Assignment 2

Due Monday, February 23, in class.
Show your work. Have fun.

1) Fun with complex numbers.  Assume j = sqrt(-1).  
   Show your work, and reduce/convert to simplest form:

   a)  (1-j)2 = 

   b)  (1-j)/(1+j) =

   c)  (-1)-1/4 =

   d)  |j2| / |(1-j)2| = abs(j^2) / abs((1-j)^2) = 

   e)  ln(ej) =

   f)  (j-j)j =
	(NOTE: (j to the minus j) to the j)

2) Fun with rectangular/polar conversion.  Convert these
   to polar, and plot them (except (c))

   a)  1 - j =

   b)  1 + (√ 3)j = 1 + sqrt(3)j =
   
   c)  x - jy =

   d)  (-1)1/4

   e)  ejΠ - 1 = exp(jPI) - 1 = 

   f)  sin(θ) + jcos(θ) = sin(theta) + jcos(theta) = 
  
3)  Use complex exponential identities to show that 

	cos(A) + cos(B) = 2 cos((A-B)/2)cos((A+B)/2)

    How does this relate to question 6 of Assignment 1?

4)  Show that Equation 4.3 from the Cook book solves the differential equation 
    for the mass/spring/damper.

5)  Find a slinky, bungy cord, big rubber band, or similar springy object.  
    Find an object of suitable (you'll see later) weight.  
    Find a timing device (stopwatch or watch with second hand).
    Attach one end of the spring to something solid and 
    relatively unmoveable (like a doorframe, your sleeping roommate,
    etc.).  Attach your suitably weighted object to the bottom
    of the spring.  Pull your object down, stretching the spring some.
    Let go, and count how many oscillations happen in a timed
    interval (say, 15 or 30 seconds).  Pull it down a measured
    amount, like 12 inches, let go, and time how long it takes 
    until the oscillations damp to 1/10 of the original amplitude
    (1.2 or so inches for an initial displacement of 12 inches).
    Try to measure or somehow deduce (look it up on the web if it's
    an object you can buy) the weight of the object.  From that, 
    your measured frequency of oscillation, and your measured 
    damping time, calculate the spring constant and damping constant 
    of your mass/spring/damper system.  Show your work.

6)  Change the constants in the masprdmp.c code to reflect your
    estimated/calculated constants (from (5) above).  Change the 
    code so that it synthesizes for a longer time (like a minute).  
    Recompile, run, inspect the resulting wave in a sound editor and 
    see if it all makes sense.  Turn in a plot of your synthesized wave
    with the axes clearly labeled.

7)  Disconnect your spring from the rigid object, and suspend your
    mass from the spring from your hand.  Move your hand up and down
    at various rates, slow, faster, really fast, noisily random.
    Report what you observe.  Is there a rate at which the mass
    oscillates in synchrony with your hand?  Why would this be?

8)  Do something really interesting with this MATLAB Code
    by Tae Hong Park, and with MATLAB in general.  
    Make a 17 second musical statement with it.
    Submit as usual and we'll listen and discuss.