PHYSICS 206a HOMEWORK #11 SOLUTIONS
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1 PHYSICS 0a HOEWORK # SOLUTIONS. Using Nwton s law of gavitation, div th acclation du to gavity xpincd by an objct of ass at th sufac of th Eath. Show that this acclation is indpndnt of th ass of th objct. Nwton s law of gavitation says that th gavitational foc xpincd by an objct of ass du to an objct of ass is givn by F G wh G is a univsal constant and is th distanc btwn th cnts of th asss. Th foc is always attactiv and acts in th diction of a lin dawn btwn th cnts of th asss. You ust oiz this! (Not just th foula all th st of it as wll.) On vy coon o is to s th ltt in that quation and to assu that it is th adius of sothing. (Just on dang of foula hunting!) This is by no ans ncssaily th cas! is th distanc btwn th cnts of th two asss. That s it. Now, if w talking about th foc xpincd by an objct sitting on th sufac of th Eath, thn, indd, is th distanc fo th cnt of th Eath to its sufac its adius. But don t just assu that is a adius! Calling th ass of th Eath and th ass of th objct, w gt F G. Accoding to Nwton s law, th objct will xpinc an acclation du to this foc if it is th only xtnal foc acting on th objct. Th lationship btwn th acclation and th foc is F a. Thus, w can wit G a. A tiny bit of algba givs G a. Thus th acclation du to gavity is indpndnt of th ass of th objct bing acclatd as was thoizd by Galilo. Substituting in nubs fo th sufac of th Eath, w gt 4 Nwton t 5.98 ts a G kiloga (.378 t) Obviously, this is th nub w v bn using all sst. Now you know wh it cos fo!. If th adius of th Eath changd and bca ½ of its cunt valu (but th ass stayd th sa), what answ would you gt fo th pvious pobl? I afaid th woding of this pobl was a littl abiguous. What I ant was that th adius changs and that th objct aind on th sufac. If th
2 adius changs but th objct stays at th old adius, nothing changs. If th adius changs and th objct ains on th sufac, th atio of th old acclation to G aold old nw ( old ) th nw acclation will b givn by. (How a nw old old 4 G nw any of you fogot to squa th ½?) So th nw acclation would b fou tis ts as gat as th old acclation w d hav g Dtin th acclation du to gavity fo an objct on th sufac of th oon. H w nd only pat th analysis w did in pobl # only using th nubs fo th ass and adius of th ooon: Nwton t 7.35 ts a G.7., kiloga (.737 t) oughly / of th acclation on th sufac of th Eath. 4. Th Clak obit is nad fo scinc fiction wit Athu C. Clak. In 945, Clak suggstd that th was an altitud abov th sufac of th Eath at which satllits would hav th sa angula spd as th Eath (Wilss Wold, Octob 945, pags ). Thus, a satllit placd in such an obit would b gosynchonous. That is, it would stay suspndd ov a singl point on th glob. By stting th foc of gavity qual to th cntiptal foc, find th adius of such an obit. (Caful: Don t confus th distanc fo th sufac of th Eath with th adius of th obit! Thy diff by th adius of th Eath.) Th a cuntly ov 300 satllits occupying such obits! Rcall ou discussion about cntiptal acclation: To say that an acclation is cntiptal is siply to say what diction it is. Th foc poviding that acclation can co fo any appopiat souc. In this cas, th foc of gavity is bing usd to cat th cntiptal acclation. Sinc w do not know, initially, th distanc fo th cnt of th Eath to such an obit, w cannot assu that th acclation will b anything lik what it is at th sufac of th Eath in fact, thy quit diffnt. (Rcall Nwton s oiginal divation of th cntiptal acclation of th oon.) W siply nd to tak th xpssion fo th acclation du to gavity and th ndd cntiptal acclation and st th qual to ach oth. v W hav G. Now, w us th angula spd lation v ω to wit this as G 3 ω. This givs G. Now, th dfinition of th angula spd is ω th angl tavld dividd by th ti it taks to tavl it ω θ. Th quint fo t
3 ou satllit to b gosynchonous is that it tavl though on full cicl in on day, so 3 θ π and t day 8400s. This givs G t. (S th nxt pobl fo so intsting insight into this intdiat sult.) 3 Solving fo, w hav 3 G t. Substituting nubs, w gt G t 3.7 Nwton t kiloga kiloga ( 8400s) t 4. ts. This is 35,800 k fo th sufac of th Eath. In English units, this is about,000 ils fo th Eath s cnt o,300 ils abov th sufac of th Eath. Ths satllits a usd piaily fo wath and counications. If you v got a satllit antnna fo you tlvision, th ason you a abl to kp you dish antnna pointd in only on diction is that th satllit it is civing a signal fo is in a Clak obit. If you go out at night and look in th diction th antnna is pointd, you vy wll ight b abl to s th satllit! Of cous, it will just look lik a di sta, but it will ain in on position as th bacound stas ov acoss th sky in th cous of th night. 5. Gnaliz th sult you found in th pvious pobl: By using gavity as th thing poviding an obiting objct with its ndd cntiptal acclation, pov Kpl s thid law. What is th constant 3 of popotionality? I.., Kpl s thid law says that T kr wh T is th piod of th obit, R is th an adius of th obit, and k is so constant. Find k. You should assu th obit is cicula. Not th quation w got in th pvious pobl along th way to th final answ. 3 At on point w found that G t. This was found siply by assuing that an objct is tavling in a cicula path and that th cntiptal foc ndd to kp it in that path is povidd by gavity. Thus, this quation is tu in gnal it is tu fo any objct in an obit, not just satllits of th Eath and not just thos in gosynchonous obits. A 3 sall anipulation givs t. Not that I v doppd th subscipt on th ass G sinc this is tu fo any objct bing obitd, not just th Eath. Fo a diffnt objct (.g., th sun) w d just substitut th appopiat ass. Th nti facto is a G constant fo a givn syst, so w can plac it by a singl sybol fo siplicity. This 3 givs t k wh k. This is what was to b found. (If you ally want a G nub, just substitut th appopiat valus fo a givn syst in this quation.)
4 . So yas ago, a (cackpot) thoy ciculatd which pdictd doo fo th Eath du to th gavitational ffct of th plants all lining up. This ffct was known as Th Jupit Effct which was th titl of th book in which this pdiction was ad. Th Eath did not, in fact, suff any of th ill ffcts pdictd. Calculat th foc of gavity xtd by Jupit on a pson on th sufac of th Eath whn th two a closst. Jupit s an distanc fo th Sun is 7.79 and its 4 ass is.9. Th Eath s an distanc fo th sun is.49. What ass of objct would hav a wight on th sufac of th Eath quivalnt to th foc of gavity of Jupit on this pson? Th foc of gavity xtd on on objct by anoth dpnds only on th two asss and th distanc btwn thi cnts. Lt s call th ass of th pson and J th ass of Jupit. I didn t giv you th Eath-Jupit distanc, but I gav you th distanc fo ach of ths to th sun. Th distanc fo Jupit to th Eath is just th distanc fo Jupit to th sun inus th distanc fo th Eath to th sun. (This is only tu whn thy a at closst appoach. At oth tis, o coplicatd goty would b ncssay.) I ad a al booboo in this pobl: I istypd th ass of 7 Jupit! It should hav bn givn as.9 a facto of 00 diffnt fo what I gav you. I do apologiz fo this. (Not that this is about 300 tis th ass of th Eath!) I ll do th calculation using th coct nubs. Plas tak a look at you solution and think about how it would chang with ths nubs instd appopiatly. Lt s tak th ass of th pson to b 0 (pick whatv valu you think asonabl o just call it ). F G W hav ( ) JupitSun J N N J p F G. Substituting nubs, w hav p Eathsun ( ) JupitSun.9 7 Eathsun 0 ( ) Fo an objct to hav this wight on Eath, it would nd to hav a ass givn by N g 3.9 N, which yilds 3.. This is th ass of 9.8 s a sall gain of sand. Whil this foc is vy sall, notic that it is still lag nough that it dosn t s unasonabl to xpct that w could asu it. Indd, a asunt of th ffct of this foc is quit fasibl. It s sall, but not copltly ngligibl! Jupit (and all th oth plants in th sola syst) dos hav an ffct on Eath. Usually, w can igno it, but it would b ipotant if w w doing calculations to pdict, fo xapl, long-t cliatological ffcts. p
5 7. What is th diffnc in th foc xpincd by a cubic t of wat du to th oon s gavitational pull whn th oon is on th sa sid of th Eath as th wat and whn th oon is on th opposit sid of th Eath as th wat? (I.., about hous lat.) What is th foc of th Eath s gavity on that quantity of wat (i.., its wight)? Rcall that tids a causd, piaily, by th gavitation of th oon pulling wat away fo th Eath. Of cous, th oon loss th coptition, but it has a significant ffct nonthlss. Th oon has an avag distanc fo th cnt of th Eath of Th adius of th Eath is E.378. (Notic that I v usd a siila sybol,, fo two vy diffnt things h: Nwton s law of gavitation spcifis that w nd th distanc btwn th cnts of th asss und considation. In this cas, I also nd a adius in od to find that distanc. Don t fall into th tap of thinking that ans adius. In this contxt, it ight an adius. But it also ight just an th distanc btwn th cnts of a pai of objcts. You will nd to dtin, fo contxt, which of ths is appopiat.) Thus, whn th wat is on th sa sid of th Eath as th oon is w will hav R. Whn th wat is on th opposit clos sid of th Eath fo th oon w will hav fa E R +. Using ths xpssions, th diffnc in foc xpincd by th wat btwn th two locations is F G G G oon oon oon wat wat wat Rclos R fa ( ) ( ) E + E Now, th ass of a cubic t of wat is 00. So w hav F G oonwat N N This is oughly quivalnt to adding on cubic cntit of wat to th sapl with th oon at its fathst location and oving on cubic cntit of wat fo it whn th oon is at its closst appoach. Fo copaison, th foc xtd by th Eath on ou cubic t of wat is W g N. So th oon s ffct is about 0. pats p illion on th wight of th wat. Still, this is sufficint to caus th tids which a ssntial to lif on Eath! 7 E
6 An intsting sid not to this is that th wok don on th wat, lifting and thn dopping it, has a dissipativ pic. Th is ngy lost vy ti a wav slaps on th sho. This ngy is bing xtactd fo th oon s obital ngy (otational kintic ngy). So th oon is slowly oving away fo th sufac of th Eath ach ya, th oon is a fw c fath away than in th pvious ya. (This otion has bn usd to pov that th Eath cannot b as old as scintific asunts indicat it to b. Unfotunatly, th folks who poulgat this poof ad an aithtic o and thi nubs a all wong as a sult. I onc found ov,000 wb sits quoting th onous sult! Sotis a littl slip on you calculato can hav fa aching ffcts.) 8. A spac shuttl has a ass of 80,000. What is th diffnc in th otational kintic ngy that nds to b givn to th shuttl if it is launchd fo a point on th quato o th noth pol in od to gt it into a cicula obit? Assu th hight of th obit is 700 k abov th sufac of th Eath. This is o ticky than difficult. Th fist thing you nd to aliz is that th spd of an objct in obit is dtind xclusivly by th hight of th obit. All obits at a givn hight tak th sa aount of ti. This can b sn fo th solution to th Clak obit pobl and th on on Kpl s law. (I v caught so of you assuing that all obits a gosynchonous lik th Clak obit. This is not tu!) As soon as w spcify th hight of th obit, w hav spcifid th gavitational acclation. Sinc th gavitational acclation is th cntiptal acclation, only on spd will allow an objct to ain in obit at a paticula hight. So th shuttl will hav th sa otational kintic ngy onc it gts into obit gadlss of whth it statd at th pol o th quato. Any diffnc btwn th otational kintic ngy ndd to b givn to th shuttl btwn th two launch locations is du to whatv diffncs in otational kintic ngy thy hav whil sitting on th gound. Now, an objct at a pol has vitually no otational kintic ngy. Its otational adius (th distanc fo th otational axis to th ass) is naly zo copad to what it is at th quato. Howv, an objct at th quato has a fai aount of otational kintic ngy: It s oving aound in a cicl with a adius of.378 t (th adius of th Eath) onc p day. So th shuttl s otational kintic ngy whn sitting at K. E. otational Iω ω th quato is 8 4 (.378 t) π 8400s 8. 9 Jouls So, a spac shuttl sitting at th quato alady has 8. billion Jouls of th ngy it nds to gt into obit. Evy littl bit hlps! This is on of th asons why launch pads a placd as clos to th quato as possibl.
7 9. It is fquntly statd that astonauts a in zo g whn in obit aound th Eath. What is th nt foc of gavity on a 80 astonaut in a 700 k obit? Why dos th astonaut float? It is cucial that you undstand that gavity dosn t gt tund off! Nith dos it gt canclld out. Th gavitational foc on astonauts in obit is dan na th sa as that which thy xpinc at th sufac of th Eath. To s this, lt s us th sult w obtaind fo th acclation at th Eath s sufac in pobl #: a G Nwton t kiloga Now, add 700 k to th adius. This givs a G Nwton t kiloga (.378 t) ( t) ts 9.8. ts 7.9 a chang of about 9%. Now, ultiply by his/h ass to find th astonaut s wight: W Nwtons. Copa this to th 784 Nwtons s s h/sh would wigh on th sufac of th Eath. So why do ths popl float? It s bcaus thy hav nough fowad vlocity that th acclation towad th cnt of th Eath that thy xpinc is siply sufficint to ak thi path cuv. In fact, th fowad vlocity is cafully tund so that thi path just cuvs nough so that thy ov in a cicl. (Actually, an llips, which ay b quit ccntic. But w ll nglct th xt cass fo now.) Th ky thing is that vything in obit at th sa hight is oving at th sa spd and acclating at th sa at. (Rad that on a coupl of tis.) So vything fo th shuttl itslf to th astonaut to tiny spcs of dust in obit at that hight a oving at th sa spd and acclating at th sa at thy all following th sa path. So, although thy ight hav vy high spd and although thy acclating alost as uch as thy would if thy jupd off a building on Eath, non of th is oving lativ to ach oth! Thy all s to float as though th w no gavity at all. Pobls # and # hav bn ovd to th nxt assignnt.
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