A particle of mass 2m is attached to a rigid support by a spring with a force constant k. At equilibrium, the spring hangs vertically downward. A particle of mass m is attached to this mass-spring combination by and identical spring with a force constant k. Find the eigenfrequencies and describe the normal modes for this system.
A star of mass M and radius R is moving with
constant velocity v through a cloud of particles of density ρ.
If all the particles which collide with the star are trapped by it, show that the mass of the star will increase at a rate
dM/dt = πρv(R2 + 2GMR/v2).
A satellite is in a circular orbit of radius r around an airless spherical planet of radius R. An asteroid of equal mass falls radially towards the planet, starting at zero velocity from a very large distance. The satellite and the asteroid collide inelastically and stick together, moving in a new orbit that just misses the planet's surface. What was the radius r of the satellite's original circular orbit in terms of R?