A $5$ kg weight is attached to a spring fixed to an inflexible beam. When the weight is at rest, the spring is $10$ m long. The weight is pulled down from its equilibrium position by $2$ m and released. It is found to oscillate with a period of $2$ seconds.

$($a$)$ What is the spring constant for this spring?

$($b$)$ What would the length of the spring be with no mass attached?

$($c$)$ What is the maximum velocity of the weight during its oscillation?

$($d$)$ The spring is found to have a Q factor of $200.$ After how many oscillations will the system have lost $90\%$ of its amplitude?

$($e$)$ What will be the total energy stored in the system at the state described in $($d$)?$

$($f$)$ The beam is attached to a rocket, which accelerates upwards at $20$ m$/$s$\mathrm{²}.$ If the experiment were repeated under these conditions, what would be the shortest length of the spring after it was released, taking into account the effects of   damping?