Relationship between the period, the mass of the central body and the radius of the orbit
Sir Isaac Newton formulated his law of gravitation using laws describing planetary motion put forward by a German astronomer, Johannes Kepler, particularly his third law.
LAWS AND PRINCIPLES
Kepler’s Third Law The square of the periodic time of a satellite’s orbit around a planet is propor- tional to the cube of the radius of the orbit (including the radius of the planet) and inversely proportional to the mass of the planet.
The derivation of this formula, using Newton’s law of universal gravitation, is required.
DERIVATION OF FORMULA T2 4P2R3 GM
Derivation: Based on the equation ‘gravitational force centripetal force’ we get:
GMm R2 mv2R
GM R2 v2R
GM R2 a
2p T b
GM R2 4p2R
T2
T2 4p2R3 GM
2 R divide by the mass
substitute v with 2p T
multiplying it out making T2 the subject of the equation
Gravitron amusement ride KEPLER’S THIRD LAW
T2 4p2R3 GM
T periodic time R radius of orbit G universal constant of gravitation
M mass of planet
SAMPLE PROBLEM 6E
Given that the Moon orbits the Earth every 27 days, and the mass of the Earth is 6 1024 kg, calculate the average radius of the Moon’s orbit around the Earth. (Gravitational constant 6.7 1011 N m2 kg2.)
JOHANNES KEPLER, 1571–1630 (GERMAN) He is best known for his three laws of planetary motion. Kepler’s laws provided one of the foundations for Isaac Newton’s theory of universal gravitation.
