Metacentric height and CG-CP coincidence

Significance of metacentric height

The metacentric height $GM$ is the distance between the centre of gravity $G$ and the metacentre $M$ of a floating or submerged body. It is the measure of stability against overturning:

• $GM>0$ (M above G): stable; the righting moment is $W\,GM\sin\theta$ for a small heel $\theta$.
• $GM=0$: neutral equilibrium.
• $GM<0$: unstable; the body capsizes.

A large $GM$ means a stiff, very stable vessel with a short rolling period (uncomfortable); a small positive $GM$ gives a gentle roll but less safety margin. It is also used in the experiment $GM=\dfrac{wx}{W\tan\theta}$ and in ship design.

When do the centre of gravity and the centre of pressure coincide?

The centre of pressure is at depth

$$h_{cp}=\bar h+\frac{I_G\sin^{2}\theta}{A\bar h}$$

This equals $\bar h$ (the depth of the centroid) when the second term is zero, i.e.

• when the plane surface is horizontal ($\theta=0$), since the pressure is uniform over the surface; or
• (approximately) when the surface is at a very large depth, since $\dfrac{I_G}{A\bar h}\to0$ relative to $\bar h$.