VISUAL PHYSICS ONLINE MODULE 5 ADVANCED MECHANICS GRAVITATIONAL FIELD: MOTION OF PLANETS AND SATELLITES SATELLITES: 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 Gavitational foce (magnitude) FG E m 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 EP G M m EP U P Kinetic enegy E K mv 1 ob G M m Total enegy G M m E EK EP Escape speed v esc GM 1
Example 1 Gavitational Foce attaction between people Chis has a mass of 75 kg and Pat has a mass of 50 kg ae standing nea each othe a distance 0.5 m apat. Ae they attacted to each othe and how stong is the attaction? Solution 1 THINK: how to appoach the poblem / type of poblem / visualize the physical situation / annotated scientific diagam / what do I know! The two people ae attacted by the gavitational foce. We can calculate the magnitude of the foce using Newton s Law of Univesal Gavitation
The answe is that the gavitational attaction is vey small 1 N answe.. What othe attaction thee maybe, physics cannot 3
Example Gavitational Foce on the Moon The Moon, the Sun and the Eath ae aligned so that both the Sun and the Eath ae at ight angles to each othe. Find the net foce acting on the Moon. G 6.67 10 N.m.kg R M M ME M E 11-8 11 3.84 10 m RMS 1.50 10 kg 7.35 10 kg 4 30 5.98 10 kg MS 1.99 10 m Solution THINK: how to appoach the poblem (ISEE) / type of poblem / visualize the physical situation / annotated scientific diagam / what do I know! Need to calculate the foces between the Moon and Eath and between Moon and Sun and then add the two foces vectoially. 4
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Example 3 Acceleation due to gavity (gavitational field stength) Compae the acceleation due to gavity (gavitational field stength) at sea level and at the top of Mt Eveest. adius of the eath height of Mt Eveest R E 6 6.371 10 m h 8848 m G 6.67 10 N.m.kg M E 11-4 5.98 10 kg Solution 3 THINK: how to appoach the poblem (ISEE) / type of poblem / visualize the physical situation / annotated scientific diagam / what do I know! Newton s Law of gavity between the Eath and an object of mass m is F G G M R E m 6
The gavitational foce can be expessed in tems of the gavitational field stength g FG m g Hence, the gavitational field stength (acceleation due to gavity) is g GM E R At sea level, - gsl 9.83 m.s R R E 6 6.371 10 m At top of Mt Eveest, R RE h 6 6.389 10 m - gme 9.77 m.s Ratio of the two acceleations due to gavity g g ME SL 9.77 0.995 9.8 A vey small diffeence, howeve, we have ignoed the exta mass that sits unde the top of Mt Eveest. 7
PUTTING SATELLITES INTO ORBIT The geate the launch velocity of an object, the geate the vetical height eached and the geate the hoizontal ange. If the launch velocity is geate enough then the acceleation due to gavity is no longe constant and we must use its dependence on the distance of the object fom the cente of the Eath. Also, if the launch velocity is lage enough, the object can be placed into obit aound the Eath o escape fom the influence of the Eath s gavitation field. Rockets and satellites ae essential devices fo ou moden wold based upon the intenet, GPS and mobile phones. Communications aound the globe between mobile phones, computes, etc use adio waves and micowaves fo the tansfe of infomation. Satellites ae used fo: adio and television tansmissions; weathe; militay applications; GPS; phones and moe. Just about all pats of the globe can tansmit o eceive electomagnetic wave communications via obiting satellites and Eath bound tansmittes and eceives (figue 1). 8
Fig. 1. Radio telescopes and satellites make it possible fo infomation to flow feely aound the globe. Thee ae 4 satellites that make up the GPS space segment. They obit the Eath about 0 000 km above us. These satellites ae tavelling at speeds of appoximately 11 000 km.h -1 and make two complete obits in less than 4 hous. GPS satellites ae poweed by sola enegy. They have backup batteies onboad to keep them unning in the event of no sola powe. Small ocket boostes on each satellite keep them flying in the coect path. GPS satellites tansmit two low powe adio signals, designated L1 and L. Civilian GPS use the L1 fequency of 1575.4 MHz in the UHF band. The signals tavel by line of sight, meaning they will pass though clouds, glass and plastic but will not go though most solid objects such as buildings and mountains. 9
LAUNCHING A ROCKET A ocket is popelled though space by a continuous explosion poduced by buning fuel and expelling the esulting hot gases out one end, i.e, chemical eactions, takes place inside the ocket and the gaseous poducts of combustion ae popelled out of the ocket with temendous a foce acting on the gas. The hot gases have a momentum in one diection, and since the total momentum of the ocket-fuel system is zeo, the ocket itself has an equal momentum in the opposite diection. Thus, the ocket moves off in the opposite diection to the expelled gases, in accodance with the Law of Consevation of Momentum. This means that the backwad momentum of the gases is exactly equal in magnitude to the fowad momentum of the ocket. This is what gives the ocket its fowad velocity. This is a consequence of Newton's Thid Law which says that fo evey eaction thee is an equal and opposite eaction; the ocket exets a foce on the gases and the gases exet a foce on the ocket popelling it fowad. 10
Note that thee is no need fo any ai to push against'' fo the ocket to wok. Newton's Thid Law assues us that ejection of an object fom the system MUST popel the system in the opposite diection. This populsive foce is efeed to as the thust of the ocket. Fig.. Rocket populsion. Newton s thid law: FRG FGR Foces act fo time inteval t impulse: FRG t FGR t Impulse = Change in momentum: pr pg ( mv) ( mv) ocket gases Momentum is conseved: p p 0 R G You should note that because at any time instant the mass of the gases is much less than the mass of the ocket, we can see that the velocity of the gases will, theefoe, be much highe in 11
magnitude than the velocity of the ocket. Although the mass of the gas emitted pe second is compaatively small, it has a vey lage momentum because its high velocity. An equal momentum is impated to the ocket in the opposite diection. This means that the ocket, despite its lage mass, builds up a high velocity. As the launch poceeds, fuel is bunt, gases expelled and the mass of the ocket deceases. This poduces an incease in acceleation, since acceleation is popotional to the applied foce (the thust) and invesely popotional to the mass. The initial acceleation is small, aound 1 m.s - but continues to build as the mass of the ocket deceases. Rocket s acceleation not constant: initially 90% mass of ocket is its fuel fuel used up mass of ocket deceases thust emains appoximately constant acceleation inceases as mass educes F a m An additional positive effect on the ocket is the deceases in aeo dynamic dag with inceasing altitude. The combination of these two factos accounts fo the incease in acceleation duing the launch of the ocket and helps the spacecaft each the high velocity that is needed fo space flight. 1
ESCAPE VELOCITY Fo a spacecaft to go on a mission to anothe planet, it is fist necessay fo the spacecaft to achieve escape velocity fom the Eath and to go into its own elliptical obit aound the Sun. The Eath obits the Sun at about 30 km.s -1. Again, it makes good sense to use this speed to help a spacecaft achieve escape velocity fo tips to othe planets. So, if the spacecaft is to go on a mission to planets beyond the Eath s obit, it is launched in the diection of Eath s obital motion aound the Sun and achieves a velocity aound the Sun geate than the Eath s 30 km.s -1. Thus, the spacecaft s obit is lage than that of the Eath and is aanged to intesect with the obit of the planet to which it is heading at a time when the planet will be at that point. Similaly, if the taget is Mecuy o Venus, the spacecaft is launched in the opposite diection to the Eath s motion though space. Then, the spacecaft achieves an escape velocity less than 30 km.s -1, whee it entes an elliptical obit aound the Sun that is smalle than the Eath s and can thus intecept eithe planet. 13
Fig. 3. Eath s obital motion aound the Sun can be helpful in launching ockets to planets in ou Sola System. Newton showed that if you climb to the top of a mountain and thow a ball, it will tavel a cetain distance and then hit the gound (A). If you could thow the ball twice as fast it would tavel even futhe (B) and if you thew it thee times as fast it would tavel futhe still. If you kept inceasing the speed by fiing it fom a supe poweful canon, and thee was no ai fiction, a point would come when the ball would be tavelling pat-way aound the wold. If the ball could be fied at just the ight speed, it would tavel completely aound the Eath and hit you in the back of the head (C). In this case, it would fall at exactly the same ate as the Eath cuves. Faste still, the ball would go into elliptical obit (D). If it was fied much faste than that, the canon ball would tavel off into space and neve etun (F) as shown in figue 4. 14
Fig. 4 Cannon ball launched fom top of a mountain with inceasing velocity. Escape velocity v esc is defined as the smallest speed that we need to give an object in ode to allow it to completely escape fom the gavitational pull of the planet on which it is sitting. To calculate it we need only to ealize that as an object moves away fom the cente of a planet, its kinetic enegy gets conveted into gavitational potential enegy. Thus, we need only figue out how much gavitational potential enegy an object gains as it moves fom the suface of the planet off to infinity. 15
Fo a ocket of mass m fied fom the suface of the Eath, the total enegy of the ocket-eath system is assumed to be constant. At the Eath s suface, the total enegy when the ocket is fied is E E E mv GM 1 E K P esc RE When the ocket has escaped the Eath s gavitational pull, we assume the ocket is an infinite distance fom the Eath and v 0, E 0 and E 0 K P E( ) E ( ) E ( ) 0 0 0 Total enegy is conseved E( suface) E( ) GM 1 E mv 0 esc RE Theefoe, the escape velocity is GM (1) vesc R E E K P The escape velocity fo a ocket fied fom a planet o moon (mass M, adius R) is GM () vesc R 16
Note that the mass m of the object has cancelled, so that the escape velocity of any object is independent of its mass. This means that if you want to thow a gain of ice o an elephant into oute space, you need to give them both the same initial velocity which fo the Eath woks out to be about 10 4 m.s -1. We can think of the gavitational field as a gavitational well suounding the Eath just like a depessed dimple in a ubbe membane. Objects of mass ae tapped in the well because of the attactive foce pulling them in towads the cente of the Eath. To escape fom the well wok must be done on the object. 17
SATELLITES and WEIGHTLESSNESS Atificial satellites cicling the Eath ae common. A satellite is placed into obit by acceleating it to sufficiently high tangential speed with the use of ockets. If the speed is too low, it will etun to the Eath. If the speed is too high, the spacecaft will not be confined by the Eath s gavity and will escape, neve to etun. Satellites ae usually placed into cicula (o nealy cicula) obits because this equies the least take-off speed (figue 5). Fig. 5. Atificial satellites launched with diffeent speeds. 18
What keeps the satellite up? Fig. 6. A moving satellite is always falling towads the cente of the Eath. A satellite is always falling towads the Eath, i.e., acceleating towads the Eath by the pull of the gavitational foce. The high tangential speed keeps the satellite fom hitting the Eath as the cuvatue of the satellite s obit as it falls matches the cuvatue of the Eath. 19
The weightlessness expeience by astonauts in a satellite obit is because they ae falling feely since the satellite is falling feely towads the Eath. The acceleation of the satellite and astonauts matches the acceleation due to gavity at that point since the only foce acting is the gavitational foce of the Eath g GM E Figue 7 shows examples of people in fee-fall and expeience the sensation of weightlessness. 0
Fig. 7. Weightlessness on Eath. 1
Weightlessness does not mean you weight is zeo. You weight is still given by the gavitational foce G M E m G M weight FG m g g E Fig. 8. The weight of the peson is both cases in equal to mg. When the peson is standing on the scales (left), the nomal foce is equal to the weight of the peson. When the peson and scales ae in fee fall (ight), the nomal foce is zeo and the peson is said to be weightless enough though thei weight is mg.
Example 4 A 60 kg peson stands on a bathoom scale while iding an elevato. What is the eading on the scale in the following cases: (1) The elevato is at est. () The elevato is going up at.0 m.s -1. (3) The elevato is going up at 4.0 m.s -1. (4) The elevato is going down at.0 m.s -1. (5) The elevato stats fom est and goes up eaching a speed of.0 m.s -1 in 1.8 s. (6) The elevato is moving up and acceleates fom.0 m.s -1 to 4.0 m.s -1 in 1.8 s. (7) The elevato stats fom est and goes down eaching a speed of.0 m.s -1 in 1.8 s. (8) The elevato is moving down and acceleates fom.0 m.s -1 to 4.0 m.s -1 in 1.8 s. (9) The elevato is moving up and slows fom 4.0 m.s -1 to.0 m.s -1 in 1.8 s. (10) The elevato is moving down and slows fom 4.0 m.s -1 to.0 m.s -1 in 1.8 s. 3
Solution 4 THINK: how to appoach the poblem / type of poblem / visualize the physical situation / annotated scientific diagam / what do I know! Visualize the situation wite down all the given and unknown infomation. Daw a diagam of the physical situation showing the inetial fame of efeence. Type of poblem foces and Newton s laws. Daw a fee-body diagam showing all the foces acting on the peson. Use Newton s nd law to give the elationship between the foces acting on the peson and the acceleation of the peson. Detemine the acceleation of the peson in each case. Solve fo the unknown quantities. + up is the positive diection m F N scale eading: foce of scale on peson a a > 0 acceleation up a < 0 acceleation down m = 60 kg g = 9.81 m.s - F G =mg weight of peson 4
The peson exets a foce on the bathoom scales and the bathoom scales exets a foce on the peson. This is an action / eaction pai. But, we ae only inteested in the foces acting on the peson which ae the weight and the nomal foce due to the scale on the peson. The scale eading F N is found fom Newton s nd law: y N G N F m( g a) N F F F F m g m a acceleation due to gavity g = 9.8 m.s - (scala quantity in this example) acceleation of peson a > 0 if diection up and a < 0 if acceleation down The weight of the peson is F G = mg = (60)(9.81) N =588.6 N We can assume when the velocity changes the acceleation a is constant and equal to the aveage acceleation a a avg v t In cases (1), (), (3) and (4) thee is no change in the velocity, hence v 0 a 0 FN FG mg 5
Theefoe, the scale eading is F N = 588.6 N o 60 kg. Fo cases (5), (6) and (10) t 1.8 s case (5) v case (6) v -1-1 0 m.s m.s -1-1 4 m.s m.s case (10) v The acceleation is -1-1 4 m.s m.s v.0 a m.s 1.11m.s t 1.8 The scale eading is N - - 60 9.81 1.11 N 655 N o 67 kg F m g a This scale eading is often called the peson s appaent weight. The peson feels the floo pushing up hade than when the elevato is stationay o moving with a constant velocity. Fo cases (7), (8) and (9) t 1.8 s case (7) v -1-1 0 m.s m.s case (8) v case (9) v The acceleation is -1-1 4 m.s m.s -1-1 4 m.s m.s v.0 a m.s 1.11m.s t 1.8 The scale eading is N - - 60 9.81 1.11 N 5 N o 53 kg F m g a The peson feels thei weight has deceased. 6
In the exteme case when the cable beaks and the elevato and the peson ae in fee-fall and the downwad acceleation is a = -g. In this case the nomal foce of the scales on the peson is F N = m(g - g) = 0 N. The peson seems to be weightless. This is the same as an astonaut obiting the Eath in a spacecaft whee they expeience appaent weightlessness. The astonaut and spacecaft ae in fee-fall and thee ae zeo nomal foces acting on the peson. The astonaut still has weight because of the gavitational foce acting on them. The acceleation does not depend upon the diection of the velocity. What is impotant is the change in the velocity. A good way to undestand this concept is to daw the appopiate motion maps 7
(1) () (3) (4) (5) + a = 0 a = 0 a = 0 a > 0 a = 0 (6) (7) (8) (9) (10) a > 0 a > 0 a < 0 a < 0 a < 0 ed aows: velocity vectos 8
Geosynchonous (geostationay) satellite A geosynchonous satellite is one that stays above the same point on the Eath, which is possible only if it is above a point on the equato. Such satellites ae used fo weathe foecasting, TV tansmissions, and communication elays. The only foce acting on the satellite is the gavitational foce. So, fo the obiting satellite, the gavitational foce must be equal to the centipetal foce assuming that the satellite moves in a cicle F F G C G M E msat msat v F C F G m v G M m sat E sat This equation has two unknowns v and. Howeve, we know that ou geosynchonous satellite evolves aound the Eath with the same peiod T that the Eath otates on its axis, namely once in 4 hous. 9
Thus, the speed of the satellite must be v T whee T 1 day 4 h (4)(3600) s 86400 s We can now solve fo and v 3 G M T E 4 7 4.3 10 m 4300 km The adius of the Eath is R E 6 6.380 10 m So, a geosynchonous satellite must obit at a distance of about 36000 km above the suface of the Eath. The obital speed v sat of the satellite is v sat GME 3070 m.s T -1 Example 5 Fo the adius fo the obit of a geosynchonous satellite given that the distance between the Eath and the Moon is 384 400 km. Solution 5 THINK: how to appoach the poblem / type of poblem / visualize the physical situation / annotated scientific diagam / what do I know! 30
How nice that the moon s appoximate peiod tuns out to be a pefect cube! A geosynchonous satellite must be 1/9 the distance to the Moon (4 000 km fom the cente of the Eath o 36 000 km above the Eath s suface which equals about 6 Eath adii high). 31
VISUAL PHYSICS ONLINE 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 3