Ch 13 Universal Gravitation

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1 Ch 13 Univesal Gavitation

2 Ch 13 Univesal Gavitation Why do celestial objects move the way they do? Keple ( ) Tycho Bahe s assistant, analyzed celestial motion mathematically Galileo ( ) Made celestial obsevations by telescope Newton ( ) Developed Law of Univesal Gavitation

3 Newton s Question What is the elationship between the acceleations of the Apple and the Moon? (The adius of the moon s obit is appoximately 3.85e8 metes; the time fo the moon to obit the eath is appoximately 27.3 days.) a c a

4 Newton s Question What is the elationship between the acceleations of the Apple and the Moon? a c = v 2 v = cicumfeence = 2π time time v = 2π time = 2π(3.85e8m) 27.3days 24hs 1day 3600s 1h a c (moon) = (1.03e3m /s)2 3.85e8m = 2.73e 3m /s2 =1.03e3m /s So, the acceleation of the apple at the suface is 9.80m/s 2, and the acceleation of the moon fa away is 2.73e-3 m/s 2. If we set these up as a atio: a towad the eath = 9.8m /s e 3m /s Newton hypothesized that the foce of eath s gavity was esponsible fo this acceleation, but how does that foce vay with distance? moon = 3.85e8m, and apple = 6.38e6m Ratio of the two distances is 3.85e8m 6.38e6m Ratio 60 1 Conclusion: Does foce possibly vay invesely with distance?

5 Law of Univesal Gavitation Evey paticle in the univese attacts evey othe paticle with a foce that is popotional to the poduct of thei masses, and invesely popotional to the squae of the distance between them. This foce acts along a line joining the two paticles. avity = G m 1 m 2 2 ˆ

6 Weighing the Eath G = Nm 2 /kg 2

7 Example 1 Calculate the mass of the Eath, given that it has a adius of 6.38e6m. (of any object) = G m eath m object 2 m object g = G m eath m object 2 m eath = g2 G = 5.98e24kg

8 Example 2 A 2000 kg space shuttle is obiting the eath at a distance of km above the eath s suface ( eath =6.38e6m). a) What is the acceleation due to eath s gavity acting on the shuttle? b) What is the acceleation due to eath s gavity acting on an astonaut hee? c) What is the weight of the 60 kg astonaut hee? d) What obital velocity must the shuttle have to maintain this distance? e) What distance above the eath is equied to maintain a geosynchonous obit?

9 Example 2 A 2000 kg space shuttle is obiting the eath at a distance of km above the eath s suface ( eath =6.38e6m). a) What is the acceleation due to eath s gavity acting on the shuttle? = F centipetal G m eath m satellite 2 a c = G m eath 2 = m satellite a c a c = 6.672e 11 Nm2 kg 2 a c =1.09m /s e24kg (6.38e6m e7m) 2 b) What is the acceleation due to eath s gavity acting on an astonaut hee? The same

10 Example 2 A 2000 kg space shuttle is obiting the eath at a distance of km above the eath s suface ( eath =6.38e6m). c) What is the weight of the 60 kg astonaut hee? = mg = ma g = (60kg)(1.09m / s 2 ) = 65.4N (about 15 pounds) d) What obital velocity must the shuttle have to maintain this distance? a c = v 2 v =, so v = a c a c v = (6.38e e7)(1.09m /s 2 ) v = 4.57e3m /s

11 Example 2 A 2000 kg space shuttle is obiting the eath at a distance of km above the eath s suface ( eath =6.38e6m). e) What distance above the eath is equied to maintain a geosynchonous obit? Need = F centipetal Need ω satellite = ω eath ω eath = 1ev 24hs = 2π 86400s $ 2π ' v = ω satellite = & ) % 86400s( G m eathm satellite v 2 = m 2 satellite G m eath = 4.23e7m $ 2π ' = & ) % 86400s( 2

12 Keple s 3 Laws 1. All planets move in elliptical obits, with the Sun at one of the focal points. 2. The adius vecto dawn fom the Sun to a planet sweeps out equal aeas in equal time intevals. T whee is the semi-majo axis. 1 3 = T

13 Example 3 A satellite moves in an elliptical obit about a lage body. At aphelion, a distance a, the satellite has a speed of v a. What is the satellite s speed at peihelion, whee it has a distance p fom the body? peihelion L i = L f a mv a = p mv p v p = a p v a aphelion

14 Foce? o Field? Paticle nea a mass expeiences a gavitational foce due to that mass. Paticle in a gavity field expeiences a gavitational foce due to that gavity field. =mg Gavity field g= /m

15 U g when g 9.80? W gavity = U i U f = ΔU ΔU = U f U i = W gavity = x f F dx x i U i - U U f + U U f U i

16 U g (new def.) U f U i = x f F dx x i U f U i = f G M Eath m 2 d i # & U U = GM m 1 f i Eath % $ f 1 i ( '

17 U g, but whee is =0? # U U = GM m 1 f i Eath % $ f 1 i & ( ' As always, we need to choose a position whee the potential enegy U will be 0. Ou custom is to let potential enegy U i =0 at a position i =. Then we can wite U = GM Eathm This is the enegy of the Mm system. It s not just the small mass that has the potential enegy.

18 Example 4 An apple is eleased fom a height of 10,000 m above the suface of the eath. How fast is it taveling ight befoe it hits the suface? (Assume no ai fiction.) U i + K i = U f + K f GMm + 0 = GMm GMm( 1 i + 1 f ) = 1 2 mv 2 v = 2GM( 1 f 1 i ) v = mv 2 # 1 2(6.672e 11)(5.98e24) 6.38e6 1 & % ( $ 6.38e6 +10,000' v = 442m /s

19 U g fo any mass(es) U = GM Eath m U = U i = Gm 1m Gm 2m Gm 1m 3 13

20 Satellites & Enegy E total = K + U i f E total = 1 2 mv 2 + G Mm This is an inteesting esult, because it shows that 1. geate causes smalle v fo elliptical obits (as Keple obseved, and; 2. The enegy of a system can be negative. (???!)

21 Satellites & Enegy Is E total positive, negative, o 0? It depends on v. E = 1 total 2 mv 2 + G Mm i f

22 Enegy Negative? Fo a bound, cicula system, F c =. mv 2 " $ % ' mv 2 # 2& = G Mm 2 = G Mm mv 2 = 1 2 G Mm E = 1 2 mv 2 G Mm E = 1 2 G Mm E = 1 2 G Mm " $ % ' # 2& G Mm = 1 2 U (Note : K = 1 2 U)

23 Example 5 What minimum escape velocity does a satellite need to have to escape Eath s gavity completely? K i + U i = K f + U f 1 2 mv 2 esc G Mm = 0 0 i v esc = 2GM i

24 Gavity between a Paticle & a Lage Mass Case 1. Spheical Shell, with paticle outside the shell = G Mm 2 ˆ fo R

25 Gavity between a Paticle & a Lage Mass Case 2. Spheical Shell, with paticle inside the shell =0 = 0 fo < R Note that the shells is not acting as some sot of gavity shield -- it s just that the sum of all the attactive foces balances out to zeo.

26 Gaphs You should be able to pedict what a gaph of F vs. looks like fo a spheical shell. R

27 Gavity between a Paticle & a Lage Mass Case 3. Spheical Solid, with paticle outside the sphee = G Mm 2 ˆ fo R

28 Gavity between a Paticle & a Lage Mass Case 4. Spheical Solid, with paticle inside the sphee = G Mm 2 ˆ M M = V V, so M M = M = M3 R 3 = G M 3 R 3 2 m 4 3 π3 4 3 πr3 ˆ = G Mm R 3 ˆ

29 Gaphs You should be able to pedict what a gaph of F vs. looks like fo a solid sphee. R

30 Example 6 Two satellites, of masses m and 3m, espectively, ae in the same cicula obit about the Eath's cente, as shown in the diagam above. The Eath has mass M e and adius R e. In this obit, which has a adius of 2R e, the satellites initially move with the same obital speed v o but in opposite diections. a. Calculate the obital speed v o of the satellites in tems of G, M e, and R e. b. Assume that the satellites collide head on and stick togethe. In tems of v o find the speed v of the combination immediately afte the collision. c. Calculate the total mechanical enegy of the system immediately afte the collision in tems of G, m, M e, and R e. Assume that the gavitational potential enegy of an object is defined to be zeo at an infinite distance fom the Eath.

31 Weid stuff to think about... It s the diffeent escape speeds equied fo diffeent planets that explains why some planets have atmosphees and othes don t. Gas molecules have speeds that depend on thei tempeatues: the geate the tempeatue, the geate the aveage speed of the molecules, and the geate the chance is that they ll have a velocity that allows them to escape the planet. Mecuy? No atmosphee Eath? Light molecules gone, heavie molecules emain Jupite? Even hydogen can t escape!

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