Chapter 36 : Equilibrium of Floating Bodies 745
As a matter of fact, metacentric height of a floating body is a direct measure of its stability.
Or in other words, more the metacentric height of a floating body, more it will be stable. In the
modern design offices, the metacentric height of a boat or ship is accurately calculated to check
its stability. Some values of metacentric height are given below :
Merchant ships = up to 1·0 m
Sailing ships = up to 1·5 m
Battle ships = up to 2·0 m
River craft = up to 3·5 m
The metacentric height of a floating body may be found out by either of the following two
methods :
- Analytical method for metacentric height, and
- Experimental method for metacentric height.
But we shall discuss only the analytical method for metacentric height.
36.7.ANALYTICAL METHOD FOR METACENTRIC HEIGHT
Consider a vessel or ship floating freely in water. Let the ship be given a clockwise rotation
through a very small angle θ (in radians) about O. As a result of this rotation, let the ship occupy a
new position shown in dotted line as shown in Fig. 36.2. We see that the immersed section has now
changed from acde to acd 1 e 1.
Fig. 36.2. Metacentric height
The original centre of buoyancy B has now changed to a new position B 1. It may be noted that
the triangular wedge aom has come out of water, whereas the triangular wedge stet has gone under
water. Since the volume of water displaced remains the same, therefore the two triangular wedges
must have equal areas.
A little consideration will show, that as the triangular wedge aom has come out of water (thus
decreasing the force of buoyancy on the left) therefore it tends to rotate the vessel in an anticlockwise
direction. Similarly, as the triangular wedge ocn has gone under water (thus increasing the force of
buoyancy on the right) therefore it again tends to rotate the vessel in an anticlockwise direction. The
combined effect of both these forces will be to form a couple, which will tend to restore or rotate the
vessel in an anticlockwise direction about O. Since the angle θ, through which the vessel is rotated is
very small, therefore the vessel may be assumed to rotate about M (i.e., metacentre).