Gavitational Potential Enegy in Geneal 6.3 To exploe such concepts as how much enegy a space pobe needs to escape fom Eath s gavity, we must expand on the topic of gavitational potential enegy, which we examined in Section 4.3 fo objects at Eath s suface. To calculate the change in gavitational potential enegy fo a mass that undegoes a vetical displacement nea Eath s suface, we developed the following equation: M F G 1 F G m m mgy whee is the change in gavitational potential enegy, m is the mass, g is the magnitude of the gavitational field constant, and y is the vetical displacement. This equation is accuate povided that the magnitude of the gavitational field stength g emains easonably constant duing y. This means that we can be faily accuate fo vetical displacements of a few hunded kilometes but inaccuate fo vetical displacements beyond that. The moe geneal poblem, howeve, is to develop an expession fo the gavitational potential enegy of a system of any two masses a finite distance apat. Recall that the law of univesal gavitation is given by F G GMm whee F G is the magnitude of the foce of gavitational attaction between any two objects, M is the mass of one object, m is the mass of the second object, and is the distance between the centes of the two spheical objects (Figue 1). To incease the sepaation of the two masses fom 1 to equies wok to be done to ovecome thei foce of attaction, just as in stetching a sping. As a esult of this wok being done, the gavitational potential enegy of the system inceases. Notice that the wok done to change the sepaation fom 1 to is equal to the change in gavitational potential enegy fom 1 to. This applies to an isolated system in which the law of consevation of enegy holds. Howeve, ecall that the wok done by a vaying foce is equal to the aea unde the foce-displacement gaph fo the inteval. The foce-sepaation gaph, with the shaded aea epesenting the wok done to incease the sepaation fom 1 to, is shown in Figue. You may not ecognize this aea as a well-known geometic shape, and you have no simple equation to detemine its aea. The mathematics fo an invese squae elationship involves calculus and is beyond the scope of this book. Howeve, instead of using the aithmetic aveage of F 1 and F, we can use the geometic aveage F, 1 F to poduce an accuate esult. Thus, to detemine the aea unde the foce-sepaation gaph fom 1 to : aea F 1 F ( 1 ) G Mm G Mm 1 ( 1 ) G Mm ( 1 ) 1 aea GM 1 m GM m This aea epesents the wok done in changing the sepaation of the two masses fom 1 to and is an expession fo the esulting change in gavitational potential enegy. Figue 1 The two masses, M and m, ae moved fom a sepaation 1 to a sepaation by a foce that just ovecomes the gavitational attaction between the masses at evey point along the path. The masses ae at est at both positions. F F 1 F wok done to incease the sepaation fom 1 to 1 Figue In this foce-sepaation gaph, the aea unde the cuve fo the inteval 1 to is equal to the wok done in inceasing the sepaation of the two masses. DID YOU KNOW? Newton s New Mathematics Newton saw the need to accuately calculate aeas, such as the aea shown in Figue. To do so, he developed a whole new banch of mathematics called calculus. At appoximately the same time, independently of Newton, Gottfied Wilhelm Leibniz (1646 1716), a Geman natual philosophe, also developed calculus. NEL Gavitation and Celestial Mechanics 85
LEARNING TIP Enegy of a System The equation fo the gavitational potential enegy between two masses gives the potential enegy of the system, such as an Eath-satellite system. Despite this fact, we often say that the potential enegy is associated with only the smalle object, in this case the satellite. Thus, E E 1 GM 1 m GM m GM m GM 1 m whee D is the change in gavitational potential enegy in joules. The tem involving 1 is changed to negative, which places the tem involving fist. Thus, the fist tem in the expession depends only on and the second tem only on 1.As, 0. Since m is now outside the gavitational field of M, the expession simplifies to 0 1 GM 1 m Thus, GM m o Eg GM m 1 GMm GMm 1 objects in contact 1 0 as Figue 3 A gaph of gavitational potential enegy as a function of fo two masses M and m Note that is the distance between the centes of two objects and that the expession is not valid at points inside eithe object. As with the law of univesal gavitation, objects must be spheical o fa enough apat that they can be consideed as small paticles. The equation fo always poduces a negative value. As inceases that is, as the masses get fathe apat inceases by becoming less negative. Also, as, 0. The zeo value of gavitational potential enegy between two masses occus when they ae infinitely fa apat; this is a easonable assumption since the point at which is the only point when the masses will have no gavitational attaction foce between them. A gaph of as a function of fo two masses is shown in Figue 3. We can show that the equation fo the change in gavitational potential enegy at Eath s suface is just a special case of the geneal situation. Nea Eath s suface 1 E and E Dy so that 1 E (because y E close to the suface of Eath) and Dy 1 Thus, GM m GM 1 m G Mm ( 1 ) 1 GM my E Howeve, fom the law of univesal gavitation, F G G Mm mg E Theefoe, mgy fo a mass nea the suface of Eath. SAMPLE poblem 1 What is the change in gavitational potential enegy of a 64.5-kg astonaut, lifted fom Eath s suface into a cicula obit of altitude 4.40 10 km? Solution G 6.67 10 11 Nm /kg M E 5.98 10 4 kg m 64.5 kg E 6.38 10 6 m 86 Chapte 6 NEL
Section 6.3 E 4.40 10 km 6.38 10 6 m 4.40 10 5 m 6.8 10 6 m On Eath s suface, 1 E (6.67 10 11 Nm /kg )(5.98 10 4 kg)(64.5 kg) 6.38 10 6 m 1 4.03 10 9 J In obit, (6.67 10 11 Nm /kg )(5.98 10 4 kg)(64.5 kg) 6.8 10 6 m 3.77 10 9 J 1 (3.77 10 9 J) (4.03 10 9 J).6 10 8 J The change in gavitational potential enegy is.6 10 8 J. Notice that even though the values fo the astonaut s gavitational potential enegy ae negative at both positions, the change in, when the astonaut s distance fom Eath inceases, is positive, indicating an incease in gavitational potential enegy. Note also that, even fo an altitude of 4.40 10 km, the appoximation assuming a constant value of g is quite good. mgy (64.5 kg)(9.80 N/kg)(4.40 10 5 m).8 10 8 J 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 0.013 times the mass of Eath.. (a) Calculate the change in gavitational potential enegy fo a 1.0-kg mass lifted 1.0 10 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 8 J. (a) 1.0 10 6 J (b) % 3. 7.91 10 3 m/s NEL Gavitation and Celestial Mechanics 87
Answes 4. (a) 1.8 10 3 J (b) peihelion; 1.8 10 3 J 5. (a) 1.56 10 10 J; 1.04 10 10 J (b) 5. 10 9 J (c) 5. 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 1.5 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.) Making Connections 5. A satellite of mass 5.00 10 kg is in a cicula obit of adius E aound Eath. Then it is moved to a cicula obit of adius 3 E. (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 E K of the ocket is zeo, its total enegy E T 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 E T E K 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 E E 3 E 4 E E E 3 E 4 E 5 E 3 E E B C E K at an altitude of 0.5 E Figue 4 This gaph of the gavitational potential enegy as a function of the altitude above Eath s suface illustates Eath s potential well. E A 88 Chapte 6 NEL
Section 6.3 along the cuve AC and E T emains constant along the line BC. The kinetic enegy deceases and, at any point, is given by the length of the vetical line fom the cuve to the hoizontal line BC. When the ocket eaches an altitude coesponding to point C, E K has deceased to zeo, and the ocket can go no highe. Instead, it falls back down, with E K and govened by the same constaints as on the upwad tip. It is an inteesting execise to detemine what minimum speed this ocket would have to be given at Eath s suface to escape the potential well of Eath. To escape, the ocket s initial kinetic enegy must just equal the depth of the potential well at Eath s suface, theeby making its total enegy zeo. This also means the ocket must each an infinite distance, whee 0, befoe coming to est. At this infinite distance, the gavitational foce is zeo and hence the ocket emains at est thee. E T E K 0 E K 1 mv GM Em E v G M E E (6.67 10 11 Nm /kg )(5.98 10 4 kg) 6.38 10 6 m v 1.1 10 4 m/s, o 11. km/s This speed is called the escape speed, which is the minimum speed needed to poject a mass m fom the suface of mass M to just escape the gavitational foce of M (with a final speed of zeo). The escape enegy is the kinetic enegy needed to give an object its escape speed. A ocket launched fom Eath with a speed geate than the escape speed moves away fom Eath, losing E K and gaining as it does so. Since its E K is geate than the depth of its gavitational potential well at any point, its total enegy will always be positive. This ocket will each an infinite sepaation distance fom Eath with some E K left. Fo a launch speed less than the escape speed, the ocket will come to est at some finite distance and then fall back to Eath. In pactice, a space vehicle does not achieve its highest speed upon launch. Its speed inceases afte launch as its ocket engines continue to be fied. If a satellite is launched fom the cago hold of an obiting space shuttle, it is aleady tavelling at the speed of the shuttle (about 8 10 3 m/s), so the small ocket engines on the satellite need to supply a elatively small amount of enegy to popel the satellite into its highe obit. A ocket whose total enegy is negative will not be able to escape fom Eath s potential well and is bound to Eath. The binding enegy of any mass is the amount of additional kinetic enegy it needs to just escape (with a final speed of zeo) to an infinite distance away. Fo a ocket of mass m at est on Eath s suface (of mass M E ), the total enegy is equal to the gavitational potential enegy: escape speed 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 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 the amount of additional kinetic enegy needed by a mass m to just escape fom a mass M E T E K 0 GM E Em E T GM E E m Thus, the binding enegy must be GM E E m to give the ocket enough enegy to escape. NEL Gavitation and Celestial Mechanics 89
An example of a bound object is a satellite moving in a cicula obit of adius in the potential well of Eath. The net foce (of magnitude F) necessay to sustain the cicula obit is povided by the foce of gavitational attaction between the satellite and Eath. Using the magnitudes of the foces, fo a satellite of mass m and obital speed v: F F G mv GM E m mv GM E m The total enegy of the satellite is constant and is given by: E T E K E T 1 mv GM E m Substituting mv GM Em into the equation: E T 1 1 GM E m E T 1 This is a vey significant esult. The total enegy of a satellite in cicula obit is negative and is equal to one-half the value of the gavitational potential enegy at the sepaation coesponding to the adius of its obit. Figue 5 shows the potential well fo Eath and the position of this obiting satellite in the well. This satellite is bound to Eath and its binding enegy is 1 GM E m. E T E E 3 E position of obiting satellite 4 E Figue 5 The gavitational potential enegy of a satellite in Eath s potential well E K In summay, the total enegy of any object in Eath s gavitational field is composed of kinetic enegy and gavitational potential enegy. The gaphs shown in Figue 6 illustate the thee geneal cases possible fo such an object. 90 Chapte 6 NEL
Section 6.3 Case 1: E T = 0, object just escapes + E E K Case : E T > 0, object escapes with a speed > 0 as E K Case 3: E T < 0, object is bound to Eath E K maximum sepaation Enegy 0 E E T Enegy 0 E E T Enegy 0 E E T E E E SAMPLE poblem A 5.00 10 -kg communications satellite is to be placed into a cicula geosynchonous obit aound Eath. (A geosynchonous satellite emains in the same elative position above Eath because it has a peiod of 4.0 h, the same as that of Eath s otation on its axis.) (a) What is the adius of the satellite s obit? (b) What is the gavitational potential enegy of the satellite when it is attached to its launch ocket, at est on Eath s suface? (c) What is the total enegy of the satellite when it is in geosynchonous obit? (d) How much wok must the launch ocket do on the satellite to place it into obit? (e) Once in obit, how much additional enegy would the satellite equie to escape fom Eath s potential well? Figue 6 Compaing the enegies of the same object given diffeent amounts of kinetic enegy at Eath s suface Solution (a) G 6.67 10 11 Nm /kg T 4.0 h 8.64 10 4 s M E 5.98 10 4 kg As fo any satellite: 4p m T F F G 3 GM E T 4p 3 (6.67 10 11 Nm /kg )(5.98 10 4 kg)(8.64 10 4 s) 4. 10 7 m The adius of the satellite s obit is 4. 10 7 m. This adius epesents an altitude of 3.58 10 4 km above Eath s suface. (b) E 6.38 10 6 m m 5.00 10 kg 4p NEL Gavitation and Celestial Mechanics 91
At the suface of Eath, GM E E m (6.67 10 11 Nm /kg )(5.98 10 4 kg)(5.00 10 kg) 6.38 10 6 m 3.13 10 10 J The gavitational potential enegy of the satellite when it is attached to its launch ocket at est on Eath s suface is 3.13 10 10 J. (c) 4. 10 7 m The total enegy of a satellite in cicula obit, bound to Eath, is given by: E T E K black hole a vey dense body in space with a gavitational field so stong that nothing can escape fom it event hoizon the suface of a blackhole singulaity the dense cente of a blackhole Schwatzschild adius the distance fom the cente of the singulaity to the event hoizon (d) 1 mv GM E m 1 GM E m (based on the theoy elated to Figue 5) 1 (6.67 10 11 Nm /kg )(5.98 10 4 kg)(5.00 10 kg) 4. 10 7 m E T.36 10 9 J The total enegy of the satellite when in geosynchonous obit is.36 10 9 J. W E E T (in obit) E T (on Eath).36 10 9 J (3.13 10 10 J) W.89 10 10 J The launch ocket must do.89 10 10 J of wokon the satellite to place it into obit. LAB EXERCISE 6.3.1 Gaphical Analysis of Enegies (p. 95) A detailed analysis of the enegies involved in launching a space vehicle and its payload fom anothe body must be caied out befoe the mission is undetaken. How can gaphing be used to analyze the enegy data elated to a spacecaft launch? (e) To escape Eath s potential well, the total enegy of the satellite must be zeo o geate. In obit, E T.36 10 9 J. Theefoe, to escape Eath s potential well, the satellite must acquie at least.36 10 9 J of additional enegy. An impotant goal of futue space missions will be to mine mineals on distant bodies, such as moons and asteoids, in the sola system. Once the mineals ae mined, some will be used fo manufactuing on the moon o asteoid, while othes will be bought back to Eath o to the Intenational Space Station fo eseach and manufactuing. You can lean about the enegies associated with this application by pefoming Lab Execise 6.3.1 in the Lab Activities section at the end of this chapte. Among the most inteesting objects in the univese ae extemely dense bodies that fom at the end of a massive sta s life. A black hole is a small, vey dense body with a gavitational field so stong that nothing can escape fom it. Even light cannot be adiated away fom its suface, which explains the object s name. The suface of a black hole is called its event hoizon because no event can be obseved fom outside this suface. Inside the event hoizon, at the vey coe of the black hole, is an unbelievably dense cente called a singulaity. The distance fom the cente of the singulaity to the event hoizon is the Schwatzschild adius, named afte Geman astonome Kal Schwatzschild (1873 1916), who was the fist peson to solve Einstein s equations of geneal elativity. Since the speed of light c is 3.00 10 8 m/s, we can use that value in the equation fo escape speed to detemine the Schwatzschild adius of a black hole of known mass. 9 Chapte 6 NEL
Section 6.3 As an example, assume that a cetain black hole esults fom the collapse of a sta that has a mass 8 times the Sun s mass. Since the minimum escape speed is v e c, we have m v e GM m v e G M GM v e G c M (6.67 10 11 Nm /kg )(8 1.99 10 30 kg) (3.00 10 8 m/s) 8.6 10 4 m 8.6 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 197, 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 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.0 10 3 -kg satellite into cicula Eath obit at an altitude of 5.0 10 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 4 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.) 1. 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 8 J (b) 3.8 10 8 J; 1.0 10 3 m/s (c) 3.8 10 8 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 1. 11.8 km 14. (a) 4.06 10 9 J (b) 4.06 10 9 J (c) 6.37 10 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 fo Eath fom the line fo the Moon. Making Connections 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 93
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?. 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. 10 3 -kg booste tank so that the tankis momentaily at est, elative to Eath, at an altitude of.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.00 10 3 kg, has an elliptical obit with an aveage altitude of 4.00 10 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.80 10 km, what is its speed at peigee? 6. A 5.00 10 -kg satellite is in cicula obit.00 10 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 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.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? Making Connections 1. 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.) 94 Chapte 6 NEL