Question $1$

a$)$ Show from first principles that the angle of inclination realized during the inclining test, can be estimated using the relation below. Symbol have their usual meanings.

$=Ta{n}^{-1}(\frac{\frac{m}{}}{\frac{[K{M}_T-KG]}{d}})$

b$)$ An inclining test is conducted on M$.$V$.$ Avenger at a displacement of $10500$ tonne and $K{M}_{T}12\mathrm{.}5$ m $.$ The $8$ m plumb line used, deflected $12\mathrm{.}5$ cm when a $15$ tonne mass was moved $7$ m transversely. After completing the test, an inspection carried out revealed that not all ship tanks were empty at the time of conducting the test. The tanks identified are: No$.2$ ballast tank, containing $109$ tonne of $1023kg{m}^3$ sea water at Kg $1\mathrm{.}0$ m $,$ and No$.1$ double bottom tank, containing $850$ tonne of $930kg{m}^3$HFO at Kg $1\mathrm{.}23$ m $.$ Meanwhile, a $25$ tonne anchor positioned at Kg $5\mathrm{.}3$ m is under construction. If each tank is $130$ cma long and $82$ cm wide, and not pressed full, determine the TRUE LIGHTSHIP KG of the ship.

Question $2$

a$)$ In a particular condition of loading, the values of GZ estimated from the cross curves of M$.$V$.$ Ascari are shown in the table below. If in two other loading conditions the ship's centre of gravity is raised and lowered by $160$ mm and $152$ mm respectively, write a report on the stability of M$.$V$.$ Ascari and predict the condition in which the ship is much safe to go to sea.

$\backslash$table$[[$Angle $($Degree$),0,15,30,45,60,75,90],[$GZ $($Meter$),0,0\mathrm{.}18,0\mathrm{.}44,0\mathrm{.}52,0\mathrm{.}41,0\mathrm{.}2,-0\mathrm{.}06]]$

b$)$ Determine the angle of lurch for MV$.$ Asears in the thind condition of $($a$)$ if she encounter wind subjected her to heeling moment, $4500CO{S}^2(kNm)$ at displacement of $6500.$