Physics HSC Course Stage 6. Space. Part 1: Earth s gravitational field

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1 Physics HSC Couse Stage 6 Space Pat 1: Eath s gavitational field

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3 Contents Intoduction... Weight... 4 The value of g... 7 Measuing g...8 Vaiations in g...11 Calculating g and W...13 You weight on othe planets Gavitational fields So, what is a gavitational field?...16 Gavitational potential enegy E p...17 Summay... Suggested answes... 3 Execises Pat Appendix Pat 1: Eath s gavitational field 1

4 Intoduction This module, entitled Space, examines the physics involved in launching a ocket into obit, obiting the Eath, etuning to Eath, and tavelling though space, both now and in the futue. Cental to most of these consideations is that the Eath exets a gavitational pull on any mass within its gavitational field. Theefoe, the module begins by examining this gavitational field and the effect it has on masses. Some of the ideas you will encounte may seem odd to you at fist, and pehaps difficult to gasp at fist eading. Do not be afaid to ead a passage a second o even a thid time, if it seems this way to you. Woked examples and poblems with suggested answes have been povided to help you though the leaning. In Pat 1 you will be given the oppotunities to lean to: define weight as the foce on an object due to a gavitational field define the change in gavitational potential enegy as the wok done to move an object fom a vey lage distance away to a point in a gavitational field E p = G mm 1 In Pat 1 you will be given the oppotunities to: pefom an investigation and gathe infomation to detemine a value fo acceleation due to gavity using pendulum motion, compute assisted technology and/o othe stategies and explain possible souces of vaiations fom the value 9.8 ms - gathe seconday infomation to identify the value of acceleation due to gavity on othe planets Space

5 analyse infomation using the expession F = mg to detemine the weight foce fo a body on Eath and the weight foce fo the same body on othe planets. Extact fom Physics Stage 6 Syllabus Boad of Studies NSW, The oiginal and most up-to-date vesion of this document can be found on the Boad s website at Pat 1: Eath s gavitational field 3

6 Weight You should ecall fom you peliminay couse studies that wheneve a mass is located within a gavitational field it expeiences a foce. It is that foce, due to gavity, that you call weight. Let s examine the idea moe closely. lift ai esistance eaction foce weight weight weight The foce on a mass due to gavity is called weight. All masses fall at the same ate within a gavitational field. Galileo Galilei suspected as much, ove fou hunded yeas ago, when he (epotedly) dopped diffeent objects fom the towe of Pisa to demonstate the fact. 4 Space

7 Howeve, diffeent masses falling at the same ate is a emakably difficult thing to pove fo two easons. Things fall vey quickly hee on Eath. Ai esistance acts diffeently on diffeent objects conside the ai esistance on a falling feathe compaed to a falling cok. Galileo woked aound these poblems by making vey smooth and highly polished, gooved amps, down which he olled balls of diffeent masses. Fom this he could demonstate his point to othe scholas that masses fall at the same ate unde the influence of gavity. Much moe ecently, in 1971, the wold witnessed a eally convincing demonstation. When on the Moon, Apollo astonaut David Scott held up a feathe and a hamme and then let them go at the same time. In the low gavity of the Moon, and in the absence of any ai, these two vey diffeent objects fell at the same slow ate and stuck the suface at the same time! To see an image of the Scott dopping hamme and feathe togethe see an image on the Physics websites page at: You should be able to ecall Newton s second law of motion. This is F=ma. That is, the net foce that is acting on an object is equal to its mass multiplied by its acceleation. Now use this idea to define a fomula fo weight as follows: weight is that foce that acts on a body within a gavitational field and is equal to the mass of the body multiplied by the ate of acceleation, called g. Hee on the suface of the Eath g is nomally taken to be 9.80 ms -. W = mg whee W = weight in newtons (N) m = mass in kilogams (kg) g = acceleation due to gavity at that place in metes pe second squaed (ms - ). You should now ty to apply this definition. Sample poblem 1 What is the weight of a peson with a mass of 65.0 kg? W=mg = = 637 N Pat 1: Eath s gavitational field 5

8 Sample poblem The same peson climbs Mt Eveest. Acceleation due to gavity at the peak, at 8.8 km above sea level, is 9.77 ms -. Calculate the peson s weight now. To solve this poblem we need to use a diffeent value fo g: W=mg = = 635 N Notice that the peson now appeas two newtons lighte than when at sea level. This is a had way to lose weight! This last example demonstates the effect that vaiations in g have upon weight. In fact, the value of acceleation due to gavity is not a constant value eveywhee, but vaies slightly aound the wold. Complete Execises 1.1 and Space

9 The value of g It is impotant that you ealise that the foce descibed as weight also fits the equation shown below, because it is the foce between two masses, one of which is the Eath. That means: mg = G mm E E whee: G = Univesal gavitational constant m = the mass of the body in kg m E = mass of the eath = x 10 4 kg E = adius of the Eath = x 10 6 m (on aveage) This simplifies to give: g=g m E E ( )( ) Substituting values: = 6 ( ) Which leads to a value: g ª 980. ms The 6378 km adius used hee fo the Eath is an aveage value so the value of g calculated, 9.80 ms -, also epesents an aveage value. Pat 1: Eath s gavitational field 7

10 Measuing g In this activity you will attempt to measue the acceleation due to gavity using the motion of a pendulum. You teache may wish to do this with you in you pactical session, but you should also attempt it at home. Fo this activity you will need: a 50 g mass o a lage nut that weighs aound 50 g about 1. m length of sting a watch with a second hand o a stop watch. You will need to find a way of hanging a one mete long sting with a 50 g mass at the end so that it can swing feely as a pendulum. In a laboatoy the pendulum appaatus would look like this: clamp bosshead etot stand sting 50 g mass caie Expeimental set up fo measuing g using a pendulum. An inteesting thing about a pendulum that swings with a small angle is that its peiod, (that is the time taken fo a complete back-and-foth swing), depends only upon the length of the sting and the value of acceleation due to gavity. 8 Space

11 The equation is as follows: l T = p g whee T = peiod of the pendulum, in s l = length of the pendulum, in m g= acceleation due to gavity, in ms - Can you eaange this equation to make g the subject of the fomula, that is, g =? Check you answe. 1 To begin with, adjust the length of you pendulum to one mete. Recod this length as exactly as you can in metes in the space below Set the pendulum swinging gently by pulling it back (no moe than 30 eithe side of vetical). Stating at exteme of the motion, time 10 full peiods, finishing back at the same point. 3 Divide this time by ten (just shift the decimal point one place to the left) and ecod this time next to the pendulum length in the table shown below. 4 Calculate a value fo g using the fomula fom above. Tial Pendulum length (m) peiod (s) g (ms - ) aveage = Pat 1: Eath s gavitational field 9

12 6 Repeat this pocess fou moe times, shotening the sting by 5 cm each time. 7 Finish off by calculating an aveage value fo g, based on all five tials. Thee is a sample set of data in the suggested answes. Compae you data to this. An altenative analysis Thee is anothe way to analyse you pendulum data. This method equies some gaph pape such as that found in the Appendix. 1 Mak out a set of axes. The x-axis will be the pendulum length in metes. The y-axis will be the peiod squaed in s. Use you data set to plot five points on you gaph. The points should all lie on a staight line. 3 Use you ule to judge a line of best fit and daw that line onto the gaph. Detemine the gadient of you line of best fit. 1 What ae the units of the gadient? You should now examine why this gaph poduces a staight line. Fist, look at the equation fo the peiod of a pendulum. Reaange this equation into the following fom: T = (expession) l 3 On you gaph, T is the y-axis and l is the x-axis, so compae the equation you just deived to the geneal equation of a staight line: y = mx You should see that the equation you deived fo the pendulum confoms to the fom of a staight line, if the axes ae selected as we have them. A diect compaison shows that the tem (expession) in you equation is equal to the gadient of you line of best fit. Use this idea now to detemine a value fo g. (expession) = gadient of line of best fit 10 Space

13 4 The pendulum method you have used in this activity often poduces vey accuate esults. Why do you think the method should be so eliable? 5 What souces of eo ae thee in this expeiment? To assist you in this question you might like to look at the section on eos in the Science esouce book. Check you answes. Vaiations in g In sample poblem above you saw that the value of g, the acceleation due to gavity, can be diffeent to the nomally accepted value of 9.80 ms -. Thee ae a numbe of factos that can cause this value to vay aound the wold. These factos ae listed below. The Eath s cust vaies in thickness and density because of such things as tectonic plate boundaies and diffeent dense mineal deposits. This can cause localised vaiations in the value of g. The Eath is flatte at the poles. This means that the poles ae close to the cente of the Eath, so that the value of g will be geate at the poles. The centifuge effect ceated by the spin of the Eath, educes the value of g. This effect is geatest at the equato. Because of these factos, the ate of acceleation due to gavity at sea level on the Eath has been found to vay fom 9.78 ms at the equato to 9.83 ms - at the poles. Of couse, as you have aleady seen, thee is one moe facto that can affect the value of g. Pat 1: Eath s gavitational field 11

14 This is the height above sea level, altitude. This is because altitude inceases the distance fom the cente of the Eath in the fomula fo g above. You can adjust the fomula fo g to allow fo altitude as follows: g me = G ( + altitude) E Thee of the factos causing a vaiation in g ae listed below. Look at the fomula fo g above. Fo each of these, state the facto fom the equation which is affected. custal vaiation flatten at the poles altitude Check you answes. Sample poblem 3 What is the value of g at the top of Sydney towe, 305 m above sea level? By how much would the weight of a kg peson educe in going fom the gound up to the summit? Solution Fistly, calculate the value of g at the top of Sydney towe. On a planetay scale this is a vey small incease in altitude, so any effect is going to be small. Stat off using the fomula given above, then substitute values into it. me g= G ( E + altitude) - ( ) = = 11 4 ( ) ( ) ms As expected, this is only a tiny decease fom the value of 9.80 ms at the gound. Next calculate the weight of the peson on the gound and at the summit. W at gound = mg = = 637 N W at summit = mg = = N That is, the peson s weight has educed by just 0. N. Put anothe way, this epesents a 0.03% eduction. Not vey much! 1 Space

15 Calculating g and W Now it is time fo you to ty to do calculations like these fo youself. Fistly, by looking back ove last few pages, locate values fo g in the locations listed below. Fo the last two loctions, howeve, you will have to calculate a value of g using the equation above. Next, calculate the weight of an 80 kg peson at each of these locations. Location g (ms - ) Weight of 80 kg peson (N) typical gound location south pole equato Mt Eveest summit a balloon 10 km high obitting space shuttle at 400 km altitude Check you answes. It s woth taking a moment to note that a peson in the obiting space shuttle still has consideable weight, in fact about 89% of the value on the gound. This may seem stange to you, since it is commonly known that an obiting spacecaft is a weightless envionment. Howeve, an astonaut in obit is only appaently weightless and the tue weight, as calculated by you above, is quite eal. The diffeence between tue weight and appaent weight will be discussed in moe detail in Pat 3 of this module. Complete Execises 1.3, 1.4 and 1.5. Pat 1: Eath s gavitational field 13

16 You weight on othe planets Have you eve wondeed what it would be like to walk aound on othe planets, with diffeent values of gavity? In the following activity you ae going to find out, at least on pape. The equation that we used ealie to detemine a value fo g on the suface of the Eath can easily be modified fo othe planets. It looks like this: g = G m planet planet The table povided below lists the five ocky planets of ou sola system. The giant gas planets ae left out, because it eally wouldn t eve be possible to locate a suface upon which you could stand. The Moon is included because astonauts have walked on its suface. Also povided in the table is infomation on the mass and diamete of each body. 1 Detemine the adius of each body, in metes. Use the fomula povided to calculate the value of acceleation due to gavity at each of these places. 3 Recod you esults in the table. 4 Find a set of scales and weigh youself. Recod you below. 5 Calculate what you weight would be on the suface of each of the planets, using the equation W = mg. 6 Recod you answes in the last column. My mass = kg 14 Space

17 Body Mass (kg) Diamete (km) Radius (m) g (m/s - ) My weight (N) Mecuy Venus Eath Moon Mas Pluto Wite a desciptive (wod) compaison of the acceleation due to gavity on Mas and on Mecuy. Compae g on the Moon to g on the Eath. What atio is it? 3 How does g on Pluto compae to g on the Moon? Check you answes. You should now attempt Execise 1.6. Pat 1: Eath s gavitational field 15

18 Gavitational fields Conside this question (you may be supised by the answe). If an astonaut in an obiting space shuttle still has a consideable weight, as you saw in the pevious sample poblem, then they have not eally escaped the Eath s gavity. How fa away, then, would the astonaut need to tavel in ode to completely escape the gavity of the Eath? Did you find that the must tavel to infinity (but not beyond!). Hee is the equation fo g again: g G m = Notice that g depends on the invese of the squae of the distance fom the cente of the Eath. This is known as an invese squae elationship. It means that as distance inceases, the value of g educes quickly. Howeve it neve actually dops to zeo until the distance equals infinity. g=g m E = 0 In othe wods, the influence of the Eath s (and any othe celestial body) gavity extends vey fa out into space but dops off in stength vey quickly with that inceasing distance. A good way to descibe this infuence is to say that it is a gavitational field. E E So, what is a gavitational field? Fom science fiction stoies you may aleady be familia with the tem foce field. It is a zone within which a paticula foce will act. Thee ae magnetic fields aound magnets, within which magnetic foces act on ion objects. Thee ae electic fields inside electic motos, within which electic foces act on electic chages. Similaly, thee ae gavitational fields aound celestial objects within which gavitational foces will act on a mass. (In fact, thee ae gavitational fields aound any mass but the mass must be huge fo the foces to become noticeable.) 16 Space

19 You can use two means to descibe the gavitational field aound the Eath. The fist is to use the value of g as an indication of the stength of the field at any paticula point within the field. (This is quite valid you should ecall that g= W, which means that it can also have the units of m newtons pe kilogam.) The second way to descibe the field is with a diagam, as shown below. g line sepaation inceases with distance fom the Eath indicating weake field The gavitational field aound the Eath. Use the space below to show that the altenative units fo g ae equivalent. The units ae ms - and Nkg -1. Check you answe. Gavitational potential enegy E p You ae now going to conside the enegy of a mass within a gavitational field. You have aleady seen that a mass within a gavitational field will expeience a foce (its weight). If that foce meets no esistance, like a floo, then the mass will fall and as it falls it speeds up and gains kinetic enegy. But enegy cannot be ceated, so whee does that kinetic enegy come fom? Pat 1: Eath s gavitational field 17

20 The answe is that the mass has enegy to begin with. This is simply because of its position within a gavitational field. This enegy is called gavitational potential enegy ; it has the symbol E p and the units joules (J). As the mass falls it has less and less gavitational potential enegy (because its being conveted into moe and moe kinetic enegy). Theefoe it is moving away fom positions of highe potential enegy and towads positions of lowe potential enegy. The next thing to conside is the location of the zeo potential enegy level. In othe wods, at what position would a mass have to be in ode to have no gavitational potential enegy at all? Give this question just a little thought and you will ealise that fo gavitational potential enegy to be zeo, the mass cannot lie anywhee within a gavitational field. You saw in the last section that in ode to be completely fee of a gavitational field a mass must lie an infinite distance away. Infinity, then, is ou zeo level: E p = 0 at. By now you may have ealised something athe cuious. If you wee to place a mass, say, ten kilometes above the suface of the Eath and let it go it would fall towad the Eath. It is falling away fom the zeo level (which is at infinity), and yet it is falling to points of lowe and lowe gavitational potential enegy. This means that its gavitational potential enegy is negative, and its most negative value will be at the suface of the Eath! This idea can be a little had to gasp at fist. The diagam below is a gaph of the gavitational potential enegy of a mass along a line fom the suface of the Eath out to infinity. + o distance gavitational potential enegy Ep Gavitational potential enegy of a mass fom Eath to infinity. 18 Space

21 It is time now fo a definition: the gavitational potential enegy E p of a mass m 1 at a point in the gavitational field of anothe lage mass m is equal to the change in gavitational potential enegy that occus when moving the mass m 1 fom the zeo level (infinity) to the point. This is defined as the wok done to move the object fom infinity (o fom a vey lage distance away) to the point. You should discuss this fo a moment. Why talk about change of E p? The eason is that you eally can only eve compae the E p of one point with the E p of anothe point, that is, calculate change in E p. To get an absolute value fo any paticula point you must compae it to the point of zeo E p. In physics language, this is the change in E p between infinity and the point in question. Now what s this about wok? When wok is done, enegy has been conveted fom one fom to anothe. If you pick up a bick and lift it up then you have done wok on it chemical enegy in the muscles of you am has been conveted into additional gavitational potential enegy of the bick. Similaly, if you lift a bick fom some point out to a vey lage distance away, then you will have done some wok to incease the E p of the bick. The amount of that wok is equal to the change of gavitational potential enegy. Also, if the bick is moved in the opposite diection, that is fom a vey lage distance away back to the oiginal point in space, then its E p is deceased. If it is deceased below zeo then it becomes negative, as expected. It can be shown mathematically that the gavitational potential enegy is given by the following equation: E p = - m1m G whee E p = gavitational potential enegy, in J m 1 = mass in the field, in kg m = mass of planet, in kg G = univesal gavitation constant = distance between the centes of the masses, in m Does this make sense to you? The ideas behind gavitational potential enegy can be confusing. Read this last section again, noting the effect of the zeo level on the sign of the potential enegy (positive o negative). Pat 1: Eath s gavitational field 19

22 1 How is change of gavitational potential enegy defined? Why is the tem change of used in the definition above? _ 3 What is the elevance of the tem a vey lage distance away in the definition? _ 4 Why does the gavitational potential enegy of a mass in a gavitational field have a negative value? _ 5 If you wee to place a gaph of the E p aound the Jupite next to the gaph shown fo the Eath, how would the two gaphs diffe? _ Check you answes. Sample poblem 5 Following is a sample poblem showing how to apply the equation fo E p. Calculate the gavitational potential enegy of a 5.00 kg meteo located at the uppe limits of the Eath s atmosphee, 10 km above the suface of the Eath. 0 Space

23 Solution In ode to pefom this calculation you need to ealise that the distance between the centes of the masses is equal to the adius of the Eath plus the altitude of the meteo. = E + altitude = = m Next, apply the equation fo gavitational potential enegy. E p =-G mm ( )( 5. 00)( ) =- ( ) 8 = J Now use the sample poblem above to solve the poblem following. Calculate the gavitational potential enegy possessed by the space shuttle in obit 400 km above the suface of the Eath. Use kg as the mass of the shuttle. Check you answes. Complete Execises 1.7, 1.8, 1.9 and In this wok unit you have leaned about the natue of the foce of gavity and gavitational fields. In the next pat you will begin to lean about ways to ovecome that gavity and beak fee fom the Eath s gavitation. Pat will focus specifically on the motion of pojectiles, that is, objects pojected upwads into the ai. Pat 1: Eath s gavitational field 1

24 Summay Weight is the foce on a mass lying within a gavitational field. W = mg The value of g at diffeent places aound the Eath s suface vaies fom the aveage value because of the Eath s shape, the Eath s spin and local vaiations in the Eath s cust. The value of g vaies with altitude above the suface of the Eath. The value of g is diffeent on othe planets due to the diffeing planetay masses and adii. A gavitational field suounds any mass, becoming significant with planetay-sized masses. It is stong close to the mass and weakens quickly with inceasing distance. Despite this, its influence extends to vey geat distances in space. The gavitational potential enegy of an object at a point in a gavitational field is equivalent to the change in gavitational potential enegy between infinity and that point. Gavitational potential enegy is defined to be the wok done to move the object fom a vey lage distance away to a specified point in a gavitational field. Space

25 Suggested answes Measuing g using a pendulum T g= 4 4p 1 p 1 = \ g T You may have had difficulty pefoming the expeiment so a sample set of esults ae povided hee: Tial Pendulum length (m) Peiod (s) An altenative analysis 1 Gadient = ise / un ª The units fo gadient ae ms -. T 4 = Ê l Ë Á p ˆ g 4p \ gadient = g 4p 4p \ g = = = 9.86 ms gadient 4 4 The pendulum method depends only upon length l and peiod T, both of which can be measued quite accuately. - Pat 1: Eath s gavitational field 3

26 5 Souces of eo in this expeiment include: the accuacy of the ule and method used to set the pendulum length You own eaction time in timing the swing. (The accuacy of a stopwatch is usually much geate than you eaction time.) Vaiations in g custal vaiation m E flatten at the poles ( E + altitude) altitude E Calculating g and W Location g (ms - ) Weight of 80 kg peson (N) typical gound location south pole equato Mt Eveest summit a balloon 10 km high obitting space shuttle, 400 km altitude Sample calculation povided fo the balloon: g=g ( E = ms me + altitude) ( )( ) = 6 3 ( ) W = mg = = N 4 Space

27 Detemining you weight on othe planets 1 The acceleation due to gavity on Mas and on Mecuy ae both vey simila appoximately 40% of g on the Eath. Acceleation due to gavity on the Moon is just 1.6 ms -. This is just 1/6th of g on the Eath. 3 Acceleation due to gavity on Pluto is much less than on the Moon appoximately /5th. So, what is a gavitational field? Since F = ma 1 newton = 1 kgms N kgm \ = = ms kg kgs Does this make sense to you? 1 The gavitational potential enegy of an object at a point in a gavitational field is equivalent to the change in gavitational potential enegy between infinity and that point. This is defined to be the wok done to move the object fom a vey lage distance away to that point. The absolute value of E p is eally just the diffeence between E p at that point and E p at the zeo level, which is at an infinite distance away. This diffeence is called the change in E p. 3 The zeo level of E p has been defined to be an infinite distance away, which is had to achieve in pactical tems, so the tem a vey lage distance away is often substituted. 4 Point close to a planet epesent lowe gavitational potential enegy levels, but the futhest point away (infinity) epesents a zeo potential enegy level. Theefoe, evey othe point must epesent a negative potential enegy level. 5 The gavitational potential enegy gaph fo Jupite will have the same shape as Eath s but will be deepe, because Jupite s geate mass ceates a stonge field. Calculation of the gavitational potential enegy of an obiting space shuttle: = E + altitude = = m Pat 1: Eath s gavitational field 5

28 E p =-G mm ( )( )( ) =- ( ) = J 6 Space

29 Execise Pat 1 Execises 1.1 to 1.10 Name: Complete the execises and etun them to you teache if you ae a distance education school student. If you ae an TAFE Open Leaning Pogam student you teache will supply you with the answes to these execises. By doing these execises you should lean whethe o not you have undestood the main concepts taught, and achieved the outcomes fo this section of the couse. You teache will send comments back to you to help you achieve any outcomes you ae not cuently achieving. Execise 1.1 a) Wite a definition of weight. b) Does an object still have weight if it is falling? Why? Why not? Execise 1. Detemine the weight of a kg gil standing: a) in an aveage sea level location whee g = 9.80 ms -. Pat 1: Eath s gavitational field 7

30 b) on top of Mt Kosciusko, 8 m above sea level, whee g = ms -. _ c) on top of Mt Eveest, whee g = 9.77 ms -. _ Execise 1.3 a) Looking at Newton s law of univesal gavitation, what vaiables affect the stength of the foce between two masses? _ b) What would be the effect on the foce of doubling the distance between the two masses? _ Execise 1.4 List fou factos that can cause the value of acceleation due to gavity to vay aound the suface of the Eath. 8 Space

31 Execise 1.5 Calculate the acceleation due to gavity at each of the following locations, as well as the weight of a kg peson. Location Altitude (m) g (ms - ) Weight of kg peson (N) Mt Kilimanjao 5889 Mt Fuji 3778 Mt Etna 379 Mt Rushmoe 1841 Execise 1.6 Calculate the acceleation due to gavity on the vaious moons of Jupite listed below, as well as the weight of a 10.0 kg fully-equipped astonaut. Moon Mass (kg) Radius (km) g on suface (ms - ) Weight of 10.0 kg peson (N) Io Euopa Ganymede Callisto Pat 1: Eath s gavitational field 9

32 Execise 1.7 a) Daw the gavitational field aound the Eath. b) A ule of field dawing says that stonge fields ae shown by dawing lines close togethe. Daw the field aound a planet twice as massive as the Eath. Execise 1.8 a) Calculate the gavitational potential enegy of a 1.00 kg ock on the suface of the Eath. Requied data can be found in this pat. 30 Space

33 b) Calculate the gavitational potential enegy of the same ock placed at the suface of Jupite. Mass of Jupite = kg and adius = km at the equato. a) At what altitude above Jupite s suface should the ock be placed so that its E p is equal to the E p it had at the suface of the Eath? Execise 1.9 Calculate the gavitational potential enegy of Euopa and Io, two moons within Jupite s gavitational field. Using the data given in Execise 1.6. Additionally, note that the adius of Euopa s obit is km and the adius of Io s obit is km. Execise 1.10 a) Wite down the definition we ae using fo change in E p. Pat 1: Eath s gavitational field 31

34 b) Explain the significance of a distance vey fa away in the definition in (a). c) Why does E p have a negative value? 3 Space

35 Appendix Pat 1: Eath s gavitational field 33

36 34 Space

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