Don't leave the assignment until the night before. You won't be able to
do it in one evening...
- Follow examples 2.3.3 or 2.3.4; expand the potential in a Taylor series
about the equilibrium position and equate the coefficient to the
quadratic term.
- Try writing
*vx=dx/dt x = d/dt(x*^{2}/2) and integrate to
get *x(t) ~ tan κt* where κ is some constant that you
can calculate.
- Should be straight forward...
- Follow example in section 3.7 or the lecture notes starting on p.85a; ignore terms that are second order in
*b*.
- Calculate
*r* for the nearest 6 atoms in terms of the equilibrium
separation *d* and the displacement *(x,y,z)*. This question
is from the text - follow the advice in the text. Only calculate
*d*^{2}V(r)/dx^{2} - the ones for the *y*
and *z* derivatives are the same by symmetry arguments. Check
your algebra using mathcad or whatever tools you have at your
disposal.
- Very close to the example 4.6.1 which was discussed in class
- Similar to previous question once you introduce a new variable
*u*
such that *x = uv* which makes *z = (u^2-v^2)/2*. At the
bottom of the loop, *u=0*. Use energy convservation to calculate
*du/dt*. Calculate *d*^{2}z/dt^{2} at the
bottom of the loop and consider the z-component of the equations of
motion to get the reaction force.

Send e-mail if you get totally stuck.