Using Newton's law of universal gravitation, show that T^2 is proportional to r^3 (where T is the orbital period of a planet around a star, and r is the distance between them).

Newton's law of gravitation is: FG=(GMm)/(r2).First of all, it's a good idea to draw a diagram of the planet and star, labelling the directions of the centripetal force and and the planet's velocity in particular, along with anything else that helps visualise the question. We know that the equation for centripetal force is FC=mω2r (from circular motion). Since this centripetal force FC and the gravitational force FG point in the same direction (from the planet to the star), we can equate them!
This gives us: (GMm)/(r2) = mω2rSubstituting in ω=2π/T, we get: (GMm)/(r2) = (4π2mr)/(T2)We can see that the two 'm's cancel out, and the 'r's combine to make r3.Do a bit of rearranging: T2 =(4π2r3)/(GM)There it is! T2 is proportional to r3; this is known as Kepler's 3rd Law of planetary motion.

JB
Answered by Jake B. Physics tutor

3383 Views

See similar Physics A Level tutors

Related Physics A Level answers

All answers ▸

If photons of wavelength 0.1nm are incident on a 2m x 2m Solar Panel at a rate of 2.51x10^15s^-1, calculate the intensity, I, of the photons on the Solar Panel.


Why is the index of refraction important for light passing between two materials?


3 resistors, R1, R2 and R3 are attached in parallel across a 6V cell with resistances 3, 4 and 5 Ohms respectively. Calculate the current across each resistor.


In one second a mass of 210 kg of air enters at A. The speed of this mass of air increases by 570 m s–1 as it passes through the engine. Calculate the force that the air exerts on the engine.


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences