A homogeneous, $4$ m long rectangular gate is held in place by a horizontal flexible cable as shown in figure

a$.$ The weight of the gate is $4000$ N $.$ The gate is hinged at point $A.$ Assume that friction in the hinge is

negligible and water is static. The width of the gate into the paper is $2$ m $.$ The density of water $$ is $1000$

$k\frac{g}{m^3},$ and the gravitational acceleration is $9\mathrm{.}8\frac{m}{s^2}.$

Figure a

Figure $b$

Answer the following questions.

$($a$)$ In figure a$,$ what are the magnitude and direction of the net hydrostatic force acting on the gate?

$($b$)$ In figure a$,$ what is the distance between the hinge and the center of pressure $($CP$)$ on the gate for the net

hydrostatic force?

$($c$)$ In figure a$,$ determine the tension in the cable.

$($d$)$ In order to reduce the tension in the cable obtained in the problem $($c$),$ the balloon of diameter $1$ m is

attached to the gate with a vertical string $($see figure b$).$ The cable tension of this problem $($d$)$ is $200$ N

smaller than that of the problem $($c$)$ because of the balloon. Only a half of the balloon's volume is immersed

in water. Assume that the balloon has a spherical shape and the weight of the balloon is $300$ N $.$ Find where

the string of the balloon should be attached to the gate $($the distance between the hinge and the string on the

gate$).$