module 10 gravitation

law of gravitation

F_g = G_nm1m2 / r²

kepler’s third law for binary stars

T² = 4π²/G(m₁+m₂)r³

kepler’s first law

planets move in ellipses with the Sun at one focus, but assume Earth’s is circular for this class

kepler’s second law

angular momentum is conserved in planetary motion (sweeps out constant areas in constant times)

kepler’s third law

T² = (4π²/G_nm_s)r_p³

for elliptical orbits, r is the

semi-major axis - ½ of the long dimension

with years and AU, the equation becomes

(time in years)² = (radius in AU)³

binary system kepler’s third law

T² = d³/(m1 + m2), where d = r₁ + r₂

velocity of a satellite

V = √Gm_e/r = √gr

gravitational potential energy as a function of radius

-Gm_sm/r

gravitational potential energy close to Earth (what we’re familiar with)

= mgy = GM_em/R_E² * y

total energy in orbital motion

½ mv_f² - GmM/r_f = -GmM/2r

grav potential energy in planetary and satellite problems is always

negative

highest speed of object in an elliptical orbit occurs at

perihelion - closest point of orbit

lowest speed of object in an elliptical orbit occurs at

aphelion - furthest point of orbit

most positive potential energy of orbital system occurs at

aphelion

highest kinetic energy of orbiting object occurs at

perihelion

total energy of an orbital system is

the same at all positions

escape speed =

√2GM_E/R_E (for earth)

schwarzschild radius for event horizon

2GM/c²

when object is thrown up, its initial kinetic energy becomes

the INCREASE in grav potential energy (GMm/R - GMm/R+h)