VISUAL PHYSICS ONLIN MODUL 5 ADVANCD MCHANICS GRAVITATIONAL FILD: MOTION OF PLANTS AND SATLLITS SATLLITS: Obital motion of object of mass m about a massive object of mass M (m << M assume M stationay w..t m) with an obital adius, obital speed v ob and peiod T G M m Gavitational foce (magnitude) FG Centipetal foce (magnitude) mv F F F C C G Obital speed v ob Angula momentum G M T L mvob constant Gavitational potential enegy P G M m P U P Kinetic enegy K mv 1 ob G M m Total enegy G M m K P Consevation of enegy 0 P K 1
How the planets move aound the Sun: Keple s Laws 1 st Law: A planet descibes an ellipse with the Sun at one focus. The distance fom the Sun to the planet vaies as the planet obits the Sun. So, we take to be aveage distance between the Sun and the planet. epesents the length of the semi-majo axis of the ellipse and is usually epesented by the symbol a. nd Law: ach planet moves so that an imaginay line dawn fom the Sun to the planet sweeps out equal aeas in equal peiods of time. This law esults fom the Law of Consevation of Angula Momentum L mv constant 3 d Law: T constant 3 3 4 3 3 T 1 1 T T G M S T scape velocity v esc GM
A satellite is an object that obits a much moe massive object. Natual satellites include the planets obiting the Sun, the moons of Jupite, and the Moon about the ath. An atificial satellite is an object put into obit fom the ath s suface using a spacecaft such a ocket o a space shuttle. Satellites ae used fo many applications and include militay and civilian ath obsevation satellites, communications satellites, navigation satellites, weathe satellites, and eseach satellites. Space stations and human spacecaft in obit ae also satellites. 3
Satellites ae placed in one of seveal diffeent types of obit depending on the natue of thei mission. Two common obit types ae a Low ath Obit (LO) and a Geostationay Obit (GO). LOs occu at a adius of between 00 and 000 km above the ath s suface with peiods vaying fom 60 to 90 minutes. The space shuttle uses this type of obit (00-50 km). LOs have the smallest field of view and fequent coveage of specific o vaied locations on the ath s suface. Obits less than 400 km ae difficult to maintain due to atmospheic dag and subsequent obital decay. They ae used mainly fo militay applications, ath obsevation, weathe monitoing and shuttle missions. xcept fo the luna flights of the Apollo pogam, all human spaceflights have taken place in LO. The altitude ecod fo a human spaceflight in LO was Gemini 11 with an apogee of 1,374.1 km. All manned space stations and most atificial satellites, have been in LO. Obital decay is the eduction in the height of an object's obit ove time due to the dag of the atmosphee on the object. All satellites in low ath obits ae subject to some degee of atmospheic dag that will eventually decay thei obit and limit 4
thei lifetimes. ven at 1000 km, as thin as the atmosphee is, it is still sufficiently dense to slow the satellite down gadually. A GO is a cicula obit in the ath's equatoial plane, any point on which evolves about the ath in the same diection and with the same peiod as the ath's otation. Geostationay obits ae useful because they cause a satellite to appea stationay with espect to a fixed point on the otating ath. As a esult, an antenna can point in a fixed diection and maintain a link with the satellite. The satellite obits in the diection of the ath's otation, at an altitude of appoximately 35,786 km above gound. This altitude is significant because it poduces an obital peiod equal to the ath's peiod of otation, known as the sideeal day. These obits allow fo the tacking of stationay point on ath and have the lagest field of view. Applications include communications, mass-media and weathe monitoing. Web investigation atificial satellite obits http://en.wikipedia.og/wiki/satellite http://en.wikipedia.og/wiki/low_ath_obit http://en.wikipedia.og/wiki/geostationay_obit Review: Cicula Motion 5
ORBITAL MOTION To place an object into a stable ath obit at a paticula adius, the launch must give it both an initial vetical and hoizontal component of velocity, elative to the ath s suface. The ocket will eventually tun so that it is tavelling hoizontal to the ath s suface. At this adius, the foce of gavity povides the acceleation needed to keep the object moving in a cicle, but a paticula obital velocity is also equied to keep the object in a stable obit (figue ). To calculate that velocity, known as the obital velocity v ob, we equate expessions fo centipetal foce F C and gavitational foce F G as follows: F F F C G C mvob centipetal foce G M m Gavitational foce F G Obital velocity of a satellite aound obiting the ath GM (3) vob Note that the velocity of a satellite as it obits aound the ath only depends on: Mass of the ath M Radius of the obit 6
It is clea fom this fomula that altitude is the only vaiable that detemines the obital velocity equied fo a specific obit. Futhe, the geate the adius of that obit, the lowe that obital velocity v ob. The obital velocity of a satellite aound othe planets is simply (4) v ob GMplanet obital velocity about any planet 7
The obital motion of the Moon about the ath We can calculate the obital velocity v ob of the Moon obiting the ath using the equation 3 fo the obital velocity o knowing the peiod T of otation of the Moon aound the ath is 7.317 days. 8
The Moon s obital velocity of was calculated to be 1.0 km.s -1. The Moon s obit is not quite cicula and the speed is only appoximately constant. The obital speed of the of the Moon vaies fom 0.970 to 1.0 km.s -1. So, ou simple models gave numeical esults which compae vey favouably with the measued values fo the obital speed of the Moon. 9
How do the planets move? Keple s Laws of Motion One of the most impotant questions histoically in Physics was how the planets move. Many histoians conside the field of Physics to date fom the wok of Newton, and the motion of the planets was the main poblem Newton set out to solve. In the pocess of doing this, he not only intoduced his laws of motion and discoveed the law of gavity, he also developed diffeential and integal calculus. Today, the same laws that goven the motion of planets, ae used by scientists to put satellites into obit aound the ath and to send spacecaft though the sola system. How the planets move is detemined by gavitational foces. The foces of gavity ae the only foces applied to the planets. The gavitational foces between the planets ae vey small compaed with the foce due to the Sun since the mass of the planets ae much less than the Sun's mass. ach planet moves almost the way the gavitational foce of the Sun alone dictates, as though the othe planets did not exist. 10
The motion of a planet is govened by Newton s Law of Univesal Gavitation G M S m (5) FG whee G is the Univesal Gavitational Constant, M S is the mass of the Sun, m is the mass of the planet and is the distance fom the Sun to the planet. G = 6.6710-11 N.m.kg M S =.010 30 kg Histoically, the laws of planetay motion wee discoveed by the outstanding Geman astonome Johannes Keple (1571-1630) based on almost 0 yeas of pocessing astonomical data, befoe Newton and without the aid of the law of gavitation. 11
Keple's Laws of Planetay Motion 1. The path of each planet aound the Sun is an ellipse with the Sun at one focus.. ach planet moves so that all imaginay lines dawn fom the Sun to the planet sweeps out equal aeas in equal peiods of time. 3. The atio of the squaes of the peiods of evolution of planets is equal to the atio of the cubes of thei obital adii (mean distance fom the Sun o length of semi-majo axis, a) (6) 3 T 1 1 4 o T a T G M 3 S 1
Keple s Fist Law A planet descibes an ellipse with the Sun at one focus. But what kind of an ellipse do planets descibe? It tuns out they ae vey close to cicles. The path of the planet neaest the Sun, Mecuy, diffes most fom a cicle, but even in this case, the longest diamete is only % geate than the shotest one. Bodies othe than the planets, fo example, comets move aound the Sun in geatly flattened ellipses. Since the Sun is located at one of the foci and not the cente, the distance fom the planet to the Sun changes moe noticeably. The point neaest the Sun is called the peihelion and the fathest point fom the Sun is the aphelion. Half the distance fom the peihelion to the aphelion is known as the semi-majo adius, a. The othe adius of the ellipse is the semi-mino adius, b. 13
The equation of an ellipse is (7) x a y ellipse b 1 Fig. 3. The path of a planet aound the Sun is an ellipse. 14
Keple's Second Law ach planet moves so that an imaginay line dawn fom the Sun to the planet sweeps out equal aeas in equal peiods of time. This law esults fom the Law of Consevation of Angula Momentum Angula momentum L mv constant whee m is the mass of the planet, is the distance fom the Sun and v is the obital (tangential) velocity of the planet. Angula momentum is conseved because the foce acting on the obital body is always diected towads the cente of the coodinate system (0,0), i.e., the Sun. Thus, this foce cannot exet a toque (twist) on the obiting body. Since thee is zeo toque acting, the obital angula momentum must emain constant. Since a planet moves in an elliptical obit, the distance is continually changing. As it appoaches neae the Sun the planet must speed up and as it gets futhe away fom the Sun it must slow down such that the poduct v constant 15
The aea of each tiangle (fo a small time inteval dt) can be expessed as A v dt v dt 1 1 1 1 1 1 1 A v dt v dt 1 1 Since angula momentum must be conseved L m v1 1 m v Theefoe, in equal time intevals, equal aeas ae swept out A 1 A 1 Fig. 4. A planet moves so that an imaginay line dawn fom the Sun to the planet sweeps out equal aeas in equal peiods of time. 16
Keple's Thid Law Fo a planet obiting the Sun with a adius, the centipetal foce esults fom the gavitational attaction between the planet and the Sun Centipetal foce = Gavitational foce mv G M m F F F F GMS v S C G C G Fo otational motion, we know that So, v f v 4 T T 4 4 T T GMS 3 3 3 3 T GM T 1 1 T T S constant Keple s 3 d Law 17
Figue 5 shows a compute simulation fo the motion of a planet aound the Sun. The dots epesent the positions of the planet at equal time intevals. Nea the aphelion, the dots ae closely spaced indicating a small speed while at the peihelion the dots ae widely spaced indicating a lage speed fo the planet. Fig. 5. Compute simulation of the motion of a planet aound the Sun. Pedict Obseve xecise Motion of planets aound a sta 18
NRGY CONSIDRATIONS Conside a satellite of mass m obiting a massive object of mass M (m << M assume M stationay w..t m) with an aveage obital adius, obital speed v and peiod T. ob The net foce acting on the satellite is the gavitational foce F G G M m The gavitational foce is esponsible fo the obit, thus the gavitational foce coesponds to the centipetal foce mv F F F C C G Hence the aveage obital speed is v ob G M T The gavitational potential of the satellite system is G M m U P P P and its kinetic enegy is K mv 1 ob GM The total enegy of the system is GM K P constant Consevation of enegy 0 P K 19
Fo a satellite in a cicula obit, the adius of obit and the obital velocity (tangential) ae both constants. In an elliptical obit, both the adius and obital velocity change duing the obit of the satellite. Howeve, the angula momentum L of the satellite emains constant L mvob constant So, if the adius inceases, the obital velocity deceases o if the adius deceases, then the obital velocity inceases. By caefully examining figue 5, you will obseve that when the satellite is at the aphelion position, the satellite is at the geatest distance fom the massive object and its speed is a minimum. When the satellite is at the peihelion position, the position closest to the massive object and smallest adius, the speed is a maximum. 0
xample A satellite of mass 500 kg is in a low obit tajectoy at an altitude of 1000 km above the ath s suface. The satellite must be moved to a highe tajectoy with an altitude of 000 m. Calculate fo both obits: the acceleation due to gavity (gavitational field stength), obital speeds, peiods, kinetic enegies, gavitational potential enegies and total enegies. How can this be achieved? What enegy must be used to shift the satellite into the highe obit? M 4 5.97 10 kg R 6 6.371 10 m 11 G 6.67310 N.m.kg 1
Solution Poblem: type / visualize / how to appoach? / scientific annotated diagam / what do you know? Acceleation due to gavity (gavitational field stength) g GM Obit #1 g = 7.33 m.s - Obit # g = 5.69 m.s -
Acceleation due to gavity deceases with inceasing altitude. The obital velocities of the satellite v ob Obit #1 Obit # GM v ob = 7.35 m.s -1 v ob = 6.90 m.s -1 Obital velocity deceases with inceasing altitude. The peiod of the satellite obits v ob GM T T v Obit #1 T = 6.30x10 3 s = 1.75 h Obit # T = 7.6x10 3 s =.1 h Peiod inceases with inceasing altitude. ob Kinetic enegies of the satellite K 1 mv ob G M m Obit #1 Obit # K = 6.76x10 10 J K = 5.95x10 10 J K deceases with inceasing altitude. 3
Gavitational potential enegies of the satellite P G M m Obit #1 Obit # P = - 1.35x10 11 J P = - 1.19x10 11 J GP inceases with inceasing altitude. Total enegies of the satellite G M m K P Obit #1 = - 6.76x10 10 J Obit # = - 5.95x10 10 J total enegy inceases with inceasing altitude. The total enegy being negative means that the satellite is bound to the ath. 4
The enegy (wok) equied to move the satellite fom obit #1 to obit # is the diffeence in the total enegies between the two obits wok W = 1 = ( - 5.95x10 10 + 6.76x10 10 ) J W = 0.81x10 10 J This enegy fo the wok equied to shift the obit must come fom the fuel that is bunt by the satellite s ockets. VISUAL PHYSICS ONLIN http://www.physics.usyd.edu.au/teach_es/hsp/sp/sphome.htm If you have any feedback, comments, suggestions o coections please email Ian Coope ian.coope@sydney.edu.au Ian Coope School of Physics Univesity of Sydney 5