---
title: "Metacentric height and CG-CP coincidence"
subject: "Fluid Mechanics"
chapter: 3
key: "semester_3/fluid_mechanics:metacentric-height-cg-cp-coincide"
source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/metacentric-height-cg-cp-coincide
site: BCT Engineering
---

# Metacentric height and CG-CP coincidence

Fluid Mechanics · Chapter 3: Hydrostatics

- Asked in: 2079 Baisakh, 2+2 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/fluid_mechanics/metacentric-height-cg-cp-coincide
- Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-3

## Question

What is the significance of metacentric height? When will the centre of gravity and centre of pressure coincide in case of plane immersed surfaces?

## Short answer

Metacentric height indicates floating stability; positive GM gives a righting moment. The centre of pressure coincides with the centre of gravity when the plane surface is horizontal (pressure uniform) or at a very large depth where the offset term becomes negligible.

## Answer

#### 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 $\bar h$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does the metacentric height measure?
   A. Stability of a floating or submerged body against overturning
   B. The depth of immersion
   C. The weight of displaced liquid
   D. The height of the centre of buoyancy
   Answer: A. GM is the distance G to M; positive means stable. (Key point: Meaning of GM)

2. What happens when GM < 0?
   A. The body is neutrally stable
   B. M is below G, an overturning couple acts and the body capsizes
   C. The body is stable
   D. The body rises
   Answer: B. GM > 0 stable, GM = 0 neutral, GM < 0 unstable. (Key point: Sign of GM)

3. What is the effect of a large GM on a ship?
   A. Unstable with long rolling period
   B. Gentle rolling and large safety margin
   C. Very stable but a short, uncomfortable rolling period
   D. No effect on rolling
   Answer: C. A stiff vessel returns quickly, so it rolls fast; a small positive GM rolls gently but is less safe. (Key point: Large GM)

4. When do the centre of gravity and the centre of pressure of a plane surface coincide?
   A. When the surface is horizontal, as pressure is uniform
   B. When the surface is vertical
   C. Only at the free surface
   D. Always, for any plane surface
   Answer: A. h_cp = h + I_G sin^2θ / (A h) equals h when θ = 0 (or approximately at very large depth). (Key point: CG and CP coincide)

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