please show all the work.QUESTION $2(30$ Marks$)$

$2\mathrm{.}1$ A circular sluice gate $AB,$ having a mass of $140kg,$ is supported by a frictionless

hinge at $A.$ A cable fixed to the center of the sluice, runs across frictionless pulleys at

$C$ and $D$ and carries a solid spherical steel ball at its other end. The relative density of

steel is $7\mathrm{.}85$ and volume of spherical steel ball is $\frac{4}{3}{r}^3($where $r$ is the radius of the

ball$).$ Determine the diameter of the spherical ball for which the gate will open if the

water level rises $2600mm$ above gate A$.(1\mathrm{.}5)$