Gravitational attraction in newtons F = G m₁m₂/r² G = 6.674e-11 m³/kgs² m₁ and m₂ are the masses of the two objects in kg r is the distance in meters between their centers (center of mass) earth radius 6,371 km = 6.37e6 meters earth mass M 5.974e24 kg earth GM = 3.987e14 moon radius 1,737,000 m moon mass 7.35e22 kg moon orbit 3.844e8 m (center to center) sun mass 1.9891e30 kg Gravitational field strength in N/kg g = G m/r² change in weight with change in distance dW = –2GMm/r³dr a = Gm/r² a is acceleration from mass attraction in m/s² m is mass of earth or other body generating the a r is radius of body in meters. G = 6.67e-11 m³/kgs² earth radius 6,371 km = 6.37e6 meters earth mass M 5.974e24 kg earth GM = 3.987e14 at earth surface, this reduces to g = GM/R² R is radius of earth M is mass of earth g = 9.8 m/s² earth, variation of g with height g = GM/r² g is acceleration from earth attraction in m/s² earth radius 6,371 km = 6.37e6 meters earth mass M 5.97e24 kg earth GM = 3.98e14 g = (3.98e14) / r² Circular orbits arise whenever the gravitational force on a satellite equals the centripetal force needed to move it with uniform circular motion. Fc = Fg Gravitational Force Fg = GMm/r² mV²/r = GMm/r² where V is orbital velocity r is distance to center of earth G is gravitational constant M is mass of earth m is mass of satellite V² = GM/r KE = ½mV² KE = ½mGM/r = ½GmM/r Gravitational Potential energy Ug = –GmM/r KE = –½Ug The kinetic energy of a satellite in a circular orbit is half its gravitational energy and is positive instead of negative. When Ug and KE are combined, their total is half the gravitational potential energy. E = KE + Ug = –½Ug + Ug = ½Ug = –GmM/2r r = R + h R is radius of earth E = –GmM/2(R+h) Average velocity v of a falling object that has travelled distance d (averaged over time): v = √(2gd) Instantaneous velocity v of a falling object that has travelled distance d on a planet with mass m and radius r (used for large fall distances where g can change significantly) v = √(2Gm((1/r) – (1/(r+h))) Instantaneous velocity v of a falling object that has travelled distance v on a planet with mass m, with the combined radius of the planet and altitude of the falling object being r, this equation is used for larger radii where g is smaller than standard g at the surface of Earth, but assumes a small distance of fall, so the change in g is small and relatively constant. v = √(2Gmd/r²) The time t taken for an object to fall from a height r to a height x, measured from the centers of the two bodies, is given by: t = (r^3/2)[ arccos√(x/r) + √((x/r)–(x²/r²)) ] / [√(2μ)] μ = G(m₁ + m₂) μ is the sum of the standard gravitational parameters of the two bodies. This equation should be used whenever there is a significant difference in the gravitational acceleration during the fall. Variation of g with altitude (h) on earth, in km g = (3.98e8)/(h+6371)² 0 km, g = 9.80 m/s² 10 km, g = 9.78 m/s² 20 km, g = 9.75 50 km, g = 9.66 100 km, g = 9.51 200 km, g = 9.23 500 km, g = 8.44 1000 km, g = 7.33 2000 km, g = 5.69 2650 km, g = 4.9 (g/2) 5000 km, g = 3.08 10000 km, g = 1.49 400000 km (moon) g = 0.002 1000000 km, g = 0.00039 10000000 km, g = 4.0e-6 100000000 km, g = 4.0e-8 1000000000 km, g = 4.0e-10 10000000000 km, g = 4.0e-12 |
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