1. **State the problem:** A vessel 76.2 m long has forward and aft drafts of 5.18 m and 5.64 m respectively. The maximum permissible draft to cross a bar is 5.5 m. We need to calculate the mass of cargo to be discharged from a position 26 m aft from amidships to allow the vessel to cross the bar safely. 2. **Calculate the initial trim:** Trim $= \text{Aft draft} - \text{Forward draft} = 5.64 - 5.18 = 0.46$ m (stern down). 3. **Calculate the mean draft:** Mean draft $= \frac{5.18 + 5.64}{2} = 5.41$ m. 4. **Find how much the draft must be reduced to meet max permissible draft:** Difference $= 5.5 - 5.41 = 0.09$ m; since max draft is less than mean draft, vessel must lift by $0.09$ m. 5. **Calculate change in trim required to reduce aft draft to 5.5 m:** To reduce aft draft from 5.64 to 5.5, aft draft must decrease by $0.14$ m. 6. **Use hydrostatic data:** Given: - TPC (tonnes per cm) = 12 t/cm = 1200 t/m - MCT1cm (moment to change trim 1 cm) = 91.5 tm - LCF = 1.52 m aft of amidships - Length (L) = 76.2 m 7. **Calculate the vessel's displacement:** Displacement $\Delta = TPC \times \text{mean draft in cm} = 12 \times 541 = 6492$ t. 8. **Calculate the initial longitudinal center of buoyancy (LCB):** LCB relative to midship is at $LCF = 1.52$ m aft. 9. **Calculate moment due to initial trim:** Moment due to trim $= \Delta \times \text{center of floatation (COF)} \times \text{trim}$, but since moment to change trim is given, use MCT directly. 10. **Find change in trim required:** Change in trim needed (from moment definition): $$\Delta \text{trim in cm} = \frac{\text{change in moment (tm)}}{\Delta}$$ But we want to use MCT to find moment for 1 cm trim change. 11. **Calculate the moment to change trim by the required amount:** Change in trim in cm $= \frac{0.14}{0.46} \times 46 = 14$ cm (approximate based on proportionality). Using exact approach: The required trim reduction to have aft draft from 5.64 m to 5.5 m is $0.14$ m = 14 cm. Moment needed $= MCT1cm \times \text{change in trim in cm} = 91.5 \times 14 = 1281$ tm. 12. **Calculate cargo mass to discharge from 26 m aft:** Load shifted moment $= \text{Mass} \times \text{distance from amidships}$. Set load shifted moment equal to required moment: $$ \text{Mass} \times 26 = 1281 \Rightarrow \text{Mass} = \frac{1281}{26} = 49.27\text{ tonnes} $$. 13. **Final answer:** Mass of cargo to be discharged $\approx 49.3$ tonnes.