A small satellite of inertial mass moves in a circular orbit of radius around Earth…
PHYSICS · Multi-part · A Level · Physics · Fields and their consequences · Fields and their consequences
Problem
A small satellite of inertial mass mi moves in a circular orbit of radius r around Earth (mass M). In the usual analysis, the satellite’s acceleration is provided by Earth’s gravitational attraction.
Answer the questions below about the role played by two potentially distinct masses.
(a) Starting from the equality of the required centripetal force and the gravitational attraction between Earth and the satellite, derive an expression for the satellite’s orbital speed v in terms of G, M, and r, explicitly showing where inertial and gravitational masses appear.
(b) Demonstrate algebraically how the equality of inertial and gravitational mass causes the satellite’s own mass to cancel from the final expression for v.
(c) Suppose instead that the satellite’s gravitational mass were larger than its inertial mass by a fractional amount ε≪1. To first order in ε, determine the fractional change in the orbital period T relative to the standard result. State any approximations you make.