We consider two springs (coils - top one thin as it has smaller spring constant and bottom one thicker as it has larger spring constant) and a mass (blue plate) in a vertical mode conguration as depicted below. The gures below in Figure 1 show on the left when the spring system is in static equilibrium. i.e. at rest. while the gure on the right has the mass displaced vertically down from static equilibrium. Here is your set of Activities: 1. Build a mathematical model for the displacement y(t) of the mass in this conguration, using Newton's Second Law with a Free Body Diagram and the named parameters above. Show that you come up with the second order differential equation (1) for y(t) the displacement from static equilibrium of the mass: my\"(t) + (61 + 02).?1'05) + (k1 + kzly) = 0 (1) where m is the mass at the point where the two masses join. k1 is the spring constant of Spring 1. k2 is the spring constant of Spring 2. c1 is the coefficient of resistance constant of Spring 1. and c2 is the coefficient of resistance constant of Spring 2. 2. Convert the second order differential equation into a system of linear first order differential equations using the 'key\" 33(t) = y'(t). 3. Use these values for the parameters in the problem. m = 10. k1 = 20. k2 = 30. c1 = 5. 62 = 5. y([}) = yo = 18, and y'(0) = v9 = 0. Solve this system using an eigenvalue and eigenvector approach completely! 4. Solve this model for y(t) from the second order differential equation you obtained in (1). 5. Using the numbers in (3) compare the eigenvalue eigenvector solution for y(t) found in (3) with that found in (4) 6. Using some tool to get a plot of the displacement, y(t). over the time interval [0. 20] sec. Two Spring One Mass Modeling Two Spring One Mass Modeling Spring 1 Spring 1 Mass Mass Spring 2 Spring 2 Figure 1. Two spring mass system with thin spring on top when the mass is at the static equilibrium, i.e. at rest (left) and the mass is displaced vertically down from the static equilibrium (right). 7. Here are some further questions to consider. Take 011 as many as assigned. a) What would happen if we changed both (:1 and (:2 (equally, differently)? b) What if both (:1 and (:2 were 0? Could they be in reality? c) What would happen if we changed both 1'11 and k2 (equally, differently)? d) What would happen if we changed yo? Say making it positive? e What would happen if we changed the mass, m? f) What would happen if we used a y'(0) NOT 0