Universal Gravitation

Transcription

1 3 Univesal Gavitation CHAPTER OUTLINE 3. Newton s Law of Univesal Gavitation 3. Fee-Fall Acceleation and the Gavitational Foce 3.3 Analysis Model: Paticle in a Field (Gavitational) 3.4 Keple s Laws and the Motion of Planets 3.5 Gavitational Potential Enegy 3.6 Enegy Consideations in Planetay and Satellite Motion * An asteisk indicates a question o poblem new to this edition. ANSWERS TO OBJECTIVE QUESTIONS OQ3. Answe (c). Ten tems ae needed in the potential enegy: OQ3. OQ3.3 OQ3.4 U U + U 3 + U 4 + U 5 + U 3 + U 4 + U 5 + U 34 + U 35 + U 45 The anking is a > b c. The gavitational potential enegy of the Eath-Sun system is negative and twice as lage in magnitude as the kinetic enegy of the Eath elative to the Sun. Then the total enegy is negative and equal in absolute value to the kinetic enegy. Answe (d). The satellite expeiences a gavitational foce, always diected towad the cente of its obit, and supplying the centipetal foce equied to hold it in its obit. This foce gives the satellite a centipetal acceleation, even if it is moving with constant angula speed. At each point on the cicula obit, the gavitational foce is diected along a adius line of the path, and is pependicula to the motion of the satellite, so this foce does no wok on the satellite. Answe (d). Having twice the mass would make the suface gavitational field two times lage. But the invese squae law says that having twice the adius would make the suface acceleation due to gavitation fou times smalle. Altogethe, g at the suface of B Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

2 Chapte OQ3.5 OQ3.6 becomes ( m/s )()/4 m/s. Answe. Switching off gavity would let the atmosphee evapoate away, but switching off the atmosphee has no effect on the planet s gavitational field. Answe. The mass of a spheical body of adius R and density ρ is M ρv ρ(4πr 3 /3). The escape velocity fom the suface of this body may then be witten in eithe of the following equivalent foms: v esc GM R and v esc G R 4πρR 3 3 8πρGR 3 We see that the escape velocity depends on the thee popeties (mass, density, and adius) of the planet. Also, the weight of an object on the suface of the planet is F g mg GMm/R, giving g GM R G ρ 4πR3 R πρgr The fee-fall acceleation at the planet s suface then depends on the same popeties as does the escape velocity. Changing the value of g would necessaily change the escape velocity. Of the listed quantities, the only one that does not affect the escape velocity is the mass of the object. OQ3.7 (i) Answe (e). Accoding to the invese squae law, /4 6 times smalle. OQ3.8 OQ3.9 (ii) Answe (c). mv / GMm/ pedicts that v is popotional to (/) /, so it becomes (/4) / / as lage. (iii) Answe (a). Accoding to Keple s thid law, (4 3 ) / 8 times lage; also, the cicumfeence is 4 times lage and the speed / as lage: 4/(/) 8. Answe. The Eath is fathest fom the sun aound July 4 evey yea, when it is summe in the nothen hemisphee and winte in the southen hemisphee. As descibed by Keple s second law, this is when the planet is moving slowest in its obit. Thus it takes moe time fo the planet to plod aound the 80 span containing the minimum-speed point. The anking is b > a > c d > e. The foce is popotional to the poduct of the masses and invesely popotional to the squae of the sepaation distance, so we compute m m / fo each case: (a) 3/ 6 8 (c) 8/4 4.5 (d) 4.5 (e) 6/ Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

3 686 Univesal Gavitation OQ3.0 OQ3. Answe (c). The Intenational Space Station obits just above the atmosphee, only a few hunded kilometes above the gound. This distance is small compaed to the adius of the Eath, so the gavitational foce on the astonaut is only slightly less than on the gound. We might think the gavitational foce is zeo o nealy zeo, because the obiting astonauts appea to be weightless. They and the space station ae in fee fall, so the nomal foce of the space station s wall/floo/ceiling on the astonauts is zeo; they float feely aound the cabin. Answe (e). We assume that the elliptical obit is so elongated that the Sun, at one focus, is almost at one end of the majo axis. If the peiod, T, is expessed in yeas and the semimajo axis, a, in astonomical units (AU), Keple s thid law states that T a 3. Thus, fo Halley s comet, with a peiod of T 76 y, the semimajo axis of its obit is 3 a ( 76) 8 AU The length of the majo axis, and the appoximate maximum distance fom the Sun, is a 36 AU. ANSWERS TO CONCEPTUAL QUESTIONS CQ3. CQ3. CQ3.3 (a) The gavitational foce is consevative. Yes. An encounte with a stationay mass cannot pemanently speed up a spacecaft. But Jupite is moving. A spacecaft flying acoss its obit just behind the planet will gain kinetic enegy because of the change in potential enegy of the spacecaft-planet system. This is a collision because the spacecaft and planet exet foces on each othe while they ae isolated fom outside foces. It is an elastic collision because only consevative foces ae involved. (c) The planet loses kinetic enegy as the spacecaft gains it. Cavendish detemined G. Then fom g GM, one may detemine the mass of the Eath. The tem weighed is bette expessed as massed. Fo a satellite in obit, one focus of an elliptical obit, o the cente of a cicula obit, must be located at the cente of the Eath. If the satellite is ove the nothen hemisphee fo half of its obit, it must be ove the southen hemisphee fo the othe half. We could shae with Easte Island a satellite that would look staight down on Aizona each moning and vetically down on Easte Island each R 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

4 evening. Chapte CQ3.4 (a) Evey point q on the sphee that does not lie along the axis connecting the cente of the sphee and the paticle will have companion point q' fo which the components of the gavitational foce pependicula to the axis will cancel. Point q' can be found by otating the sphee though 80 about the axis. The foces will not necessaily cancel if the mass is not unifomly distibuted, unless the cente of mass of the nonunifom sphee still lies along the axis. CQ3.5 CQ3.6 ANS. FIG. CQ3.4 The angula momentum of a planet going aound a sun is conseved. (a) The speed of the planet is maximum at closest appoach. The speed is a minimum at fathest distance. These two points, peihelion and aphelion espectively, ae 80 apat, at opposite ends of the majo axis of the obit. Set the univesal desciption of the gavitational foce, F g GM X m R X, equal to the local desciption, F g ma gavitational, whee M x and R x ae the mass and adius of planet X, espectively, and m is the mass of a test paticle. Divide both sides by m. CQ3.7 (a) In one sense, no. If the object is at the vey cente of the Eath thee is no othe mass located thee fo compaison and the fomula does not apply in the same way it was being applied while the object was some distance fom the cente. In anothe sense, yes. One would have to compae, though, the distance between the object with mass m to the othe individual masses that make up the Eath. CQ3.8 The gavitational foce of the Eath on an object at its cente must be zeo, not infinite as one intepetation of Equation. would suggest. All the bits of matte that make up the Eath pull in diffeent outwad diections on the object, causing the net foce on it to be zeo. The escape speed fom the Eath is. km/s and that fom the 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

5 688 Univesal Gavitation CQ3.9 Moon is.3 km/s, smalle by a facto of 5. The enegy equied and fuel would be popotional to v, o 5 times moe fuel is equied to leave the Eath vesus leaving the Moon. Ai esistance causes a decease in the enegy of the satellite-eath system. This educes the adius of the obit, binging the satellite close to the suface of the Eath. A satellite in a smalle obit, howeve, must tavel faste. Thus, the effect of ai esistance is to speed up the satellite! SOLUTIONS TO END-OF-CHAPTER PROBLEMS Section 3. Newton s Law of Univesal Gavitation P3. This is a diect application of the equation expessing Newton s law of gavitation: F GMm N m /kg N P3. Fo two 70-kg pesons, modeled as sphees, F g Gm m N m /kg m ~ 0 7 N ( m) (.50 kg) kg ( 70 kg) ( 70 kg) P3.3 (a) At the midpoint between the two objects, the foces exeted by the 00-kg and 500-kg objects ae oppositely diected, and fom F g Gm m we have F towad the 500-kg object. ( 500 kg 00 kg) N G 50.0 kg.00 m At a point between the two objects at a distance d fom the 500-kg object, the net foce on the 50.0-kg object will be zeo when ( 00 kg) 4.00 m d G 50.0 kg ( 500 kg ) G 50.0 kg d 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

6 Chapte To solve, coss-multiply to clea of factions and take the squae oot of both sides. The esult is d.45 m fom the 500-kg object towad the smalle object. P3.4 (a) The Sun-Eath distance is m and the Eath-Moon distance is m, so the distance fom the Sun to the Moon duing a sola eclipse is m m.49 0 m The mass of the Sun, Eath, and Moon ae M S kg M E kg and M M kg We have F SM Gm m ( kg) ( kg) (.49 0 m) N m /kg N ( kg) ( kg) ( m) F EM N m /kg N ( kg) ( kg) ( m) (c) F SE N m /kg N (d) The foce exeted by the Sun on the Moon is much stonge than the foce of the Eath on the Moon. In a sense, the Moon obits the Sun moe than it obits the Eath. The Moon s path is eveywhee concave towad the Sun. Only by subtacting out the sola obital motion of the Eath-Moon system do we see the Moon obiting the cente of mass of this system. 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

7 690 Univesal Gavitation P3.5 With one metic ton 000 kg, F m g Gm m g Gm ( N m /kg ) kg m/s ( 00 m) P3.6 The foce exeted on the 4.00-kg mass by the.00-kg mass is diected upwad and given by F G m m ĵ ( 4.00 kg) (.00 kg) N m /kg ĵ N ĵ ( 3.00 m) The foce exeted on the 4.00-kg mass by the 6.00-kg mass is diected to the left: F 3 G m m 3 3 ( î ) ( 4.00 kg) 6.00 kg N m /kg î N Theefoe, the esultant foce on the 4.00-kg mass is F 4 F 4 + F î ĵ 0 N ANS. FIG. P3.6 î ( 4.00 m) *P3.7 The magnitude of the gavitational foce is given by (.00 kg) (.00 kg) F Gm m N m /kg ( m) N P3.8 Assume the masses of the sphee ae the same. Using F g Gm m, we would find that the mass of a sphee is. 0 5 kg! If the sphees have at most a adius of m, the density of sphees would be at 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

8 Chapte 3 69 least kg/m 3, which is ten times the density of the most dense element, osmium. The situation is impossible because no known element could compose the sphees. P3.9 We ae given m + m 5.00 kg, which means that m 5.00 kg m. Newton s law of univesal gavitation then becomes F G m m N m 5.00 kg m N m /kg ( 0.00 m) ( m ) ( 5.00 kg)m m N 6.00 kg N m /kg Thus, o m ( 5.00 kg)m kg 0 ( m 3.00 kg) ( m.00 kg) 0 giving m 3.00 kg, so m.00 kg. The answe m.00 kg and m 3.00 kg is physically equivalent. P3.0 Let θ epesent the angle each cable makes with the vetical, L the cable length, x the distance each ball is displaced by the gavitational foce, and d m the oiginal distance between them. Then d x is the sepaation of the balls. We have Then F y 0: T cosθ mg 0 F x 0: T sinθ Gmm 0 tanθ Gmm mg ANS. FIG. P3.0 x L x Gm g d x x( d x) Gm g L x 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

9 69 Univesal Gavitation The facto Gm g is numeically small. We expect that x is vey small compaed to both L and d, so we can teat the tem (d x) as d, and (L x ) as L. We then have N m /kg x m x m ( 00 kg) ( 9.80 m/s ) ( m) Section 3. Fee-Fall Acceleation and the Gavitational Foce P3. The distance of the meteo fom the cente of Eath is R + 3R 4R. Calculate the acceleation of gavity at this distance. g GM ( N m /kg )( kg) [4( m)] 0.64 m/s, towadeath P3. The gavitational field at the suface of the Eath o Moon is given by g GM R. The expession fo density is ρ M V M 4 3 πr3, so M 4 3 πρr3 and g G 4 3 πρr3 R 4 3 GπρR Noting that this equation applies to both the Moon and the Eath, and dividing the two equations, g M ge 4 3 Gπρ M R M 4 3 Gπρ E ρ MR M ρ E 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

10 Chapte Substituting fo the factions, 6 ρ M 4 ρe and ρ M ρe P3.3 (a) Fo the gavitational foce on an object in the neighbohood of Mianda, we have m obj g Gm objm Mianda Mianda ( kg) g Gm Mianda N m / kg Mianda m m/s We ignoe the diffeence (of about 4%) in g between the lip and the base of the cliff. Fo the vetical motion of the athlete, we have y f y i + v yi + a yt m ( m/s )t t ( m)s m / 363 s (c) (d) x f x i + v xi t + a xt 0 + ( 8.50 m/s) ( 363 s) m We ignoe the cuvatue of the suface (of about 0.7 ) ove the athlete s tajectoy. v xf v xi 8.50 m/s v yf v yi + a y t 0 ( m/s )( 363 s) 7.6 m/s Thus v f ( 8.50î 7.6ĵ ) m/s m/s at 7.6 m/s tan 8.50 m/s 7.9 below the x axis. v f 8.9 m/s at 7.9 below the hoizontal 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

11 694 Univesal Gavitation Section 3.3 P3.4 (a) g g MG + a Analysis Model: Paticle in a Field (Gavitational) g y g y g y g y + g y 0 g x g x g cosθ cosθ ( a + ) / o g g x g ( î ) MG + a 3 towad the cente of mass ANS. FIG. P3.4 At 0, the fields of the two objects ae equal in magnitude and opposite in diection, to add to zeo. (c) As 0, MG( + a ) 3/ appoaches MG(0)/a 3 0. (d) (e) When is much geate than a, the angles the field vectos make with the x axis become smalle. At vey geat distances, the field vectos ae almost paallel to the axis; theefoe, they begin to look like the field vecto fom a single object of mass M. As becomes much lage than a, the expession appoaches MG( + 0 ) 3/ MG/ 3 MG/ as equied. P3.5 The vecto gavitational field at point O is given by so g Gm l î + Gm l g Gm + l î + ĵ ĵ + Gm cos 45.0 î + sin 45.0ĵ l o g Gm + l towad the opposite cone. 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

12 Chapte P3.6 (a) F GMm N ΔF GMm GMm font back Δg ΔF m GM back ANS. FIG. P3.5 ( N m /kg ) kg ( font ) font back Δg N m /kg ( 0 3 kg) ( m m) 00( kg) ( m) ( m) ( m) ( m) Δg.6 0 N/kg ANS. FIG. P3.6 Section 3.4 Keple s Laws and the Motion of Planets P3.7 The gavitational foce on mass located at distance fom the cente of the Eath is F g mg GM E m/. Thus, the acceleation of gavity at this location is g GM E /. If g 9.00 m/s at the location of the satellite, the adius of its obit must be GM E g m ( N m /kg ) kg 9.00 m/s Fom Keple s thid law fo Eath satellites, T 4π 3 GM E S, the peiod is found to be 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

13 696 Univesal Gavitation o T π 3 GM E s T s π ( m) N m /kg ( kg) h s.50 h 90.0 min P3.8 The gavitational foce exeted by Jupite on Io causes the centipetal acceleation of Io. A foce diagam of the satellite would show one downwad aow. F on Io M Io a: GM J M Io M Iov M Io π T 4π M Io T Thus the mass of Io divides out and we have Keple s thid law with m << M, M J 4π 3 GT 4π ( m) 3 d ( N m /kg )(.77 d) s and M J kg (appoximately 36 Eath masses) P3.9 (a) The desied path is an elliptical tajectoy with the Sun at one of the foci, the depatue planet at the peihelion, and the taget planet at the aphelion. The peihelion distance D is the adius of the depatue planet s obit, while the aphelion distance T is the adius of the taget planet s obit. The semimajo axis of the desied tajectoy is then a ( D + T )/. ANS. FIG. P3.9 If Eath is the depatue planet, D m.00 AU 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

14 Chapte With Mas as the taget planet, AU T.8 0 m m.5 AU Thus, the semimajo axis of the minimum enegy tajectoy is a D + T.00 AU +.5 AU.6 AU Keple s thid law, T a 3, then gives the time fo a full tip aound this path as T a 3 (.6 AU) 3.4 y so the time fo a one-way tip fom Eath to Mas is Δt.4 y T 0.7 y This tip cannot be taken at just any time. The depatue must be timed so that the spacecaft aives at the aphelion when the taget planet is located thee. P3.0 (a) The paticle does possess angula momentum, because it is not headed staight fo the oigin. (c) Its angula momentum is constant. Thee ae no identified outside influences acting on the object. Since speed is constant, the distance taveled between t A and t B is equal to the distance taveled between t C and t D. The aea of a tiangle is equal to one-half its (base) width acoss one side times its (height) dimension pependicula to that side. So bv 0 ( t B t A ) bv 0 ( t D t C ) states that the paticle s adius vecto sweeps out equal aeas in equal times. 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

15 698 Univesal Gavitation P3. Applying Newton s second law, F ma yields F g ma c fo each sta: GMM Mv ( ) o M 4v G We can wite in tems of the peiod, T, by consideing the time and distance of one complete cycle. The distance taveled in one obit is the cicumfeence of the stas common obit, so π vt. Theefoe, so, M 4v G 4v G vt π 3 ( 4.4 d) ( s/d) M v3 T πg 0 03 m/s π N m /kg kg 63.3 sola masses P3. To find the angula displacement of planet Y, we apply Newton s second law: ANS. FIG. P3. F ma: Then, using v ω, Gm planet M sta m planetv GM sta v ω GM sta 3 ω 3 x ω x 3 y ω y solving fo the angula velocity of planet Y gives ω y ω x x y y y So, given that thee ae 360 in one evolution we convet 468 to find that planet Y has tuned though.30 evolutions. ANS. FIG. P3. 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

16 Chapte P3.3 By Keple s thid law, T ka 3 (a semimajo axis). Fo any object obiting the Sun, with T in yeas and a in AU, k.00. Theefoe, fo Comet Halley, and suppessing units, (75.6) (.00) y The fathest distance the comet gets fom the Sun is 3 ANS. FIG. P3.3 *P3.4 y ( 75.6) AU (out aound the obit of Pluto). By consevation of angula momentum fo the satellite, p v p a v a, o v p v a a 89 km km km p 459 km km 6 89 km.7 We do not need to know the peiod. P3.5 Fo an object in obit about Eath, Keple s thid law gives the elation between the obital peiod T and the aveage adius of the obit ( semimajo axis ) as 4π T GM E 3 We assume that the two given distances in the poblem statements ae the peigee and apogee, espectively. Thus, if the aveage adius is + min max km km km m The peiod (time fo a ound tip fom Eath to the Moon) would be T π π 3 GM E ( m) N m /kg ( kg) s 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

17 700 Univesal Gavitation The time fo a one-way tip fom Eath to the Moon is then Δt T s day s 4.99 d P3.6 The gavitational foce on a small pacel of mateial at the sta s equato supplies the necessay centipetal acceleation: GM s m R s so ω GM s R s 3 mv R s mr s ω ω ad/s P3.7 We find the satellite s altitude fom GM J ( R J + d) 4π R J + d ( N m /kg ) kg T ( m) 3 whee d is the altitude of the satellite above Jupite s cloud tops. Then, GM J T 4π ( R J + d) 3 ( N m /kg )( kg) ( ) which gives 3 4π d d m km above the planet P3.8 (a) In T 4 π a 3 /GM cental we take a m. M cental 4π a 3 GT 4π ( m) 3 ( N m /kg )( s) kg This is a little lage than kg. 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

18 Chapte 3 70 The Eath wobbles a bit as the Moon obits it, so both objects move nealy in cicles about thei cente of mass, staying on opposite sides of it. The adius of the Moon s obit is theefoe a bit less than the Eath-Moon distance. P3.9 The speed of a planet in a cicula obit is given by (a) F ma: GM sun m Fo Mecuy, the speed is mv v GM sun v M and fo Pluto, ( N m /kg )( kg) m m/s v p ( N m /kg )( kg) m m/s With geate speed, Mecuy will eventually move fathe fom the Sun than Pluto. With oiginal distances P and M pependicula to thei lines of motion, they will be equally fa fom the Sun at time t, whee P + v P t M + v M t P M v ( M v P )t t ( m) ( m) ( m/s) ( m/s) m m /s.4 08 s 3.93 y 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

19 70 Univesal Gavitation Section 3.5 Gavitational Potential Enegy P3.30 (a) We compute the gavitational potential enegy of the satellite- Eath system fom U GM E m ( kg) ( 00 kg) N m /kg ( ) 0 6 m J, (c) The satellite and Eath exet foces of equal magnitude on each othe, diected downwad on the satellite and upwad on Eath. The magnitude of this foce is F GM E m ( kg) ( 00 kg) ( m) N m /kg 569 N P3.3 The wok done by the Moon s gavitational field is equal to the negative of the change of potential enegy of the meteo-moon system: W int ΔU Gm m 0 W int ( N m /kg )( kg)( kg) m J *P3.3 The eney equied is equal to the change in gavitational potential enegy of the object-eath system: U G Mm and g GM E so that ΔU GMm 3 3 mg ΔU ( 000 kg) 9.80 m s 3 ( m) J 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

20 P3.33 (a) The definition of density gives ρ M S 4 3 π E kg/m kg 4π m Chapte Fo an object of mass m on its suface, mg GM S m/. Thus, g GM S E ( N m /kg ) kg m/s ( m) (c) Relative to U g 0 at infinity, the potential enegy of the object-sta system at the suface of the white dwaf is U g GM Sm E ( kg) (.00 kg) N m /kg J m P3.34 (a) Enegy consevation of the object-eath system fom elease to adius : ( K + U g ) K + U g altitude h adius 0 GM m E + h mv GM m E v GM E + h / d dt f dt d d v. The time of fall is, suppessing units, v i f i i f +h Δt GM E + h / Δt d / m m m / d 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

21 704 Univesal Gavitation We can ente this expession diectly into a mathematical calculation pogam. Altenatively, to save typing we can change vaiables to u Then / Δt u ( 0 6 ) / u / du / 0 6 du 0 6. A mathematics pogam etuns the value fo this integal, giving fo the time of fall Δt s P3.35 (a) Since the paticles ae located at the cones of an equilateal tiangle, the distances between all paticle pais is equal to m. The gavitational potential enegy of the system is then U Tot U + U 3 + U 3 3U 3 Gm m ( kg) U Tot N m /kg J m Each paticle feels a net foce of attaction towad the midpoint between the othe two. Each moves towad the cente of the tiangle with the same acceleation. They collide simultaneously at the cente of the tiangle. Section 3.6 Enegy Consideations in Planetay and Satellite Motion P3.36 We use the isolated system model fo enegy: K i + U i K f + U f 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

22 Chapte which becomes mv + GM m i E f i v i + GM E 0 mv f v f o v f v GM E and v f v GM E v f m/s / ( N m /kg )( kg) m/s m / *P3.37 To detemine the enegy tansfomed to intenal enegy, we begin by calculating the change in kinetic enegy of the satellite. To find the initial kinetic enegy, we use which gives v i + h GM E ( + h) K i mv i GM E m + h ( kg) ( 500 kg) N m kg m m J Also, J. K f mv f 500 kg m s 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

23 706 Univesal Gavitation The change in gavitational potential enegy of the satellite-eath system is ΔU GM E m R i GM m E GM R E m f R i R f ( kg) ( 500 kg) ( m ) N m kg J The enegy tansfomed into intenal enegy due to fiction is then ΔE int K i K f ΔU ( ) 0 9 J J P3.38 To obtain the obital velocity, we use F mmg R o v MG R mv R We can obtain the escape velocity fom mv esc mmg R o v esc MG R v P3.39 (a) The total enegy of the satellite-eath system at a given obital altitude is given by E tot GMm The enegy needed to incease the satellite s obit is then, suppessing units, ΔE GMm i f ( ) ΔE J 469 MJ 0 3 kg 0 3 m Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

24 Chapte Both in the oiginal obit and in the final obit, the total enegy is negative, with an absolute value equal to the positive kinetic enegy. The potential enegy is negative and twice as lage as the total enegy. As the satellite is lifted fom the lowe to the highe obit, the gavitational enegy inceases, the kinetic enegy deceases, and the total enegy inceases. The value of each becomes close to zeo. Numeically, the gavitational enegy inceases by 938 MJ, the kinetic enegy deceases by 469 MJ, and the total enegy inceases by 469 MJ. P3.40 (a) The majo axis of the obit is a 50.5 AU so a 5.5 AU. Futhe, in the textbook s diagam of an ellipse, a + c 50 AU, so c 4.75 AU. Then e c a In T K s a 3 fo objects in sola obit, the Eath gives us Then ( y) K s AU 3 K s y ( AU) 3 (c) T U GMm ( y) AU ( 5.5 AU) 3 T 7 y 3 ( kg) ( kg) 50( m) N m / kg J *P3.4 Fo he jump on Eath, which gives mv i mgy f [] v i gy f ( 9.80 m/s) ( m) 3.3 m/s 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

25 708 Univesal Gavitation We assume that she has the same takeoff speed on the asteoid. Hee mv i GM m A [] R A The equality of densities between planet and asteoid, implies ρ M E 4 3 π R 3 E M A 4 3 π R 3 A M A R A Note also at Eath s suface 3 M E [3] g GM E [4] Combining the equations [], [], [3], and [4] by substitution gives v i GM A R A GM E y f GM ER A 3 R A y f ( m) ( m) R A m *P3.4 Fo a satellite in an obit of adius aound the Eath, the total enegy of the satellite-eath system is E GM E. Thus, in changing fom a cicula obit of adius to one of adius 3, the equied wok is W ΔE GM E m f + GM m E GM E m i 4 6R E GM E m 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

26 Chapte *P3.43 (a) The wok must povide the incease in gavitational enegy: W ΔU g U gf U gi GM EM p f + GM EM p i W GM E M p + y + GM E M p GM E M p + y N m kg kg 850 MJ ( 00 kg) m m In a cicula obit, gavity supplies the centipetal foce: Then, GM E M p M v p ( + y) + y M pv GM E M p ( + y) So, additional wok kinetic enegy equied is ΔW ( N m / kg ) kg m J ( 00 kg) P3.44 (a) The escape velocity fom the sola system, stating at Eath s obit, is given by v sola escape M SunG R Sun ( N m /kg ) kg 4. km/s m 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

27 70 Univesal Gavitation Let x epesent the vaiable distance fom the Sun. Then, If v v M Sun G x km s x v M Sun G.0 0 m x v M Sun G 34.7 m/s, then ( 34.7 m/s) kg ( N m /kg ) Note that at o beyond the obit of Mas, km/h is sufficient fo escape. P3.45 F c F G gives mv GmM E which educes to v GM E and peiod π v π GM E. (a) + 00 km km + 00 km km Thus, peiod π m m ( N m /kg ) kg T s 88.3 min.47 h v GM E ( N m /kg ) kg m 7.79 km/s (c) K f + U f K i + U i + enegy input gives input mv f mv i + GM m E f GM E m i [] 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

28 Chapte 3 7 i m v i π s m/s Substituting the appopiate values into [] yields: minimum enegy input J P3.46 The gavitational foce supplies the needed centipetal acceleation. Thus, GM E m mv ( + h) + h o v GM E + h (a) T π v π ( R + h) E π GM E ( + h) ( + h) 3 GM E v GM E + h (c) Minimum enegy input is ( K i U gi ) ΔE min K f + U gf This choice has the object stating with enegy with v i K i mv i π.00 day π s and Thus, ΔE min m GM E + h GM Em + h m U gi GM E m. 4π s + GM Em o R ΔE min GM E m E + h ( + h) π R E m. ( s) P3.47 (a) Gavitational sceening does not exist. The pesence of the satellite has no effect on the foce the planet exets on the ocket. 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

29 7 Univesal Gavitation The ocket has a gavitational potential enegy with espect to Ganymede U Gm m ( N m )m ( kg) ( m)kg U ( m /s )m The ocket s gavitational potential enegy with espect to Jupite at the distance of Ganymede is U Gm m ( N m )m ( kg) ( m)kg U ( m / s )m To escape fom both equies m v esc m /s km/s v esc. 0 8 m / s m P3.48 (a) Fo the satellite F v i GM E / ma; GmM E mv i gives Consevation of momentum in the fowad diection fo the exploding satellite gives: ( mv) i mv ( ) f 5mv i 4mv + m0 v 5 4 v 5 i 4 GM E / (c) With velocity pependicula to adius, the obiting fagment is at peigee. Its apogee distance and speed ae elated to and v by 4mv 4m f v f and 4mv GM E 4m 4mv f GM E 4m f 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

30 Chapte 3 73 Substituting v f v f we have v GM E v f GM E f Futhe, substituting v 5 GM E 6 5 GM E GM E f f gives 5 GM E GM E 3 f f Cleaing factions we have 7 f 5 3 f, o 7 f 3 f giving f +3 ± ( 5) 50 4 The latte oot descibes the stating point. o 4 4. The oute end of the obit has f 5 7 : f 5 7 *P3.49 The height attained is not small compaed to the adius of the Eath, so U mgy does not apply; U GM M does. Fom launch to apogee at height h, consevation of enegy gives K i + U i + ΔE mech K f + U f M pv i GM EM p GM EM p + h The mass of the pojectile cancels out, giving v i GM E GM E + h GM + h E v i GM E 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

31 74 Univesal Gavitation GM h E v i GM E ( N m /kg ) kg ( m/s) N m /kg m m ( kg) m Additional Poblems P3.50 (a) When the ocket engine shuts off at an altitude of 50 km, we may conside the ocket to be beyond Eath s atmosphee. Then, its mechanical enegy will emain constant fom that instant until it comes to est momentaily at the maximum altitude. That is, KE f + PE f KE i + PE i, o 0 GM E m max mv i GM m E i o max v i GM E + i With l + 50 km m m m and v i 6.00 km/s m/s, this gives max ( m/s) N m /kg ( kg) m m o max m. The maximum distance fom Eath s suface is then h max max m m m If the ocket wee fied fom a launch site on the equato, it would have a significant eastwad component of velocity because of the Eath s otation about its axis. Hence, compaed to being fied fom the South Pole, the ocket s initial speed would be geate, and the ocket would tavel fathe fom Eath. 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

32 P3.5 Fo a 6.00-km diamete cylinde, m and to simulate g 9.80 m/s, g v ω Chapte 3 75 ω g ad/s The equied otation ate of the cylinde is ev 0 s. (Fo a desciption of poposed cities in space, see Gead K. O Neill in Physics Today, Sept and the Wikipedia aticle on Rotating Wheel Space Station at *P3.5 To appoximate the height of the sulfu, set mv mg Ioh, with h m and g Io GM v g Io h.79 m/s.79 m s. This gives ( m) 500 m s ove 000 mi h We can obtain a moe pecise answe fom consevation of enegy: mv GMm GMm v N m kg kg v 49 m/s m m *P3.53 (a) The adius of the satellite s obit is + h m m m Then, modifying Keple s thid law fo obital motion about the Eath athe than the Sun, we have T 4π GM E s 4π ( m) N m kg ( kg) 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

33 76 Univesal Gavitation (c) o T s h s.43 h The constant tangential speed of the satellite is v π T π m s m/s 6.60 km/s The satellite s only acceleation is centipetal acceleation, so a a c v m/s m 4.74 m/s towad the Eath P3.54 If one uses the esult v GM and the elation v (π /T), one finds the adius of the obit to be smalle than the adius of the Eath, so the spacecaft would need to be in obit undegound. P3.55 The acceleation of an object at the cente of the Eath due to the gavitational foce of the Moon is given by a G M M d. At the point A neaest the Moon, a + G M M d At the point B fathest fom the Moon, a G M M d + Fom the above, we have Δg M g ( a a + ) GM M g g ( d ) d + Evaluating this expession, we find acoss the planet Δg M g ( kg) N m kg 9.80 m/s ( m m) m m ANS. FIG. P Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

34 Chapte 3 77 P3.56 (a) The only foce acting on the astonaut is the nomal foce exeted on him by the floo of the cabin. The nomal foce supplies the centipetal foce: F c mv and n mg This gives mv mg v g v (9.80m/s )(0.0m) v 7.00 m/s Since v ω, we have ω v 7.00 m/s 0.0 m ad/s ANS. FIG. P3.56 (c) Because his feet stay in place on the floo, his head will be moving at the same tangential speed as his feet. Howeve, his feet and his head ae tavelling in cicles of diffeent adii. If he stands up without holding on to anything with his hands, the only foce on his body is adial. Because the wall of the cabin nea the tavele's head moves in a smalle cicle, it moves at a slowe tangential speed than that of the tavele's head so his head moves towad the wall if he is not caeful, thee could be a collision. This is an example of the Coiolis foce investigated in Section 6.3. Holding onto to a igid suppot with his hands will povide a tangential foce to the tavele to slow the uppe pat of his body down. 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

35 78 Univesal Gavitation P3.57 (a) Ignoing ai esistance, the enegy consevation fo the object- Eath system fom fiing to apex is given by, ( K + U g ) K + U g i mv i GmM E f 0 GmM E + h whee mv esc GmM E. Then v i v esc v esc + h v esc v esc v i v esc + h v R + h E i v esc h h v esc v esc v i v esc R v E i v esc v i v esc + v i v esc v i h ( m) 8.76 km/s. km/s ( 8.76 km/s) m The fall of the meteoite is the time-evesal of the upwad flight of the pojectile, so it is descibed by the same enegy equation: v i v esc R E + h h v esc + h.5 07 m. 0 3 m/s m / s m m v i m/s 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

36 Chapte 3 79 P3.58 (a) Ignoing ai esistance, the enegy consevation fo the object- Eath system fom fiing to apex is given by, ( K + U g ) K + U g i mv i GmM E f 0 GmM E + h whee mv esc GmM E. Then v i v esc v esc + h v esc v esc v i v esc + h v R + h E i v esc h h v esc v esc v i v esc R v E i v esc v i v esc + v i v esc v i The fall of the meteoite is the time-evesal of the upwad flight of the pojectile, so it is descibed by the same enegy equation. Fom (a) above, eplacing v i with v f, we have v f v esc v f v esc R v E esc + h R E + h v f v esc h + h (c) With v i << v esc, h R v E i R v E i v esc GM E ageement with 0 v i + g h 0. But g GM E., so h v i g, in 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

37 70 Univesal Gavitation P3.59 (a) Let R epesent the adius of the asteoid. Then its volume is 4 3 π R3 and its mass is ρ 4 3 π R3. Fo you obital motion, F ma gives Gm m R m v R Gρ4π R3 3R v R solving fo R, 3v R Gρ4π / m/s N m /kg 4π 00 kg/m m ρ 4 3 π R3 ( 00 kg/m 3 ) 4 3 π ( m) kg (c) v π R T T π R v π ( m) s 3.5 h 8.5 m/s (d) Fo an illustative model, we take you mass as 90.0 kg and assume the asteoid is oiginally at est. Angula momentum is conseved fo the asteoid-you system: L i L f 0 m vr Iω 0 m vr 5 m π R T asteoid m v 4π m R 5 T asteoid T asteoid 4π m R 5m v 4π kg kg / ( m) ( 8.50 m/s) s 6.5 billion yeas Thus you unning does not poduce significant otation of the asteoid if it is oiginally stationay and does not significantly affect any otation it does have. This poblem is ealistic. Many asteoids, such as Ida and Eos, ae oughly 30 km in diamete. They ae typically iegula in 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

38 Chapte 3 7 shape and not spheical. Satellites such as Phobos (of Mas), Adastea (of Jupite), Calypso (of Satun), and Ophelia (of Uanus) would allow a visito the same expeience of easy obital motion. So would many Kuipe Belt objects. P3.60 (a) The two appopiate isolated system models ae consevation of momentum and consevation of enegy applied to the system consisting of the two sphees. Applying consevation of momentum to the system, we find m vi + m v i m v f + m v f M v f + M v f v f v f (c) Applying consevation of enegy to the system, we find K i + U i + ΔE K f + U f 0 Gm m i + 0 m v f GM M R Mv f Mv f v f GM R GM 6R v f GM 3R v f + m v f + ( M)v f Gm m f (d) Combining the esults fo pats and (c), v f 6v f GM 3R GM 3R v f GM ( M ) 4R v 3 G M R v 3 G M R P3.6 (a) At infinite sepaation U 0 and at est K 0. Since the system is isolated, the enegy and momentum of the two-planet system is conseved. We have 0 m v + m v Gm m d [] 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

39 7 Univesal Gavitation and 0 m v m v [] because the initial momentum of the system is zeo. Combine equations [] and []: v m G d m + m G and v m d( m + m ) The elative velocity is then v v ( v ) G( m + m ) d The instant befoe the collision, the distance between the planets is d +. Substitute given numeical values into the equation found fo v and v in pat (a) to find Theefoe, v m/s and v m/s K m v J and K m v J P3.6 (a) The fee-fall acceleation poduced by the Eath is g GM E GM E (diected downwad) Its ate of change is dg d GM E ( ) 3 GM E 3 The minus sign indicates that g deceases with inceasing height. At the Eath s suface, dg d GM E 3 Fo small diffeences, Thus, Δg Δ Δg h GM E 3 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

40 Chapte 3 73 Δg GM Eh 3 ( kg) 6.00 m ( m) 3 (c) Δg N m /kg m/s P3.63 (a) Each bit of mass dm in the ing is at the same distance fom the object at A. The sepaate contibutions Gmdm to the system enegy add up to GmM ing. When the object is at A, this is (c) ( N m /kg )( 000 kg)( kg) m 7.04 N + ( m) When the object is at the cente of the ing, the potential enegy of the system is ( 000 kg) ( kg) N m / kg m Total enegy of the object-ing system is conseved: J ( K + U g ) K + U g A B J ( 000 kg )v B J *P3.64 v B J 000 kg The oiginal obit adius is / 3. m/s a m m m The oiginal enegy is E i GMm a ( kg) ( 0 4 kg) ( m) N m kg.90 0 J 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

41 74 Univesal Gavitation We assume that the peigee distance in the new obit is m. Then the majo axis is a m m m and the final enegy is E f GMm a ( kg) ( 0 4 kg) N m kg.48 0 J m The enegy input equied fom the engine is E f E i.48 0 J (.90 0 J).4 0 J P3.65 Fom the walk, π m. Thus, the adius of the planet is Fom the dop: so, which gives m π m Δy gt g( 9. s).40 m m/s MG g.40 m 9. s M kg P3.66 The distance between the obiting stas is d cos30 3 since cos30 3. The net inwad foce on one obiting sta is Gmm cos30 + GMm d + Gmm cos30 mv d Gmcos30 + GM 4π 3 T G m 3 + M 4π 3 T ANS. FIG. P Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

42 Chapte 3 75 solving fo the peiod gives 4π 3 T G M + m/ 3 3 T π G M + m/ 3 P3.67 (a) We find the peiod fom T π v / π m m/s 0 8 y s We estimate the mass of the Milky Way fom M 4π a 3 GT kg o about 0 4 kg 4π ( m) 3 ( N m /kg ) s, Note that this is the mass of the galaxy contained within the Sun s obit of the galactic cente. Recent studies show that the tue mass of the galaxy, including an extended halo of dak matte, is at least an ode of magnitude lage than ou estimate. (c) A sola mass is about 0 30 kg: 0 4 / The numbe of stas is on the ode of 0. P3.68 Enegy consevation fo the two-sphee system fom elease to contact: (a) Gmm R Gmm + mv + mv Gm R v v Gm R The injected momentum is the final momentum of each sphee, / mv m / Gm R / Gm 3 R / 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

43 76 Univesal Gavitation If they now collide elastically each sphee eveses its velocity to eceive impulse mv Gm 3 mv mv R P3.69 (a) The net toque exeted on the Eath is zeo. Theefoe, the angula momentum of the Eath is conseved. We use this to find the speed at aphelion: and m a v a m p v p / v a v p p a m/s m/s K p mv p ( kg) ( m/s) J U p GmM p ( kg) ( kg) N m /kg J.47 0 m (c) Using the same fom as in pat, K a J and U a J (d) Compae to find that K p + U p J and K a + U a J. They agee, with a small ounding eo. P3.70 Fo both cicula obits, F ma: GM E m mv v GM E ANS. FIG. P Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

44 Chapte 3 77 (a) (c) The oiginal speed is v i ( N m / kg ) kg m m m/s The final speed is v i ( N m / kg ) kg m m/s The enegy of the satellite-eath system is Oiginally, K + U g mv GM Em E i N m /kg m GM E GM E GM Em ( kg) ( 00 kg) ( m) J (d) Finally, ( kg) ( 00 kg) ( m) E f N m /kg J (e) (f) Thus the object speeds up as it spials down to the planet. The loss of gavitational enegy is so lage that the total enegy deceases by E i E f J J J The only foces on the object ae the backwad foce of ai esistance R, compaatively vey small in magnitude, and the foce of gavity. Because the spial path of the satellite is not pependicula to the gavitational foce, one component of the gavitational foce pulls fowad on the satellite to do positive wok and make its speed incease. 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

45 78 Univesal Gavitation P3.7 The centipetal acceleation of the blob comes fom gavitational acceleation: v M G c 4π T GM c T 4π 3 Solving fo the adius gives N m /kg obit 9 km ( s) kg P3.7 Fom Keple s thid law, minimum peiod means minimum obit size. The teetop satellite in Poblem 38 has minimum peiod. The adius of the satellite s cicula obit is essentially equal to the adius R of the planet. 4π /3 F ma: GMm R ( GρV R 4π R ) RT Gρ 4 3 πr3 4π R 3 T mv R m R πr T The adius divides out: T Gρ 3π T 3π Gρ P3.73 Let m epesent the mass of the meteooid and v i its speed when fa away. No toque acts on the meteooid, so its angula momentum is conseved as it moves between the distant point and the point whee it gazes the Eath, moving pependicula to the adius: ANS. FIG. P3.73 L i L f : m i v i m f v f m v f m 3 v i v f 3v i 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

46 Chapte 3 79 Now, the enegy of the meteooid-eath system is also conseved: ( K + U g ) K + U g i : f v i ( 9v i ) GM E mv i + 0 mv f GM m E GM E 4v i : v i GM E 4 P3.74 If we choose the coodinate of the cente of mass at the oigin, then ( 0 M m ) M + m and M m (Note: this is equivalent to saying that the net toque must be zeo and the two expeience no angula acceleation.) Fo each mass F ma so m ω MGm and M d ω MGm d ANS. FIG. P3.74 Combining these two equations and using d + gives ( + )ω ( M + m)g with d ω ω ω and we find T π ω T 4π d 3 G M + m 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

47 730 Univesal Gavitation P3.75 The gavitational foces the paticles exet on each othe ae in the x diection. They do not affect the velocity of the cente of mass. Enegy is conseved fo the pai of paticles in a efeence fame coasting along with thei cente of mass, and momentum consevation means that the identical paticles move towad each othe with equal speeds in this fame: U gi + K i + K i U gf + K f + K f Gm m + 0 Gm m + i f m v + m v ( N m /kg )( 000 kg) 0.0 m ( N m /kg )( 000 kg).00 m J 000 kg / v m/s + ( 000 kg)v Then thei vecto velocities ae ( ) î m/s and ( )î m/s fo the tailing paticle and the leading paticle, espectively. P3.76 (a) The gavitational foce exeted on m by the Eath (mass M E ) acceleates m accoding to g GM E. The equal-magnitude foce exeted on the Eath by m poduces acceleation of the Eath given by g Gm. The acceleation of elative appoach is then g + g Gm + GM E ( kg + m) ( m) N m /kg +.77 m/s m kg and (c) Hee m 5 kg and m 000 kg ae both negligible compaed to the mass of the Eath, so the acceleation of elative appoach is just.77 m/s. 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.

48 Chapte 3 73 (d) Substituting m kg into the expession fo (g + g ) above gives g + g 3.70 m/s (e) Any object with mass small compaed to the Eath stats to fall with acceleation.77 m/s. As m inceases to become compaable to the mass of the Eath, the acceleation inceases, and can become abitaily lage. It appoaches a diect popotionality to m. P3.77 Fo the Eath, F ma: GM s m Then GM s T 4π 3 mv m π T Also, the angula momentum L mv m π is a constant fo the T LT Eath. We eliminate between the equations: πm LT GM s T 4π πm 3/ L gives GM s T / 4π πm Now the ates of change with time t ae descibed by o which gives GM s T dt / dt + G dm s dt T / 0 dt dt dm s dt ΔT Δt dm s dt ΔT T M s ΔT Δt T M s y s s y kg/s y kg 3/ 04 Cengage Leaning. All Rights Reseved. May not be scanned, copied o duplicated, o posted to a publicly accessible website, in whole o in pat.