Problem $3$: $(20$ Points$)$

A free vibration test is conducted on an empty elevetad water tank. A cable attached to the tank applies a lateral $($horizontal force$)$ of $80$ kN and pulls the tank horizontally by $5$ cm $.$ The cable is suddenly cut and the resulting free vibration is recorded. At the end of four complete cycles, the time is $2$ second and the amplitude is $2\mathrm{.}5$ cm $.$

From these data, compute the following:

a$)$ Damping ratio

b$)$ Natural period of the undamped vibration,

c$)$ Stifness

d$)$ Weight

c$)$ Damping coefficient

f$)$ Number of cycles required for the displacement amplitude to decrease $0\mathrm{.}5$ cm