(^746) „„„„„ A Textbook of Engineering Mechanics
Let l = Length of the ship,
b = Breadth of the ship,
θ = Very small angle (in radians) through which the ship is
rotated about O, and
V = Volume of water displaced by the ship.
Since the angle θ is very small, therefore
am =cn =
2
bθ
and volume of wedge of water aom =
11
22 22 2
bbb
am l l
⎛⎞⎛⎞θ
⎜⎟⎜⎟×=×
⎝⎠⎝⎠

2
8
blθ
...(i)
∴ Weight of this wedge of water

2
8
wb lθ
...(Q w = Sp. wt. of water)
Similarly, weight of wedge con of water

2
8
wb lθ
We know that arm of the couple

2
3
b
∴ Moment of the restoring couple

(^232)
83 12
wb l wb l
b
θθ
×= ...(ii)
and moment of the disturbing force =wV × BB 1 ...(iii)
Equating these two moments, i.e.(ii) and (iii),
3
12
wb lθ
=w × V × BB 1 ...(iv)
Substituting the value of
3
12
lb
= I (i.e., moment of inertia of the plane of the ship) and
BB 1 = BM × θ in equation (iii),
wIθ =w × V (BM × θ)
∴ BM =
Moment of inertia of the plane
Volume of water displaced
I
V

Now metacentric height, GM =BM ± BG
Note: +ve sign is to be used if G is lower than B, and –ve sign is to be used if G is higher
than B.