Buoyancy and Stability: An Introduction

Ever since people set out to sea in ships, the issues of buoyancy and stability have been of importance. In spite of this, the treatment it receives in textbooks is often lacking. Following is an overview of the subject; basic understanding of the principles is essential in performing the experiment and interpreting the results.



Figure 1

Buoyancy is ultimately what makes things float, such as the buoy in Figure 1. This is true whether the material the boat is made of is lighter than water (like the balsa wood rafts Thor Heyerdahl and his crew crossed the Pacific with in 1947) or heavier than water. The latter would include objects from the buoy shown to the ships of the U.S. Navy.

The basic concept is very simple: for anything placed in a fluid medium, the upward force the medium exerts on the body is equal to the weight of the fluid the body displaces. This is not only true of bodies placed in water; it is also true of those in air. The difference is that, for those in air, the weight of the air displaced is usually not enough to “float” the aircraft. A notable exception are dirigibles such as the “Goodyear blimp,” which is filled with helium, a gas lighter than air. Another lighter-than-air gas used is hydrogen. This is very combustible, as everyone was reminded of when the Hindenburg caught fire in New Jersey in 1938.

Most buoyancy applications are marine ones, and it is those we will concentrate on in this experiment. We will also concentrate on rectangular forms and flat-bottomed vessels, which simplifies the math somewhat. However, these principles can be extended to just about any floating craft.

Using a flat-bottomed craft also makes it easier to understand why displacing a fluid works. Consider first the following: how the force of the fluid on the flat hull of a craft varies with depth1:


Figure 2: Illustrating Water Pressure Increasing in Proportion to Draught

For a fluid at rest, the hydrostatic pressure increases linearly with depth, thus




Figure 3 Inadequate Freeboard

where p is the hydrostatic pressure, γ is the unit weight of the water, and D is the depth from the water’s surface to the bottommost point of the vessel, usually called the draught. This distance from the water line to the top of the rectangle (the gunwale) is called the freeboard; the results of inadequate freeboard can be seen in Figure 3.

In any case, for a vessel of beam (width) W and a length L the volume it displaces is given by the equation



Combining and rearranging these two equations,



For the boat to float, it has to be in static equilibrium, and so the downward force of the weight of the boat Wboat must equal the upward force Fbuoyant. Therefore,



So we’ve established a relationship between the weight of the boat and the volume of water it displaces. The “far right” hand side only applies to boats with a flat bottom and straight sides.

What this means is that there are three ways we can weigh an existing boat:

  1. We can simply weigh it on a scale. For small boats this isn’t too difficult; larger ones can be tricky.  We can then estimate how far it will sink into the water.

  2. We can measure the freeboard, then obtain D and, knowing L, W and the unit weight of water, we can compute the weight of the boat. This works easily for rectangular boats; for real boats, you have to determine the relationship between the actual waterline and the displacement, then see where the actual waterline ends up.

  3. We can use an overflow method, which is okay for small experiments (like Archimedes used) but not so hot on a larger scale.  But this illustrates our concept.

Procedure for determining volume of water displacement2:


Figure 4 Displacement





Buoyancy is a fairly straightforward concept, although it may be a little hard to grasp up front. Stability—the ability of the ship to resist overturning—is a little more difficult, although it’s obviously important, as the following diagram of a ship with waves coming at the beam shows3.


Figure 5 Showing the Transverse Movements of an Easy-Rolling Vessel Among Waves, and Also of a Raft

Let’s define (or recall) a couple of terms.

Centre of Gravity: this is easy, mathematically this is the centroid of the mass or weight of the ship. An illustration of this is below.


Figure 6: Centre of Gravity of a Ship

Centre of Buoyancy: this is a little trickier, this is the centroid of the cross-sectional area of the ship under the water line, as shown below.


Figure 7 Centre of Buoyancy of Box-Shaped Vessel

As you can see, for a box-shaped vessel which is not listing (i.e., leaning at an angle) or has no squat (i.e., not angled along the length of the boat) the centre of buoyancy is located halfway down the draught of the vessel, halfway across the beam, and dead amidships.

The centre of gravity and the centre of buoyancy are not necessarily at the same place; in fact, they are usually different. That difference determines both the stability of the ship and, literally, how it rolls.

We know that motor vehicles with high centres of gravity (such as off-road vehicles) are more prone to turn over in use than those with lower centres of gravity. Ships are the same; we need to have a way to decide how stable a ship is and whether there is a point that a ship becomes unconditionally stable or unconditionally unstable.

As long as a ship is upright, and both the centre of gravity and the centre of buoyancy are in the centre of the ship in all respects, it is theoretically possible for a ship never to turn over. As a practical matter this is impossible; even very large ships like cruise ships, which use their size to resist roll in most wave situations, are going to roll some. Below is a diagram which shows the centre of gravity and the centre of buoyancy for a ship which is upright and which is inclined 14º.


Figure 8 Vessel Floating Upright (left,) and Inclined 14 Degrees (right)

We need to look at this carefully and note the following:

  • The point G is the centre of gravity of the ship.

  • The point B or B’ is the centre of buoyancy of the ship. In the course of inclination the centre of buoyancy will change because the shape of the cross-section under the waterline changes; this is fairly simple to calculate for rectangular ships and more complicated for curved hull shapes.

  • The point M is the metastatic point of the ship. The distance GM is called the metastatic height of the ship.

  • If point G is below point B or B’, the ship is unconditionally stable; it will not turn over unless G and B’ is changed by taking on water, shifting cargo in the ship, etc.

  • If point G is below point M, the ship is conditionally stable, and if point G is above point M, the ship is unconditionally unstable.

The reason for this last point is simple: the ship above is rolling in a clockwise direction. The resisting moment of the buoyancy, calculated by (GZ)(Wbuoyant) is counter-clockwise, as the buoyant force is upward. This is true as long as G is below M. If G moves upward above M, then the now driving moment (GZ)(Wbuoyant) turns clockwise, the same direction as the rolling of the ship, and the ship will generally turn over4.

Thus the location of M, abstract as it may seem, becomes a critical part of the design of a ship. But how is it done? There are two methods we will discuss here of determining the metastatic height of a ship.

Determining Metastatic Height

Theoretical Method

This method uses the following formula to determine the location of the metacentre:



For a rectangular vessel, the moment of inertia is the same as we used in mechanics of materials, i.e., LW3/12, and is applied as follows:


The displacement volume was given earlier. We then compute the distance between the metacentre M and the centre of buoyancy B as follows:


Note carefully that this is NOT the metacentric height GM; it is then necessary to subtract the distance from the centre of buoyancy to the centre of gravity from this result to obtain GM. This is done as follows:


Figure 9: Computing GM From the Height of the Metacentre Above the Centre of Buoyancy

It’s worth noting here that the location of point M is independent of the centre of gravity and dependent upon the geometry of the ship and its volume under the water line (or total weight.)

Timing the Roll

This method is sort of an “old salt’s” rule of thumb method. First, let’s define the roll time. The roll time is the time it takes for a ship to start from rest at an angle of roll (port or starboard,) roll to the opposite side, and return to the original orientation. This can be approximated by the equation5




  • tr = roll time of ship, seconds
  • GM = metastatic height of ship, meters or feet
  • W = beam of ship, meters or feet
  • C = constant based on units of GM and B
    = 0.44 for units of feet
    = 0.80 for units of meters

Solving for metastatic height,



This is significant for another reason: another rule of thumb used in yacht design is that the roll time is seconds should be between 1 and 1.1 times the beam W of the boat in meters6. Yachts with shorter roll times tend to “check” or quickly come back to centre when in rough seas; this can be a hard experience for passengers and make a complete mess of stowed cargo. Yachts with longer roll times will come over like they’re about to capsize, and then slowly roll back around. The usual result of this is seasickness and a miserable ride.

1Walton, T. (1899) Know Your Own Ship. Charles Griffin and Company, London, England. Much of the material that follows on buoyancy and stability comes from this work.

2Keep in mind that the unit weight of sea water is greater than fresh. Why is this so?

3Seamanship pointer: if you’re in a boat and are facing high waves, wake, etc., best way to take them is to point the blow into the direction the waves are coming from, not take the waves on the beam.

4Whether it actually does gets into issues of freeboard, rate of roll, etc., which are beyond this presentation or experiment.

5Nudelman, N. (1992) Yacht Design Course. Lesson 6: Stability. Westlawn Institute of Marine Technology, Stamford, CT.

6Gerr, D. (1992) The Nature of Boats. International Marine Publishing, Camden, ME. Much of the material on this method of determining GM is taken from this source.

Chicago Yacht Club Regatta, Late 1940’s

Following are some photographs from a Chicago Yacht Club regatta on Lake Michigan in the late 1940’s.

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I can’t be more specific about the date; however, one of Chet’s signature achievements at the Club took place in June 1946 when he, as Chairman of the Power Yacht Committee, helped to instigate the Commodore Fleet Review.  That helped launch his bid to become Commodore of the Club in 1950, fifty years after his father had held the post.  Given the large number of sail boats shown, it’s probably another event, but I’m pretty sure it’s in that era.


We’ve Added the Marine Documents from vulcanhammer.net, and a site for Fluid Mechanics Laboratory

Thing continue to progress at Chet Aero Marine (the expanded name for the site.)


First: we’ve added the marine design, construction and maintenance documents from vulcanhammer.net.  That site, which celebrates its twentieth anniversary next year, was  basically two sites: geotechnical and marine.  Given the material on the sea already here, it was a natural to add these, and let vulcanhammer.net concentrate on the geotechnical information.


The second is that we’ve added a page for my Fluid Mechanics Laboratory course. I’ve maintained sites on the courses I’ve taught for many years, this adds to those.  Hopefully the information found there will be of assistance to more than my students.

Chet Aero, an Old Site with a New Look

It’s been a long time in coming, but chet-aero.com–the site dedicated to my family’s adventures in both sky and sea–is now a WordPress “blog.”  It’s overdue, but some explanation about it is in order.


The original chet-aero site dates from 2004, first published around the time of another contentious American presidential election.  At the time, I had had many years of maintaining the other sites, vulcanhammer.net (with its geotechnical and marine downloads) and Positive Infinity (more of an opinion site.)  In digging through the family archives, it was clear that my grandfather’s aviation career was not only of interest to us but a fascinating piece of American history.  Moreover, it was my desire to move my web design forward.  At the time I was moving from Microsoft Front Page (remember that?) to Adobe GoLive.  Unfortunately, when you maintain a site for a long time without changing locations or basic formats, you get a great deal of legacy crud in the code.  chet-aero.com (which was both the URL and the name for the site) was to be a new effort with a new look.  (Additionally it was the first site done completely from my Mac, to which I had switched two years earlier.)

Placeholder Image

Although I’ve made many contacts and associations through the years, chet-aero.com never quite “went viral” the way I wanted it to.  There are several reasons for this.

The first is that it was a static site (with active elements) in a era when the long migration to active sites and platforms (such as WordPress) was under way.  This was the time when the blogosphere was in full swing (as Dan Rather found out the hard way.)  I was slow in adapting to that: Positive Infinity didn’t start its long march towards an active site until the following year and only adopted WordPress in 2006.

The second is that the downloads section–which put vulcanhammer.net on the map–never carried the site the way it had elsewhere, probably due to the specific subject matter.

The third relates to the second: the subject matter, although an interesting part of American history, is a small one.  Adding the yachts in 2010 didn’t really help that much.

In going to WordPress, I’ve been able to address many issues that badly needed it:

  1. The site is “mobile device friendly,” an integral feature of WordPress blogging.  That’s important now that search engines prefer mobile-capable sites.
  2. The photos–probably the strongest feature of the site–are much better handled by WordPress’ fb-style elements and slide shows.  Traditional HTML was clumsy in that regard.  That also enabled me to ditch the old Flash elements–which were one of the nicer things about the site when they first went up–for the WordPress slide shows.
  3. The menuing–something I was never satisfied with in the old site–is automattic.
  4. The old mov videos are finally a thing of the past.
  5. I’ve corrected spelling numerous spelling errors, some of which go back to the site’s infancy.

So take a look at it.  The URL chet-aero.com will soon point to the site; for several reasons it won’t be fully activated here.  Thank you for your support in the past and God bless.