1. **State the problem:** A ship 55 m long has drafts 3.65 m aft and 2.75 m forward. Two loads (27 t and 70 t) are added at 18 m aft and 19 m forward of amidships respectively. We need to find the new drafts forward and aft after adding these loads. 2. **Calculate initial mean draft:** Mean draft $T_m = \frac{3.65 + 2.75}{2} = 3.20$ m 3. **Calculate initial trim:** Trim by stern (difference in drafts): $\text{Trim} = 3.65 - 2.75 = 0.90$ m 4. **Calculate moment of inertia arm from amidships:** Ship length $L = 55$ m Distance of center of flotation (C.F.) from amidships $F = 1.46$ m forward (positive forward) 5. **Calculate moments introduced by initial draft and loads:** The position of C.F. from amidships implies initial moments balance around a point 1.46 m forward. 6. **Calculate initial displacement ($\Delta$) and TPC:** $\Delta = 2351$ t at mean draft 4.65 m TPC (tons per cm immersion) $= 6.12$ t/cm 7. **Calculate change in displacement ($\Delta W$):** Total load added $= 27 + 70 = 97$ t 8. **Calculate new mean draft ($T_m^{new}$):** Change in draft in cm $= \frac{\Delta W}{TPC} = \frac{97}{6.12} \approx 15.85$ cm New mean draft $= 3.20 + 0.1585 = 3.3585$ m 9. **Calculate moments due to added weights:** Position of 27 t load is 18 m aft of amidships (negative position) $= -18$ m Position of 70 t load is 19 m forward of amidships (positive position) $= +19$ m Calculate moment of added weights about amidships: $$ M = (27)(-18) + (70)(19) = -486 + 1330 = 844 \text{ t m}\ $$ 10. **Calculate change in trim:** Moment to change trim relates to TPCM (tons per cm moment): $$ TPCM = \frac{TPC \times L^2}{100} = \frac{6.12 \times 55^2}{100} = \frac{6.12 \times 3025}{100} = 185.73 \text{ t m/cm}\ $$ Trim change in cm: $$ \Delta \text{Trim} = \frac{M}{TPCM} = \frac{844}{185.73} \approx 4.54 \text{ cm} $$ 11. **Calculate new trim:** Initial trim $= 0.90$ m $= 90$ cm New trim $= 90 + 4.54 = 94.54$ cm (trim by stern) 12. **Calculate new drafts forward and aft:** Let forward draft $T_f$ and aft draft $T_a$ satisfy: $$ T_m^{new} = \frac{T_f + T_a}{2} = 3.3585 $$ $$ \text{Trim} = T_a - T_f = 0.9454 $$ Solving: $$ T_a = T_f + 0.9454 $$ $$ \frac{T_f + T_f + 0.9454}{2} =3.3585 \Rightarrow 2T_f + 0.9454 = 6.717 $$ $$ 2T_f = 6.717 - 0.9454 = 5.7716 \Rightarrow T_f = 2.8858 \text{ m} $$ $$ T_a = 2.8858 + 0.9454 = 3.8312 \text{ m} $$ **Final answer:** New forward draft $= 2.89$ m (approx.) New aft draft $= 3.83$ m (approx.)