A spring with spring constant k is fixed to the ceiling at one end and a weight of mass m is suspended at the other end, causing the ceiling to vibrate periodically.Take the position O of the natural length of the figure as the origin and x as the vertical downward direction.Assuming that the magnitude of the gravitational acceleration is g and the vertical displacement of the ceiling at time t is given by L sin$($t$),$ answer the following questions.$($Let L and be positive constants.$)$

$($i$)$ Find the equation of motion satisfied by the displacement x$($t$)$ of the weight.

$($ii$)$ The equation in $($i$)$ can be viewed as a system of periodic external forces F acting on a weight suspended by a spring from a fixed ceiling.Find this F$.$

$($iii$)$ If the angular frequency of the ceiling is $$k$/$m$.$ At t $=0,$ find the solution of the equation of motion when the weight is at rest, in a position of balance with gravity.

$($iv$)$ When $=$k$/,$ find the general solution of $($i$)$ and state what the weight oscillates from that result.