Assigned Jan 29 2008, Due Feb 5 2008
  9.2 (two sketches required, 
       draw them in "top down" fashion,
       with the rotation in the plane of the paper.) 
  9.14, 9.15

  Coriolis Force Problem 1) 
   An aircraft is flying 800 km/h in latitude 55 degrees N
   (theta = 90 degrees - latitude).
   Find the angle through which it must tilt its wings to
   compensate for the horizontal component of the Coriolis
   force.

  
  Coriolis Force Problem 2)
   Coriolis force for a projectile:  If a projectile is
   fired due east from a point on the Earth's surface that
   is at latitude lambda (colatitude = pi/2 - lambda) with
   a velocity of magnitude V0, and at an the angle of 
   inclination with respect to the horizontal of alpha,
   show that the lateral deflection when the projecile
   strikes the ground is:
   
    d = 4 (V0^3/g^2) * [Omega*sin(lambda)*sin^2(alpha)*cos(alpha)]

   where Omega is the rotation frequency of the Earth.

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Assigned Feb 7 2008 Due Feb 14 2008 
10.14, 
10.16 (angular momentum conserved before it hits, energy conserved after)
10.22, 10.36
(delayed to Feb 19)

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Assigned Feb 20 2008, Due Feb 28 2008

10.42
10.44
10.50
10.52 (the problem refers to free precession rate found in 
       Eq. 10.06, it should say Eq. 10.96)

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RIDGID BODY ROTATION Take Home Problems:
Assigned Feb 25 2008    Due March 4 2008


1) A double-mass pendulum is made of a massless rod
of length b, with two masses attached to it.  The
first mass (m_1) is attached at the end (at b).  

The second mass (m_2) is attached halfway up (at b/2).  

The top of the rod (at 0)  is attached to the ceiling, 
free to swing any way it wants.

Take the body fixed axes as 
e_1: along the length of the pendulum
e_2: in the plane, perpendicular to e_1
e_3: out of the page.

a) Find the moment of inertia tensor.

b) Now assume the pendulum only swings in the plane
of the page (making an angle \theta with the vertical).
Write down the angular velocity (\omega) vector, and
then the angular momentum (L) vector.  
(remember, L=  I \omega).

c) Gravity acts in the "down" direction.  Write
down the gravity vector in e_1,e_2,e_3 basis.
Now calculate the torque (r x F) due to gravity
and set it equal to L-dot.  (It might be less messy
to do the two masses separately, and then add).

d) Write down a differential equation for small oscillations
of this pendulum (ie, sin(theta) is approx = theta).

e) [extra credit] You can of course do this problem with
Lagrangian Mechanics.  Write down the Lagrangian, and show
that you can get the same equation of motion (in theta)
as in parts (c, d).

---------

Assigned March 6, 2008, Due March 13, 2008
13.6
13.10
13.12
13.26
---------------------------------------------
Assigned March 13, Due March 25
11.18
11.22
11.24  
11.26
11.28
---------------------------------------------
Assigned March 27, Due April 3
14.8
14.11
14.14
14.18
14.20: You can determine r_min from angular momentum L
       and energy of the projectile.  Recall the angular
       momentum relations we used in lecture.  
---------------------------------------------
=============================================
HAMILTONIAN, NORMAL MODES, SCATTERING 
Take home exam due thurs April 17

See figures for probs 1, 2 here:
http://www.physics.rutgers.edu/ugrad/382s08/382_exam_fig.jpg

1) A massless string passes over a small massless pulley
and carries mass 2m on one end.  On the other end of the
string is a mass m.  Below this mass m, is another mass
m, supported by a spring with spring constant k.

Find the Hamiltonian for this system, using x, the
distance of the first mass m beneath the pulley. 
and y, the extension of the spring, as generalized
coordinates.

If the system is released from rest with the spring
unextended,  find the positions of the masses at any
later time.
  
------
2) A spring of negligible mass and spring constant k,
supports a mass m.  Beneath this mass is attached a
second identical spring and another mass m.  

Use x and y as displacements of the masses from 
equilibrium.  Consider only vertical motion. 

How many normal modes are there?

What are the frequencies of oscillation for the
various normal modes?
---------
A beam of particles strikes a wall containing
2*10^29 atoms/m^3. (this is density or mass/V)
Each atom behaves as a hard sphere of radius
3*10^-15m. 

Find the thickness of the wall such that exactly
half the incident particles go through without
scattering.

What thickness would be needed to stop all but
one particle in 10^6?
==========================================
Assigned April 22 Due April 29
15.8
15.26
15.28
15.50
15.76