$1.$ The barge's length, breadth, draft, and depth are $120$ meters, $15\mathrm{.}6$ meters, $4\mathrm{.}5$ meters, and $6\mathrm{.}8$ meters, respectively. The initial vertical center of gravity is $\backslash (\backslash$mathrm$\{$KG$\}\_\{0\}=5\mathrm{.}2\backslash )$ meters. The barge is divided into $12$ equal tanks which are shown in Figures $2$ and $3.$ The height of the tanks is $2\mathrm{.}5$ meters. All tanks are filled with seawater. What will be the heeling angle if half of the water in $\backslash (7^\{\backslash$text $\{$th $\}\}\backslash )$ and $\backslash (9^\{\backslash$text $\{$th $\}\}\backslash )$ tanks are off$-$loaded?

Aft

Figure $1.$ Perspective view of the barge.

Port

Starboard

Figure $2.$ Top view of the barge and the numbers of tanks

$2.$ All the tanks are converted into cargo holds after all the water is off$-$loaded. The bulkheads between the cargo holds $3-4$ and $5-6$ are removed and two cargo holds that span from starboard to port are formed. Then, these two cargo holds are partially loaded with grain. The initial righting moment arms of the barge are given in the table after loading. The grain moves in the cargo hold due to the severe weather conditions and the height of the grain increases by $1\mathrm{.}3$ meters on the starboard side of the cargo hold. Calculate the final righting moment arms and the loll angle. $(\backslash (\backslash$rho$\_\{\backslash$text $\{$main $\}\}=0\mathrm{.}8\backslash$mathrm$\{$t$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash ))$

$3.$ The draft of the barge is $3\mathrm{.}8$ meters and the $\backslash (\backslash$mathrm$\{$KG$\}\_\{0\}=4\mathrm{.}6\backslash )$ meters when all the tanks are empty. What will the final drafts be at the corners of the barge if $500$ tonnes of cargo is placed on the main deck that is $40$ meters forward of the midship and $3\mathrm{.}5$ meters to the port of the centerline?

$4.$ The draft of the barge is $3\mathrm{.}8$ meters and the $\backslash (\backslash$mathrm$\{$KG$\}\_\{0\}=4\mathrm{.}6\backslash )$ meters when all the tanks are empty. Analyze the damaged stability of the barge when tanks $6$ and $8$ are damaged together and find the heeling angle. Use both methods and compare the results.

SUBMIT YOUR HOMEWORK IN WRITTEN FORM IN THE CLASS ON JANUARY $\backslash (3^\{\backslash$text $\{$RD $\}\},2025\backslash ).$ DO NOT UPLOAD TO OIS.

IF YOU SOLVE THESE PROBLEMS ALONE, YOU WILL BE READY FOR THE FINAL EXAM.

Figure $3.$ Forward view of the barge and the height of the tanks