Don't leave the assignment until the night before. You won't be able to do it in one evening...
1. 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.
2. Try writing vx=dx/dt x = d/dt(x2/2) and integrate to get x(t) ~ tan κt where κ is some constant that you can calculate.
3. Should be straight forward...
4. Follow example in section 3.7 or the lecture notes starting on p.85a; ignore terms that are second order in b.
5. 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 d2V(r)/dx2 - 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.
6. Very close to the example 4.6.1 which was discussed in class
7. 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 d2z/dt2 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.