Consider a weight held in place by a spring. Initially suspended at rest, the height ofThe spring constant is kThis represents how "stiff" the spring is and is constant. The

frictional losses areisthe weight is x=0The mass of the weight ismcharacterized by the constant b . The force applied to the weight after t=0 f(t) . This leads to the ODE m \frac{d^{2} x}{d t^{2}}+b \frac{d x}{d t}+k x=f(t) 2. (65 points) Imagine the spring is vertical. The weight is held by a platform atx=0 initially. If m=0.1kg , b=0.1 kg/ s , and k=1.0 kg/s . The platformis removed at t=0 O What are the initial conditions for this system? b. (15 points) What is the new steady-state position? That is, where willt - o ? Show this using the final-value theorem.the mass settle as c. (15 points) What is the maximum distance the mass will move from itsinitial position? (Hint: what is the force applied to the weight?)(15 peints) Whatvalue vwill recuultming d. (15 points) What spring constant value will result in the system comingto (and remaining within) 1% of its new steady-state position withoutovershooting that value as quickly as possible? e. (15 points) Imagine the spring system is now horizontal.represents how far the spring is from its resting position. If 1 N of forceis instantaneously transferred to the weight (in thedirection)described above att=0 ,what is the response? That is, what isx(t) when the weight is flicked with 1 N of force? Assume the weightis only free to move in one direction.