## Equilibrium

Equilibrium of a body floating in still water

A body floating freely in still water experiences a downward force acting on it due to gravity. If the body has a mass m, this force will be mg and is known as the weight. Since the body is in equilibrium there must be a force of the same magnitude and in the same line of action as the weight but opposing it. Otherwise the body would move. This opposing force is generated by the hydrostatic pressures which act on the body, Figure 5.1. These act normal to the body's surface and can be resolved into vertical and horizontal components. The sum of the vertical components must equal the weight. The horizontal components must cancel out otherwise the body would move sideways. The gravitational force mg can be imagined as concentrated at a point G which is the centre of mass, commonly known as the centre of gravity. Similarly the opposing force can be imagined to be concentrated at a point B.

Consider now the hydrostatic forces acting on a small element of the surface, da, a depth y below the surface.

Pressure = density X gravitational acceleration X depth = pgy

The normal force on an element of area da = pgy da

If y is the angle of inclination of the body's surface to the horizontal then the vertical component of force is:

(pgy da)cos y = pg (volume of vertical element)

Integrating over the whole volume the total vertical force is:

pg V where V is the immersed volume of the body.

This is also the weight of the displaced water. It is this vertical force which 'buoys up' the body and it is known as the buoyancy force or simply buoyancy. The point, B, through which it acts is the centroid of volume of the displaced water and is known as the centre of buoyancy.

Since the buoyancy force is equal to the weight of the body, m = pV. In other words the mass of the body is equal to the mass of the water displaced by the body. This can be visualized in simple physical terms. Consider the underwater portion of the floating body to be replaced by a weightless membrane filled to the level of the free surface with water of the same density as that in which the body is floating. As far as the water is concerned the membrane need not exist, there is a state of equilibrium and the forces on the skin must balance out.

### Underwater volume

Once the ship form is defined the underwater volume can be calculated by the rules discussed earlier. If the immersed areas of a number of sections throughout the length of a ship are calculated, a sectional area curve can be drawn as in Figure 5.2. The underwater volume is:

If immersed cross-sectional areas are calculated to a number of waterlines parallel to the design waterline, then the volume up to each can be determined and plotted against draught as in Figure 5.3. The volume corresponding to any given draught T can be picked off, provided the waterline at T is parallel to those used in deriving the curve.

A more general method of finding the underwater volume, known as the volume of displacement, is to make use of Bonjean curves. These are curves of immersed cross-sectional areas plotted against draught for

64 FLOTATION AND INITIAL STABILITY

Cross-sectional area

Cross-sectional area

Figure 5.2 Cross-sectional area curve

each transverse section. They are usually drawn on the ship profile as in Figure 5.4. Suppose the ship is floating at waterline WL. The immersed areas for this waterline are obtained by drawing horizontal lines, shown dotted, from the intercept of the waterline with the middle line of a section to the Bonjean curve for that section. Having the areas for all the sections, the underwater volume and its longitudinal centroid, its centre of buoyancy, can be calculated.

When the displacement of a ship was calculated manually, it was customary to use what was called a displacement sheet. A typical layout is shown in Figure 5.5. The displacement from the base up to, in this o

## How To Have A Perfect Boating Experience

Lets start by identifying what exactly certain boats are. Sometimes the terminology can get lost on beginners, so well look at some of the most common boats and what theyre called. These boats are exactly what the name implies. They are meant to be used for fishing. Most fishing boats are powered by outboard motors, and many also have a trolling motor mounted on the bow. Bass boats can be made of aluminium or fibreglass.

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