Circular Motion: Kepler's Third Law

Gravity & Circular Satellite Orbits As the moon orbits the Earth in a circular path, it is undergoing circular motion For it to be undergoing circular motion, there must be a centripetal force acting on it, keeping it in motion This force is gravity

Equating the formula for gravity and the equation for centripetal force: GMm/d² = mv²/r The distance from the Earth to the Moon is the same as the radius of rotation (=R) GMm/R = mv² v² = GM/R (dividing across by the mass of the planet) v² ∝ 1/R (as G and M are both constants)

Kepler's Third LawThe square of the periodic time of a satellite's orbit around a planet is proportional to the cube of the radius of the orbit (including the radius of the planet) and inversely proportional to the mass of the planet

T² = 4π²R³/GM  T = periodic time R = radius of orbit G = universal constant of gravitation M = mass of planet

Derivation of formula Gravitational force = centripetal force GMm/R² = mω²R GM/R² = ω²R ... (divide by mass) GM/R² = (2π/T)²R ...  (substitute ω with 2π/T) GM/R² = 4π²R/T² ... (multiplying out the brackets) T² = 4π²R³/GM

New Page