A solid body with a triangular cross$-$section floats in a fluid as shown below. The cross$-$section of the body is unchanged over its length $($L$)=8$ metres into the page. The height of the body $($h$)=3\mathrm{.}4$ metres and its maximum width $($c$)=2\mathrm{.}2$ metres. The uniform density of the solid body $(\backslash$rho s$)=450$ kg$/$m$3$ and

the density of the fluid $(\backslash$rho f$)=1000$ kg$/$m$3.$

$($a$)$: Determine the weight force $($FW$)$ of the floating body in kilo$-$newtons.

$($b$)$: Determine the buoyancy force $($FB$)$ acting on the floating body in kilo$-$newtons.

$($c$)$: Determine the submerged volume of the body $($in m$3)$

$($d$)$: Noting that the submerged volume can be written as the difference between the total volume of the floating object and the volume of the floating body above the free surface, determine the value of h$1$ in the figure above in metres.

$($e$)$: Determine the submerged depth $($h$2)$ of the floating object in metres.

$($f$)$: Determine the width of the triangle $($c$1)$ at the waterline cross$-$section $($in metres$).$

$($g$)$: Noting that the distance $(\_$centroid$)$ from the base of the floating object to the centroid of the submerged volume can be written as: $($see photo$).$Determine the distance $($GB$)$ between the centre of gravity $($G$)$ and the centre of buoyancy $($B$)$ of the floating body $($in metres$).$

$($h$)$: Determine the second moment of area about the waterline cross$-$section of the body $($in m$4).$ Note: Adopt the combination of L and c$1$; that generates the smallest second moment

of area. This is because the smallest second moment of area is critical in terms of stability.

$($i$)$: Determine the distance $($BM$)$ between the centre of buoyancy $($B$)$ and metacentre $($M$)$ in metres.

$($j$)$: Determine the metacentric height $($GM$)$ for the floating body in metres.

$($k$)$: State the value of the metacentric height necessary for neutral stability $($in metres$).$

$2($l$)$: Classify the stability of the floating object as stable, neutral or unstable.

ANSWERS:

a$)132\mathrm{.}1$

b$)132\mathrm{.}1$

c$)13\mathrm{.}46$

d$)2\mathrm{.}522$

e$)0\mathrm{.}8785$

f$)1\mathrm{.}632$

g$)0\mathrm{.}7158$

h$)2\mathrm{.}895$

i$)0\mathrm{.}2151$

j$)-0\mathrm{.}5000$

k$)0\mathrm{.}000$

l$)$ Unstable

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