Assignment 11

Problem 1:

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.

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Solution:

Problem 2:

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

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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

Solution:

Problem 3:

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).

Solution:

Problem 4:

imageA 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?

Solution:

Problem 5:

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.)

Solution: