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Unformatted text preview: Gravity in a nut shell
m For a small object and a large object, the things you need to know to figure out weight, escape, and orbit questions are: weight escape and orbit questions are: M F = GmM/r2 / PE = GmM/r KE = mv2 Force Centrip= mv2/r The weight of an object sitting on the Earth. How much does Tom Brady weigh on the field? F = GmM/R2 G = 6.673 1011 m3 kg1 s2 m R M Thomas Edward Patrick "Tom" Brady, Jr. American football quarterback for the New England Patriots Weight: 225 lb (102 kg = m ) Mother Earth: M = 6 x 1024 kg, R = 6400 km = 6.4 x 106 m. g,
Gravitational force on Tom Brady standing on football field: F GmM/R (6.673 F = GmM/R2 = (6.673 1011 x 102 x 6 x 1024)/(6.4 x 106)2 102 x 6 x 10 )/(6.4 x 10 Google version: = (6.673e11*102*6e24)/(6.4e6)^2 = 997 newtons. = 997 newtons How many pounds in a newton? 1 pound force = 4.45 newtons So, Brady weighs 224 lbs! (Wikipedia is right) What about escape velocity? Can Michael Jordan escape the court? Energy to remove an object from the surface of a planet = kinetic energy of the object escaping GmM/R = mvesc2 Jordan s m crosses out so: Jordan's m crosses out so: vesc = (2GM/R)
m R M For Earth:
vesc=? ? vesc = (2* 6.673 1011 *6 x 1024 / 6.4 x 106) 1/2
vesc = 11,186 m/s = 11.2 km/s = 7 mps = 25,200 mph Orbiting a Planet How fast does Bolt have to run? How fast does Bolt have to run?
In this case, we need to balance centripetal and gravitational ti t l d it ti l forces: FC = FG = mv2/r = GmM/r2 = F /r = GmM/r Bolt's m crosses out, and we want him run, jump, and orbit j p only a few inches off the ground, so r is approximately R (r ~ R) and: V = (GM/R)
orbit v m r R M vorbit = (6.673 1011 *6 x 1024 / 6.4 x 106)1/2
vorbit = 7 909 m/s = 7.91 km/s = 4.94 mps = 17,800 mph If Bolt takes off runing and orbits the planet, g how long do we have to wait until we see him coming up behind us? ? We just need to know how far he has We just need to know how far he has v to go, and how fast he is orbiting. The distance is just the circumference: C = 2R = 2*6400 = 40,200 km His speed is v = 7.91 km/s So the time t = C/v = 5100 seconds or about 85 minutes. or about 85 minutes p The space station is a little farther from earth, and takes about 90 minutes to orbit the earth.
m r R M One final Zen Moment
The time it takes (period) for a small object to orbit a planet is the The time it takes (period) for a small object to orbit a planet is the circumference over the speed: P = C/v We know that C = 2r, and v = (GM/r)1/2 , ( /) So P = C/v = 2r/(GM/r)1/2 = 2r*r1/2/(GM)1/2 = (r3)1/2*2/(GM)1/2 If I square both sides, then P2 = r3 * (2)2/(GM) r (2)
This is just Kepler's Law for any orbiting object! If you plug in the mass of the sun for M, and use r = 1 AU, you will find that P = 1 year! f th f M d 1 AU ill fi d th t P 1 ! Gravity in a nut shell Gravity in a nut shell Ida: M = 4.2 x 1016 kg M m Dactyl: m = 3 x 109 kg Dactyl is in an orbit about 90 km from Ida. From what we presented above, you should be able to find the weight of Dactyl if it sat on Ida, how long it takes Dactyl h i h f l if i d h l i k l to orbit Ida, and how fast it needs to go to escape Ida. ...
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This note was uploaded on 03/18/2012 for the course ASTRO AST1002 taught by Professor Jamesbrooks during the Spring '12 term at Florida State College.
- Spring '12