Physics 2A Chapter 11 - Universal Gravitation Fall 2017

Transcription

1 Physcs A Chapte - Unvesal Gavtaton Fall 07 hese notes ae ve pages. A quck summay: he text boxes n the notes contan the esults that wll compse the toolbox o Chapte. hee ae thee sectons: the law o gavtaton, ccula obts, and potental enegy. hee ae no new ules o Chapte, but t s mpotant to use the pope tools o each poblem. Unvesal Gavtaton In hs yea away om college n the ealy 660 s, Isaac Newton nvented calculus, ceated the Laws o Moton, and deved the acceleaton o an object n ccula moton. He then tuned hs attenton skywad... and ealzed that the moon was n ccula moton. (echncally, the moon s path s ellptcal, as ae the obtal paths o the planets. Newton knew ths, and he developed the necessay explanaton o ths ellptcal moton. Ccula moton s a specal, smpled case o ellptcal moton.) Accodng to hs Laws o Moton, the change n the moon s decton meant the moon was acceleatng. hs acceleaton must be caused by a oce, and the oce actng on the moon must come om somethng extenal (.e. the moon cannot apply a oce to tsel.) Newton concluded that the Eath must be pullng on the moon wth a oce that does not eque physcal contact. He logcally assumed that ths oce must be the same oce o gavty that we ae amla wth nea the suace o the Eath. Newton was the st peson n hstoy to undestand that gavty s not estcted to the suace o the Eath. He developed the law o unvesal gavtaton, whch govens not only the nteacton between the Eath and moon, but between the sun and planets, and, n act, all objects. hs law clams that any two objects apply a oce o gavty to each othe. he oce s equal on both objects (accodng to Newton s hd Law) and opposte n decton (.e. t pulls the objects towad each othe.) he magntude o the oce s popotonal to the mass o each object and nvesely popotonal to the dstance between them squaed. We can wte ths algebacally as: F Law o Unvesal Gavtaton Whee G s the unvesal gavtaton constant: G 6.67 x 0 - N-m /kg And: M and m ae the masses o the two objects. s the dstance between the centes o the two objects. Page o 5

2 Physcs A Chapte - Unvesal Gavtaton Fall 07 We can now consde how ths new denton connects wth ou pevous denton o gavty. Nea the suace o the Eath, we st used g (n Chapte ) as the acceleaton due to gavty. We then (n Chapte 4), acknowledged that gavty must be a oce (because t causes acceleaton) and we dened the oce o gavty as: F mg hs pevous denton s stll vald, and we can use t wth ou new denton o gavty. I we use M o the mass o a planet, whethe Eath o any othe planet, and m o an object nea the planet, we can set the two expessons o oce equal and we get: o: mg g Acceleaton due to gavty Note that: hs expesson allows o a way to calculate g o any planet at any pont. he pont o whch we calculate g can be on the suace o away om the planet. he dstance s om the cente o the planet to the pont we ae nteested n. o calculate g at the suace o the planet, s the adus o the planet. he ony o Newton s dscovey was that whle he knew the adus o the Eath and how stong g was at the suace o the Eath, he dd not know G no the mass o the Eath. In essence, n Newton s tme thee wee two unknowns n ths expesson. he value o G was unknown untl Btsh scentst Heny Cavendsh, n the late 700 s, calculated t om data he collected n an expement that nvolved the attacton o lead sphees. Cavendsh subsequently calculated the mass o the Eath and ceated an oppotunty o astonomes to detemne the masses o the othe planets and the sun. Ccula Obts As pevously mentoned, nealy all obtal paths o natual objects (n contast to man-made satelltes) ae ellptcal. But consdeng the smple case o ccula obts has seveal advantages. Fst, the algeba s much smple. Second, the concepts nvolved ae smla and ease to study o ccula obts. Page o 5

3 Physcs A Chapte - Unvesal Gavtaton Fall 07 hd, many obts o man-made satelltes actually ae ccula, so the study o ccula obts does have pactcal applcaton. he study o ccula obts combnes two smple pncples: the acceleaton o an object n ccula moton s towad the cente o the ccle (wth a coespondng oce pullng the object that decton), and the oce actng on an obtng object s the oce o gavty. We can expess these two deas as: F mv F and hese two oces must be equal to each othe, so we can set the expessons equal. Beoe we do ths, a vey mpotant obsevaton s necessay: In the st expesson, s the adus o the ccula path. In the second expesson, s the dstance between the centes o the two objects. In geneal, these ae not the same thng! he cente o an obtng object s path s the cente o mass o the two objects, not the cente o the lage object. hs leads to two possble stuatons:. I M s vey lage (at least 000 tmes) compaed to m, then the can be consdeed the same.. I M s not vey lage compaed to m, then s deent n these two expessons. Note that the st stuaton apples to all man-made satelltes n obt aound the Eath, but t does not apply to the nteacton o the Eath and moon; the mass o Eath s only about 80 tmes the mass o the moon.) Fo ths specal stuaton, I M s vey lage compaed to m, e.g. o man-made satelltes n obt aound a planet: mv v o We can also wte the speed n tems o the dstance and the peod,.e. the tme the satellte takes to complete one obt: v π and π Page 3 o 5

4 Physcs A Chapte - Unvesal Gavtaton Fall 07 o: 3 4π In summay, Ccula Obts, M s vey lage compaed to m π v 3 4π v Notce that thee ae thee paametes o the obtng object: v s the speed o the obtng object s the adus o the obt s the peod, the tme o the object to complete one obt Also note that M s the mass o the planet (o moon, o sun...) beng obted. he mass o the obtng object (.e. the satellte) does not appea n these expessons, as t was elmnated n the algeba above. Potental Enegy In Chapte 7, we dened gavtatonal potental enegy as the negatve o the wok done by gavty. When an object ses, gavty does negatve wok on the object, takng knetc enegy om the object. Gavty then stoes ths enegy as potental enegy, whch t etuns to the object late (when the object alls.) Snce the oce o gavty s constant nea the suace o the Eath, we could wte the wok done by gavty and the denton o potental enegy as: W g -mg h U g mg h U g mgh We can now do the same thng o objects that ae not nea the suace o the Eath. he man deence o these objects s that the oce o gavty s not constant; snce the oce changes as the object moves close to Eath o uthe om Eath, the wok done by gavty must be taken n vey small bts. he total wok done by gavty when an object moves towad o away om Eath can be ound by addng all the vey small bts along the path o the object. As gavty acts n a adal decton (.e. dectly towad the cente o the Eath) we can dene the ntal and nal poston o an object as and and small steps between these two postons as d (.e. a tny bt o dstance n the decton.) We can then wte: Page 4 o 5

5 Physcs A Chapte - Unvesal Gavtaton Fall 07 tny bt o wok done by gavty: dw Fg d d he total wok done by gavty s just the ntegal o ths expesson,.e. the sum o all the tny bts : W g d he change n potental enegy s the negatve o the wok done by gavty: U g Fom ths we can dene a new expesson o potental enegy that depends on (nstead o h ): U g Gavtatonal Potental Enegy Impotant notes: wll neve be zeo! Remembe that s the dstance between the objects centes. U appoaches zeo as gets vey lage,.e. appoaches nnty. I the objects ae vey a away, then the potental enegy s essentally zeo. U s always negatve, whch s stange... but t woks. I the objects ae vey a away, U s zeo... and t deceases (.e. negatve) as they get close. Enegy poblems n Chapte ae exactly the same as those n othe chaptes. hese knd o poblems nvolve the speed and poston o an object. he ules ae the same: daw two pctues (ntal and nal), label nomaton (speed and poston), and wte the enegy equaton (ntal K + U, plus the wok done on the object, equals nal K + U.) he only deence s the expesson o U. Page 5 o 5