AP Phyic Gaiy Wapup You ge hee equaion o wok wih, and only hee. Eeyhing ele you hae o deelop fo hee hee equaion (alo hae o ue oe ohe one). Hee hey ae: F a G c G = Newon law of gaiy. 1 = Cenipeal acceleaion equaion. Σ F = FNe = a Second law. Hee wha you hae o be able o do: 1. You hould undeand he unifo cicula oion of a paicle o you can: a. Relae he adiu of he cicle and he peed o ae of eoluion of he paicle o he agniude of he cenipeal acceleaion. You hae hi equaion o wok wih: a c = In hi equaion, i he linea peed, o i faily eay o elae i o he adiu () o o he cenipeal acceleaion (a c ). You ju ue he old equaion a i i gien. If ge bigge, hen he cenipeal acceleaion ge alle, &c. The ae of eoluion i he ae ha he hing oae. Thi would be he nube of oaion diided by he ie i ook o do he. Thi i called he angula elociy, ω. Unfounaely, you don go no equaion fo hi. Ju eebe ha any ae i iply a quaniy diided by ie, o he angula elociy i iply: θ ω = Whee θ i he angula diplaceen, which will o likely be he nube of eoluion ha hae been ade? Wha happen o he cenipeal acceleaion if he elociy i inceaed and you wan o keep he ae adiu? If he adiu inceae, bu he elociy ay he ae, wha happen o he cenipeal acceleaion? Tha kind of hing. 186
You hould be able o deie an equaion ha elae linea peed o angula elociy. We did ha in you gaiy handou. The Phyic Teache howed you how o deelop he equaion: = π ω Baically, he elociy i popoional o he angula elociy. So whaee happen when he linea peed inceae alo happen if he angula elociy inceae. You igh be able o ge away wihou deeloping he equaion a all. Uing hi concep, i i a iple ae o elae he ae of eoluion of he paicle o he agniude of he cenipeal acceleaion. b. Decibe he diecion of he paicle elociy and acceleaion a any inan duing he oion. The acceleaion i alway owad he cene. The elociy i alway angen o he objec pah. Pah of Obi a c Moon Plane Fo any ype of cicula oion, uch a he oon obiing a plane, he diecion of he cenipeal acceleaion i owad he cene and he elociy diecion i a angen o he pah. c. Deeine he coponen of he elociy and acceleaion eco a any inan and kech o idenify gaph of hee quaniie. Thi i a iple ae ue igonoey o eole he acceleaion and o elociy eco ino i coponen. Again, gaphing hee quaniie i quie iple. 187
5 1 a c 4 3 Aboe i a dawing of an objec ha i following a cicula pah. The elociy and cenipeal acceleaion eco ae hown a fie diffeen poiion. To he igh of ha i a gaph of elociy V ie fo he ae oion. Pleae now figue ou how he gaph wok. The elociy i eihe only up and down o only igh o lef a he fou cadinal poiion, 1, 3, 4, and 5. So he peed i eihe all x o all y. Fo all ohe poiion on he pah, hee will be boh a hoizonal and eical coponen fo he elociy. Can you ee how he peed i he u of he quae of he coponen? Think abou ill you can. Hee i a gaph of he cenipeal acceleaion V ie fo he ae e of poin. 4 x 5 3 y 1 a c 4 3 a y 5 a x 1. Suden hould be able o analyze iuaion in which a body oe wih pecified acceleaion unde he influence of one o oe foce o hey can deeine he agniude and diecion of he ne foce, o of one of he foce ha ake up he ne foce, in iuaion uch a he following: 188
a. Moion in a hoizonal cicle (e.g., a on a oaing ey-go-ound, o ca ounding a banked cue). The ain hing hee i uing he econd law and he cenipeal acceleaion o find he cenipeal foce. F = a ac = Fc = Fc = The cenipeal foce can hen be ued o calculae he ficional foce acing on a ca aeling in a cicle o oe uch hing. In a peiou uni we looked a eeal poble along hi line. The cenipeal foce will be ued in eeal ohe ecion. One ha eally equie i i when we udy a chaged paicle oing hough a unifo agneic field. You ll ee how hi wok lae on. b. Moion in a eical cicle (e.g., a winging on he end of a ing, ca olling down a cued ack, ide on a Fei wheel). Daw a fee body diaga. Analyze he foce. The cenipeal foce will alway end up being he ne foce acing on he body and will alway be dieced o he cene of he cicula pah. Thee ae wo foce woking on he eical cicle he weigh of he objec and he cenipeal foce. A he ey op fo he inan of ie ha he body i up hee, he u of he eical foce u be zeo ince he body i oing ideway and no up and down. So you can ole fo he enion which will un ou o be The idea hee i ha he cenipeal foce u equal he weigh of he objec. Fo hi you can calculae an equaion fo he iniu elociy a body need o ael along a eical cicula pah. The equaion i: = g 3. You hould know Newon Law of Unieal Gaiaion o you can: a. Deeine he foce ha one pheically yeical a exe on anohe. Thi i pie. You ju ue Newon law of gaiy. b. Deeine he engh of he gaiaional field a a pecified poin ouide a pheically yeical a. Thi i anohe lice of eally good pupkin pie wih whipped cea on op (yuy). Again, eely apply good old Newon law of gaiy. 4. You hould undeand he oion of a body in obi unde he influence of gaiaional foce o you can, fo a cicula obi: a. Recognize ha he oion doe no depend on he body a, decibe qualiaiely how he elociy, peiod of eoluion, and cenipeal acceleaion 189
depend upon he adiu of he obi, and deie expeion fo he elociy and peiod of eoluion in uch an obi. Thi i a deiaion pecial. The Phyic Teache deonaed how o do all of hi in a peiou uni. You hae o deie he equaion fo cenipeal foce and e i equal o he foce of gaiy fo Newon law of gaiy. Then you ole fo he elociy. Thi gie you he obial elociy. = G 1 x To find he peiod, you ue he equaion fo elociy, = and he equaion fo he obial elociy we ju found. The diance x i he cicufeence of he cicle. Which i π. Pu he ogehe and you end up wih: = π 3 G We did a nube of poble ha equied deeloping and uing hee equaion. AP Te Exaple: Fo 1997: To udy cicula oion, wo uden ue he hand-held deice hown, which coni of a od on which a ping cale i aached. A polihed gla ube aached a he op ee a a guide fo a ligh cod aached he ping cale. A ball of a 0.00 kg i aached o he ohe end of he cod. One uden wing he ball aound a conan peed in a hoizonal cicle wih a adiu of 0.500. Aue ficion and ai eiance ae negligible. a. Explain how he uden, by uing a ie and he infoaion gien aboe, can deeine he peed of he ball a i i eoling. Meaue adiu and calculae cicufeence. Ue op wach o eaue he peiod x (ie of one eoluion). Ue = o find he peed of he ball. b. How uch wok i done by he cod in one eoluion? Explain how you aied a you anwe. No wok. The oion i pependicula o he foce. 190
c. The peed of he ball i deeined o be 3.7 /. Auing ha he cod i hoizonal a i wing, calculae he expeced enion in he cod. FT = FC F = a ac = FC = F C ( 0.00kg )( 3.7 ) = = 0.500 5.5 N d. The acual enion in he cod a eaued by he ping cale i 5.8 N. Wha i he pecen diffeence beween hi eaued alue of he enion and he alue calculaed in pa (c)? 5.8 N 5.5 N % diff = ( 100% ) = 5.5% 5.5 N e. The uden find ha, depie hei be effo, hey canno wing he ball o ha he cod eain exacly hoizonal. On he picue of he ball below, daw eco o epeen he foce acing on he ball and idenify he foce ha each eco epeen. Explain why i i no poible fo he ball o wing o ha he cod eain exacly hoizonal. The enion in he ing,, poide he cenipeal foce. g i he weigh of he ball. Thee i alway a downwad foce acing on he ball, i weigh, g g. So hee u be a coponen of he enion in he oppoie diecion ince he ball i no oing in a eical diecion. The u of he eical foce u be zeo. f. Calculae he angle ha he cod ake wih he hoizonal. g ( 0.00kg )( 9.8 ) θ = in = in = 19.8 T kg 5.8 1 1 o Fo 001: A ball of a M i aached o a ing of lengh R and negligible a. The ball oe clockwie in a eical cicle, a hown o he igh. When he ball i a poin P, he ing i hoizonal. Poin Q i a he boo of he cicle and poin Z i a he op of he cicle. Ai eiance i negligible. Expe all algebaic anwe in e of he gien quaniie and fundaenal conan. P Z M Q Side View 191
(a) On he figue below, daw and label all he foce exeed on he ball when i i a poin P and Q, epeciely. P g Q g (b) Deie an expeion fo in, he iniu peed he ball can hae a poin Z wihou leaing he cicula pah. FC = g bu he enion i zeo FC = g F = a Fc = ac Fc = Deie he equaion fo cenipeal foce. = g = g = g = Rg (c) The axiu enion he ing can hae wihou beaking i T ax. Deie an expeion fo ax, he axiu peed he ball can hae a poin Q wihou beaking he ing. The Tenion in he ing i he upwad foce and he weigh i he downwad foce, he u of hee wo foce i a o: a = TMax g Bu he acceleaion acing on he ye i he cenipeal acceleaion. = TMax g = TMax g ( ) R = TM ax g = TM ax Mg M ( ) ( ) (d) Suppoe ha he ing beak a he inan he ball i a poin P. Decibe he oion of he ball iediaely afe he ing beak. The ball would go aigh up, lowing down a i i acceleaed downwad by gaiy, i will op, hen fall down, acceleaing a i goe. I oion i angenial o he pah, which i aigh up. 19
Fo 1999: 5. A coin C of a 0.0050 kg i placed on a hoizonal dik a a diance of 0.14 fo he cene, a hown below. The dik oae a a conan ae in a couneclockwie diecion a een fo aboe. The coin doe no lip, and he ie i ake fo he coin o ake a coplee eoluion i 1.5. a. The figue o he igh how he dik and coin a iewed fo aboe. Daw and label eco on he figue below o how he inananeou acceleaion and linea elociy eco fo he coin when i i a he poiion hown. b. Deeine he linea peed of he coin. a The coin ake one eoluion i a ie of 1.5. So he diance i ael in ha ie i he cicufence of he cicle i ake, which i π. x π π ( 0.14) = = = = 0.586 ( 1.5 ) c. The ae of oaion of he dik i gadually inceaed. The coefficien of aic ficion beween he coin and he dik i 0.50. Deeine he linea peed of he coin when i ju begin o lip. Fc = f = μg = μg = ( 0.14 )( 0.5) 9.8 0.88 = d. If he expeien in pa (c) wee epeaed wih a econd, idenical coin glued o he op of he fi coin, how would hi affec he anwe o pa (c)? Explain you eaoning. Thee would be no change. The a cancel ou. 193
Fo 1995: Pa of he ack of an aueen pak olle coae i haped a hown below. A afey ba i oiened lengh-wie along he op of each ca. In one olle coae ca, a all 0.10 kiloga ball i upended fo hi ba by a ho lengh of ligh, inexenible ing. a. Iniially, he ca i a e a poin A. i. On he diaga o he igh, daw and label all he foce acing on he 0.10-kiloga ball. ii. Calculae he enion in he ing. g = 0 = g = 0.10 kg 9.8 = 0.98 N The ca i hen acceleaed hoizonally, goe up a 30 incline, goe down a 30 incline, and hen goe aound a eical cicula loop of adiu 5 ee. Fo each of he fou iuaion decibed in pa (B) o (E), do all hee of he following. In each iuaion, aue ha he ball ha opped winging back and foh. 1) Deeine he hoizonal coponen T h of he enion in he ing in newon and ecod you anwe in he pace poided. ) Deeine he eical coponen T of he enion in he ing in newon and ecod you anwe in he pace poided. 3) Show on he adjacen diaga he appoxiae diecion of he ing wih epec o he eical. The dahed line how he eical in each iuaion. b. The ca i a poin B oing hoizonally o he igh wih an acceleaion of 5.0 /. Th = a = 0.10 kg 5.0 0.49 = T = g = 0.10 kg 9.8 0.98 = c. The ca i a poin C and i being pulled up he 30 incline wih a conan peed of 30 /. T = 0 h T = g = 0.10 kg 9.8 0.98 = d. The ca i a poin D oing down he incline wih an acceleaion of 5.0 /. o Th = acoθ = 0.10 kg 5.0 co 30 = 0.43 N g 194
The u of he foce in he y diecion u equal g T = a T = g a The acceleaion in he eical diecion i a θ a ainθ. a V. Theefoe: T V T g o TV = g ainθ = 0.10 kg 9.8 0.10 kg 5.0 in 30 0.73 N = e. The ca i a poin E oing upide down wih an inananeou peed of 5 / and no angenial acceleaion a he op of he eical loop of adiu 5. T H = 0 In he eical diecion, we hae a cenipeal F C, he enion, and he weigh of he ball. The cenipeal foce u equal he u of he ohe wo foce. FC = TV g TV = FC g = g ( 5 ) 0.10 kg 5 TV = ( 0.10 kg) 9.8 T 1.5 V = N 195
Moe Foolih Tiia: A jogge' heel ike he gound 1,500 ie pe ile. A lifeie upply of all he iain you need ha a a of only abou 5 g (eigh ounce). 3 ou of 4 opoei wea eyeglae. Copue ue, on aeage, blink een ie pe inue. Dak cicle unde he eye ae an inheied ai. The ii ebane conol he aoun of ligh ha ene you eye. In 199, when EuoDiney fi opened in Fance, pak iio bea up oe of he coued chaace (oh pleae no Snow Whie!) becaue a he ie o people had been again he pak being buil. Unil he 1960' en wih long hai wee no allowed o ene Dineyland. Eiffel Towe wa he alle ucue in he wold befoe he conucion of he Epie Sae Building in 1930. A B-5 bobe aiplane cahed ino he 79h floo of he Epie Sae Building on July 8, 1945. Sunbea ha hine down hough cloud ae called cepucula ay. The Apollo 11 Luna Module had only 0 econd of fuel lef when i landed on he oon in 1969. Kie flying i a pofeional po in Thailand. A noal aindop fall a abou 7 ile pe hou. A penny while ha ix finge hole. The huan eye ee eeyhing upide-down, bu he bain un i igh ide up. A hell coniue 1 pecen of an egg' weigh. A ilicon chip a quae-inch quae ha he daa poceing capaciy of he oiginal 1949 ENIAC copue, which occupied a ciy block. All he di fo he foundaion o build he foe Wold Tade Cene in NYC wa duped ino he Hudon Rie o fo he couniy now known a Baey Ciy Pak. Rice pape doe no conain ice o any ice poduc. The heigh of he Eiffel Towe aie a uch a ix inche depending on he epeaue. A Apil 000, Hong Kong had 39,000 faxline - one of he highe ae of buine fax ue in he wold. The only pa of he huan body ha ha no blood upply i he conea. I ake i oxygen diecly fo he ai. 196