2. Text V

3. The Law of Buoyancy and Naval Architecture

I Fill in the gaps with the terms given: metacentric, deadweight, magnitude, displacement, hydrostatic principles, submerged, law of buoyancy.

1 The A. ______, discovered by the ancient Greek mathematician and inventor Archimedes, states that any body completely or partially submerged in a fluid (gas or liquid) at rest is acted upon by an upward, or buoyant, force the magnitude of which is equal to the weight of the fluid displaced by the body. The volume of displaced fluid is equivalent to the volume of an object fully immersed in a fluid or to that fraction of the volume below the surface for an object partially submerged in a liquid. The weight of the displaced portion of the fluid is equivalent to the B _________ of the buoyant force. The buoyant force on a body floating in a liquid or gas is also equivalent in magnitude to the weight of the floating object and is opposite in direction; the object neither rises nor sinks. For example, a ship that is launched sinks into the ocean until the weight of the water it displaces is just equal to its own weight. As the ship is loaded, it sinks deeper, displacing more water, and so the magnitude of the buoyant force continuously matches the weight of the ship and its cargo.

2 If the weight of an object is less than that of the displaced fluid, the object rises, as in the case of a block of wood that is released beneath the surface of water or a helium-filled balloon that is let loose in air. An object heavier than the amount of the fluid it displaces, though it sinks when released, has an apparent weight loss equal to the weight of the fluid displaced. In fact, in some accurate weighings, a correction must be made in order to compensate for the buoyancy effect of the surrounding air.

3 The buoyant force, which always opposes gravity, is nevertheless caused by gravity. Fluid pressure increases with depth because of the (gravitational) weight of the fluid above. This increasing pressure applies a force on a C. ____ object that increases with depth. The result is buoyancy. The basis of naval architecture is found in Archimedes' principle. This law of buoyancy determines not only the draft at which a vessel will float but also the angles that it will assume when in equilibrium with the water.

4 A ship may be designed to carry a specified weight of cargo, plus such necessary supplies as fuel, lubricating oil, crew, and the crew's life support. These combine to form a total known as D ______. To deadweight must be added the weight of the ship's structure, propulsion machinery, hull engineering (nonpropulsive machinery), and outfit (fixed items having to do with crew life support). These categories of weight are known collectively as lightship weight. The sum of deadweight and lightship weight is E ______—that is, the weight that must be equaled by the weight of displaced water if the ship is to float. Of course, the volume of water displaced by a ship is a function of the size of that ship, but in turn the weight of water that is to be matched by displacement is also a function of the ship's size. The early stages of ship design, therefore, are a struggle to predict the size of the ship that the sum of all weights will require. The naval architect's resources include experience-based formulas that provide approximate values for making such predictions. Subsequent refinements usually produce accurate predictions of the ship's draft—that is, the depth of water in which the finished ship will float.

5 Accurately predicting a ship's draft is a necessary result of correctly applied F______ but is far from sufficient. If the many items of weight on a ship are not distributed with considerable precision, the ship will float at unwanted angles of heel (sideways inclination) and trim (endwise inclination). Nonzero trim angles may lift the tips of propeller blades above the surface, or they may increase the possibility that the bow will slam into waves during heavy weather. Nonzero heel angles (which tend to be much greater than trim angles) may make all human activity aboard difficult; moreover, they are dangerous because they reduce the margin against capsizing. In general, the avoidance of such inclinations requires an extension of Archimedes' principle to the first moments of weights and volumes: the collective first moment of all weights must equal the first weight moment of the water displaced.

6 A figure depicting the static stability of a ship shows the cross section of a ship that is floating at heel angle θ, caused by the placement of a weight (w) a certain distance (d) from the centre line. At this angle, the upsetting moment, calculated as w × d × cos θ, is equaled by the righting moment Δ × GZ, (Δ is the symbol for displacement, and GZ is the distance from the centre of gravity [G] to the centre of buoyancy [Z]). Under these conditions, the ship is said to be in static equilibrium. If w is removed, the upsetting moment will become zero, and the righting moment will return the ship to its upright position. The ship is therefore judged to be stable. The moment will act in the stable direction only as long as the point M (the “metacentre,” the point where the buoyant force intersects the midplane) is above G (the centre of gravity of the ship and its contents). If M is below G, the forces of weight and buoyancy will tend to increase the angle of heel, and the equilibrium will be unstable. The distance from G to M, taken to be positive if M is above G, is called the transverse G ______ height.

7 A value for G_____ height is usually found only for the zero heel condition; hence, it is an accurate measure of stability only for small disturbances—for example, ones that cause heeling of no more than about 10°. For larger angles, the “righting arm,” GZ, is used to measure stability. In any stability analysis, the value of GZ is plotted over the entire range of heel angles for which it is positive, or restoring. The resultant curve of statical stability shows thereby the angle beyond which the ship cannot return to upright and the angle at which the restoring moment is at a maximum. The area of the curve between its origin and any specified angle is proportional to the energy required to heel the ship to that angle.