PHYSICS 321 - Classical Mechanics I

Spring 2010

Homework Set #7 (due 3/24/10)

(but note that the represented material may be on the midterm exam of 3/19!)

1. Thornton and Marion, problem 4-1. You can solve this problem by calculating either the force on the particle or the potential energy. To get the answer in the form shown in the problem, you must perform a Taylor series expansion of (1+y)^-1/2, where y = (x/l)². Note that the rest length of the springs is l-d, not just l.

2. Thornton and Marion, problem 4-2. Draw the phase space diagram (x-dot vs. x) right below the potential energy plot, as is done in Figure 4-5 in the book. This way one can see the correspondence between the two diagrams. No algebra needs to be invoked - a sketch should suffice!

3. Thornton and Marion, problem 4-6.

4. Thornton and Marion, problem 4-7. The pendulum starts from rest at the horizontal position of x = x₀.

5. What is the asymptotic value of the mapping function x_n+1 = 0.6sin(πx_n)? Here, x is restricted to the range 0<x<1 and the angle is, of course, in radians. (Depending on your starting value, it reaches it in less than ~10 iterations.)

6. What are the two asymptotic values for the mapping function x_n+1 = 0.73sin(πx_n)?