Pactice Undestanding Concepts 1. Detemine the gavitational potential enegy of the Eath Moon system, given that the aveage distance between thei centes is 3.84 10 5 km, and the mass of the Moon is 0.0123 times the mass of Eath. 2. (a) Calculate the change in gavitational potential enegy fo a 1.0-kg mass lifted 1.0 10 2 km above the suface of Eath. (b) What pecentage eo would have been made in using the equation mgy and taking the value of g at Eath s suface? (c) What does this tell you about the need fo the moe exact teatment in most nomal Eath-bound poblems? 3. With what initial speed must an object be pojected vetically upwad fom the suface of Eath to ise to a maximum height equal to Eath s adius? (Neglect ai esistance.) Apply enegy consevation. Answes 1. 7.64 10 28 J 2. (a) 1.0 10 6 J (b) 2% 3. 7.91 10 3 m/s NEL Gavitation and Celestial Mechanics 287
Answes 4. (a) 1.8 10 32 J (b) peihelion; 1.8 10 32 J 5. (a) 1.56 10 10 J; 1.04 10 10 J (b) 5.2 10 9 J (c) 5.2 10 9 J LEARNING TIP Apo and Pei The pefix apo means away fom and geo epesents Eath, so apogee efes to the point in a satellite s elliptical obit fathest fom Eath. Futhemoe, since helios epesents the Sun, aphelion efes to the point in a planet s elliptical obit fathest fom the Sun. The pefix pei means aound, so peihelion efes to the point in a planet s obit closest to the Sun. What does peigee mean? 4. The distance fom the Sun to Eath vaies fom 1.47 10 11 m at peihelion (closest appoach) to 1.52 10 11 m at aphelion (fathest distance away). (a) What is the maximum change in the gavitational potential enegy of Eath duing one obit of the Sun? (b) At what point in its obit is Eath moving the fastest? What is its maximum change in kinetic enegy duing one obit? (Think about enegy consevation.) 5. A satellite of mass 5.00 10 2 kg is in a cicula obit of adius 2 aound Eath. Then it is moved to a cicula obit of adius 3. (a) Detemine the satellite s gavitational potential enegy in each obit. (b) Detemine the change in gavitational potential enegy fom the fist obit to the second obit. (c) Detemine the wok done in moving the satellite fom the fist obit to the second obit. Apply enegy consevation. Escape fom a Gavitational Field We have seen that any two masses have a gavitational potential enegy of GM m at a sepaation distance. The negative value of this potential enegy is chaacteistic of a potential well, a name deived fom the shape of the gaph of the gavitational potential enegy as a function of sepaation distance (Figue 4). Fo example, a ocket at est on Eath s suface has the value of, given by point A on the gaph in Figue 4. Since the kinetic enegy of the ocket is zeo, its total enegy would also be epesented by point A, and the ocket would not leave the gound. Howeve, suppose the ocket is launched at a speed such that its kinetic enegy is epesented by the distance AB on the gaph. Now its total enegy is epesented by point B, and the ocket begins to ise. As its altitude inceases, inceases suface of Eath altitude above Eath's suface 0 2 3 4 2 3 4 5 3 2 B C at an altitude of 0.5 Figue 4 This gaph of the gavitational potential enegy as a function of the altitude above Eath s suface illustates Eath s potential well. A 288 Chapte 6 NEL
Case 1: = 0, object just escapes + Case 2: > 0, object escapes with a speed > 0 as Case 3: < 0, object is bound to Eath maximum sepaation Enegy 0 Enegy 0 Enegy 0
Section 6.3 As an example, assume that a cetain black hole esults fom the collapse of a sta that has a mass 28 times the Sun s mass. Since the minimum escape speed is v e c, we have m v 2 e GM m 2 v 2 e 2G M 2 GM v 2 e 2G c 2 M 2(6.67 10 11 Nm 2 /kg 2 )(28 1.99 10 30 kg) (3.00 10 8 m/s) 2 8.26 10 4 m 82.6 km Since light cannot escape fom a black hole, the only way a black hole can be detected is indiectly. Mateial that is close enough to the black hole gets sucked in, and as it does so, the mateial emits X ays that can be detected and analyzed. The celestial mechanics analyzed in this chapte is not a complete pictue. You will lean moe about high-speed and high-enegy paticles when you study Einstein s special theoy of elativity in Chapte 11. DID YOU KNOW? Fist Black Hole Discovey In 1972, Pofesso Tom Bolton, while woking at the Univesity of Toonto s David Dunlap Obsevatoy in Richmond Hill, Ontaio, was investigating a point in space, Cygnus X-1, because it was a souce of X ays. It tuned out to be one of the most significant discoveies in astonomy: a blackhole. This was the fist evidence to suppot the existence of blackholes, which wee peviously hypothetical objecs. Pactice Undestanding Concepts 6. Does the escape speed of a space pobe depend on its mass? Why o why not? 7. Jupite s mass is 318 times that of Eath, and its adius is 10.9 times that of Eath. Detemine the atio of the escape speed fom Jupite to the escape speed fom Eath. 8. The Moon is a satellite of mass 7.35 10 22 kg, with an aveage distance of 3.84 10 8 m fom the cente of Eath. (a) What is the gavitational potential enegy of the Moon Eath system? (b) What is the Moon s kinetic enegy and speed in cicula obit? (c) What is the Moon s binding enegy to Eath? 9. What is the total enegy needed to place a 2.0 10 3 -kg satellite into ciculaath obit at an altitude of 5.0 10 2 km? 10. How much additional enegy would have to be supplied to the satellite in question 9 once it was in obit, to allow it to escape fom Eath s gavitational field? 11. Conside a geosynchonous satellite with an obital peiod of 24 h. (a) What is the satellite s speed in obit? (b) What speed must the satellite each duing launch to attain the geosynchonous obit? (Assume all fuel is buned in a shot peiod. Neglect ai esistance.) 12. Detemine the Schwaztschild adius, in kilometes, of a black hole of mass 4.00 times the Sun s mass. Answes 7. 5.40:1 8. (a) 7.63 10 28 J (b) 3.82 10 28 J; 1.02 10 3 m/s (c) 3.82 10 28 J 9. 6.7 10 10 J 10. 5.80 10 10 J 11. (a) 3.1 10 3 m/s (b) 1.1 10 4 m/s 12. 11.8 km 14. (a) 4.06 10 9 J (b) 4.06 10 9 J (c) 6.37 10 2 J Applying Inquiy Skills 13. Sketch the geneal shape of the potential wells of both Eath and the Moon on a single gaph. Label the axes and use colou coding to distinguish the line foath fom the line fo the Moon. 14. (a) Calculate the binding enegy of a 65.0-kg peson on Eath s suface. (b) How much kinetic enegy would this peson equie to just escape fom the gavitational field of Eath? (c) How much wokis equied to aise this peson by 1.00 m at Eath s suface? (d) Explain why one of NASA s objectives in designing launches into space is to minimize the mass of the payload (including the astonauts). NEL Gavitation and Celestial Mechanics 293
SUMMARY Gavitational Potential Enegy in Geneal The gavitational potential enegy of a system of two (spheical) masses is diectly popotional to the poduct of thei masses, and invesely popotional to the distance between thei centes. A gavitational potential enegy of zeo is assigned to an isolated system of two masses that ae so fa apat (i.e., thei sepaation is appoaching infinity) that the foce of gavity between them has dopped to zeo. The change in gavitational potential enegy vey close to Eath s suface is a special case of gavitational potential enegy in geneal. Escape speed is the minimum speed needed to poject a mass m fom the suface of mass M to just escape the gavitational foce of M. Escape enegy is the minimum kinetic enegy needed to poject a mass m fom the suface of mass M to just escape the gavitational foce of M. Binding enegy is the amount of additional kinetic enegy needed by a mass m to just escape fom a mass M. Section 6.3 Questions Undestanding Concepts 1. How does the escape enegy of a 1500-kg ocket compae to that of a 500-kg ocket, both initially at est on Eath? 2. Do you agee o disagee with the statement, No satellite can obit Eath in less than about 80 min? Give easons. (Hint: The geate the altitude of an Eath satellite, the longe it takes to complete one obit.) 3. A space shuttle ejects a 1.2 10 3 -kg booste tank so that the tankis momentaily at est, elative to Eath, at an altitude of 2.0 10 3 km. Neglect atmospheic effects. (a) How much wokis done on the booste tankby the foce of gavity in etuning it to Eath s suface? (b) Detemine the impact speed of the booste tank. 4. A space vehicle, launched as a luna pobe, aives above most of Eath s atmosphee. At this point, its kinetic enegy is 5.0 10 9 J and its gavitational potential enegy is 6.4 10 9 J. What is its binding enegy? 5. An atificial Eath satellite, of mass 2.00 10 3 kg, has an elliptical obit with an aveage altitude of 4.00 10 2 km. (a) What is its aveage gavitational potential enegy while in obit? (b) What is its aveage kinetic enegy while in obit? (c) What is its total enegy while in obit? (d) If its peigee (closest position) is 2.80 10 2 km, what is its speed at peigee? 6. A 5.00 10 2 -kg satellite is in cicula obit 2.00 10 2 km above Eath s suface. Calculate (a) the gavitational potential enegy of the satellite (b) the kinetic enegy of the satellite (c) the binding enegy of the satellite (d) the pecentage incease in launching enegy equied fo the satellite to escape fom Eath 7. (a) Calculate the escape speed fom the suface of the Sun: mass 1.99 10 30 kg, adius 6.96 10 8 m. (b) What speed would an object leaving Eath need to escape fom ou sola system? 8. The mass of the Moon is 7.35 10 22 kg, and its adius is 1.74 10 6 m. With what speed must an object be pojected fom the its suface to each an altitude equal to its adius? 9. A blackhole has a Schwatzschild adius of 15.4 km. What is the mass of the blackhole in tems of the Sun s mass? Applying Inquiy Skills 10. Mas is a planet that could be visited by humans in the futue. (a) Geneate a gaph of Mas potential well (using data fom Appendix C) fo a spacecaft of mass 2.0 10 3 kg that is launched fom Mas. Daw the gaph up to 5 M. (b) On you gaph, daw (i) the line fo the kinetic enegy needed fo the caft to just escape fom Mas (ii) the line of the total enegy fom Mas suface to 5 M 11. (a) What is the theoetical Schwatzschild adius of a blackhole whose mass is equal to the mass of Eath. Expess you answe in millimetes. (b) What does you answe imply about the density of a blackhole? 12. How would the amount of fuel equied to send a spacecaft fom Eath to the Moon compae with the amount needed to send the same spacecaft fom the Moon back to Eath? Explain. (Numeical values ae not equied.) 294 Chapte 6 NEL