A $12$ m long retractable bridge system has a weight of $100$ kg $.$ The radius of

gyration relative to the pivot point of the bridge is $0\mathrm{.}5$ m $.$ The bridge is held in

suspension under normal conditions by a spring connected at the center of

the bridge.

When necessary, the bridge is opened by applying a moment $M$ at the pivot

point. The nominal length of the spring is $4\mathrm{.}75$ m $,$ and the spring constant is

$k=500\frac{N}{m}.$

In the position shown in the figure, the spring is stretched by $1\mathrm{.}25$ m under

the effect of the bridge's weight. Just before the bridge reaches the fully open

position, its angular velocity is $=0\mathrm{.}4ra\frac{d}{s}.$

Calculate the constant moment M that allows the bridge to open.