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Assessment Task
Q1 The Fibonacci numbers are computed according to the following relation: F n+1 = Fn + F n?1 with F 0 = F 1 =1., compute the ratio.
i) Create a script m-file to compute the ratio Fn Fn-1 for the first 50 Fibonacci numbers.
ii) Plot a graph of this ratio as a function of n. What do your results show?
iii) It is claimed that this ratio approaches the value of the golden mean 1 + 5 /2.
What is the smallest value of n that gives 0.1% accuracy of the golden mean?
Q2 A simple mass-spring system subject to damping is vibrating under the action of a
periodic force F cos t. The equation of motion is
where x is the displacement of the mass, k is the damping parameter and ? is the natural frequency of the system.
i) Create a function m-file to solve this second-order differential equation with the system of first-order differential equations.
ii) Consider the unforced motion when F = 0 and ? = 3.8 with the initial conditions
x 0 = 1, x 0 = 0. Use the numerical solver ode45 to solve the differential equation for the cases k = 0.0, 0.1, 0.2, 0.4, 1.0, 2.0 over the range 0 to 4?. Plot the six graphs on one figure. Interpret the results physically for the different values of k.
iii) Consider the forced motion when F = 2.0 and ? = 3.8 with the initial conditions x 0 = 1, x 0 = 0. Use the numerical solver ode45 to solve the differential equation for the cases when k = 0.0, k = 1.0 over the range 0 to 4?. Plot the two graphs on one figure. State the physically important difference between the results of parts (ii) and (iii).
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