How would I derive Kepler's third law from Newton's law of gravitation and the equations of circular motion?

Kepler's third law states that the square of the period of the orbit is directly proportional to the cube of the radius of the orbit (T^2=kr^3) where r is some constant to be determined. This can be determined using:

Newton's law of gravitation: F=GMm/r^2 Centripetal Force: F=mw^2r, where w is the angular velocity in rad/s.

By equating these 2 equations and cancelling out any terms possible we arrive at GM=r^3w^2. The angular velocity can be described as the angle a body has travelled through in a period of time. Assuming a full circular orbit this would be equal to 2pi radians in a period of T. Therefore w=2pi/T. This can be substituted in to obtain T^2=(4*pi^2/GM)*r^3. Therefore the constant of proportionality equals 4pi^2/GM.

MW
Answered by Matthew W. Physics tutor

7947 Views

See similar Physics A Level tutors

Related Physics A Level answers

All answers ▸

What is the Rutherford scattering experiment and what did it tell us about the nature of the atom?


If a stationary observer sees a ship moving relativistically (near the speed of light), will it appear contracted or enlarged? And by how much.


What is the angular velocity of a person standing on the surface of the earth. Give your answer in radians per second


How can an object be accelerating when it's velocity is constant, and how does centripetal acceleration work.


We're here to help

contact us iconContact ustelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

MyTutor is part of the IXL family of brands:

© 2025 by IXL Learning