Answer Happy

1. (Ideal mass-spring system with no rubber band) Choose a value of kį such that 12 < ki < 13 and study solutions of the equation day +kiy = 10. diž Examine solutions using both their graphs and the phase portrait. Are solutions pe- riodic? If so, approximate the period of the solutions. Be specific about the physical interpretation of the solutions for different initial conditions.

2. (Mass-spring system with damping but no rubber band) In Part 1, b = 0. Now repeat your analysis for day dy +b+kıy= 10 dt2 dt using the same value of kų as in Part 1 and various values of b. In particular. try b = 1.0 and b = 10.0. Describe how solutions change as b is adjusted. In fact, there is a particularly important b-value between b = 1.0 and b = 10.0 that separates the b = 1.0 behavior from the b= 10.0 behavior. This "bifurcation" value of b is difficult to locate numerically, but try your best.

diz 3. (Mass-spring system with rubber band but no damping) Once again let b = 0, but now add the rubber band to the system. That is consider the equation dy +kıy + kah(y) = 10. Use the same value for ki as you used in Part 1 and choose a value of ky such that 4.5 < k2 < 5.0. Repeat the analysis described in Part 1 for this equation. 4. (Damped mass-spring system with rubber band) We now add damping into the sys- tem in Part 3 and obtain dy dy + + kıy+kzh(y) = 10. dt Repeat the analysis described in Part 2 for this equation. d12 Your report: Address each of the previous items. You may provide illustrations from the computer, but remember that although a good illustration is worth 1000 words, 1000 illustrations are worth nothing. Make sure you use your conclusions about the solutions of these equations to describe how the mass oscillates.