0 m XCU: () 5 ( 5 ly)( 9 0 m/ly)( 00 0 m/s) 0 k 0 G ( 0 N m /k ) (b) No, the object hs mss ey much ete thn 50 sol msses G 0 (c) 0 m, c c which does fit VALUA: he chwzschild dius of blck hole is ppoimtely the sme s the dius of ecuy's obit ound the sun
- Chpte π IDNIFY: he clumps obit the blck hole hei speed, obit dius nd obitl peiod e elted by / π hei obit dius nd peiod e elted to the mss of the blck hole by he dius of the blck G G hole's eent hoizon is elted to the mss of the blck hole by c UP: 00 0 m/s h 9 0 s c 00 0 m/s (00 0 m/s)(9 0 s) XCU: () 0 m π π / π π π ( 0 m) (b) ies 0 k G G ( 0 N m /k )(9 0 s) (c) G ( 0 N m /k )( 0 k) c (00 0 m/s) 9 9 0 m VALUA: he blck hole hs mss tht is bout 0 sol msses 9 IDNIFY: Use q() to find ech ittionl foce ch foce is ttctie In pt (b) pply consetion of eney UP: Fo pi of msses m nd m with seption, U G XCU: () Fom syety, the net ittionl foce will be in the diection 5 fom the -is (bisectin the nd y es), with mnitude (0 k) (0 k) F ( 0 N m /k )(0050 k) + sin 5 9 0 N ((050 m) ) (050 m) (b) he initil displcement is so le tht the initil potentil my be tken to be zeo Fom the wok-eney (0 k) (0 k) theoem, m Gm + Cncelin the fcto of m nd solin fo, nd usin the (050 m) (050 m) 5 numeicl lues ies 0 0 m/s VALUA: he esult in pt (b) is independent of the mss of the pticle It would tke the pticle lon time to ech point P 50 IDNIFY: Use q() to clculte ech ittionl foce nd dd the foces s ectos () UP: he loctions of the msses e sketched in Fiue 50 ection poes tht ny two spheiclly syetic msses intect s thouh they wee point msses with ll the mss concentted t thei centes Fiue 50 he foce dim fo m is ien in Fiue 50b cosθ 000 sinθ 000 XCU: Fiue 50b ( 0 N m /k )(00 k)(0500 k) (00 m) F G ( 0 N m /k )(00 k)(0500 k) (500 m) F G F 0 0 F 5 0 N, y 0 0 0 N 0 5 0 N
0 cos (0 0 N)(000) 5 0 N F F θ 0 y + sin + (0 0 N)(000) + 0 0 N F F θ 0 0 + 5 0 N 5 0 N 05 0 N F F F y y + y 0 + 0 0 N + 0 0 N F F F Fiue 50c F nd its components e sketched in Fiue 50c F F + F y F ( 05 0 N) + ( + 0 0 N) 0 0 0 N 0 F Fy + 0 0 N tn θ ; θ 0 F 05 0 N Gittion -5 VALUA: Both sphees ttct the thid sphee nd the net foce is in the second qudnt (b) UP: Fo the net foce to be zeo the foces fom the two sphees must be equl in mnitude nd opposite in diection Fo the foces on it to be opposite in diection the thid sphee must be on the y-is nd between the othe two sphees he foces on the thid sphee e shown in Fiue 50d 00y 00(00 m y) Fiue 50d ( 00 + 00) y (00 m) 00 nd y 9 m XCU: F net 0 if F F G G y (00 m y) 00 00 y (00 m y) hus the sphee would he to be plced t the point 0, y 9 m VALUA: Fo the foces to he the sme mnitude the thid sphee must be close to the sphee tht hs smlle mss 5 IDNIFY: τ F sinφ he net toque is the sum of the toques due to ech foce UP: Fom mple, usin Newton's thid lw, the foces of the smll st on ech le st e 5 F 0 N nd F 0 N Let counteclockwise toques be positie XCU: () he diection fom the oiin to the point midwy between the two le sts is ctn( 000 m), which is not the nle ( ) found in the emple 000 m (b) he coon lee m is 000 m, nd the foce on the uppe mss is t n nle of 5 fom the lee m he net toque is τ + F(00 0 m)sin 5 F(00 0 m) 5 0 N m, with the minus sin indictin clockwise toque VALUA: (c) hee cn be no net toque due to ittionl fields with espect to the cente of ity, nd so the cente of ity in this cse is not t the cente of mss Fo the cente of ity to be the sme point s the cente of mss, the ity foce on ech mss must be popotionl to the mss, with the sme constnt of popotionlity, nd tht is not the cse hee 5 IDNIFY: he ity foce fo ech pi of objects is ien by q() he wok done is W Δ U UP: he simplest wy to ppoch this poblem is to find the foce between the spcecft nd the cente of mss of the eth-moon system, which is 0 m fom the cente of the eth he distnce fom the spcecft to the cente of mss of the eth-moon system is 0 m (Fiue 5) m 5 0 k m 59 0 k,
- Chpte XCU: () Usin the Lw of Gittion, the foce on the spcecft is N, n nle of 0 fom the eth-spcecft line A B (b) U G U 0 nd 0 m fo the spcecft nd the eth, nd the spcecft nd the moon Gm ( 0 N m / k )(59 0 k + 5 0 k)(50 k) 0 m W U U + + 9 W 0 J Fiue 5 5 IDNIFY: Apply consetion of eney nd consetion of line momentum to the motion of the two sphees UP: Denote the 5-k sphee by subscipt nd the 00-k sphee by subscipt XCU: () Line momentum is conseed becuse we e inoin ll othe foces, tht is, the net etenl foce on the system is zeo Hence, m m (b) Fom the wok-eney theoem in the fom Ki + Ui Kf + Uf, with the initil kinetic eney K i 0 nd U G, Gm m ( ) f i m + m eliminte in fo of nd simplifyin yields ubstitution of numeicl lues ies Usin the consetion of momentum eltion m m to Gm, + m m f i 5 with simil epession fo 0 m/s, 0 0 m/s he mnitude of the eltie 5 elocity is the sum of the speeds, 0 0 m/s nd is popotionl to thei cceletion, nd (c) he distnce the centes of the sphees tel ( ) m, o When the sphees finlly mke contct, thei centes will be distnce of m pt, o + + 0 m, o + + 0 m hus, m 0, nd m he point of contct of the sufces is m 0 9 m fom the initil position of the cente of the 50 k sphee m VALUA: he esult / cn lso be obtined fom the consetion of momentum esult tht, m t eey point in the motion 5 IDNIFY: Apply q() UP: m 59 0 k XCU: olin q () fo, Gm π ( d)(, 00 s/d) 5 ( 0 N m /k )(59 0 k) 5 0 m, π fom which 0 m VALUA: he esult we clculted is in ey ood eement with the obit dius ien in Appendi F
Gittion - 55 IDNIFY nd UP: () o sty boe the sme point on the sufce of the eth the obitl peiod of the stellite must equl the obitl peiod of the eth: d( h/ d)(00 s/ h) 0 s q() ies the eltion between the obit dius nd the peiod: / π π XCU: nd Gm Gm Gm ( 0 s) ( 0 N m /k )(59 0 k) 0 m π π his is the dius of the obit; it is elted to the heiht h boe the eth s sufce nd the dius by h+ hus h 0 m 0 m 59 0 m of the eth VALUA: he obitl speed of the eosynchonous stellite is π / 00 m/s he ltitude is much le nd the speed is much less thn fo the stellite in mple (b) Conside Fiue 55 cosθ θ 0 m 0 m Fiue 55 A line fom the stellite is tnent to point on the eth tht is t n nle of boe the equto he sketch shows tht points t hihe ltitudes e blocked by the eth fom iewin the stellite 5 IDNIFY: Apply q() to elte the obitl peiod nd P, the plnet's mss, nd then use q() pplied to the plnet to clculte the stonut's weiht 5 UP: he dius of the obit of the lnde is 55 0 m + 0 0 m XCU: Fom q(), π nd G 5 π π (55 0 m+ 0 0 m) G ( 0 N m /k )(5 0 s) P P 0 k o bout hlf the eth's mss Now we cn find the stonut s weiht on the sufce fom q() (he lndin on the noth pole emoes ny need to ccount fo centipetl cceletion) w G m ( 0 N m /k )( 0 k)( 5 k) ( 0 0 m) p p, N VALUA: At the sufce of the eth the weiht of the stonut would be 9 N 5 IDNIFY: Fom mple 5, the escpe speed is G Use ρ / V to wite this epession in tems of ρ UP: Fo sphee V π XCU: In tems of the density, is, ρ the tio ( ) ( π )( )( )( ) π ρ nd so the escpe speed is / 0 N m /k 500 k/m 50 0 m m/s VALUA: his is much less thn the escpe speed fo the eth,,00 m/s G 5 IDNIFY: Fom mple 5, the escpe speed is Use ρ / V to wite this epession in tems of ρ On eth, the heiht h you cn jump is elted to you jump speed by h Fo pt (b), pply q() to uop UP: Fo sphee V π
- Chpte πgρ XCU: ρ /( π ), so the escpe speed cn be witten s qutin the two epessions π h fo nd squin ies h ρg, o, whee 90 m/s is fo the sufce of the eth, not πρg the steoid stimte h m (ible fo diffeent people, of couse), km Fo uop, G πρg ( m/s ) ρ 0 0 k/m πg π(5 0 m)( 0 N m /k ) VALUA: he eth hs ee density 5500 k/m he ee density of uop is bout hlf tht of the eth but little le thn the ee density of most steoids 59 IDNIFY nd UP: he obseed peiod llows you to clculte the nul elocity of the stellite eltie to you You know you nul elocity s you otte with the eth, so you cn find the nul elocity of the stellite in spce-fied efeence fme ω ies the obitl speed of the stellite nd Newton s second lw eltes this to the obit dius of the stellite XCU: () he stellite is eolin west to est, in the sme diection the eth is ottin If the nul speed of the stellite is ω nd the nul speed of the eth is ω, the nul speed ω el of the stellite eltie to you is ω ω ω el s ω el ( e)/( h) ( ) e/h ω ( ) e/h ωs ωel + ω ( ) e/h 0 d/s!! F m sys G m Gm Gm nd with ω this ies ; 0 0 m ω his is the dius of the stellite s obit Its heiht h boe the sufce of the eth is h 9 0 m VALUA: In pt () the stellite is eolin fste thn the eth s ottion nd in pt (b) it is eolin slowe lowe nd ω mens le obit dius (b) Now the stellite is eolin opposite to the ottion of the eth If west to est is positie, then ω el ( ) e/h 5 ωs ωel + ω ( ) e/h 0 d/s Gm ies 0 m nd h 59 0 m ω 0 IDNIFY: Apply the lw of ittion to the stonut t the noth pole to clculte the mss of plnet hen!! π pply F m to the stonut, with d, towd the cente of the plnet, to clculte the peiod Apply q() to the stellite in ode to clculte its obitl peiod UP: Get dius of X: ( π ),50 km nd 0 0 m Astonut mss: m ω 9 N 9 k 90 m/s Gm X m (95 N)(0 0 m) 5 XCU: w, whee w 950 N 05 0 k Gm ( 0 N m /k )(9 k) Apply Newton s second lw to stonut on scle t the equto of X F Fscle md, so π m π (9 k)(0 0 m) h F Fscle 950 N 500 N nd 5 0 s h 00 s π π (0 0 m+ 0 0 m) (b) Fo the stellite, 90 0 s hous 5 Gm ( 0 N m /k )(05 0 k) X 950 N VALUA: he cceletion of ity t the sufce of the plnet is X 95 m/s, simil to the 9 k lue on eth he dius of the plnet is bout twice tht of eth he plnet ottes moe pidly thn eth nd the lenth of dy is bout one-thid wht it is on eth
Gm IDNIFY: Use nd follow the pocedue specified in the poblem UP: 0 m mh XCU: he fctionl eo is ( + h)( ) G / /( + h) Gm ( ) Gittion -9 Usin q() fo the fctionl diffeence is ( + h)/ h/, so if the fctionl diffeence is % h (00) 0 m VALUA: Fo h km, the fctionl eo is only 00% q() is ey ccute fo the motion of objects ne the eth's sufce IDNIFY: Use the mesuements of the motion of the ock to clculte, the lue of on ono hen use π this to clculte the mss of ono Fo the ship, F md nd UP: ke +y upwd When the stone etuns to the ound its elocity is 0 m/s, downwd G m c 00 0 m he dius of ono is 0 m he ship moes in n obit of dius π π 0 m + 00 0 m 0 m XCU: () 0 y + 0 m/s, y 0 m/s, y nd t 00 s y 0 y + yt ies y 0 y 0 m/s 0 m/s nd 00 m/s t 00 s (00 m/s )( 0 m) 5 m 55 0 k G 0 N m /k (b) F md ies G m nd Gm ( 0 m) π π π π Gm Gm / / 5 ( 0 N m /k )(55 0 k) 55 0 s 5 h VALUA: 50 nd m m, so 00, which ees with the lue clculted (50) in pt () IDNIFY nd UP: Use q() to clculte the ity foce t ech loction Fo the top of ount eest wite h+ nd use the fct tht h<< to obtin n epession fo the diffeence in the two foces XCU: At cmento, the ity foce on you is F G At the top of ount eest, heiht of F G G ( + h) ( + h/ ) h 00 m boe sel leel, the ity foce on you is h h ( + h / ), F F F F h 0% F VALUA: he chne in the ittionl foce is ey smll, so fo objects ne the sufce of the eth it is ood ppoimtion to tet it s constnt IDNIFY: Apply q(9) to the pticle-eth nd pticle-moon systems UP: When the pticle is distnce fom the cente of the eth, it is distnce the moon fom the cente of
-0 Chpte m m XCU: () he totl ittionl potentil eney in this model is U Gm + (b) ee ecise 5 he point whee the net ittionl foce nishes is 0 m + m / m Usin this lue fo in the epession in pt () nd the wok-eney theoem, includin the initil potentil eney of Gm( m/ + m/( )) ies km s (c) he finl distnce fom the eth is not, but the th-moon distnce minus the dius of the moon, o 0 m Fom the wok-eney theoem, the ocket impcts the moon with speed of 9 km/s VALUA: he spcecft hs ete ittionl potentil eney t the sufce of the moon thn t the sufce of the eth, so it eches the sufce of the moon with speed tht is less thn its lunch speed on eth 5 IDNIFY nd UP: Fist use the dius of the obit to find the initil obitl speed, fom q(0) pplied to the moon XCU: Gm/ nd + h 0 m + 500 0 m 9 0 m ( 0 N m /k )(5 0 k) hus 55 0 m/s 9 0 m Afte the speed deceses by 00 m/s it becomes 55 0 m/s 00 m/s 5 0 m/s IDNIFY nd UP: Use consetion of eney to find the speed when the spcecft eches the lun sufce K + U + W K + U othe Gity is the only foce tht does wok so W othe 0 nd K K+ U U XCU: U Gm m/; m m m U Gm/ m m m + G (/ / ) And the mss m diides out to ie + Gm (/ / ) m m 0 m/s( km/000 m)(00 s/ h) 00 km/h VALUA: Afte the thuste fies the spcecft is moin too slowly to be in stble obit; the ittionl foce is le thn wht is needed to mintin cicul obit he spcecft ins eney s it is cceleted towd the sufce IDNIFY: 0 mens the ppent weiht is zeo, so UP: he dius of the eth is 0 m XCU: d 90 m/s π d π 50 0 s, which is 5 min, o bout n hou nd hlf d VALUA: At the poles, would still be 90 m/s IDNIFY nd UP: Apply consetion of eney ust use q(9) fo the ittionl potentil eney since h is not smll comped to As indicted in Fiue, tke point to be whee the he is elesed nd point to be just boe the sufce of the eth, so + h nd Fiue XCU: K + U + Wothe K + U Only ity does wok, so W othe 0 K 0, K m G G U G, U G h+
hus, G m G h+ Gm Gmh Gm ( + h ) + h ( + h) ( + h) Gmh ( + h) Gittion - VALUA: If h, Gm/, which equls the escpe speed In this limit this eent is the eese of n object bein pojected upwd fom the sufce with the escpe speed If h#, then Gm h/ h, the sme esult if used q() fo U G IDNIFY: In obit the totl mechnicl eney of the stellite is U G W f i UP: U 0 s Gm XCU: () he eney the stellite hs s it sits on the sufce of the th is i he eney it hs Gm when it is in obit t dius is f he wok needed to put it in obit is the diffeence between Gm these: W f i (b) he totl eney of the stellite f wy fom the th is zeo, so the dditionl wok needed is Gm Gm 0 VALUA: (c) he wok needed to put the stellite into obit ws the sme s the wok needed to put the stellite fom obit to the ede of the uniese 9 IDNIFY: At the escpe speed, K + U 0 UP: At the sufce of the eth the stellite is distnce 0 m fom the cente of the eth nd π distnce 50 0 m fom the sun he obitl speed of the eth is, whee 5 0 s is the π cosφ obitl peiod he speed of point on the sufce of the eth t n nle φ fom the equto is, whee, 00 s is the ottionl peiod of the eth m m s XCU: () he escpe speed will be G + 5 0 m/s kin the simplifyin ssumption tht the diection of lunch is the diection of the eth s motion in its obit, the speed eltie to the π π (50 0 m) cente of the eth is 5 0 m/s 0 m/s (5 0 s) (b) he ottionl speed t Cpe Cnel is the sufce of the eth is 0 m/s (c) In Fench Guin, the ottionl speed is 0 m/s ( 0 m) cos 5 π 09 0 m/s, so the speed eltie to,00 s 0 m/s, so the speed eltie to the sufce of the eth is VALUA: he obitl speed of the eth is le fction of the escpe speed, but the ottionl speed of point on the sufce of the eth is much less 0 IDNIFY: Fom the discussion of ection, the foce on point mss t distnce fom the cente of spheiclly syetic mss distibution is the sme s thouh we emoed ll the mss t points fthe thn fom the cente nd concentted ll the eminin mss t the cente UP: he mss of hollow sphee of density ρ, inne dius nd oute dius is ρ π( ) Fom Fiue 9 in the tetbook, the inne coe hs oute dius 0 k/m he oute coe hs inne dius 0 m, oute dius he totl mss of the eth is m 59 0 k nd its dius is 0 m 0 m, inne dius zeo nd density 0 m nd density 0 k/m
- Chpte XCU: () F G m (00 k)(90 m/s ) 90 N (b) he mss of the inne coe is m ρ π( ) ( 0 k/m ) π( 0 m) 9 0 k he mss of the oute coe is m nd oute coes contibute to the foce inne inne ( 0 k/m ) π ([ 0 m] [ 0 m] ) 0 k Only the inne oute (9 0 k + 0 k)(00 k) F (c) Only the inne coe contibutes to the foce nd F (d) At 0, F 0 ( 0 N m /k ) 0 N ( 0 m) (9 0 k)(00 k) ( 0 N m /k ) N ( 0 m) VALUA: In this model the eth is spheiclly syetic but not unifom, so the esult of mple 0 doesn't pply In pticul, the foce t the sufce of the oute coe is ete thn the foce t the sufce of the eth IDNIFY: q() eltes obitl peiod nd obitl dius fo cicul obit 0 UP: he mss of the sun is 99 0 k π XCU: () he peiod of the steoid is Insetin 0 m fo ies G ies peiod of y (b) If the peiod is 59 y, then 90 0 m y nd 5 0 m (c) his hppens becuse 0 5, nothe tio of intees o once eey 5 obits of the steoid nd obits of Jupite, the steoid is t its peijoe distnce olin when y, 0 m VALUA: he obit dius fo Jupite is 0 m nd fo s it is 0 m he steoid belt lies between s nd Jupite he mss of Jupite is bout 000 times tht of s, so the effect of Jupite on the steoids is much le IDNIFY: Apply the wok-eney eltion in the fom W Δ, whee K + U he speed is elted to the obit dius by q(0) UP: m 59 0 k XCU: () In moin to lowe obit by whtee mens, ity does positie wok, nd so the speed does incese (b) ( ) / / Gm, so ( ) / / Δ Δ Gm Δ Gm Note tht positie decese in dius imilly, the kinetic eney is ( ) ( ) Δ K ( / )( G/ ) Δ nd ( / ) W Δ U +Δ K ( Gm m/ ) Δ / Δ U Gm m Δ (c) Gm 0 m/s, \ ( ) Δ Δ / Gm / 9 m/s, ( )( ) Δ K Gm m Δ / 0 0 J, K / m / Gm m/, nd so 9 Δ U Δ K 0 J, nd 0 / 95 0 J Δ is ien s Gm m (fom q(5)), 0 0 J W Δ K (d) As the tem buns up suests, the eney is coneted to het o is dissipted in the collisions of the debis with the ound VALUA: When deceses, K inceses nd U deceses (becomes moe netie) IDNIFY: Use q() to clculte F Apply Newton s nd lw to cicul motion of ech st to find the obitl speed nd peiod Apply the consetion of eney epession, q(), to clculte the eney input (wok) equied to septe the two sts to infinity () UP: he cm is midwy between the two sts since they he equl msses Let be the obit dius fo ech st, s sketched in Fiue he two sts e septed by distnce, so F G /( ) G / Fiue
(b) XCU: F md / ( / ) so G/ G Gittion - And π/ π / G π / G (c) UP: Apply K + U + Wothe K + U to the system of the two sts epte to infinity implies K 0 nd U 0 XCU: U G / K + ( )( G/ ) G / hus the eney equied is W ( K + U ) ( G / G / ) G / othe VALUA: he close the sts e nd the ete thei mss, the le thei obitl speed, the shote thei obitl peiod nd the ete the eney equied to septe them!! IDNIFY: In the cente of mss coodinte system, cm 0 Apply F m to ech st, whee F is the π ittionl foce of one st on the othe nd d π UP: llows to be clculted fom nd IDNIFY: () he dii nd e mesued with espect to the cente of mss, nd so, nd / / (b) he foces on ech st e equl in mnitude, so the poduct of the mss nd the dil cceletions e π π equl: Fom the esult of pt (), the numetos of these epessions e equl, nd so the denomintos e equl, nd the peiods e the sme o find the peiod in the syetic fom desied, thee e mny possible outes An elent method, usin bit of hindsiht, is to use the boe epessions to elte the G π ( + ) peiods to the foce F, so tht equilent epessions fo the peiod e nd ( + ) G π ( + ) π ( + ) π( + ) Addin the epessions ies ( + ) o G G G ( + ) (c) Fist we must find the dii of ech obit ien the speed nd peiod dt In cicul obit, ( 0 m/s)( d)(,00 s/d) π, o 0 hus 0 m π α nd π ( 0 m/s)( d)(,00 s/d) 0 β 0 m Now find the sum of the msses Use α α ββ, nd π π ( α + β) the fct tht α β( α + β), insetin the lues of, nd the dii his ies G 0 0 π ( 0 m + 0 m) 0 ( α + β) 0 k α + β ince [( d)(,00 s/d)] ( 0 N m /k ) 0 9 /, 0 k, o 0 0 k nd β α α β α (d) Let α efe to the st nd β efe to the ( / ) (0/) (0), β α β α α α α α 0 β 0 k blck hole Use the eltionships deied in pts () nd (b): ( ) α + β G α + β Fo onoceotis, insetin the lues 9 fo nd nd β ies α 9 0 m, α 0 km/s nd fo the blck hole β 0 m, β km/s VALUA: ince is the sme, is smlle when is smlle 5 IDNIFY nd UP: Use consetion of eney, K + U + W K U othe + he ity foce eeted by the sun is the only foce tht does wok on the comet, so W othe 0 XCU: K m, U G/, 5 0 m K m U Gm m/, 0 50 0 m 0 0 m/s π
- Chpte m Gm m/ m Gm m/ + Gm + Gm 0 m/s VALUA: he comet hs ete speed when it is close to the sun IDNIFY: Apply consetion of eney UP: Let m be the mss of s nd be the mss of the sun he subscipts nd p denote phelion nd peihelion G m G m XCU: m mp, o p G 50 0 m/s p p VALUA: We could insted use consetion of nul momentum Note tht t the etemes of distnce (peiheleion nd phelion), s elocity ecto must be pependicul to its dius ecto, nd so the mnitude of the nul momentum is L m ince L is constnt, the poduct must be constnt, nd so (9 0 m) (9 0 m/s) 50 0 m/s s hs le speed when it is close to the sun p p (0 0 m) () IDNIFY nd UP: Use q(), pplied to the stellites obitin the eth the thn the sun XCU: Find the lue of fo the ellipticl obit: + p + h + + hp, whee h nd h p e the heihts t poee nd peiee, espectiely + ( h + h )/ p 0 m + (00 0 m + 000 0 m)/ 5 0 m π π(5 0 m) G ( 0 N m /k )(59 0 k) (b) Consetion of nul momentum ies p p p 0 m + 00 0 m 5 5 p 0 m + 00 0 m 9 0 s (c) Consetion of eney pplied to poee nd peiee ies K + U Kp + Up m Gm m/ m Gm m/ P p Gm (/ / ) Gm ( )/ P p p p But p 5, so Gm ( )/ 55 0 m/s, p p p 0 m/s (d) Need so tht 0, whee K + U t peiee: m Gm / p p Gm m p / p 0 his mens n incese of t poee: Gm/ ( 0 N m /k )(59 0 k)/ 0 m 0 0 m/s 0 0 m/s 0 m/s 0 m/s ( 0 N m /k )(59 0 k)/0 0 m 0 m/s his mens n incese of 0 m/s 55 0 m/s 5 0 m/s VALUA: Peiee is moe efficient At this point is smlle so is le nd the stellite hs moe kinetic eney nd moe totl eney G, whee nd e the mss nd dius of the plnet UP: Let mu nd U be the mss nd dius of Unus nd let U be the cceletion due to ity t its IDNIFY: poles he obit dius of ind is h+ U, whee sufce of Unus XCU: () Fom the lue of t the poles, m ( m/s ) ( 55 0 m) ( 0 N m /k ) G U U U 09 0 k h 0 0 m is the ltitude of ind boe the
Gm / / 0 m/s (b) ( ) U U U (c) Gm / 000 m/s Gittion -5 VALUA: (d) No Both the object nd ind e in obit toethe ound Unus, due to the ittionl foce of Unus he object hs dditionl foce towd ind 9 IDNIFY nd UP: Apply consetion of eney (q) nd sole fo W Only othe h+ is ien, so use q(0) to elte nd K + U + W K + U XCU: othe U G/, whee m is the mss of s nd + h, whee is the dius of s nd h 000 0 m ( 0 k)(000 k) 0 U ( 0 N m /k ) 0 0 J 0 0 m + 000 0 m U G/, whee is the new obit dius U ( 0 k)(000 k) ( 0 N m /k ) 0 J 0 0 m + 000 0 m 0 Fo cicul obit Gm / (q(0), with the mss of s the thn the mss of the eth) Usin this ies K m m( Gm / ) Gm m/, so K U K 0 90 0 J hen K U Wothe K U U + nd K + + + ies U 9 + 5 0 J W K K + U U + 9 0 othe ( ) ( ) (5 0 J 90 0 J) ( 0 W + 9 9 9 othe 5 0 J 0 0 J 0 J 0 0 0 J + 0 J) VALUA: When the obit dius inceses the kinetic eney deceses nd the ittionl potentil eney inceses K U so K + U U nd the totl eney lso inceses (becomes less netie) Positie wok must be done to incese the totl eney of the stellite 0 IDNIFY nd UP: Use q() to clculte 0,000 y(5 0 s/ y) 9 0 s π XCU: q():, Gm 0 m π Gm Gm π VALUA: he ee obit dius of Pluto is 59 0 m (Appendi F); the semi-mjo is fo this comet is le by fcto of 5 liht yes liht yes(9 0 m/ liht ye) 0 m he distnce of Alph Centui is le by fcto of 00 he obit of the comet etends well pst Pluto but is well within the distnce to Alph Centui IDNIFY: Intete dm ρdv to find the mss of the plnet Outside the plnet, the plnet behes like point mss, so t the sufce G/ UP: A thin spheicl shell with thickness d hs olume dv π d he eth hs dius 0 m XCU: Get : dm ρdv ρ π d he density is ρ ρ0 b, whee ρ0 ρs ρ 0 50 0 k/m t the cente nd t the sufce, ρ 0 0 k/m, so b π ρ0 ρs 0 ( ρ0 b) πd ρ0 πb π ρ0 π π ρ0 + ρs nd 5 0 k hen ( ) G Gπ ρ 0 + ρs πg ρ 0 + ρs 50 0 k/m π ( 0 m)( 0 N m /k ) + 0 0 k/m 9 m/s
- Chpte VALUA: he ee density of the plnet is ρ 55 0 k/m Note (5 0 k) V π π( 0 m) tht this is not ( ρ0 + ρs)/ IDNIFY nd UP: Use q() to clculte the foce between the point mss nd smll sement of the semicicle XCU: he dius of the semicicle is L/ π Diide the semicicle up into smll sements of lenth dθ, s shown in Fiue Fiue d ( / L) dθ ( / π) dθ df! is the ity foce on m eeted by d df y 0; the y-components fom the uppe hlf of the semicicle cncel the y-components fom the lowe hlf he -components e ll in the + -diection nd ll dd md df G md Gmπ df G cosθ cos θ dθ L Gm cos Gm F π π π π () df θ dθ π L π L π Gm F L VALUA: If the semicicle wee eplced by point mss t, the ity foce would be Gm/ π Gm/ L his is π / times le thn the foce eeted by the semicicl wie Fo the semicicle it is the -components tht dd, nd the sum is less thn if the foce mnitudes wee dded IDNIFY: he diect clcultion of the foce tht the sphee eets on the in is slihtly moe inoled thn the clcultion of the foce tht the in eets on the sphee hese foces e equl in mnitude but opposite in diection, so it will suffice to do the ltte clcultion By syety, the foce on the sphee will be lon the is of the in in Fiue 5 in the tetbook, towd the in UP: Diide the in into infinitesiml elements with mss d ( Gm) d XCU: ch mss element d of the in eets foce of mnitude on the sphee, nd the + Gmd Gmd -component of this foce is + + + ( ) / Gm / +, in the -diection he sphee ttcts the in with foce of the sme mnitude Gm VALUA: As >> the denominto ppoches nd F, s epected IDNIFY: Use q() fo the foce between smll sement of the od nd the pticle Intete oe the lenth of the od to find the totl foce heefoe, the foce on the sphee is ( ) /
Gittion - UP: Use coodinte system with the oiin t the left-hnd end of the od nd the -is lon the od, s shown in Fiue Diide the od into smll sements of lenth d (Use fo the coodinte so not to confuse with the distnce fom the end of the od to the pticle) Fiue XCU: he mss of ech sement is d d ( / L) ch sement is distnce L + fom mss m, so Gm d Gm d the foce on the pticle due to sement is df ( L + ) L ( L + ) 0 Gm 0 d Gm 0 F df L L L L ( L + ) L L + Gm Gm ( L + ) Gm F L L+ L ( L+ ) ( L+ ) VALUA: Fo >> L this esult become F Gm/, the sme s fo pi of point msses 5 IDNIFY: Compe F to Hooke s lw UP: he eth hs mss m 59 0 k nd dius 0 m XCU: Fo F k, U k Gm m U his is he foce hee is in the sme fom, so by nloy ( ) lso ien by the intel of F fom 0 to with espect to distnce G (b) Fom pt (), the initil ittionl potentil eney is qutin initil potentil eney nd finl kinetic eney (initil kinetic eney nd finl potentil eney e both zeo) ies Gm, so 90 0 m/s VALUA: When 0, U( ) 0, s specified in the poblem IDNIFY: In qs() nd () eplce by + Δ nd by + Δ Use the epession in the hint to simplifyin the esultin equtions UP: he eth hs m 59 0 k nd 0 m h+, whee h is the ltitude boe the sufce of the eth π XCU: () theefoe G / ( ) π π Δ π Δ π Δ +Δ +Δ + + + G G G G G ince, π Δ Δ G, nd theefoe ( ) ( Δ ) Δ +Δ + nd Δ ( ) G G π, Δ πδ G G G Δ ince π (b) ttin with (q()), π /, nd G (q(0)), find the elocity nd G ( 0 N m /k )(59 0 k) peiod of the initil obit: 0 m/s, nd 0 m π / 559 s 95 min We then cn use the two deied equtions to ppoimte Δ nd Δ :
- Chpte (00 m) πδ π π(00 m) Δ 0 s nd Δ πδ 005 m/s Befoe the cble beks, the (559 s) 0 m/s shuttle will he teled distnce d, d (5 m ) (00 m ) 5 m t (5 m) (005 m/s) s min It will tke minutes fo the cble to bek (c) he I is moin fste thn the spce shuttle, so the totl nle it coes in n obit must be π dins moe ( ) thn the nle tht the spce shuttle coes befoe they e once in in line themticlly, t Δ t π ( +Δ) Usin the binomil theoem nd nelectin tems of ode t ( Δ) t Δ Δ Δ ( ) ( ) ΔΔ, + t + π heefoe, t π ince π nd Δ Δ, t, s Δ π Δ Δ π π Δ + + Δ π Δ Δ + t π π (559 s) ws to be shown t 5 0 s 900 d 9 y It is hihly doubtful the shuttle cew would Δ (0 s) suie the conessionl heins if they miss! VALUA: When the obit dius inceses, the obitl peiod inceses nd the obitl speed deceses IDNIFY: Apply q(9) to the tnsfe obit UP: he obit dius fo th is 50 0 m nd fo s it is 0 m Fom Fiue 9 in the tetbook, ( + ) XCU: () o et fom the cicul obit of the eth to the tnsfe obit, the spcecft s eney must incese, nd the ockets e fied in the diection opposite tht of the motion, tht is, in the diection tht inceses the speed Once t the obit of s, the eney needs to be incesed in, nd so the ockets need to be fied in the diection opposite tht of the motion Fom Fiue in the tetbook, the semimjo is of the tnsfe obit is the ithmetic ee of the obit dii of the eth nd s, nd so fom q(), the eney of the spcecft while in the tnsfe obit is intemedite between the eneies of the cicul obits etunin fom s to the eth, the pocedue is eesed, nd the ockets e fied inst the diection of motion (b) he time will be hlf the peiod s ien in q (), with the semimjo is equl to ( + ) 9 0 m so π (9 0 m) t 0 s dys, 0 ( 0 N m /k )(99 0 k) which is moe thn months ( 0 s) (c) Duin this time, s will pss thouh n nle of (0 ) 59, nd the spcecft ( d)(, 00 s/d) psses thouh n nle of 0, so the nle between the eth-sun line nd the s-sun line must be VALUA: he peiod fo the tnsfe obit is 5 dys, the ee of the obitl peiods fo th nd s!! IDNIFY: Apply F m to ech e UP: Denote the obit dius s nd the distnce fom this dius to eithe e s δ ch e, of mss m, cn be modeled s subject to two foces, the ittionl foce fom the blck hole nd the tension foce (ctully the foce fom the body tissues), denoted by F Gm XCU: he foce eqution fo eithe e is F mω ( + δ ), whee δ cn be of eithe sin ( + δ ) eplce the poduct mω with the lue fo δ 0, mω Gm/, nd sole fo F: + δ Gm F ( Gm) δ ( ( δ/ ) ) + + ( + δ ) Usin the binomil theoem to epnd the tem in sque bckets in powes of δ /, Gm Gm F + δ ( ( δ/ ) ) ( δ) kn his tension is much le thn tht which could be sustined by humn tissue, nd the stonut is in touble (b) he cente of ity is not the cente of mss he ity foce on the two es is not the sme VALUA: he tension between he es is popotionl to thei seption 9 IDNIFY: As suested in the poblem, diide the disk into ins of dius nd thickness d UP: ch in hs n e da π d nd mss d da d π