A satellite of mass m moves in a circular orbit of variable radius r about a planet of mass M (m≪M).
(i) Write the expressions for its kinetic energy K(r), potential energy U(r) and total mechanical energy E(r) as functions of r.
(ii) By differentiating your expression for E(r), obtain an expression for the infinitesimal change in total mechanical energy dE when the orbital radius is increased by an amount dr.
(iii) Hence show that the work that must be supplied to slowly raise the orbit from radius r to r+Δr (with Δr≪r) is approximately 2r2GMmΔr.
(iv) Discuss qualitatively how this result explains why successively higher orbits require progressively less additional energy per unit increase in radius.