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.

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.

A length of wire of mass M is bent into an arc of a circle of radius R, subtending an angle of ø at the center. A particle of mass m is placed at the center. What force does the wire apply on the particle?