Universal Gravitation

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1 Univesal Gavitation Keple s Thee Laws of Planetay Motion IB 1 Law 1: All planets obit the Sun in elliptical paths with the Sun at one focus. Law : An imaginay line joining any planet to the Sun sweeps out equal aeas in equal time intevals. Law 3: The squae of the obital peiod of any planet is popotional to the cube of its aveage obital adius. T T Fomula 3 k 3 T 3 Newton s Law of Univesal Gavitation Evey paticle in the Univese attacts evey othe paticle with a foce that is diectly popotional to the poduct of the masses and that is invesely popotional to the squae of the distance between them. Two appoximations used in deiving the law: 1. Masses ae consideed to be point masses. Point mass: infinitely small object (adius = 0) whose mass is m. The foce between two spheical masses whose sepaation is lage compaed to thei adii is the same as if the two sphees wee point masses with thei masses concentated at the centes of the sphees. Sun Mean Eath-Sun distance = 1.50 x m Eath Mean adius = 6.96 x 10 8 m Mean adius = 6.37 x 10 6 m 1

2 Fomula m1 m Fg Gm m Fg = 1 Gavitational Constant : G = 6.77 x N m /kg Point mass Extended spheical body IB 1 Re Re 3Re 4Re Newton s Deivation of Keple s Thid Law What povides the centipetal foce fo obital motion? gavitation Deivation F c ma c GMm mv GM v but v T GM T GM 4 T T 4 3 GM Application weighing the Sun T yea days hous s 3 4 GM x10 m 6.67x10 30 MS.0x10 kg accepted value=1.99x10 30 kg M s What is the esultant gavitational foce on the Eath fom the Sun and Moon, as shown below? Aveage Eath-Sun distance = 1.50 x m Aveage Eath-Moon distance = 3.84 x 10 8 m Sun Mass = 1.99 x kg Eath Mass = 5.98 x 10 4 kg Moon Mass = 7.36 x 10 kg

3 Gavitational Field Stength IB 1 Gavitational field stength at a point in a gavitational field: the gavitational foce exeted pe unit mass on a small/point mass Symbol: g Fomula: g = F g / m Units: N/kg (m/s ) Deiving fomula fo gavitational field stength at any point above the suface of a planet g = F g /m g = (GMm/ )/m g = GM/ Point mass Type: vecto Deiving fomula fo gavitational field stength at the suface of a planet 1. What is the gavitational field stength of the Eath at its suface? g = GM/ g o = G M p / R p g at the suface of the Eath g o = G M E /R e. What is the gavitational field stength at an altitude equal to the adius of Eath? Extended spheical body g o g atio g/g o = R E / Re Re 3Re 4Re 3

4 Aveage Eath-Moon distance = 3.84 x 10 8 m IB 1 Eath Mass = 5.98 x 10 4 kg Moon Mass = 7.36 x 10 kg 3. a) What is the esultant gavitational field stength at a point midway between the Eath and Moon? b) What is the esultant gavitational foce acting on a kg space pobe at this location? Aveage Eath-Moon distance = 3.84 x 10 8 m Eath Mass = 5.98 x 10 4 kg Moon Mass = 7.36 x 10 kg 4. a) Is thee a point whee the esultant gavitational field stength of the Eath and Moon is zeo? If so, whee? b) What is the esultant gavitational foce acting on a kg space pobe at this location? 4

5 Gavitational Potential Enegy Diffeence in gavitational potential enegy between any two points in a gavitational field: IB 1 ΔE P = mgδh This diffeence is path independent. 1. Same ΔE P between any two points no matte what path is taken between them.. Wok done in moving a mass between two points in a gavitational field is independent of the path taken. 3. ΔE P is zeo between any two points at the same level no matte what path is taken. 4. ΔE P is zeo fo any closed path (a path that begins and ends at same point). Old fomula fo gavitational potential enegy: E p = mgh Base level: infinity E P = -400 J discuss poblems with definition 1) g vaies above suface ) abitay base level E P = -100 J Gavitational PE at infinity: zeo E P = 0 Gavitational Potential Enegy of a mass at a point in a gavitational field: the wok done in binging a small point mass in fom infinity to that point in the gavitational field Deivation of gavitational potential enegy fomula 5

6 E p = - GMm/ E P = -Gm 1 m / Fomula not valid inside planet Symbol: V Units: J Fomula: Type: scala What is the gavitational potential enegy of a 5000 kg satellite: Potential enegy vs. distance: IB 1 E p at suface E p = -GM p m/r p a) on the suface of the Eath? b) obiting the Eath at an altitude of 00 km? c) How much does the potential enegy of the satellite incease when it is put into this obit? Gavitational Potential Gavitational potential at a point in a gavitational field: wok done pe unit mass to bing a small point mass in fom infinity to that point in the gavitational field Fomulas: Diffeence in gavitational potential: Gavitational Potential vs. distance ΔV = W/m ΔV = ΔE p /m Gavitational potential at a point: V = E p /m V = -GM/ Symbol: V Units: J/kg Type: scala V at suface V o = -GM p /R p 6

7 1. What is the gavitational potential due to the Eath s gavitational field: IB 1 a) at the suface of the Eath? b) At a location thee Eath adii fom the cente of the Eath? c) What is the change in potential in moving fom the suface to this new location? d) What is the minimum amount of enegy needed to lift a 5000 kg satellite to this location?. What is the net gavitational potential at a spot midway between the Eath and the Sun? Sun Eath 3. Deive an expession fo the gavitational potential at the suface of a planet in tems of the gavitational field stength. 7

8 Escape Speed IB 1 Escape Speed: minimum initial speed an object must have at the suface of a planet in ode to escape the gavitational attaction of the planet Tavel to infinity Just make it means velocity is zeo at infinity means E K is zeo at infinity as well as E P E o = E p + E k E f = E p Planet Assumptions: planet is isolated ignoe ai esistance Deivation: Note: 1. Diection of tavel is ielevant ΔE p is path independent. independent of mass of ocket 3. Moe speed (E K ) is needed in eal life since ai fiction is not negligible at lowe altitudes 1. What is the escape speed fo Eath?. If the Eath became a black hole, how lage would it be? 8

9 Satellite Motion IB 1 Peiod of a satellite: Natual Satellites Acceleation of a satellite: Atificial Satellites Obital speed of a satellite: T T 3 k 3 T 3 a c = v / v = π /T a c = 4π /T a c = g = GM/ v = π /T F c ma c GMm mv GM v GM v Weightlessness Fee fall Obital motion Deep space 1. Compae the motion of satellites A and B. A faste, less time (smalle peiod) B slowe, moe time (longe peiod) T /R 3 = constant tue fo all satellites T = kr 3/. What happens to the equied obital speed if: a) the mass of the satellite inceases? Nothing speed is independent of mass b) the satellite is boosted into a highe obit? Satellite would obit at a slowe (tangential) speed 3. What would happen to a satellite if it encounteed appeciable ai esistance? Slow down, dop to lowe obit, and speed up, encounte even moe ai molecules (dense), cycle continues spial to Eath 9

10 Enegy of Obiting Satellites Compae the enegies of the two obiting satellites. IB 1 Gavitational Potential Enegy Kinetic Enegy Total Enegy Enegy Deivations Gaphs of the enegies of an obiting satellite Gavitational Potential Enegy Kinetic Enegy Total Enegy R E Compaisons: A 1500 kg satellite is to be put into obit aound the Eath at an altitude of 00 km. a) How much potential enegy will the satellite have at this altitude? d) What is the obital speed of the satellite? b) How much kinetic enegy will the satellite need to obit at this altitude? e) What is the minimum amount of enegy needed to lift the satellite fom the suface of the Eath to this altitude? c) What is the total amount of enegy the satellite has at this altitude? 10

11 Compaisons IB 1 Equipotential suface: a suface on which the potential is the same eveywhee 1. The gavitational foce does no wok as a mass moves on along equipotential suface.. The wok done in moving a mass between equipotential sufaces is path independent. 3. The wok done in moving a mass along a closed path is zeo. one point mass two point masses 11

12 On the diagam at ight: IB 1 a) Sketch the gavitational field aound the point mass. b) Sketch equipotential sufaces aound the point mass. What is the elationship between the gavitational field and the equipotential sufaces? Pependicula Field lines point in diection of deceasing potential Gavitational Potential Gadient gadient: ate of change with espect to something slope o deivative gavitational potential gadient: the gavitational field is the negative gadient of the gavitational potential with espect to distance Fomula deive g = -ΔV/Δ A B -80 J/kg -70 J/kg What is the aveage gavitational field stength between equipotential sufaces A and B if they ae 5.0 m apat? 1

13 Pactice Questions IB 1 1. a) Calculate the gavitational foce the Sun exets on the Eath b) Compae this to the gavitational foce that the Eath exets on the Sun.. a) Calculate the stength of the gavitational field of the Sun at a location one million kilometes fom the Sun. b) What is the Sun s gavitational foce at this point? 3. a) Calculate the stength of the Sun s gavitational field at the suface of the Eath. b) Explain why the net gavitational field stength at the suface of the Eath can be appoximated as due solely to the Eath s gavitational field. 13

14 IB 1 4. a) Calculate the esultant gavitational field at a spot midway between the Eath and Sun. b) Compae the contibutions fom the Sun and the Eath to this esultant field. c) What is the gavitational foce acting on a 5000 kg space pobe at this location? 5. A 5000kg satellite obits Mas at a distance of 1000 km. a) What is the gavitational potential at the suface of Mas? Mass of Mas: 6.4 x 10 3 kg Mean planetay adius: 3.37 x 10 6 m Mas b) How much gavitational potential enegy does the satellite have on the suface of Mas? c) What is the gavitational potential at obiting altitude? d) How much gavitational potential enegy does the satellite have at this altitude? e) What is the minimum enegy needed to lift the satellite to this altitude? 14

15 IB 1 6. A kg satellite is placed in a low altitude obit. a) If the altitude is sufficiently low, what is the appoximate adius of the satellite s obit? b) Calculate the satellite s obital speed. c) Calculate the obital peiod of the satellite. d) Calculate the gavitational potential enegy of the satellite. e) Calculate the kinetic enegy of the satellite. f) Calculate the total enegy of the satellite. g) What is the minimum amount of enegy needed to lift the satellite into this obit? 15

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