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 block of mass M is rigidly connected to a massless circular track of radius a on a frictionless horizontal table as shown.

A particle of mass m is confined to move without friction on the
circular track which is vertical.

(a) Set up the Lagrangian, using θ as one of the coordinates.

(b) Find the equations of motion.

(c) In the limit of small angles, solve the equation of motion
for θ as function of time

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(R^{2} + 2GMR/v^{2}).

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?

A two electron system has (in its center of mass frame) energy E and angular momentum L. What is the closest distance the electrons can approach each other? (Ignore radiation.)