A particle of mass m is located on the outside of a uniform sphere of mass M at a distance r from its center. Find the potential energy of gravitational interaction between the particle and the sphere.

An artificial satellite (mass m) of a planet (mass M) revolves in a circular orbit whose radius is n times the radius R of the planet. The satellite experiences a slight resistance due to cosmic dust. The resistance force is dependent on the satellite velocity as F = a v², where a is a constant. Calculate how long the satellite will remain in orbit before it falls on to the planet's surface.

Assume the Earth to be a uniform sphere of mass M and radius R. Find the pressure P inside the sphere caused by the gravitational compression as a function of r, the distance from the center.