Metacentric Height Calculation (Math Problem Sample)

Metacentric Height
A rectangular body 5 m long, 3m wide, 1.2 m height is immersed to a depth of 0.8 m in water. Find the metacentric height.
Solution
Given;
Dimension of the block = 5m x 3m x1.2m
Depth of immersion = 0.8 m
Distance AB = ½ X Depth of immersion
= ½ x 0.8 m = 0.4 m
Density of sea water = 1000 kg/m3
Meter-centric GM is given by GM
Centre of gravity to be 0.6 m above the bottom of the block
GM= I/∀-BG
I=Moment of Inertia
I= 1/12 〖×5 ×3〗^2 m^4= 45/4 m^4
∀ =Volume of the block submerged in water
=5 ×3 ×0.8=12.0 m^3
BG = AG – AB = 0.6-0.4 = 0.2 m
GM= 45/4 × 1/12-0.2= 45/48-0.2=0.7375 m
metacentric height. = 0.7375 m