The small dam shown in Fig. A$1$ is made up of a fixed vertical wall combined with a curved plate that is pinned at its ends $\backslash ($ B $\backslash )$ and $\backslash ($ C $\backslash ).$ The curved plate has the shape of an arc that is a quarter of a circle of radius $4$ m $.$ The length of the fixed wall $($shaded$)$ is $3$ m and the width of the dam $($into the paper'$)$ is $10$ m $.$

Figure A$1$

$($i$)$ Determine the gauge pressure on the water side of the curved plate at A which is mid$-$way between B and C$.$

$(3$ marks$)$

$($ii$)$ Determine the horizontal and vertical components of the hydrostatic force acting on the plate.

$(11$ marks$)$

$($iii$)$ Determine the magnitude and direction of the total hydrostatic force acting on the plate.

$(3$ marks$)$

$($iv$)$ Determine the depth of the centre of action of the hydrostatic force acting on the plate.

$(5$ marks$)$

$($v$)$ Atmospheric pressure is of the order of $\backslash (10^\{5\}\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$m$\}^\{2\}\backslash ).$ Explain carefully why it is reasonable to use gauge pressures when calculating the force on the plate in this application.

$(3$ marks$)$