Homework #4

Due by 5pm Wednesday September 30^th. YES that is peculiar. I am overlapping this homework with the mid-term because it allows me to set a wider range of problems and to reduce the leap between this homework and the next. It does, however, mean that I have to place an extra amount of trust in you folk since he usual collaboration is allowed/encouraged on the homework but I am the only persom you may talk to about the exam.

NOTE there will be no new homework next week. Instead we have our first mid-term! The mid-term will be open-book take-home with limited help from me and will include material from chapters 1, 2, 3, and 6,

Problems mostly from the book

1. 3-43) A classic kind of show that this gives SHO for small oscillations problem.
2. Consider a lightly damped, unit frequency SHO, β = 0.01. Find and plot the output when the oscillator is driven by a square wave acceleration (Force/mass) given by
   F(T)/m = +1 {2nπ <= t < (2n+1)π} and F(t)/m = -1 {(2n+1)π <= t < 2nπ}.
   Note that this driving force is periodic with period 1.
   I would like you to try to solve this both analytically and numerically (I suggest Maple) and to compare your results. (Worth two problems.)
3. 6-2) You know the answer, now prove it!
4. 6-4) And, as you know, a geodesic is the shortest distance curve. Hint, with a suitable choice of variables you should be able to make this problem look exactly like the previous one and thus use the previous result.
5. 6-5) This is another version of the soap bubble surface.
6. 6-6) Another one to make you work through a chapter example. In this problem you ned to realise that the initial conditions are different from those in the book example and figure out how that changes things. Hint: you can simplify the integral with half-angle formulae.
7. 6-11) To start you thinking about constraints.