TE Enriched Sample (E)

Orbital motion under gravity 65 3.2 A Circular orbits Let us derive Kepler’s third law for circular orbits using Newton’s law of universal gravitation and Newton’s second law of motion. For a planet of mass m moving around the Sun of mass M in a circular orbit of radius r , the gravitational force acting on it is F G r Mm 2 = This force provides the centripetal force for the circular motion of the planet. Therefore, G r Mm r mv 2 2 = ( MC: DSE 2020 Q1.2)( SQ: DSE 2013 Q1b, 2017 Q1a, 2019 Q1aii) The Sun and the planet attract each other with equal but opposite forces. However, the motion of the Sun is negligible since it is much more massive than the planet. Hence the speed v of the planet is v r GM = orbital speed The orbital period T is T v r r GM r GM r 2 2 2 3 # r r r = = = Taking the square of both sides of the equation, we have T GM r 4 2 2 3 r = This is actually Kepler’s third law T 2 = ka 3 , with k GM 4 2 r = . B Elliptical orbits Kepler’s third law for elliptical orbits is similar to that for circular orbits. The difference lies in replacing the orbital radius r with the semi-major axis a : Recall that the semi-major axis a reduces to the radius r for a circular orbit. T GM a 4 2 2 3 = r SI units must be used when applying this formula while AU and years are used when applying T 2 = a 3 . Fig. 3.11 A planet moving around the Sun in a circular orbit v GMm r r 2 planet (mass m ) Sun (mass M ) circular orbit F = Teaching notes The orbital period = circumference of orbit orbital speed (Ref: DSE 2012 Q1.3) Teaching notes Ss should be able to derive Kepler's third law for circular orbits from centripetal force (or acceleration) equation. This includes more complicated cases, e.g. changing orbit or a system with many planets (DSE 2019 Q1aii, 2020 Q1.2). Sample © United Prime Educational Publishing (HK) Limited, Pearson Education Asia Limited 2023 All rights reserved; no part of this publication may be reproduced, photocopied, recorded or otherwise, without the prior written permission of the Publishers.